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In the context of extended classical mechanics, an important aspect of negative apparent mass (−Mᵃᵖᵖ) and how it interacts with positive matter mass (Mᴍ) as the electron accelerates, particularly when approaching high velocities. To reflect this, we need to focus on the dynamics between the electron’s matter mass and apparent mass, and how these interplay as the electron approaches the speed of light, eventually making the matter mass negligible and the apparent mass dominant. This leads to the effective mass transitioning toward negative values, which could imply a shift from gravitational attraction to antigravitational effects.
Structural Implications of Negative Apparent Mass:
As the negative apparent mass −Mᵃᵖᵖ becomes dominant, it exerts an increasing pressure on the positive matter mass of the electron, which can cause the structural integrity of the electron to be compromised.
The pressure exerted by the negative apparent mass could overwhelm the electron's normal structure, potentially leading to its disintegration or transformation into a state where the traditional concept of "matter" no longer applies in the usual sense.
The key insight here is that as the electron accelerates to high speeds, its matter mass Mᴍ becomes negligible, and the negative apparent mass −Mᵃᵖᵖ becomes dominant.
This transition leads to the effective mass becoming negative, which shifts the electron’s behaviour from gravitational attraction to antigravity.
As the kinetic energy increases, it is no longer just a result of the matter mass, but instead is primarily driven by the negative apparent mass, which could result in the electron reaching speeds near c and transitioning to a state where its structural integrity is challenged by the forces acting on it.
Electron Transition from Matter to Antimatter:
Transition from Matter to Antimatter:
As the electron's velocity increases toward the speed of light, the negative apparent mass (−Mᵃᵖᵖ) becomes dominant, reducing the effective mass (Mᵉᶠᶠ).
When the velocity approaches c, the matter mass (Mᴍ) effectively becomes negligible compared to the negative apparent mass. In this state, the electron could experience antigravitational effects as a result of its negative effective mass.
This leads to the electron being subjected to forces that no longer attract it to gravitational sources, but instead, these forces would push it away from those sources. This is an antigravity effect.
Structural Integrity and Breakdown:
The most critical point is that, as the negative apparent mass grows, it exerts a counteracting pressure on the structure of the electron.
This pressure is not simply a force acting against gravitational attraction; it is a fundamental change in the dynamics of the electron's existence, transitioning it from matter to something that could potentially behave like antimatter under the extreme conditions.
Gravitational Bound Systems:
In any gravitationally bound system (such as a galaxy), as an object’s speed increases and it approaches c, it becomes increasingly difficult for the object to maintain its matter mass structure.
At the limiting point, when negative apparent mass dominates, the matter mass of the electron would no longer be able to counteract the pressure from the negative apparent mass, leading to the breakdown of its structural integrity.
Thus, the electron would no longer behave as conventional matter; its behaviour would be governed by its negative effective mass, and its structure could potentially collapse or dissipate under these extreme conditions. This breakdown explains why no matter can survive as matter within a gravitationally bound system at light's speeds, where negative apparent mass takes over and results in antigravity.
In essence, the application of force to accelerate matter to light's speeds in a gravitationally bound system results in a transition from a gravitationally attractive state to a repulsive, antigravitational state governed by negative effective mass.
Conclusion:
The framework of extended classical mechanics provides a novel lens to understand the transition of an electron from matter-like behaviour to an antimatter-like state. As the electron accelerates toward the speed of light, its positive matter mass (Mᴍ) diminishes, and the negative apparent mass (−Mᵃᵖᵖ) becomes dominant. This transition redefines its effective mass (Mᵉᶠᶠ), leading to a shift from gravitational attraction to antigravitational effects. The interplay of these mass components, under extreme conditions, challenges the structural integrity of the electron, potentially transforming it beyond the traditional concept of matter. These findings elucidate a critical mechanism by which matter, under intense forces and velocities, could evolve into a state exhibiting antimatter-like properties, driven by the dominance of negative effective mass.
Description of Mathematical Terms:
1. c (speed of light): A fundamental constant in physics, representing the maximum speed within a gravitationally bound system at which information or matter can travel in a vacuum, approximately 3 × 10⁸ m/s.
2. F (force): A vector quantity representing the interaction that changes the motion of an object, calculated in extended classical mechanics as  F = (Mᴍ − Mᵃᵖᵖ)⋅aᵉᶠᶠ.
3. KE (kinetic energy): The energy an object possesses due to its motion, driven by both matter mass (Mᴍ) and negative apparent mass (− Mᵃᵖᵖ) in this context.
4. Mᵃᵖᵖ (apparent mass): A concept in extended classical mechanics representing the negative contribution to effective mass, arising from kinetic energy or other dynamic effects.
5. Mᵉᶠᶠ (effective mass): The net mass of a system combining matter mass (Mᴍ) and apparent mass (Mᵃᵖᵖ), expressed as Mᵉᶠᶠ = Mᴍ − Mᵃᵖᵖ. It governs the dynamic response to forces.
6. Mᴍ (matter mass): The intrinsic positive mass of an object, such as an electron, representing its rest mass without motion effects.
7. Mᴍ,ᴘᴇ (matter mass potential energy): The contribution to energy arising from the object's position within a potential field, linked to its intrinsic mass (Mᴍ).
8. Mᵃᵖᵖ,ᴋᴇ (apparent mass kinetic energy):The kinetic energy associated with the negative apparent mass, highlighting the dominant role of Mᵃᵖᵖ at high velocities.
9. PE (potential energy): Energy stored in an object due to its position within a gravitational or other force field, related to Mᴍ.
10. v (velocity): The speed and direction of motion of an object. In this context, v approaches c, leading to significant effects on Mᴍ, Mᵉᶠᶠ, and F.
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Jose Oreste Mazzini ’s First Question:
"What experiment has been done to verify the existence of negative mass?"
1. The Context of Negative Apparent Mass:
The concept of negative apparent mass (−Mᵃᵖᵖ) differs fundamentally from intrinsic negative mass. It arises as a contextual property, emerging from the equations of effective mass (Mᵉᶠᶠ) under extreme conditions. The term "apparent" signifies that this property is not an inherent attribute of the particle but is instead influenced by external factors.
Key insights include:
Apparent mass: A dynamic result influenced by energy, momentum, and the interplay with external forces, not a static characteristic of matter.
Negative apparent mass: Emerges under specific conditions, particularly when the energy contributions from potential and kinetic dynamics surpass the rest mass energy.
This theoretical framework aligns with phenomena where gravitational dynamics deviate from classical predictions, including dark energy interactions.
Reference: Observational research by A.D. Chernin et al., "Dark Energy and the Structure of the Coma Cluster of Galaxies," supports the interpretation of dark energy dynamics in systems where apparent mass plays a role.
2. "Matter to Antimatter" Transition:
The proposed transition from matter to antimatter-like behaviour under extreme conditions is unconventional but extends the understanding of particle dynamics. When negative apparent mass dominates, the following occurs:
Structural disintegration: Negative apparent mass exerts pressure that challenges the electron's structural integrity. This pressure increases as the electron's velocity approaches the speed of light (c), rendering its rest mass negligible. This is a consistent mathematical prediction of physical consequences.
Transition dynamics: The effective mass (Mᵉᶠᶠ) becomes dominated by (−Mᵃᵖᵖ), leading to repulsive (antigravitational) effects. The electron no longer adheres to conventional matter dynamics.
Antigravity effects: As negative apparent mass dominates at light's speeds, repulsion from gravitational sources occurs. This behaviour aligns with the theoretical underpinnings of antimatter-like states in extreme conditions.
3. Gravitational Bound Systems and Structural Breakdown:
The inability of matter to survive as conventional matter at light's speeds in gravitationally bound systems highlights the interplay between −Mᵃᵖᵖ, and matter M:
Increasing speed and Mᵃᵖᵖ: As the electron accelerates, Mᵃᵖᵖ grows while matter M diminishes. A tipping point is reached where structural forces are overwhelmed.
Collapse or dissipation: At this point, the electron ceases to behave as traditional matter. Instead, it transitions to state resembling antimatter, characterized by antigravitational interactions.
4. Supporting Evidence and Theoretical Alignment:
While direct experimental validation of negative apparent mass remains an open frontier, theoretical consistency with extended classical mechanics offers promising pathways for exploration:
Alignment with dark energy dynamics: The interpretation of negative apparent mass mirrors the influence of dark energy on cosmic expansion, as shown in the work of A.D. Chernin et al.
High-energy phenomena: Observations of high-energy particles near black holes or data from particle accelerators could provide indirect evidence of these transitions.
5. Transition to Antimatter-like Behaviour:
The transition described is not conventional antimatter (as defined in particle physics, with opposite charge but identical mass). Instead, it represents a novel state governed by:
Negative effective mass: This leads to repulsion from gravitational sources, creating antigravity effects.
• Dynamic behaviour under extreme at light's speeds, conventional properties of matter cease to apply, resulting in a fundamentally different state of existence.
Conclusion:
The theoretical framework for the "Matter to Antimatter" transition provides a robust model for understanding high-energy dynamics and structural transformations under extreme conditions. While experimental validation is pending, its consistency with extended classical mechanics and alignment with observed phenomena (e.g., dark energy effects) support its plausibility. Further research and experimentation are essential to substantiate these claims and deepen our understanding of particle behaviour near the speed of light.
Regards,
Soumendra Nath Thakur
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In light of my paper "Dark Matter is Just Gravity, Only Normal Matter is the Truth" (Indian Journal of Advanced Physics, October 2023), which demonstrated that dark matter is essentially gravity, a pertinent discussion arises about dark energy. Could dark energy, like normal energy, simply be a force we already understand, causing the universe's expansion and the movement of objects within it? This parallels the understanding of dark matter not as a mysterious entity but as a familiar force affecting planetary and galactic movements.
The question now is whether the research community on platforms like ResearchGate agrees with this perspective. Why do we tend to label unexplained phenomena as "dark" or mysterious, when perhaps they are simply manifestations of forces we encounter daily? By shedding light on these concepts, we might discover that what we perceive as "dark" is actually well within our grasp of understanding. Can we collectively agree that embracing the light of scientific inquiry could dispel the so-called darkness surrounding these cosmic forces?
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No, dark energy is just a misconception by astronomers and theorists. It doesn't exist. The problem is the Hubble distance formula which is a off by about 15%, at a redshift of z = .6., and by factor of more than 2 at a redshift of 14. Of course the major problem is mainstream cosmology which is almost entirely wrong. It will continue on a down-slide with the James Webb for another 2-3 years, following that replacement ideas will begin to appear.
There is no such thing as dark energy, dark matter, Inflation, not even the expansion of space. It will all be seriously questioned, proven wrong, and / or replaced within the next decade.
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A universe model compatible with VSLT
The research of Halton Arp and Eric Lerner supports a stationary universe, and the intrinsic nature of redshift, i.e. not linked to its presumed expansion (Doppler effect).
I also believe that the universe is stationary. Which doesn't mean static. Galaxies are not eternal, but have a limited average lifespan (of the order of a few tens of billions of years). Galaxies were not all born at the same time, but each galaxy has its own moment of birth. Two different galaxies have different birth moments (often billions of years apart).
A galaxy is born from PLASMA released by a supermassive rotating black hole when it EXPLODES.
The fact that black holes explode is confirmed by the GAMMA RAY BURST phenomenon. These are beams of gamma rays so intense that billions of suns are needed to generate them.
How do these rays originate?
The only possible explanation is that the electromagnetic radiation that composes them was IMPRISONED in a black hole which, EXPLODING, released it into space.
Rotating black holes are essential for the existence of a galaxy, because they provide the gravitational force that allows the galaxy's stars to rotate in a plane passing through its nucleus. All galaxies have stars that rotate in a plane. This is evidence that the galaxy's core is super-massive and rotating black hole.
This black hole absorbs not only the radiation that goes beyond its event horizon, but also all the MATTER that passes through it.
This matter is endowed with kinetic energy which is released to the black hole. The TEMPERATURE of the latter is therefore destined to increase over time.
It follows that inside the black hole matter can only exist in the PLASMA state.
When the black hole explodes, this plasma is projected into space together with the radiation which, due to the very high temperature, will have a very high frequency (gamma rays).
From the plasma projected into space, a new generation of neutral atoms will arise (due to Coulomb attraction), stars, a new super-massive rotating black hole and a new galaxy.
The atoms of a newborn galaxy emit light at the fastest possible speed. And since the speed is proportional to the wavelength, with the maximum possible wavelength.
As time passes, the atoms emit light at a slightly decreasing speed. In fact, atomic electrons move in a sea of ​​NEUTRINOS. These latter particles, like electrons, have mass, although infinitesimal. They therefore interact with atomic electrons, slowing them down. And since the speed of electrons is proportional to the speed of light, it follows that v, and therefore c, decreases slightly over time.
The redshift is therefore not linked to the distance between galaxies, but to their age difference.
Our galaxy is one of the oldest in the visible universe. This is the reason why almost all galaxies appear red (they are younger. They emit light with a great speed and therefore a long wavelength).
The reason why redshift appears to be linked to distance is that to observe distant galaxies they need to be very BRIGHT, i.e. very YOUNG.
As time passes, the atoms present in galactic clouds are pushed towards the outside of the galaxy by the "radiation pressure" emitted by the stars in the galaxy. They thus form a shell of matter that absorbs the photons coming from the stars of the galaxy (at a surface temperature of around 5700 °K) and re-emits them at a much lower temperature (around 2.75 °K)
This is the origin of the cosmic radiation we observe. This radiation comes from the shell of NORMAL matter that surrounds OUR galaxy, placed at such a distance that the temperature of the photons that hit it goes from 5700 °K to 2.75 °K.
In fact, the temperature map of the cosmic background radiation presents symmetries around the GALACTIC PLANE.
This shell is present in all older galaxies, while it is absent in young, newly formed galaxies.
It is responsible for the fact that for older galaxies the number of visible stars is much lower than that of stars actually present, number deduced from the deflection of light passing near the galaxy which provides its effective mass.
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There are many connections between the constants Giuseppe Pipino so as soon as one varies some others have to vary in a way that compensates as you say. Basically, almost all have to change i any one changes when you account for all the experimental evidence. Look at the work of Wetterich for a more comprehensive analysis.
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How are earthquakes in the world of quantum mechanics?
What causes an Earthquake?
When I put the above question into social media, the response on all the sites was the same: an earthquake occurs when the rock underground suddenly breaks along a fault line. The sudden release of energy causes the seismic wave that makes the ground shake. When two blocks of rock or two plates rub against each other until one of the rocks or plates break, is when the earthquake occurs. This is the general answer for the cause of an earthquake.
My question is: what is it that caused these two rocks/plates to rub against each other in the first place? And what caused the rock underground to break at the fault line?
We really don’t know what caused it or for what reason an earthquake suddenly takes place. There is no straight forward answer for this question.
My theory of a Quantum Mechanics Universe has the answer for this phenomenon and can be described by Quantum Mechanics Unification Gravity.
My suggestion at this point is to read my Quantum Mechanic Gravity (QMG) to understand the concept.
Let me explain:
The Earth has four major layers: the inner core, the outer core, the mantle and the crust. The inner core of Earth consists of harder and heavier elements. As we travel to the surface, the heavy elements become lighter. The character of a heavier element is that it can take more heat than lighter elements.
The Earth is about 10 minutes in light years away from the Sun and the Sun is shining on the Earth 24/7. The character of the Sun’s energy wave or sun’s mass-less particles are traveling through the earth and nothing can stop it. Light is thermo-dynamic energy (thermodynamic is the relationship between heat and exertion).
Where does all this energy go and what is it used for?
Most of the energy is used or wasted over the surface of the Earth; but the energy wave that is shining perpendicular to the Earth has a better chance to travel into the Earth where eventually it meets at the center (inner core). The energy of the Sun is interchanged to quantum mechanics energy in heavier elements in the center of the solid hard core of the Earth.
NOTE: This is additional hard evidence that sunlight does not carry any mass. If the sunlight had mass, the inner core of the Earth would not be hot, because mass cannot travel through a harder mass.
The center of the Earth consists of heavy elements of the periodic table that first absorb this energy wave of heat from the Sun. Secondly, the surrounding core (outer core) is another heavy metal family that also absorbs this heat, until both, inner core and a part of the outer core become a melting pot of bubbling heavy elements, which then cause a new heavy metal of the inner ring of the outer core (close to the center of the Earth) to join this hot melting pot.
The quantum mechanics gravity causes the heavier elements to sink more into the center; meanwhile more sunlight energy is still being absorbed. It gets to a point that this boiling pot has limited space to expand and the heat generates a sudden expansion which creates a high pressure in the center of the Earth in both the inner and outer cores. The stress of this built up energy and pressure under the tectonic plates create the underground shaking and earthquakes as a result of the release of this heat and pressure. At first the heat and pressure is released through volcanoes but it also causes the plates to shift/break and create earthquakes.
We all know too well of the horrific results of an earthquake or a sudden volcanic eruption. Hundreds or in some occasions thousands of lives are sacrificed because of these two catastrophic events, but unfortunately the Earth is not aware of the loss of these lives, it is trying to prevent a greater global disaster.
Everything in the Universe that has conscience movement has intelligent life. That applies to everything from an atom up to the Earth itself and beyond.
Here, believe it or not, the Earth acts similarly to a human. As when we eat, our body is trying to save some of the energy for future needs, and the Earth is doing the same thing. It keeps this energy inside the inner core for two reasons, one it is part of its nature or growing to be completed, and keep the heat for a crucial moment that may arise.
This procedure of transferring heat to the center core is the natural behavior of any spherical atomic element in the space of the Universe, because in general the heat always transfers with outer chemical elements into the core of that planet. This course of action is the formation for the creation of a planet.
We learned from my paper on Gravity, how we walk on the Earth and how the Earth creates an electromagnetic force field in conjunction with a space wave. This friction of the Earth and space causes the Earth, after a long period of time, to lose its momentum in space of both its rotation and its revolution around the Sun by a very small fraction of its speed. To re-energize its momentum and movement to coordinate with the solar system and galaxy’s movement, it needs this energy from the Sun, inside at the center core, to generate its speed back to normal. The Earth is an intelligent planet and has been performing this procedure for billions of years. It wants to keep all its elements in good working order, especially with today’s demand of humankind which uses so many natural resources which change the distribution of the weight on Earth and pollutes the Earth in all sectors; here the Earth must create more of the heavier elements in the center to maintain this momentum.
The space of the Universe is very clear, but when there is wave and temperature that exist, friction also exists. We experience this phenomenon with all the satellites in orbit around the Earth. Sometimes, from the control room on Earth, for similar reasons, we must boost the satellite’s speed or get it back to its original speed. Otherwise they would lose their orbit and momentum and sometimes they fall back to the Earth. For the same reason, if the earth was going mechanically (the Big Bang theory) around the Sun, it would stop spinning after a period of time.
An earthquake is based on this same principle; when micro gravity is trying to pull heavier matter to the center it pushes the lighter elements up as we have observed with volcanoes. Probably, by knowing this phenomenon we should be able to predict the region of earthquakes better.
Here I should mention that the inner core of the Earth helps the Earth to keep the magnetic field of the North and South Poles in an orderly manner as well.
At the end of this segment, I should mention that the nature of the Earth is that it has had volcanic eruptions and earthquakes all its lifetime, for the natural release of this excess heat and also to support the vegetation life as well, by producing carbon-dioxide. In a sense, in layman’s terms, the Earth is breathing, by naturally absorbing the heat from the Sun and placing the heavier elements towards the center and pushing lighter elements to the surface of the Earth, through volcanoes and earthquakes.
Suad Mohammed Ali added a reply:
At the moment, there are several hypotheses in geophysics that explain especially dangerous processes of the earth's crust movements - sudden outbursts of rocks and gas from a rock mass from the point of view of classical physics. Despite the fact that various macroscopic systems can be accurately described using classical mechanics and electrodynamics, a real mechanism and a working model of this phenomenon cannot be built. Consequently, to develop a model of sudden outbursts of rocks and gas, it is necessary to apply new approaches and methods, different from the description of macroscopic systems. This article describes a quantum version of the process of the ejection of rocks from a rock mass. In particular, we described the mechanism of the Coulomb explosion that occurs in the rocks of the earth's crust with a sharp change in rock pressure and built a model of the sudden release of rocks and gases. In our opinion, the quantum processes described by us can be sources not only of sudden outbursts and rockslide but also sources of more formidable phenomena - earthquakes and volcanic explosions.
Hasan Altawil added a reply:
In quantum mechanics, earthquakes can be understood metaphorically in terms of fluctuations and disturbances at the quantum level, but they are not directly related to classical seismic events. Quantum fluctuations, for instance, involve temporary changes in energy in a point in space, somewhat analogous to how stress builds and is released in an earthquake. However, quantum mechanics primarily deals with phenomena at the subatomic scale, making the analogy only conceptual rather than literal.
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  • Subject: Invitation to Join Dailyplanet.Club and Response to Your Quantum Mechanics and Earthquakes Discussion
  • Dear Abbas Kashani,
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  • I would like to invite you to join us at www.Dailyplanet.Club, where we are building a forward-thinking community of researchers, innovators, and sustainability advocates. Your intriguing discussion on quantum mechanics and earthquakes caught my attention, and I believe your unique perspective on quantum mechanics' role in natural phenomena would be an excellent addition to our conversations.
  • Regarding your theory of Quantum Mechanics Unification Gravity explaining earthquakes, it provides a fresh lens through which we can understand the complex interactions within Earth's core. Your exploration of quantum processes at the core of the Earth and their relationship to tectonic shifts is fascinating and could contribute to predicting earthquake regions more accurately.
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Does "dark matter" make up large proportions of those galaxies?
Newtonian gravity behaves differently at very large scales of mass and distance, i.e., galaxy scales, in contra-indication to the assumption that massive quantities of invisible, or "dark matter" make up large proportions of those galaxies.
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Preston Guynn added a reply
Your discussion statement question is:
  • "Does 'dark matter' make up large proportions of those galaxies? Newtonian gravity behaves differently at very large scales of mass and distance, i.e., galaxy scales, in contra-indication to the assumption that massive quantities of invisible, or 'dark matter' make up large proportions of those galaxies."
The phrase "Newtonian gravity" refers to a very specific equation relating mass and acceleration, so saying it behaves differently under some condition is not a correct usage of the phrase. Newtonian gravity is Newtonian gravity, and it gives incorrect results at scales greater than the solar system. There is a significant body of research on modified Newtonian gravity, and you can find it by searching on the phrase or "MOND".
Your question"Does dark matter make up large proportion of those galaxies?" is the question that numerous branches of research are investigating either experimentally or theoretically. First of course is the search for any experimental evidence of any matter that couples gravitationally but not via the electromagnetic field. No evidence of any such matter has been found. Second is that there is no such matter expected from current models such as the so called standard model of physics.
Even if there were some type of matter that couples gravitationally but not via electro-magnetic coupling, the number of non-conforming physical observations cannot be solved by such matter. The galaxies not only have a rotation that is unexplained by GR, but the galaxies interacting in clusters, and the clusters of galaxies interacting in superclusters could not simultaneously be described by such matter regardless of its distribution patterns. Additionally, gravitational lensing observed due to galaxies and clusters of galaxies could not be described by GR simply by applying such conjectured matter. The number of non-conforming observations cannot be solved by adding matter or energy, so general relativity should be abandoned as a dead end. Newtonian gravity does not apply, and no known modification of Newtonian gravity describes all the observed interactions. Modern physics will only progress when GR is abandoned and my research based on special relativity is adopted. See
Article The Physical Basis of the Fine Structure Constant in Relativ...
Article Thomas Precession is the Basis for the Structure of Matter and Space
For some insights on dark matter see :
Article Cold Dark Matter and Strong Gravitational Lensing: Concord o...
Abbas Kashani added a reply
Dear and respected Preston Gan
Researcher in Guynn Engineering
United States of America
You answered my question very well. Thank you very much for your excellent and technical explanations. You made me proud and I am happy for you because you are a great scientist. Thank you Abbas
Jouni Laine added a reply
According to my theory, the influence of quantum entanglement on spacetime curvature could provide an alternative explanation for the gravitational effects attributed to dark matter in galaxies. Traditional models suggest that large proportions of invisible “dark matter” are required to account for the observed gravitational behavior at galaxy scales. This is because, under Newtonian gravity, the visible mass of galaxies cannot account for the gravitational forces observed, leading to the hypothesis that there must be additional, unseen mass—dark matter.
However, my research proposes that quantum entanglement could be influencing spacetime curvature in a way that mimics the effects of this “missing” dark matter. If quantum entanglement can alter the curvature of spacetime, it might enhance the gravitational pull within galaxies without requiring massive quantities of unseen matter. This would mean that the observed discrepancies at galactic scales could be due to quantum entanglement effects rather than vast amounts of dark matter.
In this view, while dark matter has been the dominant explanation, it might be possible that the gravitational anomalies are instead the result of entanglement-induced modifications to spacetime. This theory could offer a new perspective on why Newtonian gravity appears to behave differently at large scales, suggesting that the need for dark matter could be reconsidered in light of quantum effects on gravity.
Abbas Kashani added a reply
Dear Johnny Line, greetings and respect
You answered my question very well. Thank you very much for your excellent and technical explanations. You made me proud and I am happy for you because you are a great scientist. Thank you Abbas
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It is conceivable that the constant "G" varies according to where you are, but the only way to prove that is to be somewhere so far from here that we will never be able to prove it, which makes it a novel but scientifically pointless proposition (if there is no way to prove something, it cannot be considered scientifically reasonable because then you can invent thousands of explanations, only one of which (if any) that can be correct, which is a doomed explanation). "G" is certainly a constant everywhere within 30 thousand light-years from us, and there will never be any way to measure its value even at that distance, let alone hundreds of thousands or millions of light-years distant. So at the moment I would say that "dark matter" almost certainly exists IN GALAXIES, and possibly BETWEEN GALAXIES IN RICH CLUSTERS OF GALAXIES. However, whether it exists in the huge amounts posited by cosmologists EVERYWHERE is certainly "up in the air" in every sense of the phrase. And I'm reasonably certain that "dark energy" is a fantasy made up to explain something that doesn't need explaining.
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Does "dark matter" make up large proportions of those galaxies?
Newtonian gravity behaves differently at very large scales of mass and distance, i.e., galaxy scales, in contra-indication to the assumption that massive quantities of invisible, or "dark matter" make up large proportions of those galaxies.
… Read more
  • 717 kB27.pdf
Preston Guynn added a reply
Your discussion statement question is:
  • "Does 'dark matter' make up large proportions of those galaxies? Newtonian gravity behaves differently at very large scales of mass and distance, i.e., galaxy scales, in contra-indication to the assumption that massive quantities of invisible, or 'dark matter' make up large proportions of those galaxies."
The phrase "Newtonian gravity" refers to a very specific equation relating mass and acceleration, so saying it behaves differently under some condition is not a correct usage of the phrase. Newtonian gravity is Newtonian gravity, and it gives incorrect results at scales greater than the solar system. There is a significant body of research on modified Newtonian gravity, and you can find it by searching on the phrase or "MOND".
Your question"Does dark matter make up large proportion of those galaxies?" is the question that numerous branches of research are investigating either experimentally or theoretically. First of course is the search for any experimental evidence of any matter that couples gravitationally but not via the electromagnetic field. No evidence of any such matter has been found. Second is that there is no such matter expected from current models such as the so called standard model of physics.
Even if there were some type of matter that couples gravitationally but not via electro-magnetic coupling, the number of non-conforming physical observations cannot be solved by such matter. The galaxies not only have a rotation that is unexplained by GR, but the galaxies interacting in clusters, and the clusters of galaxies interacting in superclusters could not simultaneously be described by such matter regardless of its distribution patterns. Additionally, gravitational lensing observed due to galaxies and clusters of galaxies could not be described by GR simply by applying such conjectured matter. The number of non-conforming observations cannot be solved by adding matter or energy, so general relativity should be abandoned as a dead end. Newtonian gravity does not apply, and no known modification of Newtonian gravity describes all the observed interactions. Modern physics will only progress when GR is abandoned and my research based on special relativity is adopted. See
Article The Physical Basis of the Fine Structure Constant in Relativ...
Article Thomas Precession is the Basis for the Structure of Matter and Space
For some insights on dark matter see :
Article Cold Dark Matter and Strong Gravitational Lensing: Concord o...
Abbas Kashani added a reply
Dear and respected Preston Gan
Researcher in Guynn Engineering
United States of America
You answered my question very well. Thank you very much for your excellent and technical explanations. You made me proud and I am happy for you because you are a great scientist. Thank you Abbas
Jouni Laine added a reply
According to my theory, the influence of quantum entanglement on spacetime curvature could provide an alternative explanation for the gravitational effects attributed to dark matter in galaxies. Traditional models suggest that large proportions of invisible “dark matter” are required to account for the observed gravitational behavior at galaxy scales. This is because, under Newtonian gravity, the visible mass of galaxies cannot account for the gravitational forces observed, leading to the hypothesis that there must be additional, unseen mass—dark matter.
However, my research proposes that quantum entanglement could be influencing spacetime curvature in a way that mimics the effects of this “missing” dark matter. If quantum entanglement can alter the curvature of spacetime, it might enhance the gravitational pull within galaxies without requiring massive quantities of unseen matter. This would mean that the observed discrepancies at galactic scales could be due to quantum entanglement effects rather than vast amounts of dark matter.
In this view, while dark matter has been the dominant explanation, it might be possible that the gravitational anomalies are instead the result of entanglement-induced modifications to spacetime. This theory could offer a new perspective on why Newtonian gravity appears to behave differently at large scales, suggesting that the need for dark matter could be reconsidered in light of quantum effects on gravity.
Abbas Kashani added a reply
Dear Johnny Line, greetings and respect
You answered my question very well. Thank you very much for your excellent and technical explanations. You made me proud and I am happy for you because you are a great scientist. Thank you Abbas
Forrest Noble added a reply
18 hours ago
No ! Dark Matter, like Dark Energy, is simply a 'place holder' for an unknown source of energy which cannot presently be explained excepting via speculation and related hypotheses. If either or both do not exist, their replacement will do damage to, or also cause the replacement of mainstream cosmology, by far simpler but presently unrecognized alternative(s).
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Dark matter is widely believed to make up a significant proportion of galaxies, with estimates suggesting it comprises about 85% of all matter in the universe. This understanding is based on observations of gravitational effects that can't be explained by visible matter alone. However, it's important to note that dark matter hasn't been directly detected, and some scientists propose alternative explanations for these gravitational effects, such as modifications to our understanding of gravity or quantum effects. For now, the dark matter hypothesis remains the most widely accepted explanation among astrophysicists for the observed gravitational behavior in galaxies and larger cosmic structures.
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Does "dark matter" make up large proportions of those galaxies?
Newtonian gravity behaves differently at very large scales of mass and distance, i.e., galaxy scales, in contra-indication to the assumption that massive quantities of invisible, or "dark matter" make up large proportions of those galaxies.
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Dear Johnny Line, greetings and respect
You answered my question very well. Thank you very much for your excellent and technical explanations. You made me proud and I am happy for you because you are a great scientist. Thank you Abbas
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Hello,
I would like to know how to calculate the mass of an elliptical galaxy. If someone could explain the commonly used methods and formulas, it would be greatly appreciated. Additionally, could anyone provide me with the masses of the following galaxies:
M87 (NGC 4486) , M86 (NGC 4406), M60 (NGC 4649), M49 (NGC 4472), NGC 1316, NGC1399 and NGC 4889?
Thank you in advance for your help.
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Dear Michal Michalowski
Thank you very much for your helpful response. Your suggestions on using velocity dispersion for ellipticals and rotational velocity for rotation-dominated galaxies, as well as considering strong lensing for modeling total mass, are extremely helpful. I appreciate your time and expertise in addressing my question.
Best regards
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Is there a galactic rotation anomaly? Is it possible to find out the speed and time of the galactic rotation anomaly?
Abstract: Orbital speeds of stars, far from centre of a galaxy, are found roughly constant, instead of reductions predicted by current gravitational theories (applied on galactic and cosmological scales). This is called the anomalous rotation of galaxies. This article intends to show that constant angular speeds of all macro bodies in a galaxy are natural phenomenon and there is no mystery about it.
Keywords: Galaxy, Stable galaxy, rotational anomaly.
A planetary system is a group of macro bodies, moving at certain linear speed in circular path around galactic centre. Central body of planetary system is by far the largest and controls mean linear speeds of all other members. Gravitational attractions between macro bodies of planetary system cause perturbations in their directions of motion, resulting in additional curvatures of their paths. When perturbed paths of smaller macro bodies are related to central body in assumed static state, we get apparent orbital paths of planetary bodies. They appear to revolve around static central body in elliptical/circular paths. Apparent orbital paths are unreal constructs about imaginary static state of central body. They are convenient to find relative positions of macro bodies in the system and to predict cyclic phenomena occurring annually. In reality, planetary bodies do not orbit around central body but they move in wavy paths about the central body. Central and planetary bodies move at a mean linear speed along their curved path around galactic centre.
Perturbations of orbital paths of macro bodies in planetary system are related directly to their matter-content and inverse square of distance from central body. Distance from central body has greater effect of magnitudes of perturbations. Hence, normally, paths of planetary bodies at greater distance from central body are perturbed by lesser magnitudes. Curvatures and thus angular speeds of their apparent orbits reduce as distance from central body increases. Since planetary system has no real spin motion, this is an imaginary phenomenon. However, many learned cosmologists seem to take spin motion of planetary system as real phenomenon and consider that members of all spinning group pf macro bodies should behave in similar manner, i.e. angular (spin) speed of members should reduce as their distance from centre of system increases.
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Stable galaxy consists of many macro bodies revolving around its centre. This group can be considered as a spinning fluid macro body, rotating at a constant angular speed. Gravitational collapse initiates spin motion of galactic cloud and maintains constant spin speed of outer parts of stable galaxy. Centre part of galaxy, which is usually hidden, may or may not be spinning. We can observe only visible stars and their angular speeds about galactic centre. Linear motions of macro bodies, caused by gravitational attractions towards other macro bodies in the system, have two components each. One component, due to additional linear work invested in association with it, produces macro body’s linear motion, in a direction slightly deflected away from centre of circular path. Other component, towards centre of its circular path, is caused by additional angular work invested in association with it. This component produces angular motion of macro body.
All matter-particles in a fluid macro body, spinning at constant speed, have constant angular speeds. Consider a matter-particle at O, in figure 1, moving in circular path AOB. XX is tangent to circular path at O. Instantaneous linear speed of matter-particle is represented by arrow OC, in magnitude and direction. It has two components; OD, along tangent XX and DC, perpendicular to tangent XX and away from centre of circular path. This component, DC, represents centrifugal action on matter- C particle due to its motion in circular path. In
📷order to maintain constant curvature of path, X D O X matter-particle has to have instantaneous A linear (centripetal) motion equal to CE E
toward centre of circular path. If magnitudes B Figure 1 and directions of instantaneous motions are as shown in figure 1, matter-particle maintains its motion along circular path AOB at constant angular speed.
Should the matter-particle increase its instantaneous linear speed for any reason, both components OD and DC would increase. Component OD tends to move matter-particle at greater linear speed along tangent XX. Outward component DC tends to move matter-particle away from centre of its circular path. The matter particle tends to increase radius of curvature of its path. This action is usually assigned to imaginary ‘centrifugal force’. In reality expansion of radius of curvature of path is caused by centrifugal component of linear motion. Reduction in centripetal action also produces similar results.
Should the matter-particle decrease its instantaneous linear speed for any reason, both components OD and DC would reduce. Component OD tends to move matter-particle at lesser linear speed along tangent XX. Reduction in outward component DC tends to move matter-particle towards centre of its circular path. The matter particle tends to reduce radius of curvature of its path. Reduction of radius of curvature of path is caused by reduction in centrifugal component of linear motion. Increase in centripetal action also produces similar results.
In other words, matter-particle regulates its distance from centre of its circular path so that its angular speed remains constant. This is the reason for action of centrifuges. As linear speeds of matterparticles increase, they move outwards, in an effort to maintain their angular speed constant.
Additional work, done for linear motion of a matter-particle and additional work, done for its angular motion are entirely separate and distinct. Additional work for linear motion of a matter-particle can produce only linear motion and additional work for angular motion can produce only angular motion. In the case, explained above, increased in linear speed of matter-particle is considered. That is, additional work invested in association with matter-particle is of linear nature. It can only increase its linear motion. As no additional work for angular motion is invested matter-particle cannot change its angular speed. Instead, matter-particle is compelled to move away from centre of its rotation, so that it can increase magnitude of linear motion while keeping magnitude of angular motion constant.
Similarly, increase in centripetal effort invests additional work required for angular motion of matterparticle. Matter-particle tends to increase magnitude of its angular motion. Curvature of its path
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increases by reducing its distance from centre of circular path. Matter-particle tends to move towards centre of circular path, so that it can increase its angular speed while keeping its linear speed constant.
Every macro body in a stable galaxy behaves in a manner similar to matter-particle, represented in figure 1. They tend to position themselves in the system, so that their linear and angular speeds match corresponding works associated with them. Macro bodies strive to maintain their angular speeds constant by keeping appropriate distance from centre of rotation. Macro bodies towards the central region may experience additional centripetal effort. They might increase their angular motion and move towards central point to merge with black hole present there. In due course of time, macro bodies on outer fringes move away from galaxy and destroy its stability.
In a galaxy, various macro bodies arrive at their relative position gradually by error and trial, during which their relative positions and linear and angular speeds are stabilized. Galaxy, as a whole, stabilizes only when constituent macro bodies have reached their steady relative positions and motions. In order to maintain stability, it is essential to maintain relative positions of all constituent macro bodies by having constant and equal angular speeds and linear speeds corresponding to their distances from galactic centre. Change in relative position or linear or angular speed of even one macro body is liable to destabilize the galaxy.
As and when superior 3D matter-particles at the fringe of galaxies attain linear speeds approaching speed of light, they break-down into primary 3D matter-particles and produce halo around equatorial region. Halos of neighbouring stable galaxies interact to prevent their translational movements and maintain steady state of universe.
Therefore constant angular speeds of constituent macro bodies of stable galaxies are their natural states. There are no mysteries or anomalies about them. This phenomenon is mystified by those who consider imaginary spin motions of planetary systems are real. Therefore, assumptions of dark matter, time dilation, modification of gravitational laws, etc and complicated mathematical exercises are irrational and unnecessary to prove non-existing rotation anomaly of galaxies.
Conclusion:
Galactic rotation anomaly is a non-existing phenomenon derived from imaginary spin motions of planetary systems about their central bodies in assumed static states. Constant angular speeds of stars in a galaxy confirm static state of galactic center (in space), rather than produce an anomaly.
Reference:
[1] Nainan K. Varghese, MATTER (Re-examined), http://www.matterdoc.info
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Abbas Kashani > "Is there a galactic rotation anomaly?"
There is a galactic rotation anomaly, but only according to officially accepted theories of gravity and the (Big Bang) theory of the formation of the galaxies inferred for a finite, closed and a created (in the finite past) universe.
But all these theories based on causality and theology are wrong! The dialectical and scientific view is that the universe is Infinite, Eternal and Ever-changing, mediated by dialectical chance and necessity. Gravity is a dialectical contradiction of the unity of the opposites of attraction and repulsion (due to inherent free motion of matter particles, vis viva). In short (human) time scale, new galaxies are seen to be formed through the dissipation and/or ejection of matter in the form of stars, star clusters or even a large part of the galaxy as quasars from the existing galaxies.
So, the observed high orbital velocities of the starts, star clusters etc. at the periphery of the galaxies and of the planets at the periphery of the planetary systems within the galaxies is just a natural phenomena and there is no anomaly!
"Ambartsumian, Arp and the Breeding Galaxies" : http://redshift.vif.com/JournalFiles/V12NO2PDF/V12N2MAL.pdf
KEPLER -NEWTON -LEIBNIZ -HEGEL Portentous and Conflicting Legacies in Theoretical Physics, Cosmology and in Ruling https://www.rajpub.com/index.php/jap/article/view/9106
"THE CONCEPTUAL DEFECT OF THE LAW OF UNIVERSAL GRAVITATION OR ‘FREE FALL’: A DIALECTICAL REASSESSMENT OF KEPLER’S LAWS":
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Hello all,
I wanted to know, can I use galaxy (USA, Europe or Australia) platform for analyzing the shotgun data, and can it be used for publication purpose as well?
Thanks :)
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The use of trdtional scientfic rules remain the most accurate scientfis one.....while Galaxy .org remain supportive
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AGN (Active Galactic Nuclei) are high energy galaxies powered by supermassive blackholes. The evolution of a galaxy is characterized by parameters such as luminosity and redshift. The redshift is a measure of how far away a celestial object such as a star is away from earth as a frame of reference.
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gravitational Redshift is not used to determine distance. It is instead due to the Genera Theory of Relativity.
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"The structure and function of the kidneys is altered by space flight, with galactic radiation causing permanent damage that would jeopardise any mission to Mars, according to a new study led by researchers from UCL"
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Yes. Astronauts because of the thrill of exploration, rich people partly for the thrill, and partly as a way to show off how insignificant it is to them to throw away tens of millions of dollars for whatever they want to do. Astronauts, of course, are depending on NASA and similar agencies to figure out ways to reduce the dangers of space travel, and space tourists aren't really out "there" long enough for it to significantly increase their risk of disease, because if the required physical shows that they suffer from a problem that MIGHT be significantly affected, they aren't allowed to go, regardless of how much money they offer (those providing the rides don't want even one customer to die, as it would be bad for business).
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Will artificial intelligence help analyze images taken by space supertelescopes and help identify other life forms on distant exoplanets?
Will generative artificial intelligence technology help analyze images taken by space supertelescopes and identify other life forms on distant exoplanets millions of light years away located in other planetary systems, constellations, galaxies?
Space supertelescopes, including one of the most modern and powerful space telescopes, which is the James Webb supertelescope, take many images of galaxies, suns, nebulae, etc., millions of light years distant. In distant galaxies, of which there are millions if not more in the Universe, there are many constellations numbering in the billions, planetary systems that contain many exoplanets. Many of these billions of exoplanets orbiting other suns in other planetary systems are similar in many ways to our plaenta Earth. For many thousands or millions of these exoplanets, the similarity of chemical element composition, physico-chemical conditions, temperature levels, chemical composition of inorganic compounds, atmospheric processes, surface formation, possible presence of water or highly alternative chemical-physical structures, etc. with what is found on Earth may be so great that it is highly likely that life is or has been found on many of these exoplanets. Most likely, these are different life forms to those we know. The dissimilarity of these life forms is determined by different conditions of physical and chemical processes, different composition of chemical elements, different chemical compounds, different atmospheric processes, different temperature ranges, different calendar of rotation around other suns, etc. Perhaps on some of these exoplanets where other life forms arose other intelligent beings also arose. Perhaps on some of these exoplanets where evolved life created other intelligent beings are also present advanced civilizations created by said other intelligent beings. Humanity has been searching for many years for answers to questions about the possible existence of other forms of life, other intelligent beings, other civilizations on distant exoplanets.For years, space supertelescopes have been involved for this purpose, which successively over time as space exploration technology advances, take more and more perfect photographs of more and more distant celestial bodies, galaxies, constellations, planetary systems, exoplanets. This produces a huge amount of data contained in the thousands or millions of photographs taken in this way. It would take many years for a human to analyze such a large amount of data contained in these photographs. Industry 4.0/5.0 technologies, including Big Data Analytics and generative artificial intelligence, can help analyze these large data sets contained in the aforementioned many photographs.
I described the applications of Big Data technologies in sentiment analysis, business analytics and risk management in an article of my co-authorship:
APPLICATION OF DATA BASE SYSTEMS BIG DATA AND BUSINESS INTELLIGENCE SOFTWARE IN INTEGRATED RISK MANAGEMENT IN ORGANIZATION
I described the key issues of opportunities and threats to the development of artificial intelligence technology in my article below:
OPPORTUNITIES AND THREATS TO THE DEVELOPMENT OF ARTIFICIAL INTELLIGENCE APPLICATIONS AND THE NEED FOR NORMATIVE REGULATION OF THIS DEVELOPMENT
In view of the above, I address the following question to the esteemed community of scientists and researchers:
Will the technology of generative artificial intelligence help to analyze images taken by space supertelescopes and to identify other forms of life on exoplanets millions of light years away located in other planetary systems, constellations, galaxies?
Will artificial intelligence help analyze images taken by space supertelescopes and identify other life forms on distant exoplanets?
Will artificial intelligence help identify other life forms on distant exoplanets?
What do you think about this topic?
What is your opinion on this issue?
Please answer,
I invite everyone to join the discussion,
Thank you very much,
Best wishes,
Dariusz Prokopowicz
The above text is entirely my own work written by me on the basis of my research.
In writing this text, I did not use other sources or automatic text generation systems.
Copyright by Dariusz Prokopowicz
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AI can assist in sifting through vast amounts of data to identify patterns that may indicate the presence of life, such as changes in atmospheric composition or irregularities in planetary characteristics. It can also help in modeling and simulating various conditions that might support life.
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Wondering if I should get an individual subscription to Article Galaxy, seems like a useful tool for accessing and organizing papers. Let me know your thoughts.
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Article Galaxy is user friendly and fast, has a responsive support team and is overall flexible with a Reprints feature, which allows easy ordering of articles. Additionally, in a 2019 review, Article Galaxy was evaluated as a useful tool for researchers, universities, and large enterprises. It covers content, technology, and overall value. With all said, there are still some disadvantages, such as personalized access and quick sourcing from Reprints Desk, it lacks a built-in reader and offline access.
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Today, I published an article about First Motion. It is based on the notion that all matter is always on the move; how there is no matter that exists at a standstill.
Then, viewing Einstein's GR, this motion is not incorporated in his work. As a result, we long for an ether, or a spacetime with properties of its own, to explain all motions of matter.
For instance, there needs to be more gravity to hold a galaxy together if we follow Einstein's GR. Yet with First Motion, all known gravity is indeed all we need.
Will you engage me in a discussion on this subject matter? Did Einstein miss out on the most important motion that matter is involved in?
Find attached the article I published today. Note that Einstein's mathematical framework is not in question. Rather, the First Motion turns that framework inside out (and remains correct therefore). All of a sudden, there is no need anymore for all that extra gravity to hold a galaxy together.
Thank you for your help in discussing this question.
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Thank you for this reply, Larissa, but your good information is addressing the surrounding realities and not the exact mechanism as proposed. The specifics of the exact proposal appear not understood.
Indeed, the universe is not a medium in this proposal (i.e. the universe itself does not have a gravitational field).
The proposal is therefore fully focused on the behaviors of matter while this is moving through space. To appreciate this, one should pause all other scientific thoughts about space and spacetime. The focus is truly based on the mechanisms of matter.
Four distinct motions for matter moving through space are identified, and one of these motions is declared as non-gravitational, whereas the three other motions are identified as gravity informed.
Among these four motions of matter, the one not based on gravity is the prime mover; it is the 'sent-off' push that the materialization process (Big Bang) caused to happen and that is ongoing. It is the fastest speed matter is involved in.
The visualization of just planet Earth in these four motions is (going from smallest/slowest to largest/fastest):
4. Earth spinning
3. Earth revolving around Sun
2. Earth circling together with Milky Way in its circular motion
1. Earth speeding in a single direction, away from where it all started for this matter/energy.
That First Motion (fourth in the list) applies to all matter in the universe. There exists no matter in the universe at a standstill. One of the motions is a straight line through space, the other motions are circular/elliptical in essence.
So thank you once more for replying and providing good information. Yet the specific point the article is trying to show is that matter already by itself is involved in gravitational and non-gravitational behaviors.
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Dear Friends, awesome finding. The 127th finding in my TOU (Theory of Universality) is that the electrons are space-waves just like gravitons. If we take the mass of the graviton as 2.89186 x 10^-38 kg, the speed of graviton as 4.586509 x 10^6 m/s, the surface speed of electron in hydrogen atom as 2.2 x 10^6 m/s and the surface speed of electron as SQRT(-25.963) x linear speed. Now, if consider the gravitons and the electrons in the space-time frame of Milky way Galaxy, and keeping in mind that the electron has 3 degrees of freedom of movement, the mass of electron works out to be 9.11941635 x 10^-31 kg; which agrees with the observed value of 9.1093837 x 10^-31 kg. To be published in next Annexure.
Pl join :
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It will be published in the next Annexure of my TOU.
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In "Remembering V.M. Slipher’s work at Lowell" (April 2020 - https://www.astronomy.com/science/remembering-v-m-sliphers-work-at-lowell/), Astronomy magazine's editor David J. Eicher writes that redshift indicates the velocity of recession, and also says USA astronomers Vesto Melvin (known as V. M.) Slipher and Edwin Hubble * discovered the expanding universe. My opinion is that the Editor is right to give V. M. Slipher credit for a major cosmic accomplishment but mistaken about the nature of that accomplishment. I also think Slipher and Hubble didn't discover the expanding universe because redshifted spectral lines don't indicate recession of astronomical bodies. Rather, Slipher found the first proof of Albert Einstein's General Relativity Theory - beating Arthur Eddington's 1919 eclipse measurements, and even beating the full General Relativity theory's 1915 publication by at least a year.
* Edwin Hubble (1889-1953), the astronomer credited with discovering cosmic expansion, remained doubtful about the expansion interpretation for his entire life. He believed “expanding models are a forced interpretation of the observational results.” (E. Hubble, “Effects of Red Shifts on the Distribution of Nebulae”, Ap. J., 84, 517 [1936]) According to astronomer Allan Sandage, "Hubble believed that his count data gave a more reasonable result concerning spatial curvature if the redshift correction was made assuming no recession. To the very end of his writings he maintained this position, favouring (or at the very least keeping open) the model where no true expansion exists, and therefore that the redshift "represents a hitherto unrecognized principle of nature." (Sandage, Allan, "Edwin Hubble 1889–1953", The Journal of the Royal Astronomical Society of Canada, Vol. 83, No.6 [1989])
V. M. found evidence of Relativity's gravitational redshift, which was found by Einstein 8 years before his full theory. The term "gravitational redshift" refers to the shift of wavelength of a photon to longer wavelength (the red side in an optical spectrum) when electromagnetic waves radiate within a gravity field. Relativity says gravity results from the curvature of space-time. Therefore gravity is spacetime - and the more spacetime there is between Earth and a star or galaxy, the greater is the gravitational redshift. The redshift has nothing to do with velocity but increases with distance. We should not be surprised that light waves do not follow exactly the same Doppler effect as sound waves, since they have a different form. Light and other electromagnetic waves are transverse while sound waves are called compressional or longitudinal. Andromeda galaxy's apparent approach is related to Slipher's finding that "spiral nebulae" (as galaxies were called 110 years ago) are rotating. A large, close galaxy like M31 (Andromeda) would appear to be approaching us because it isn't far enough away to send us light that's significantly redshifted; but a huge number of its stars are currently approaching us as they orbit Andromeda's centre, and therefore sending us blueshifted light.
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The statement you make, and I quote, "Relativity says gravity results from the curvature of space-time. Therefore gravity is spacetime," appears to me to be overgeneralized because it does not refer to the mathematical proofs for the gravitational laws of inertial motion at all. Nor does it refer to the difference between measuring gravitational force and measuring energy production relative to the velocity of light squared.
Isaac Newton wrote that Force equals Mass multiplied by Acceleration. Albert Einstein showed that there is a different way to express the same idea so that it is mathematically more precise and accurate. Newton maintains a God-like overview perspective, a telescopic point of view, if you wish, while Einstein takes into account, as you mention, the curvature of space, as a result of his discovery of the photoelectric effect law.
Newton's mathematical calculations only provide a general approximation, while Einstein's mathematical equations achieve factual accuracy and precision, not only because they take into account the curvature of space, but also because they are based on points of observation inside of the spacetime continuum, that is, they adopt a submicroscopic perspective.
Best regards,
Nancy Ann Watanabe
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Light should be dispersion in the gravitational field:
Therefore, to have a right measurement of the numbers and spectrogram of the celestial objects and to have an accurate measurement of the size and distance of a celestial object, light dispersion in gravitational field need be considered.
For example, as one galaxy with multi-wavelengths was observed as several galaxies with one wavelength, it should result in that the numbers and spectrogram of the galaxy is wrong. Further, it should result in that the conclusion about the origin and the element making up of the galaxy is wrong.
And, the observed size and distance of the galaxy are not accurate.
Therefore, astronomical observation need be reconsidered as Light dispersion in gravitational field is known.
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Dear Zhu, Yin Zhu
'Light should be dispersion in the gravitational field'
Yes of course
Here's a theory of gravity:
(This Hungarian text is full of grammatical errors, and there are also conceptual errors! The language was poorly proofread. During the linguistic proofreading, such a correction was made, The essence of a well conceived concept has failed. All this was revealed during the translation into Romanian. etc.)
This theory prediction:
The light dispersion in the gravitational field of rocky plnets decreases as the light approaches to their surface!'
with the help of the light dispersion, the gravitational field of planets can be described mathematically!. (With this concept, under certain conditions, the gravitational influence of exoplanets can be determined!)
This phenomenon can be used to predict earthquakes!
)
So seems well that the mentioned writing ccould be valuable even it has the present erroneous form.
The next was written by my former co-author onto base of conluhion:
'’No quark
No, contradict you!
Our theory extended
It has Newton and Einstein too..’
Regards,
Laszlo
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Does NFW profile work for any galaxy?
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You're welcome!
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In your opinion, what is the essence of life in the context of the Universe, i.e. in the context of other life forms potentially occurring on many distant exoplanets relative to the known life occurring on planet Earth?
How do you think very different forms of life might exist on distant exoplanets relative to the life forms we know?
Given what we know and what we don't know about the cosmos, how different do you think life forms might be on distant exoplanets relative to the known life forms found on planet Earth?
Considering how different environmental, climatic, geological conditions, the composition of elements and inorganic and possibly also organic compounds, etc. on distant exoplanets may be under many or even all of the categories known to us, how different life forms may exist on these other planets located many millions of light years from our solar system?
According to some astrophysicists, what we do not know about the cosmos is estimated to be 96 per cent. Included in this is, for example, the essence of dark matter beyond the Universe as we know it and dark matter causing the expansion of the Universe as we know it. Life in its essence is defined in an ambiguous way. Depending on whether the definition refers only to life forms found on planet Earth or to possible other life forms that may exist on distant exoplanets, the definition of life is not necessarily the same.
Depending on what role the billions-of-years-long process of evolution of life forms on planet Earth has played in the development of life forms and the resolution of the question of the randomness of the emergence of life on different exoplanets or the intentionality of the evolutionary process aimed, for example, at the to the creation of more and more complex forms of life, forms of life increasingly adaptable to specific, changing environmental conditions of a specific exoplanet, increasingly better adapted to different environments, and to the possibly intentional or accidental bringing about of intelligent beings, including beings forming organised civilisations, changing the environment of the planet and aiming at space exploration, colonisation of other exoplanets similar to their home planet. These eventualities to be resolved that have not been resolved are many. Consequently, defining the essence of life in the context of possibly other extraterrestrial life forms on many exoplanets is not uniform. In the context of potential completely other, unknown forms of extraterrestrial life that probably occur on many distant exoplanets, life can be defined as a process of spontaneous, self-contained, independent, organised processing of matter based on energy acquired from the environment and forming, through this process, more complex structures of specific chemical compounds, elements of matter available on a specific planet and adapting to the more or less variable environment of the planet, etc. It is likely that many questions will be answered when the first evidence of the existence of other forms of extraterrestrial life occurring on many distant exoplanets emerges. There are already more than 5 500 confirmed existing exoplanets, and there is already fragmentary information about another 9 000, also suggesting their existence. This knowledge has been building up very rapidly over the past decade or so. On some of Jupiter's moons, there are deep water oceans many kilometres deep beneath the icy crust, in which specific but as yet unknown chemical compounds and perhaps certain other forms of life exist. For example, there are planets in our Solar System with largely different environments to planet Earth. For example, the atmosphere of Venus contains mainly sulphuric acid. But does this rule out the existence of some firm, less organised, procariot-type life forms. Not necessarily.
In view of the above, I address the following questions to the esteemed community of scientists and researchers:
Given what we know and what we do not know about the cosmos, how much different environmental, climatic, geological conditions, composition of elements and inorganic and possibly also organic compounds etc. on distant exoplanets can be, how much different life forms can exist on these other planets located many millions of light years from our solar system?
Given what we know and what we don't know about the cosmos, how very different life forms might exist on distant exoplanets to the known life forms found on our planet Earth?
How do you think very different life forms might exist on distant exoplanets to the life forms we know?
What do you think is the essence of life in the context of the Universe, i.e. in the context of other life forms potentially occurring on many distant exoplanets in relation to the known life occurring on planet Earth?
In your opinion, what is the essence of life in the context of the millions of planet Earth-like exoplanets found in the Universe?
What is the essence of life in the context of the Universe?
What do you think about this topic?
What is your opinion on this subject?
Please respond,
I invite you all to discuss,
Counting on your opinions, on getting to know your personal opinion, on an honest approach to the discussion of scientific issues and not the ready-made answers generated in ChatGPT, I deliberately used the phrase "in your opinion" in the question.
The above text is entirely my own work written by me on the basis of my research.
I have not used other sources or automatic text generation systems such as ChatGPT in writing this text.
Copyright by Dariusz Prokopowicz
Thank you very much,
Best regards,
Dariusz Prokopowicz
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I don't know the best answer for it. But good answer is Earth-like exoplanets are our colonial target to sustain by terraforming with the help of asteroid impact to create magnetosphere -air-aqua-atmosphere like our Earth. Simply,Earth is the prototype to examine and validate this feasibility.
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The standard model of cosmology presents a rather dismal picture of the future of the universe with dark energy and accelerated expansion leading to a cold death with no stars. The results of the JWST will be able to show the spectrum of light from the distant galaxies and cosmologists confidently predict that there will only be hydrogen and helium. When the results show that these galaxies are much older than the universe we will have to look for a new model of the evolution of the universe.
The key to understanding the evolution of the universe is to recognise that it is finite with a space boundary. When we think we are looking back to the beginning of time we are really looking out towards the observation horizon. The first galaxy formed around 126 billion years ago and the number of galaxies increases by a factor of 20 every 14 billion years.
In this model of the evolution of the universe the Cosmic Microwave Background is coming from radiating matter held at the event horizon of the universe. This has the effect of ensuring that all the matter formed within the event horizon stays within the event horizon. The expansion rate of the universe is constant. Renewal comes from galaxy mergers and dark matter provides neutrons for star formation.
This is a much more optimistic view of the future of the universe which can continue in a stable state for the indefinite future.
Richard
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Suresh Wanayalaege I don’t agree with the model of the proton as three quarks. In the Space Time Wave theory the proton is three wavelengths of a looped wave in the medium of space.
Richard
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Black Holes out of a galaxy: do they exist???
🔴➣➣The question is as follow.
Are there black holes (also binary or system of more than two elements) outside the confines of a galaxy{*}, in the spaces between one galaxy and another??? 
{*}Galaxy is not meant only the Milky Way but any type of galaxy. In what way can be identified and/or measured these hypothetical extragalactic black holes???
🔴➢➢Il quesito è il seguente. 
Esistono buchi neri (anche binari o sistemi a più di 2 unità) al di fuori dei confini di una galassia{*}, negli spazi tra una galassia e l'altra??? 
{*}Galassia non viene intesa la sola Via Lattea ma qualsiasi tipo di galassia.
in che modo possono essere individuati e/o misurati questi ipotetici buchi neri extragalattici???
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🟥➢➢Moreover there are other related topics in this multiple RG Open question:
Are the singularities of the universe correctly counted??
Is the mass of the singularities of the Universe correctly evaluated and evaluable??
What are major singularities (ex: black holes) made of??
Time**, in this Universe, does not flow constantly but is directly a function of the density of the Universe itself measured at the various evolutionary stages??
**When this Universe is very expanded (example ... > 10 billion years) then time would proceed much faster than when the Universe was much less expanded (example ... < 1 billion years). Does this mean that counting time with the current average density of the Universe would give a distorted estimate of the overall age of the Universe?? Could the first billion years of the universe have lasted billions of current years due to the very high density of the universe itself at that age??
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Previous POSTS:
►https://www.facebook.com/SalVi.SalvatoreVicidomini/posts/2378526012179048
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According to the paper below❌ then one could well guess-speculate that the brightest and oldest QUASARs that we observe today are the result of mega collisions between numerous galaxies in formation and with central hypermassive black holes. Mega-collisions occurred at the dawn of the formation of galaxies, when the size of the Universe was much smaller than the current one (high density of the galaxies themselves). This would have created such colossal black holes very early that today they would represent a completely different entity from the common hyper-massive black holes. Would these immense black holes, all extra-galactic, be the size of a medium-small sized Galaxy??
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❌Galaxy interactions are the dominant trigger for local type 2 quasars.
Monthly Notices of the Royal Astronomical Society, Vol.522.
ArXiv:2303.15506
DOI:10.1093/mnras/stad455.
ABSTRACT
The triggering mechanism for the most luminous, quasar-like active galactic nuclei (AGN) remains a source of debate, with some studies favouring triggering via galaxy mergers, but others finding little evidence to support this mechanism. Here, we present deep Isaac Newton Telescope/Wide Field Camera imaging observations of a complete sample of 48 optically selected type 2 quasars – the QSOFEED sample (⁠L[OIII]>108.5L⊙
⁠; z < 0.14). Based on visual inspection by eight classifiers, we find clear evidence that galaxy interactions are the dominant triggering mechanism for quasar activity in the local universe, with 65+6−7
per cent of the type 2 quasar hosts showing morphological features consistent with galaxy mergers or encounters, compared with only 22+5−4
per cent of a stellar-mass- and redshift-matched comparison sample of non-AGN galaxies – a 5σ difference. The type 2 quasar hosts are a factor of 3.0+0.5−0.8
more likely to be morphologically disturbed than their matched non-AGN counterparts, similar to our previous results for powerful 3CR radio AGN of comparable [O III] emission-line luminosity and redshift. In contrast to the idea that quasars are triggered at the peaks of galaxy mergers as the two nuclei coalesce, and only become visible post-coalescence, the majority of morphologically disturbed type 2 quasar sources in our sample are observed in the pre-coalescence phase (61+8−9
per cent). We argue that much of the apparent ambiguity that surrounds observational results in this field is a result of differences in the surface brightness depths of the observations, combined with the effects of cosmological surface brightness dimming.
CONCLUSIONS
Our deep imaging observations of nearby type 2 quasars provide strong evidence that galaxy interactions are the dominant triggering mechanism for quasars in the local universe, consistent with the results for other samples of nearby radio-loud and radio-quiet quasars that have been observed to a similar surface brightness depth. Much of the apparent ambiguity of the results in this field is likely to be due to differences in the surface brightness depths of the observations combined with the effects of cosmological surface brightness dimming. Clearly, it is important that these factors are given full consideration in future studies of quasar triggering.
Beyond the dominance of galaxy interactions, there appears to be a wide range of circumstances under which luminous, quasar-like AGN are triggered. Although our results indicate that the gas flows associated with galaxy interactions can provide sufficient mass infall rates to the central SMBH to trigger quasar activity even well before the two nuclei have coalesced, some objects are triggered in a post-coalescence phase. Moreover, a minority of our sample are disc galaxies that appear undisturbed in deep imaging observations. Therefore, secular processes may sometimes be capable of triggering quasar activity, even if this is not the dominant mechanism at low redshifts.
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With the James Webb Telescope, will we learn the answer to age-old questions about the existence of extraterrestrial life forms, other life forms, intelligent other living beings, other civilisations operating on exoplanets billions of light years away, located in other planetary systems, in other constellations, other stellar constellations, located in other galaxies, galactic nebulae, etc.?
For centuries, man has been searching for an answer to the question of whether life originated and developed only on planet Earth, or whether it originated beyond Earth and came to Earth, e.g. whether life originated and developed only on planet Earth, whether it originated beyond Earth and came to Earth, e.g. in the form of simple microorganisms in the rocks of meteorites falling to Earth, whether it also developed in many other places in the Universe, whether life also developed, used to function and/or still functions, whether other forms of life developed, whether other intelligent life forms developed, whether these other intelligent life forms have created other civilisations on other exoplanets many millions of light years distant, located in other planetary systems, in other constellations, other star constellations, located in other galaxies, galactic nebulae, etc.? ? For several decades now, as man has been building ever more perfect space telescopes and listening for sounds from distant corners of the Universe, the possibilities of answering these questions have been gradually increasing. On the one hand, cosmologists, astronomers, researchers into astrophysics, the biosphere of the cosmos, etc., are trying to estimate the number of exoplanets which are at a similar distance from their suns as the planet Earth, have a sufficiently abundant amount of various elements and chemical compounds, and have the potential for the emergence of at least simple forms of life. On the other hand, it has still not been possible to hear a sound from space that would be evidence of the existence of another civilisation. It has still not been possible to photograph exoplanets millions of light years away in such a way, with sufficiently close-ups in the photographs, that traces of other life forms can be seen. But technological advances in the possibilities of space exploration and research are continuing relentlessly. Recently, the most advanced space telescope to date, called the James Webb telescope, was placed in Earth's orbit. The James Webb Telescope is currently the most perfect device ever created by man to photograph distant cosmic objects, distant galaxies, constellations, planetary systems and also exoplanets, millions and billions of light years away. This telescope is superior to the previously used Hubble Telescope for infrared observations. The James Webb Space Telescope was built between 2007 and 2021 a space telescope for observing and taking pictures of distant space objects mainly in the infrared. In a sense (observing the Cosmos only in the infrared range), the James Webb Space Telescope is intended to be the successor to the Hubble Space Telescope. Overseen and largely funded by NASA, the project was developed in collaboration with ESA and the CSA. With the James Webb Telescope, it is possible to photograph and study exoplanets located in other planetary systems, in other constellations and to photograph distant galaxies up to 13 billion light years away from Earth. As a result, the James Webb Telescope is already being described as a kind of cosmic time machine, as it takes pictures of galaxies billions of light years away, the photographed image of which shows these galaxies as they looked those billions of years ago. The James Webb Space Telescope was placed into Earth's orbit at the end of 2021, and to date has photographed many cosmic objects that were previously virtually invisible to humans in any way. The date for the launch of this telescope has been postponed several times since 2007, because due to the global financial crisis of 2008 and other economic issues, there was not enough money to complete the project in the previous years before 2021. The cost of building the James Webb Space Telescope was estimated to be around USD 10 billion. In view of the above, the James Webb Space Telescope is the most powerful telescope ever built by man to be placed in Earth's orbit and the most perfect device for taking photographs of space objects billions of light years away. Perhaps, thanks to the James Webb telescope, man will finally get an answer to the above questions.
In view of the above, I address the following question to the esteemed community of scientists and researchers:
With the James Webb Telescope, will we learn the answer to the age-old questions regarding the existence of extraterrestrial life forms, other life forms, intelligent other living beings, other civilizations operating on exoplanets billions of light years away, located in other planetary systems, in other constellations, other stellar constellations, located in other galaxies, galactic nebulae, etc.?
What do you think?
What is your opinion on this subject?
Please respond,
I invite you all to discuss,
Thank you very much,
Best regards,
Dariusz Prokopowicz
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Dariusz Prokopowicz I think that is unlikely that the JWST will actually detect evidence of life on other planets. I do think that the JWST observations together with rational scientific analysis will show that the Big Bang story of the evolution of the universe must be discarded.
If you think about the hypothetical cloud of gas and dust condensing into stars and then galaxies it just doesn’t match what we observe in the JWST observations of well formed galaxies with a look back time in excess of 13 billion years.
Freed from the constraints of the Big Bang theory we can develop a much better understanding of the formation of the solar system which will in turn give a fresh appreciation of how unique and unlikely is the planet Earth.
link
Richard
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The Andromeda galaxy is approximately 2.5 million light years away and is approaching the Milky Way galaxy at around 110 km/s. We know that the motion of galaxies is affected in two ways. One is the expansion of space and the other is the acceleration due to gravity.
If we take the hypothesis that the Milky Way and Andromeda were relatively at rest at some time in the past we can envisage what would then happen. Initially the expansion of space would take the galaxies further apart. Then the gravitational attraction between the two galaxies would accelerate the galaxies towards each other so that eventually the velocity due to this acceleration would exceed the effect of the expansion pf space. The galaxies would then move towards each other. We know that they are now at a distance of 2.5 million light years travelling at 110 km/s so we can model their past history.
The problem is that the calculation shows that the galaxies started at rest 53.9 billion years ago.
This is fine in a model in which the first galaxy formed 126 billion years ago:
But it does present a problem if you think the universe is 13.8 billion years old.
The spreadsheet model tracking the motion of Andromeda is also useful because it illustrates the dynamics of the combined effect of the expansion of space and gravitational acceleration. This is helpful when explaining how spiral galaxies form from two spherical regions of gas and dark matter. This also solves the angular momentum problem first identified by Fred Hoyle which questions the cause of the rotation of spiral galaxies.
Richard
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“…The relative motion of the Milky Way galaxy and the Andromeda galaxy suggests that the universe must be at least 53.9 billion years old.
I would be really interested to see if anyone can resolve the Andromeda problem in the context of the Big Bang theory in which the universe is only 13.8 billion years old.….”
- generally speaking the age of a galaxy is rather well estimated by its stars nomenclature, for what is observable in Milky Way and Andromeda seems as rather convincingly it follows that they are ~ 10 billions of years old. More aged – “old” galaxies in observable Matter are not too far from this value, and, though seems nobody can too in detail describe what are galaxies if are, say, yet ~20 billions of years old, now seems as rather rational to suggest from cosmological observations, that such galaxies would be rather dim objects in Space, while the galaxies above are well seen.
- are relevant to this thread question.
Cheers
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Aberration of starlight, was discovered and measured by Bradley at the end of '700.
Bradely detected a constant angle of tilting of the telescope 20", the tangent of it corresponds to v/c (he was aware of the estimation of the speed of light made by Romer) with v = 30km/s is the speed of Earth around Sun.
Aberration is due to a change of position of the observer in regard to light rays while they cross the telescope. Because the telescope moves a distance vh/c during the time it takes light to travel down the tube, the tube must be tilted in the direction of motion for the light to move along the optical axis of the tube.
An indirect measurement of Earth's speed rotation was found by Bradley.
Bessel in 1838 managed to implement the stellar parallax and needed to deal with aberration. He excluded that the motion of the stars could make any difference, due to their distance hence corroborating Bradely's explanation of the phenomenon.
In '800 when the Luminiferous Aether was in vogue, many including Stokes tried to provide an explanation with aether dragged but it was not found compatible.
Einsteins in 1905 provided an explanation of it with the Lorentz Transformations. The angle as v/c comes from a first approximation of the formula of aberration in SR, where v is the speed of the observer in the reference frame of the source.
This demonstration considers also the fact that since stars are quite far away so rays come parallell as if they come from infinite distance.
There is also an aberration due to rotation of Earth, but in this case it is quite small since the speed of rotation of earth is lower than .5 km/s on its surface hence the value of the aberration due to it is about 60 times smaller, depending also on the position on earth of the telescope.
Only in 1925 Hubble discovered that we are part of a galaxy and after that it was clear that the relative speed of earth and stars can reach 400 km/s.
So why stellar aberration should depend only on the speed of the earth around the Sun (and around itself) but is virtually independent of the speed of Sun in the galaxy?
The satisfactory answer given by some partecipants seems to be:
Bradley made a certain observation with a certain inclination of the telescope on Earth's orbit such that he would only spot close enough stars which are virtually in the same restframe as the solar system.
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"So why stellar aberration should depend only on the speed of the earth around the Sun (and around itself) but is virtually independent of the speed of Sun in the galaxy?"
To simplify what Sergey Sheludko and Jonathan Doolin has explained - in order to measure annual aberration (due to the Earth's orbital motion) you need to wait half a year (or so) to have the Earth moving in a different direction, so you can measure the change in the positions of stars. In order to measure the galactic aberration (due to the Sun's orbital motion around the galactic centre) you need to wait half a galactic year, so around 100 million years. This is not practical, so if you wait just a couple of years the direction of motion of the Sun is almost the same, so the change in the positions of stars due to this effect is negligible.
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The other day i saw a picture of a galaxy inside galaxy. I can create a galaxy through a radar graph but couldn't in creating a galaxy inside galaxy. Is there any theory to support this phenomenon?
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The ring galaxy in the linked image has been known of for years and years, as well as the mechanism of its formation by another galaxy running head-on into the core of the galaxy in the image, stripping it of most of its gas and causing the shock-wave from the collision to form a ring around the previous core, which now looks like an elliptical galaxy. There are quite a few of these around, and in some cases, the galaxy that ran into the ring galaxy is still close enough to it to tell that it was the culprit. Hoag's Galaxy is unusual, though, in that few ring galaxies have such a perfectly circular ring; usually there is some kind of distortion, and of course if not seen "face-on", as in the case of this galaxy, the circle looks like an ellipse.
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Construction and Destruction words are complex and conjugate quantities such as life and death , begining and end etc. Birth of a child and formation- of Universe and Destruction of a child and Universe concerned to Quantum and Quantization process.
My researches on the field of Cosmology have discussed very well such secret issues , especially a research paper " Quantum and Quantization (Theory of Everything") .
Spiral galaxies differs from other galaxies.Their formation and Destruction dynamics are different.But one thing is sure that this all concerned to Quant and Quantization process.
Here I am attaching my gew papers to solve such problems.
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We know that the massive stars at the ends of their lives turn up to be black holes, and we know that in the center of each galaxy there's a massive or super massive black hole. Does this mean that we can say that the galaxy is a remnant of that star after it exploded?
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Yeah.. The central black hole of galaxies are supermassive.. when star explode it turns into black hole but there are few stars which turns into supermassive black hole depends on the mass of the star..
Basically these supermassive black hole surrounds all near space stuff to its gravitational field even stars , small black holes , pulsars.. which makes whole galaxy.
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Hello,
The NASA/IPAC Extragalactic Database gives two velocities for each galaxy: The velocity (helio) and velocity (CMB).
I need just to know, if possible, the difference between these two velocities.
Thanks in advance.
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Dear honorable Forrest Noble,
Thank you very much for your response. I have other questions for you, if possible.
1/ Is the trajectory of the proper galaxies movement known?
2/ In a group of galaxies such as the M87 group for example, do the galaxies orbit around the center of the group?
3/ The CMB velocity of a galaxy, represents the velocity of its gravitation around the cluster ?, its proper velocity ?, or the velocity of its movement related to the Hubble constantt?.
Best regards
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Hello everyone!
If anyone working with galaxy cheminformatics server for virtual screening of compound library could help me out please let me know that:
I've screened a library of compounds through vina tool of galaxy by giving input files: protein.pdbqt, conf.txt and library.sdf (with valid chunk sizes). I opted output format .pdbqt instead of .sdf assuming to visualize the poses more efficiently. Like for sdf output, the results can be compiled as csv from galaxy, is there any way to do the same for pdbqt output format as well? Since I want all the ligands in a single tabular form to shortlist on the basis of binding energy/docking score.
please suggest some alternative.
Thank you in advance!
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No, you have to screen them by selecting sdf as a output file parameter.
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I am still an undergraduate and a newcomer to this field, so I very much appreciate any Professor answering my question.
Thank you,
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No, this is not possible. To study a galaxy in detail, one has to observe in different wavelength (optical, radio, ...), and for this different telescopes are needed. Even when you restrict to optical wavelengths, you will need different auxiliary instruments like special spectrographs etc., which may not be all available at one telescope. Also, observing time on large telescopes is very limited and valuable, and you do not need a large telecopes for all kinds of observations. Therefore, when a smaller telescope fullfils your demands you do not have to apply for time at a large telescope (which will also not be allocated if this large telescope is not needed for your observing plan). - By the way, it is not possible to determine "all" parameters of a galaxy. The nearer a galaxy is, the more details can be observed.
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I have a dataset (fastq files) of 15 fastq flies uploaded on Galaxy bioinformatics portal. I deleted them some time back to free some space. Now I want to retrieve these files. I can see these files as deleted, but not able to restore or download them. Is there any way to get these files back on my portal.
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Sure! Note the
x shown, y deleted
stats displayed just beneath the history title? Clicking on the "y deleted" part will unhide the deleted datasets in your history bar. The ones you haven't purged yet should have an Undelete it link, which will do just what it says
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I am analyzing my small RNA seq data on Galaxy, I need to remove all rRNA reads from my data. I downloaded a rRNA reference genome for mice and tried mapping with Bowtie2, but it kept failing. Apparently my rRNA reference file had multiple duplicate names. Where can I get a rRNA reference genome from?
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Hi Sanat Bhadsavle , you can download the entire SILVA LSU/SSU rRNA Database (LSU= large subunit rRNA, SSU= small subunit rRNA). For practical reasons, just merge both LSU/SSU FASTA files. Now you can (additionally) isolate all entries that belong to mice. The SILVA entries contain the whole taxonomy in the name.
The next step, map your RNA-Seq data against our SILVA database, I would recommend hisat2 over bowtie2. The resulting SAM or BAM (mapping) file can be filtered to extract all entries that do not map to your reference.
samtools view -f 4 mapping.bam > unmapped.sam
will store all unmapped reads in the unnmapped.sam file
Cheers
Roman
See here for more information:
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Have any of you used Galaxy (https://usegalaxy.org/) to analyze NGS RNAseq sequencing data to search for new splicing forms of RNA? I am looking for a pipeline for data analysis in Galaxy. Which programs are the best to use to find new splicing forms?
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Hi Robert,
The steps you need for such an analysis are:
1. align your data with a splice-aware genome alignment tool (I prefer the STAR aligner)
2. check for splice junctions that are present in your data but not in other annotations
You can perform step 1 using Galaxy (there is "RNA STAR" and other such aligners). The second step however is a bit more custom and as such not present as program. I suppose you could try to compare the junctions file from your alignment to a genome annotation file that you can download from ENSEMBL/UCSC/NCBI. You could bedtools as a tool for this, but it will be pretty tedious.
If you only have a few junctions/genes that you are interested in, you could also look "manually".
If you have some programming knowledge, I advise you to use Whippet (https://github.com/timbitz/Whippet.jl) for this kind of analysis. Their documentation is pretty good and it has a feature for finding novel splice junctions.
Best regards
Alexander
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Question1 : I would like to know if possible the distance separating the elliptical galaxy M60 (NGC 4649) and some spiral galaxies belonging to its group such as: NGC 4019, NGC 4037, NGC 4049, NGC 4064, NGC 4116 and NGC 4123 ?
Question2 : Same question for galaxy M86 (NGC 4406) and members of its group: NGC 4435, NGC 4438, NGC 4458, NGC 4461, NGC 4473 and NGC 4477.
Question3 : Same question for galaxy M87 (NGC 4486) and members of its group: NGC 4206, NGC 4262, NGC 4298, NGC 4302, NGC 4307 and NGC 4313.
I thank you in advance for any information that can help me.
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Here’s a way to do this:
Step 1: Look up each of the objects in your list by name from an online extragalactic database, such as this one: https://ned.ipac.caltech.edu/byname.
Step 2: Note the RA (right ascension) and DEC (declination) of each object, and convert the values to radians.
Step 3: Note the ‘Hubble Distance’ of each object. Also called the cosmological ‘proper distance’, this is what enters Hubble’s law (v = H_0. D). It is basically the present distance to the object from Earth, calculated by ‘freezing’ the expansion of the universe.
Step 4: Since the Hubble distance vector satisfies Hubble’s law, the following must hold:
D_12 = sqrt( (D_1)^2 + (D_2)^2 – 2.D_1.D_2*cos(theta) )
where D_1 and D_2 are the proper distances to objects #1 and #2 respectively ; theta is the angle between the unit vectors pointing to these objects ; and D_12 is the proper distance between these two objects that you wish to find.
You can find ‘theta’ like this:
r_1 = [sin(DEC_1), cos(DEC_1)cos(RA_1), cos(DEC_1)sin(RA_1)]
r_2 = [sin(DEC_2), cos(DEC_2)cos(RA_2), cos(DEC_2)sin(RA_2)]
theta = arccos( r_1.r_2 / (|r_1| |r_2|) )
where r_1.r_2 is a dot product between two unit vectors, r_1 and r_2, pointing towards the objects #1 and #2 respectively, constructed in 3D spherical coordinates.
Repeat these steps for every pair of objects that you wish to find the proper distance between. Hope this answers your question!
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I want to perform RNA-Seq data analysis for DEG's, by taking RAW reads from the NCBI-SRA database, of DENV1, DENV2, DENV3, DENV4. I want to perform this analysis on a galaxy web server. I'm a bit confused about the datasets from SRA. My confusion is, in this accession no from GEO-database- GSE69602, there is a total of 116 data are present. and I took only Total cell lysate data. In total cell lysate, there are two biological replicates at each time interval, like 6hr, 12hr, 24hr, 48hr, 72hr, and the other one is mock. I performed one analysis by taking two biological replicates of 72 hr and two mocks. workflow is, FastQC-Trimmomatic-RNA-STAR, StringTie, DEseq2. I want to know that is the right way or I'm doing anything wrong & if I have to take all the data from the respective time intervals, what is the protocol to specify those data at DEseq2?
All datas are singel-end data,
if you need to see my galaxy history I can share it with you.
A big thank you in advance
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Hi, The workflow created by you is correct you can also try EdgeR for Differential expression.
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Do you think that man will ever leave our solar system?
Please, answer, comments.
I invite you to the discussion.
Best wishes
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Well, it seems like a utopia to me, but why not? Primitive man did not imagine that airplanes were created and one day could fly and it was achieved. So perhaps, within centuries, this purpose can be achieved as well.
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Will as a result of the continuation of technological progress in the twenty-first century more perfect telescopes or other astronomical instruments that will allow to know what is on the surface of the nearest exoplanets, and above all the guilty star systems similar to the Earth exoplanets located in other planetary systems circulating around other suns?
Please reply
Best wishes
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I think may be
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I have been investigating astronomical reference system inversion effects for an article on astrophysics. I need to find examples of "shrunk space".
This would manifest itself as an increase in star counts per unit volume of space. It should be found at the center of large galaxies like the Milky Way and M87. Inversion effects would be accompanied by intense radio and infrared emission.
Stars are normally separated by a few light years. I am looking for examples where the interstellar distances are much less than expected, and preferably in the central portions of large galaxies.
Can anyone help?
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The observed quantity astronomers use to define the light within a given radius or annulus of a galaxy is called surface brightness, a 2-D quantity. To determine the 3-D stellar density within a given radius or annulus requires a model since the light detected is the integrated light from the front to the back of a galaxy. The type of galaxies you seek are called high surface brightness (HSB) galaxies. These are found among spiral, normal elliptical, giant elliptical and compact dwarf elliptical galaxies. Some are luminous radio galaxies, some not. There is a very large volume of papers on these objects, as well as on low surface brightness (LSB) galaxies within these morphological types. There are also some HSB globular clusters found in other galaxies. BTW, the Milky Way is not an HSB by any measure. You very likely have a lot of reading to do to extract from data tables the kinds of objects of interest. Much of this data has been available since the mid 1990s.
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Supposedly Pioneer 10 will fly to the nearest Alfa Centauri constellation for about 10,000 years.
Will humanity manage to build a new generation of space ships that will be able to overcome such huge distances in the galaxy many times faster?
When could this happen?
Please, answer, comments. I invite you to the discussion.
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Dear Yoshinari Minami,
Thanks for answering the question:
If and when will humans be able to explore other planetary systems?
Thank you very much for providing interesting publications describing important issues of the discussed issues.
Thank you, Regards,
Dariusz Prokopowicz
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Do you think that there is life beyond our Solar System?
Please, answer, comments.
I invite you to the discussion.
Best wishes
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Extraterrestrial life is hypothetical life which may occur outside Earth and which did not originate on Earth. Such life might range from simple prokaryotes (or comparable life forms) to intelligent beings and even sapient beings, possibly bringing forth civilizations which might be far more advanced than humanity. The Drake equation speculates about the existence of sapient life elsewhere in the universe. The science of extraterrestrial life in all its forms is known as astrobiology. https://en.m.wikipedia.org/wiki/Extraterrestrial_life
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Why in spite of the many years of listening to radio waves emitted from various parts of the cosmos, did not there appear such, which would mean the possibility of existence in another cosmos of intelligent creatures?
For several dozen years, various astronomical programs have been running radio-frequency listening programs of various emission ranges to diagnose those that could be a testimony that somewhere in another constellation there is or has existed the civilization of other intelligent beings.
However, up to now, it has not been possible to diagnose such waves that could confirm the existence of other intelligent beings in the cosmos and thus other, developed forms of life.
Why has not it worked so far?
Why in spite of the many years of listening to radio waves emitted from various parts of the cosmos, did not there appear such, which would mean the possibility of existence in another cosmos of intelligent creatures?
Please, answer, comments. I invite you to the discussion.
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Dear Gerges Francis Tawdrous,
Thanks for the links to interesting publications on the topics discussed in this discussion. Yes, you indicated the key determinants related to the question: What contract can we obtain from other civilizations?
Thank you, Regards,
Dariusz Prokopowicz
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"THIS IS AN ABSOLUTELY SCIENTIFIC QUESTION"
Planets considered habitable by researchers are located in a region relatively far from the star only so that water is in liquid form. However, it can't be too far away to freeze water. That's how we've been doing research to look for life and habitable planets.
However, could it be that this is not absolutely true?
Extremophiles here on Earth find ways to survive in unimaginable places.
Tell us your original opinion, without the "Google Genius" or other people's opinions, just be yourself!
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In fact, when searching for life in other planets, scientists follow the principle (follow the water), in order to reduce and limit the search process, not necessarily liquid surface water, but even the water in the ground because even the water under the surface of the earth can contain life, and indeed, because not only higher or sophisticated organisms need it, but even microorganisms, in our study there is what is called the water activity. Every microorganism needs this percentage when it is less than that, it cannot continue its life activities... So in order to be water exists, there must be an atmosphere that pushes it to the surface, as well as there is sufficient heat in the interior of the planet to keep water liquid underground. In addition, the existence of life is also related to the distance of the planet from the sun. In our solar system, the possibility of life was limited to an earlier time only on Mars because of the evidence of the presence of liquid water on its surface for long periods, and there may also be water under the surface due to its volcanic activity... In fact, research and studies are still ongoing to determine the harshest conditions in which a living creature can live in space, So scientists send water bears and they are the toughest creatures in terms of endurance on earth to explore that... On the other hand, what the creatures on the surface of the earth need may be different from others, this is very possible, because the organisms usually adapt to their surroundings and this is what we find with the microorganisms there whoever kills them is salt, and there are organisms that halophiles, the same is true of acid and heat, even there are organisms that live in volcanoes, did we ask ourselves why?? (my supervisor always said in the sense that they agreed to live in a very harsh atmosphere in order not to die). So from this, we conclude that it is possible to find forms of life in the harshest conditions in which we cannot expect them to exist... My sincere gratitude to everyone.
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There are several tools such SExtractor, GALFIT, ALADIN, SALSAJ etc., used by different researchers for the photometric analysis. Which tool do you recommand for the photometric analysis of the galaxy structures?
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For decomposition either Galfit or Imfit. Galfit is more frequently used, there is even a Facebook page devoted to it, where you can get an answer to any question within a day. But, you will also need a tool for 2D fitting of light profile - IRAF's ellipse beeing the golden standard. It has been recently rewritten in Python and is a part of a Photutils package (https://photutils.readthedocs.io/en/stable/isophote.html). Eventually, you need to overplot 3D (Galfit) results over 2D points (ellipse fitting). That's why you need to master both.
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Does the Solar System move in a meaningful pattern?
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Thank you so much for the LINK to this article!
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  • Full Article TextIntroduction and Background Available Data and Approximations Used in its Analysis Forward Models of Galaxies that Presume Nested Orbits Inverse Models of Galactic Rotation Discussion and Conclusions Author Contributions Funding Acknowledgments Conflicts of Interest References
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Open AccessReviewDebated Models for Galactic Rotation Curves: A Review and Mathematical Assessment by📷Anne M. Hofmeister* and📷Robert E. CrissDepartment of Earth and Planetary Science, Washington University, St. Louis, MO 63130, USA*Author to whom correspondence should be addressed.Galaxies 2020, 8(2), 47; https://doi.org/10.3390/galaxies8020047Received: 25 April 2020 / Revised: 26 May 2020 / Accepted: 27 May 2020 / Published: 1 June 2020(This article belongs to the Special Issue Debate on the Physics of Galactic Rotation and the Existence of Dark Matter)Citation ExportDownload PDFBrowse FiguresAbstract Proposed explanations of galactic rotation curves (RC = tangential velocity vs. equatorial radius, determined from Doppler measurements) involve dramatically different assumptions. A dominant, original camp invoked huge amounts of unknown, non-baryonic dark matter (NBDM) in surrounding haloes to reconcile RC simulated using their Newtonian orbital models (NOMs) for billions of stars in spiral galaxies with the familiar Keplerian orbital patterns of the few, tiny planets in our Solar System. A competing minority proposed that hypothetical, non-relativistic, non-Newtonian forces govern the internal motions of galaxies. More than 40 years of controversy has followed. Other smaller groups, unsatisfied by explanations rooted in unknown matter or undocumented forces, have variously employed force summations, spin models, or relativistic adaptations to explain galactic rotation curves. Some small groups have pursued inverse models and found no need for NBDM. The successes, failures, and underlying assumptions of the above models are reviewed in this paper, focusing on their mathematical underpinnings. We also show that extractions of RC from Doppler measurements need revising to account for the effect of galaxy shape on flux-velocity profiles and for the possible presence of a secondary spin axis. The latter is indicated by complex Doppler shift patterns. Our findings, combined with independent evidence such as hadron collider experiments failing to produce non-baryonic matter, suggest that a paradigm shift is unfolding.Keywords: spiral galaxies; rotation curves; data analysis; gravitational models; non-baryonic matter; inverse methods1. Introduction and Background Motions of stars, planets, and other matter in space have been scrutinized for centuries. Diverse observations involve visible light, due to our intrinsic human capabilities and to photons in this spectral region being easy to quantify, e.g., with photomultiplier tubes. In the past century, the question has repeatedly arisen as to whether material not observable with visible light exists and affects remotely observed motions. The phrase Dark Matter (DM) originated circa 1900 with Poincare’s discussion of a lecture by Kelvin. In the 1930′s, Zwicky called on ordinary baryonic material not luminous in the visible, to explain motions of galaxies in the Coma Cluster. In the decades shortly thereafter, the problem was viewed as a mass discrepancy (see [1]).The historical definition of DM changed greatly following Rubin and Ford’s [2] discovery that rotation curves (RC = tangential velocity v as a function of equatorial radius r) of Andromeda do not exhibit the familiar (Keplerian) orbital patterns of planets in our Solar System. Many subsequent papers provided similar data and centered on explaining the distributions of mass that could simulate the measured RC profiles. Most researchers accepted the Solar System analogy as valid, despite the recognized fact that the motions of our rather tiny planets are essentially independent of each other, so that the reduced 2-body problem holds. In contrast, galaxies have billions of stars that interact and occupy any given orbital radius (a many body problem), effectively constituting a continuous density distribution. This improper Solar System analogy led to the view that spherical haloes of DM surround spiral galaxies [3]. Because astronomers of this era were using the radio region of the electromagnetic (EM) spectrum to probe H atoms in galaxies, it was clear that the proposed massive haloes could not be ordinary matter. Consequently, haloes inferred from simulations of RC were alleged to consist of non-baryonic dark matter (NBDM), a hypothetical, ad hoc substance that does not interact with light.where G is the gravitational constant, and k obtained during fitting is near unity. Importantly, k = 1 describes the reduced 2-body orbital problem, but not systems with a graded mass distribution.Inference of NBDM haloes was immediately contested by an eminent astronomer. G. Burbidge pointed out in 1975 that considering the orbits of dwarf satellite galaxies gave a reasonably mass for the Milky Way galaxy [4]. He stated, “there is no unambiguous dynamical evidence which demonstrates that galaxies have very massive haloes”. Burbidge [4] further pointed out that certain assumptions underlie the formulation then used for the dynamical mass, which is also Min:Mdyn=krv2orbitG where k ~1(1)In 1983 the majority approach, henceforth referred to as NOMs (Newtonian Orbital Models), was challenged by Milgrom [5]. In its simplest form, MOdified Newtonian Dynamics (MOND) recapitulates Equation (1), but assumes that the variable k differs significantly from unity because of a deviation in Newton’s law. Although the majority of astronomers have followed the approach of Rubin, Faber, and others, MOND has garnered significant support from the astronomical community, as demonstrated by a 2015 special issue of the Canadian Journal of Physics [6].Milgrom’s model continues to draw support primarily because NBDM haloes have not been detected (Section 1.1). However, independent evidence for the proposed force modification, which is unrelated to general relativity, has not been found to our knowledge. Relativistic models for galactic rotation also exist, but involve fewer researchers (e.g., [7]), perhaps due to computational difficulties.The philosophies of the NBDM and MOND camps are irreconcilable. This impasse, which is tied to the importance of cosmological models to astronomy [1], has drawn interest from researchers in other fields to generate alternative classes of models. The methodologies of those highly skilled in computation [8,9,10,11,12,13] differ greatly from the analytical methods of the present authors [14,15], yet these very different approaches have led to the same conclusion: Specifically, the dynamics of galaxies are satisfactorily explained by Newton’s law of gravitation and no massive NBDM haloes are needed.Section 1.2 discusses the models applied to galactic rotation in general terms. This permits us to set a framework for evaluation, as outlined in Section 1.3. 1.1. Evidence Independent of Rotation Cuve Modeling Observational astronomers have directed substantial efforts towards detecting matter surrounding spiral galaxies. Large collaborations and expenditures are involved.Radio wavelengths (λ) probe H atoms and cosmic rays (nuclei of H and He atoms) in and around galaxies. Isophotes (e.g., Figure 1) reveal that this baryonic gas extends beyond the visible central regions of spiral galaxies, where density decreases in both radial and azimuthal directions. Further information is provided in a 2015 special issue of Galaxies [16] and in a review [17]. This distribution of baryons has characteristics similar to those of an atmosphere around a star or planet, such as having a much lower density than the central body.📷Figure 1. Distributions of baryonic matter in edge-on spiral galaxies: (a) Visible image of ultrathin UCC3697 (whitish). The blue overlay images the surrounding neutral H atom gas, detected using the L-band, which includes emissions at 21 cm. Thin red arrow indicates the z-axis. Yellow X indicates a secondary spin axis, pointing into the page with the sense shown by the curved yellow arrows. This secondary spin explains the form of the warp (see Section 2.2 ). Publicly available from NRAO/AUI/NSF [23]; (b) Radio contours of NGC5907 in the C- and L-bands, with an optical image, are publicly available at https://www.queensu.ca/changes/. The L-band range. The CHANG-ES project [24] is described in Irwin et al. 2012, AJ, 144, 43 [25] and details about this first data release are provided by Wiegert et al. 2015, AJ, 150, 81 [26].In contrast, NBDM has not been detected in special searches, as exemplified by recent exploration of the short wavelength end of the EM spectrum (γ-rays of ~10 TeV) [18] which focused on dwarf spheroidal satellites orbiting the Milky Way. Although earlier NOMs multicomponent fits suggested compositions exceeding 99% DM [19], none was observed. Neutrinos are now dismissed as composing haloes [20].Failures of observational astronomy to provide independent evidence for non-baryonic haloes led to efforts in an entirely different direction: Although experimental particle physics cannot establish the constituency of haloes, this approach could potentially discover non-baryons and reveal their properties. However, three different, well thought out experiments in 2019 using large hadron colliders provided no evidence for such particles. Consequently, weakly interacting massive particles (WIMPs) are now ruled out [20].At present, definitive tests and viable candidates for NBDM particles are lacking. Proposals now invoke a hypothetical particle (axions) from 1980s literature [21], which has not been documented in particle physics experiments. Consequently, the proposals involve gravitational lensing models [22]. This type of evidence is indirect and gravitational, and so does not differ in essence from the NOMs approach.Independent support for NBDM haloes is now limited to cosmological and other models. Cosmological models require large, but lower amounts of NBDM (e.g., [1]). However, model-to-model comparisons have nothing to do with the scientific method, and provide insufficient proof that either model is valid. 1.2. Types of Models Two fundamentally different approaches, termed forward and inverse methods, are used to investigate physical phenomena, including galactic RC, as follows:Forward (or direct) problems are familiar. These are solved by inserting some inputs into a formulation (e.g., into an equation or computer program), which then returns a result that can be compared to a measurement [27]. The latter is the case for NOM analyses, where the density or mass distributions for each of several different geometric shapes are assumed, and then velocity curves are calculated and compared to observations (Table 1, top). Forward methods are a focus in physical science and are emphasized in education, more now than ever, due to computational advances, commercialization, and proliferation of powerful, readily available software.Table 1. Comparison of forward and inverse approaches to RC analysis. 1.📷In the less familiar inverse problem, the nature of a remote source is deduced from its output or response [27]. This approach is rarely taught, and has been applied to RC by three different research groups (Table 1, bottom), one of which provides an analytical solution. Groetsch [27] mentions several successful solutions to inverse problems in astronomy in his monograph, including the important work of Ambartsumian [28] and Kepler’s historic deduction of the laws of planetary motion.Analytical solutions to inverse problems are unique and therefore are extremely important, but may be conceptually and analytically difficult to obtain, and commonly are not even possible. Unique analytical inverse models are increasingly confused with statistical solutions that have become popular recently [29], due to the computational advances mentioned above. Statistical solutions need not be unique. 1.3. Purpose and Goals This paper reviews and evaluates the available models of galactic rotation, focusing on spiral types. In view of the diverse morphologies of galaxies, actual motions are more complex than the snapshot of tangential velocities revealed by Doppler shift measurements. Dynamical galactic models are but approximations to this reality and invariably contain flaws. Moreover, RC curves are obtained from the Doppler data using models which have their own approximations. Section 2 provides evidence for two systematic errors existing which artificially flatten RC. Our evaluations focus on the mathematical physics of both forward (Section 3) and inverse models (Section 4), in hopes of providing the most impartial evaluation that we can muster. Section 5 summarizes. The developments over the last 5 years (e.g., Table 1) suggest that a paradigm shift of RC analysis is beginning. 2. Available Data and Approximations Used in its Analysis Most remote sensing involves processing some type of EM radiation. Images are important, because galaxies are classified according to their morphology, and shape affects gravitation. Contour maps can be based on intensity over some particular wavelength range, but can also be referenced to the departure from some average or reference value. The latter is the essence of Doppler measurements. 2.1. Shapes from Images The shape of a containerless body is defined by its isodensity contours. Face-on photographs of spiral galaxies document radial symmetry, with luminous density decreasing outwards (e.g., Figure 2a,b). Spiral arms indicate a secondary angular dependence that does not greatly perturb the radial dependence of measured velocities [36].📷Figure 2. Geometry of spiral galaxies, as approximated in oblate spheroid and disk models: (a) Visual image of nearly face-on NGC 7742, type SA(r), by the Hubble Heritage Team (AURA/STScI/NASA), and publicly available from http://hubblesite.org/newscenter/archive/releases/1998/28/image/a/). This particular ring galaxy is counter-rotating [37]; (b) Schematic of graded density in the equatorial plane; (c) Geometry of a spinning, oblate spheroid; (d) Side view of homeoids of constant density and shape which nest to form the oblate; (e) Geometry and gravitational forces (white arrows) for a disk of finite thickness and radius. Although the horizontal gravitational forces would be balanced by the centrifugal pseudo-force (solid arrow), the force along z is unopposed at any finite distance above the equatorial plane; (f) The basis of ring models: perspective view of concentric cylinders of various density (gray) rotating about the z-axis. Density and velocity gradients are co-linear; (g) Perspective view of stacked disks rotating about the z-axis, which describes density varying vertically; (h) Side view of the assembly of a thick disk with graded density. Nested hollow pillboxes are the required shape for isodensity contours in thick disk geometry.Spiral galaxies have finite thickness perpendicular to the equatorial plane, which needs to be accounted for in extracting RC from Doppler measurements. Our concern in this section is how the assumed shape for the cross section impacts RC derivations. 2.1.1. Density Contours of the Oblate Spheroid Historically, oblate spheroids were recognized as the required shape for galaxies [38,39]. The basis is Newton’s and Maclaurin’s quantification of how spin affects the shape of a large, homogeneous self-gravitating object [40]. The problem of Earth’s inhomogeneous, layered configuration was recently solved [41]. The oblate shape persists, despite this dense object experiencing additional frictional forces from relative motions among its layers, due to the strength of Newtonian attraction.Numerous edge-on images at virtually all wavelengths demonstrate that spiral galaxies have the expected oblate gravitational shape (Figure 1, see [24,42]). The geometry of an oblate spheroid (Figure 2c) is represented by its ellipticity (e) which is defined via the ratio of the minor axis (c) to the major (a):e=(1−c2/a2)12/(2)Any orientation of an oblate spheroid provides an elliptical cross section: for views down the special z-axis, e = 0. The surface of the oblate body is described by:z2=(1−e2)(a2−r2)(3)Binney and Tremaine’s [43] incorrect definition of z2 = (1 − e2)(r2 − a2) in their equation 2.114 provides imaginary numbers for z inside the oblate body. Formulae for spheroids in their tables 2.1 and 2.2 are also incorrect, perhaps due to this mistake.Thin internal shells of constant density are known as homeoids, after Newton. Constant density is linked to homeoids being equipotential. When viewed down the z-axis, homeoids have circular outlines (Figure 2b). In the z-r plane, homeoid surfaces have an elliptical shape (Figure 2d), which is defined by equations similar to the above, except that a is where the homeoid surface intersects the equatorial plane. Homeoids have constant density which means that the density where any given homeoid intersects the z-axis is identical to the density where it intersects the r-axis. These intersections are linked via (2) and (3). The oblate body and its nested internal homeoids are presumed to have the same ellipticity [15,44]. Equations for nesting in accord with constant force surfaces are given in [45]: this case may describe distant material orbiting about the dense center.Contours at various wavelengths for spiral galaxies are elliptical in nearly edge-on orientations (e.g., Figure 1), thus corroborating the oblate shape. Greater variation is seen for tilted spiral galaxies, which arises from various structural elements (such as bars and double nuclei) diverging from axial symmetry, shadowing in perspective views, and interactions with neighbors ([24,42]). Nonetheless, the basic oblate spheroid shape is evident in thousands of images (e.g., [42]). 2.1.2. Flat Oblate Spheroids are not Thin Disks After ~1963, the disk shape (Figure 2e) was considered to describe spiral galaxies [46]. This digression stems from Perek’s [47] erroneous assertion that density in the axial (z) direction is independent of ρ in the r direction for a spheroid, which contradicts Newton’s homeoid theorem [48]. The mathematical description of the ellipse (3) links ρ(z) to ρ(r) for each and every position.Oblate objects and disks differ fundamentally. Oblate spheroids are inherently 3-dimensional, whereas a thick disk can be cast as the vertical extension of 2-dimensional surface, i.e., a volume element (dV) of a flat, thin disk of constant thickness is 2πrdr times its height (H), whereas dV of a sphere is 4πr2dr, and that for a spheroid is simply the latter times a constant geometric factor involving its ellipticity. Basically, spherical and cylindrical geometrical elements differ by a factor of r. Cylindrical geometry is ideal for the analysis of fields about long, thin wires [49], but fields around a flat disk are far more complicated (Section 3.1.5).Density along the z-axis for any given homeoid depends on e and ρ for that homeoid at the equatorial plane (Figure 2c,d). In contrast, for cylindrical shapes, ambiguity exists regarding possible components, depending on whether rotation is coherent vertically (Figure 2f) or horizontally (Figure 2g). Using rings implicitly assumes coherent vertical rotation of concentric cylinders, as is evident in the mathematics applied to Doppler shifts [50] (see below) and to certain disk models (Section 3 and Section 4). This is a problem because density being independent of z creates a discontinuity at some height, H (Figure 2f). Instead, for thin disks, isodensity surfaces must be hollow pillboxes (Figure 2h). This case resembles the homeoids, but has sharp corners that are not possible in a stable, self-gravitating body. A force decomposition appropriate to the cylindrical coordinate system (Figure 2e) shows that the centrifugal acceleration of rotation could balance the radial component against the gravitational draw to the center, but the downwards force at finite z above the equatorial plane is unopposed. Consequently, if sharp corners were somehow created in a real galaxy, these would quickly round-up into the gravitationally stable shape of the oblate spheroid. 2.2. Rotation Curves from Doppler Shifts Sofue and Rubin [36] summarize data analysis procedures used to obtain RC from Doppler shifts of the approaching and receding galactic limbs. Advances made since 2001 are detailed in the many subsequent papers on this subject. However, the basics (summarized below) have not changed. Regarding the Milky Way, observations are from the inside, which requires a slightly different approach: for details, see Sofue [51,52]. 2.2.1. General Approach in Data Analysis Optical determinations of Doppler shifts trace motions of stars, whereas those of gas emission lines (e.g., Hα, HI, and CO) reveal motions of unconsolidated material. Spectral lines emitted by hydrogen gas clouds permit velocity measurements at great distances from galactic centers, whereas CO is useful for their inner zones, if dust does not interfere [36].If a galaxy is perfectly flat and exactly face-on, its Doppler shifts from spin about the z-axis cannot be measured. Measurements must be corrected to account for the galactic plane being inclined by angle (i) to the line of sight (Figure 3a,b), and so the derived RC depict the dependence of velocity on radius along the apparent major axis (Figure 3e). The tilted orientation superimposes structural elements for regions that are thick and/or dense, which creates difficulties in establishing RC near galactic centers [36], or for nearly edge-on galaxies (i > 80° [35]), or along the minor axis (Figure 3c,d). Uncertainties are high for i < 40° [35] due to the presentation.📷Figure 3. Pictorial essay of an RC extraction by deBlok et al. [53] for a tilted, intermediate sized spiral galaxy from high-resolution study of the 21 cm band using the Very Long Array: (a) Detailed, visual image of NGC 2403, type SAB(s)cd, by Göran Nilsson and The Liverpool Telescope group, which is publicly available at https://commons.wikimedia.org/w/index.php?curid=63317782. Red lines link this image with the Doppler measurements; (b) Image at 3.6 μm from IRAC Spitzer, showing the multiple spiral arms, which is on the same scale as the Doppler measurements; (c) Velocity field derived from fitting Hermite polynomials to the natural-weighted data cube, where the thick line indicates the systemic velocity and the difference between contours is listed; (d) The thin disk model of the Doppler measurements with the same grayscales and contours as the data; (e) Extracted RC (black curve) along the major axis with the inclinations stated on a velocity-position diagram (gray). Excellent agreement is obtained between the approaching and receding limbs. Panels (b) to (e) are modified after de Blok et al. (2008) High-resolution rotation curves and galaxy mass models from THINGS. Astrophys. J. 136, 2648–2719 [53] with permissions by the AAS; (f) Color version of the Doppler measurements with superimposed rotation curves. The vertical dashed line is approximately at 13 kpc. Reproduced with permissions from Springer Nature: Nature Astronomy 2, 615-616, Is there a universal alternative to dark matter? De Blok, W.J.G., https://doi.org/10.1038/s41550-018-0547-4, copyright August 1, 2018 [54].where φ is the positional angle of the major axis and x0 and y0 depict the galactic center. An iterative procedure is required due to the forms of these equations [50]. A high degree of smoothing exists, as seen in the ring model of Figure 3d. Also, the desired parameter VC is multiplied by sin(i), which lumping induces ambiguity [55], unless the inclination is independently determined.Measurements consist of a velocity field (Figure 3c,f, Section 2.2.3) denoted V(x,y) where x and y are sky coordinates. The tilted ring model of Bergeman [50] is routinely applied, which assumes that the velocity field measurements describe the equatorial plane and that the disk is thin and is composed of rings (Figure 2f):V(x,y)=V0+VC(r)sin(i)cos(θ)(4)where V0 is the (systemic) velocity of the center away from the Sun and VC is the circular velocity at a specified mean equatorial radius (r) of the ring. The azimuthal angle (θ) is defined by:cos(θ)=−(x−x0)sin(ϕ)+(y−y0)cos(ϕ)r; sin(θ)=−(x−x0)cos(ϕ)−(y−y0)sin(ϕ)rcos(i)(5) 2.2.2. Two Examples Progressive improvements in resolution have increased the accuracy of Doppler measurements considerably over in the last few decades. Two recent studies exemplify modern determinations.Data for NGC 2403 (Figure 3) were discussed in detail above because optimal conditions exist for obtaining its RC [35]. That the RC for its two limbs nearly match (Figure 3e,f) indicates regular, circular motions. Velocity fields of NGC 2403 from HI lines have been measured and analyzed many times, with increasing resolution and sensitivity. The results of deBlok et al. [53] show that agreement is good, except beyond 13 kpc. Variation at large radius may relate to statistical analysis of the peak profiles (Section 2.2.3).The Southern Pinwheel (NGC 5236) possesses a much more complicated velocity field (Figure 4). Earlier HI studies probed the central regions, which appeared more “normal,” but still indicated asymmetry. The unusual single arm appears to be rotating with the galaxy, despite its large distance of ~45 kpc from the center. Section 2.2.4 argues that the complexity is due to a second axis of spin.📷Figure 4. Pictorial essay of an RC extraction for a giant spiral galaxy with an unusually large baryonic atmosphere and a twisted velocity field: (a) Detailed, visual image of M83, which is type SAB(s)c with a double nucleus, by William Blair, NASA, ESA, and the Hubble Heritage Team (STScI/AURA), publicly available at http://www.spacetelescope.org/images/heic1403a/. Red lines link this image with the area mapped in Doppler measurements. Yellow arrow marks one possible 2nd axis of spin, from symmetry of the arms; (b) Integrated HI distribution overlaid onto the B-band image of M83; (c) The mean velocity field, plotted on the same scale; (d) Position-velocity diagram along the major axis with the inclinations stated; (e) The minor axis. Panels (b) to (e) are a high-resolution study of the 21 cm band which used the Australia Telescope Compact Array and are modified after supplement figure A50 in “The Local Volume HI Survey (LVHIS)” by B.S. Koribalski et al. (2018) Mon. Not. R. Astron. Soc. 478, 1611–1648 [56]. 2.2.3. How the Geometrical Control on Density Contours Affects Velocity-Flux Systematics Geometrical constraints on the superposition of elements have not been accounted for in analyses of the velocity fields. The peak in flux-velocity diagrams is assumed to represent VC for the radius being sampled (e.g., [53,57,58,59]; Figure 5a). This is untrue for both the oblate shape and the thin disk approximation in the regions even where elements are not superimposed, as follows:📷Figure 5. Velocity distributions in a spiral galaxy: (a) Measured profile, modified after C. Carignan, (1985) Light and mass distribution of the magellanic-type spiral NGC 3109, Astrophys. J. 299, 59–73 [57], with permissions by the AAS. In this panel only, velocity increases to the right. Blue line shows the expected equatorial velocity for a disk. Red and green lines = the expected equatorial velocity for an oblate; (b) Schematic constructing the expected profile (upper section) for a thin disk consisting of rotating rings, where the lower section shows a vertical slice through the galaxy indicating the systematic changes in density and velocity with equatorial radius and a line of sight (LOS). Light lines show the flux-velocity correlations inside the disk, where the colored curve estimates attenuation of the emitted flux along the LOS. For a ring-disk geometry, the equatorial velocity would be midway between the minimum and maximum velocities; (c) Schematic constructing the expected profile (upper section) for an oblate shape consisting of rotating homeoids (lower section). The top and bottom velocity-flux correlations are identical, but are offset for clarity. These will sum, but emissions from the bottom are attenuated more. The red curve approximates the attenuated sum. The equatorial velocity would be the maximum observed, if resolution is very high, but more likely a small tail will exist, representing beam smearing.Along the line of sight crossing the outer sections, a very thin disk constituted of orbiting rings would have velocity and density (detected as emitted flux) that both increase as radius decreases (Figure 5b). For an ideal flat and thin disk where all emissions are received under optically thin conditions, the flux profile would be triangular, and the velocity at the equatorial plane (Veq) would be the average of the highest and lowest velocities along the LOS (Figure 5b). However, as radius decreases, conditions become increasingly optically thick. Light is attenuated as it emerges from within the galaxy, in such a manner that emissions originating below the equatorial plane are highly attenuated, due to this light crossing large amounts of superjacent matter. The blue curve in Figure 5b sketches the expected attenuation pattern. The velocity associated with the maximum flux depends on the particulars of the gas distribution. The same holds for the velocity associated with the average flux. Instead, Veq will be represented by the average of the extrema detected, as long as some emitted light is received from the innermost ring crossing the LOS. Due to the asymmetry of the peaks (Figure 5a), using statistics overestimates |Veq|, if the thin disk geometry were a reasonable approximation.Rotational velocity and emission intensity along the LOS crossing an oblate object are correlated in a different manner than for an idealized thin disk. Each homeoidal shell of an oblate object (Figure 2d) rotates coherently. Consequently, the velocity-density gradient is parallel to r only exactly on the equatorial plane. Along a random LOS towards the outside of a spiral galaxy, the maximum tangential velocity is reached when the emitted light originates from the equatorial plane of the spiral (Figure 5c). Under optically thin conditions, the velocity-flux profile is approximately triangular, and has a sharp termination at Veq. Without attenuation, triangular profiles describe each of each of the top and bottom sections of the galaxy. Attenuation will make the termination less vertical, but as long as some light is received from the equatorial plane, the maximum velocity is the equatorial velocity, and an asymmetric flux-velocity profile is expected (red curve in Figure 5c). Attenuation will move the peak to lower velocities than the cutoff associated with the equatorial plane. How much depends on the gas distribution.Reported flux-velocity profiles are in accord with the asymmetric pattern expected for oblate bodies. Other examples from Carignan [57], Gentile et al. [59], Chemin et al. [58], and deBlok et al. [53] have a pattern similar to the red curve in in Figure 5c, for all but the central regions and for the minor axis. Moreover, the steep side of the roughly triangular flux-velocity profiles is always located towards the highest absolute velocity (after subtracting the system velocity), regardless of whether the galactic limb is approaching or receding (Figure 5a), as expected for the oblate. For the skewed velocity-flux distribution of NGC 2403 (not shown), the representative value for the equatorial plane is ~205 km s−1, not ~185 km s−1 from the position of the flux peak. This ~+10% correction is near the center of the RC curves. Data were not provided to the outside. However, a correction of < +2% is indicated for the outskirts by the flux profile of ESO 79-G14 [59] and the lower resolution data (Figure 5a) point to this systematic error decreasing with radius. Hence, RC are not as flat as currently reported.Thus, use of the position of maximum flux underestimates |VC| by variable amounts. We cannot reconstruct RC with the information reported in Doppler studies of galaxies. Re-analyzing raw data is needed.Bergeman’s [50] construction is well-reasoned. The geometry is described as rings in a disk. This construction depicts infinitesimally thin coaxially rotating cylinders, which are sufficiently short to be optically thin vertically, i.e., emissions from Veq = VC are presumably being measured. Bergeman’s equations, here (3) and (4), should thus describe spiral galaxies except near the center and for the minor axis, where elements are superimposed. Another factor merits consideration. 2.2.4. Why Multiple Spin Axes Should Exist Spin of a spiral galaxy about one preferred axis produces a pattern similar to that of Figure 3f or Figure 6a (top). More complex patterns (e.g., that of M83 in Figure 4c) have been ascribed to the effects of galactic warping. Alternatively, a 2nd axis of spin could exist. This new proposal is supported by many observations, including the image of Figure 1a, as follows:📷Figure 6. Indication of multiple spin axes: (a) Schematics of a single rotation of a tilted spiral (top) and a simple secondary rotation for a face-on spiral; (b) Wire diagrams of the oblate spheroid and Jacobi’s triaxial ellipsoid, with spinning about the short axes added to images created by Ag2gaeh and publicly available at https://en.wikipedia.org/wiki/Ellipsoid#/media/File:Ellipsoide.svg under a Creative Commons Attribution Share Alike 4.0 International license; (c) The mean velocity field of Circinus, showing superimposed spins around two axis (black arrows), which are nearly parallel as presented to the LOS, modified after figure 13b in “The Local Volume HI Survey (LVHIS)” by B.S. Koribalski et al. (2018) Mon. Not. R. Astron. Soc. 478, 1611–1648 [56].First, Doppler shifts are observed for face-on galaxies, but should not be, if only a single spin axis exists. An idealized pattern is shown in Figure 6a, bottom. The 2nd axis should be secondary as the faster rate should accompany spin about the short z-axis defined by high symmetry of the oblate (Figure 6b). The fast and slow Doppler shifts would be superimposed, as suggested by the pattern for Circinus (Figure 6c).Second, perpendicular spin axes are supported theoretically by Jacobi’s 1822 demonstration that the triaxial ellipsoid shape (Figure 6b) is an equilibrium configuration for a self-gravitating fluid body of uniform density that is rotating with a constant angular velocity about its shortest axis. Many have discussed this problem subsequently (e.g., Routh and Green near 1900). Solutions should also exist for motions involving rotation at different rates about each of the short and moderate length axes (Figure 6b), and for variable density, since focoidal shells of the triaxial ellipsoid behave analogously to Newton’s homeoids. Spin about a 2nd axis should be linked to flattening, as deduced for the primary axis by Maclaurin and Todhunter, but may alternatively result from gravitational interactions with neighbors.Third, any projection or section through an oblate or triaxial ellipsoid defines an ellipse. Such foreshortening along the LOS contributes ambiguities, as demonstrated by difficulties in classifying nearly edge-on presentations. Nonetheless, evidence for triaxiality in spirals is common. The largest spiral class is SB, where the central bar clearly breaks the axial symmetry of the oblate. Spiral arms of M83 exhibit 2-fold symmetry (Figure 4a), discussed further below. 2.2.5. Evidence for Multiple Spin Axes in Doppler Patterns Perpendicular spin axes are essential to the polar ring morphology, where the spin axis of the outer ring is at 90° to the special axis of the central spiral [60].The complex pattern of Doppler shifts for M83 (Figure 4c) reveals spin about 2 perpendicular axes, such that the secondary axis spins more slowly. Rotation about the apparent minor axis (Figure 4e) occurs at substantial speeds, since the maximum tangential velocities in this direction are only a factor of 2 lower than those along the major axis. Substantial departure from axial symmetry is evident in the visual image (Figure 4a): the double nuclei roughly define the orientation of the secondary spin axis. The large arm of H gas which is approaching the observer seems to be a singlet. Perhaps the equatorial plane of the gas is tilted with respect to the central, visible spiral, so much of the receding arm is either in front of the plane of visible galaxy or behind, making this feature difficult to observe. Figure 1a shows a 3-dimensional structure that is consistent with a 2nd spin axis.Behavior similar to that of M83 and Circinus are common among large galaxies [56]. Although Circinus (Figure 6c) is classified as type SA, its images are obscured by the plane of the Milky Way. The arms of this galaxy suggest some reduction from axial symmetry. In fact, some reduction is required even for highly symmetric arms. For example, 3 arms requires the existence of a 3-fold axis of rotation, and so a turn of 120° returns the same image, but axial symmetry is described by an n-fold axis, where n→∞, and a turn of any angle returns the same image. Importantly, 2-fold rotational symmetry (e.g., a bar shape) is consistent with a triaxial distortion of the oblate, as can be seen in comparing Figure 4a to Figure 6b. 2.2.6. Evidence for Multiple Spin Axes in Rotation Curves RC of M83 indicate that rotational motions exist about both the major and minor apparent axes. However, most studies only report RC for the major axis, which is more reliable due to separation of elements, so we focus on this axis.We proposed above that existence of an unrecognized secondary spin axis causes tangential velocities to be increasingly overestimated as r increases. Comparing RC of differently shaped spirals provides a test. Our task is simplified by RC also being categorized by shape. Importantly, these generic shapes for measured RC shown in Figure 7b are likely affected by the processing error covered in Section 2.2.3. As shown in Figure 7a, this processing error and the existence of a 2nd spin axis will both make measured RC appear flatter than they actually are. Hence, measured dV/dr being negative at large radius means that any correction will make the slope more negative, so this category would not be perturbed by either correction. It should represent uniaxial spin since the decrease is expected at very large radius, where the point mass approximation becomes reasonable. Support is provided by RC for the Milky Way, which is analyzed from the inside and so only spin around its c-axis can be detected. The Milky Way has negative dV/dr for its distal regions [51,52].📷Figure 7. Evaluation of RC for possible multiple spin axes: (a) Schematic of how systematic errors will affect rotation curves. Blue = typical measured RC with a flat trend. Red arrows = corrections for asymmetric peaks in flux-velocity profiles. Green arrows = corrections for a 2nd spin axis. Both effects make the slopes of rotation curves decrease more strongly at high radius, as shown in the orange and green RC; (b) Simplified representation of commonly observed rotation curve types, after the classification scheme used by Wiegert and English [61] and others. Gray rectangle = the galactic region used to determine the dV/dr; (c) Histogram of the list of non-interacting galaxies in Table 3 of [61]. Types with a ring are indicated by (r) and an ellipse. The box shows the morphological types. The double arrow indicates that dV/dr being negative correlates with axial symmetry, but anti-correlates with symmetry lowering elements such as bars.On this basis, we categorized the non-interacting spirals in Table 3 of [61] based on slope in the outer half of each listed spiral (Figure 7b). Please note that the RC profiles shown are much simpler than measured RC and that even the examples of Figure 3 and Figure 4 are much smoother than is common.Flat slopes, posited as an distinct galactic category (Figure 7c), may represent negative dV/dr (Figure 7a), since asymmetries in flux-velocity peak shape would make the slope negative, irrespective of whether a 2nd axis exists.Triangulum was listed in table 3 of [61], but due to its proximity to the much larger Milky Way and Andromeda galaxies, it is quite possible that the RC of small M83 is perturbed, just as orbits of their dwarf satellites are controlled by the alignment of the two giants. Thus, Triangulum is distinguished in the histogram of Figure 7c.From Figure 7c, measured negative dV/dr is connected with ringed spirals, and thus with axial symmetry. Most of type SA have negative slopes, which is compatible with the overall uniaxial symmetry of this morphological type. Negative dV/dr is not observed in barred types lacking rings, which is consistent with observed departure from uniaxial symmetry.Edge-on galaxies have a negative slope. This is expected because the presence of 2nd spin axis that is pointing along the LOS will have no effect, whereas the presence of a 2nd spin axis that is perpendicular to the LOS will alter the Doppler shifts most near the center, where elements greatly overlap. This deduction is based on envisioning horizontal compression of Figure 6a. The results of Figure 7c point to the triaxial symmetry of many galaxies, as demonstrated in their visual images, affecting measured Doppler shifts. However, before the effect of triaxiality can be quantified, the raw data should be reprocessed to correct for the maximum velocity best representing the equatorial plane. Also, galaxies that are both isolated and non-interacting must be considered. 2.2.7. Summary and Prognosis Determinations of accurate RCs, and therefore of effects such as non-circularity among the motions, are hampered because rings (actually coaxial cylinders: Figure 2f) are assumed and viewed as thin. In contrast, images of spiral galaxies show that density grades outwards from the center in all directions. Hence, the equatorial velocity is not represented by the peak in the velocity-flux diagram associated with each data acquisition, but instead is the maximum velocity along the LOS. Low resolution studies are more affected by this correction than high (~30% for ~1985, Figure 5a but ~10% for ~2008 [54,59]. The centers of galaxies have a larger systematic error than the outskirts, since it is less clear what the true equatorial velocity is, given the larger traverse associated with each data collection point, shadowing of elements, and stronger attenuation.We have shown that the view of spirals as disks, rather than as oblate spheroids, has also greatly influenced the processing of the Doppler data that underlies rotation curves. Barred spirals visually appear to be triaxial ellipsoids, rather than oblate spheroids. Triaxial geometry necessitates that the motions of the stars are not circular. Irregularities in the Doppler patterns confirm that non-circular motions exist. Although these have been interpreted as the galaxy being warped, a self-gravitating body with 2-axes of rotation is much more likely (e.g., Figure 1a). If triaxiality is low, the short c-axis of any spiral galaxy should dominate its spin, and so the paths remain nearly circular. With low triaxiality, RC are slightly perturbed from axial symmetry, which explains the data summarized in Figure 7. Above, we inferred that the flat curves commonly observed for spiral galaxies contain a systematic error. The velocity should decrease, as observed for spirals with uniaxial symmetry but without interactions and perturbations by neighboring galaxies, and as expected at great distance from the center. When this systematic error exists, it is in the same direction as the processing error identified above.Triaxial ellipsoid shapes are common, based on SB being the most populous galactic morphology. Whether the effect is simply a perturbation, involving slow rotation about the 2nd axis, cannot be ascertained until the processing of the Doppler shifts is revisited. If the triaxiality is large, then the key assumption underlying RC determinations is invalid, and values of v(r) must be analyzed with trepidation.For several reasons, the remainder of this report focuses on models for the circular motions about the c-axis. First, spirals with rings have axial symmetry and therefore should have circular motions, which can be properly analyzed in the future. Second, behaviors can be predicted for circular motions and compared to available data. Third, once the Doppler data are reanalyzed, robust models are needed to understand motions inside spiral galaxies with demonstrably circular motions. 3. Forward Models of Galaxies that Presume Nested Orbits Most models of RC treat the motions of stars and gas in a galaxy as being orbits. Interior potentials and forces differ from the exterior, but because these match on the surface of a self-gravitating object, considering orbits permits progress to be made towards understanding galactic rotation. This section focusses on general behavior.Forward modeling is used in orbital models, where the density and shape are assumed in simulating RC (Table 1). The basic shapes considered are the point mass, oblate spheroids (which are spheres when e = 0), and the flat disk, which has some unrecognized problems. 3.1. Synopsis and Evaluation of the Mathematical Underpinnings of Forward Orbital Models Equating the centripetal force for a test particle or ring of mass m to the gravitational attractive force recapitulates the early, 1970s explanations of v(r) measurements. Importantly, the moment of inertia is mr2 for any particle in a circular path, or for a thin ring or thin cylindrical shell about the special axis. Forces around a central point or outside of a spherical distribution of matter are described by:Fcentrip=mv2s; Fsphere=−GMinms2 for s>a(6)The variable s, the radius of a sphere, is used here to emphasize that such orbits are not restricted to any single plane. Equating the two forces in (6) yields Burbidge’s result (1) for the endmember case of k = 1. Such Keplerian orbits exist only around a central point mass or a spherical mass distribution, as follows: 3.1.1. Spheres vs. Point Masses In the early studies of galactic rotation, the finding that their inner zones spun like a record was viewed as a puzzle. This view stems from confusing the physics of a mass that is restricted to a small central region with the physics of a mass that is distributed gradually over a spherical (or spheroidal) volume of interest.According to Newton, a tiny particle at radius r within a spherical distribution of matter is attracted to the matter inside r as if all that interior matter of total mass Min were positioned at the very center. Furthermore, Newton proved that shells of matter outside that radius exert no net force on the interior particle. This particle is equivalent to the test mass in an orbital problem. Hence, modeling a galaxy in the limiting case of a spherical mass distribution requires that Min grows with r up to the body radius a, whereupon growth stops. The result is Keplerian orbits when r > a, whereas for r < a, a velocity profile for a homogeneous object is linear, even if randomly oriented orbits are considered.That velocity profiles are linear inside a homogeneous sphere or spheroid is trivial to demonstrate mathematically. For a particle or ring orbiting inside a homogeneous, spherical distribution of mass, the forces in (6) are equal, but an additional constraint of Min = 4πρ r3/3 applies. This substitution yields:v=r4πρG3−−−−−√(7)Equation (7) establishes the linear relation between velocity and radial distance for this particular orbital case. Instead, if spherical shells of the object are rotating coherently about a defined axis (Section 4), the moment of inertia differs from that of the ring by a factor of ⅔, but the velocity profile is nevertheless linear, and the object is again predicted to spin like a record (Figure 8).📷Figure 8. Forward models of rotation curves for a test particle orbits, comparing a large, central point mass (dot-dashed curve) to orbits both in and around oblate bodies with homogeneous density but varying ellipticity (various black curves). Gray curve = a sphere, for which e = 0: this same pattern was obtained for Coulombic forces in and around a sphere with uniformly distributed charge (e.g., figure 28–9 in [49]). The dashed curve represents an axial ratio of c/a = 0.2, whereas the dotted curve depicts c/a = 0.1. For variable density, the maximum would be rounded rather than a corner, and thus would resemble the idealized RC curves in Figure 7b. Please note that this approach assumes orbiting points or rings, not co-rotating spheroidal shells.Our analysis is analogous to Thomson’s atomic model, where the charge of the nucleus is distributed uniformly in a tiny sphere. The electrical force vs. radius ([49], pp. 695–697) accelerates a circulating electron with the same functional dependence as the case for the sphere in Figure 8.A discontinuous change in dv/dr at the surface is expected for a body which abruptly terminates. However, galaxies become progressively rarefied as r increases. A rotation curve for a spherical galaxy would have a smooth peak, rather than a sharp corner as in Figure 8, because this density gradation can be considered to be the summation of progressively larger spheres, each with constant, but reduced, density.To first order, measured RC have features resembling a spherical distribution of mass when orbits and Newton’s findings for a sphere are considered. Tangential velocities first increase from v = 0 at r = 0 towards some maximum value, and then decrease. This pattern recapitulates the RC pattern associated with a single spin axis and with a ringed morphology (Figure 7). However, with a spherical distribution, orbits need not be limited to the equatorial plane, but can possess the iconic picture of electrons, whose circular orbits crisscross about an atomic nucleus.The Keplerian pattern can persist to the limit of r→0 only in the case of a central point mass. This case aptly describes our Solar System because the Sun’s radius is tiny compared to planetary orbits. However, for a Keplerian pattern to describe a galaxy, the mass at the center must be far larger than the mass of all the other stars combined. Because spirals are flat, most of their mass lies outside ~½ of their visible radii: see Section 4. 3.1.2. Oblate Shapes The geometry of a galaxy does not alter Fcentrip of its constituents from (6) but greatly affects the pull of gravity in its interior. Spin is a symmetry breaking mathematical operation that creates a special axis. A body with a tiny difference between its c and a axes is an oblate spheroid, which lacks the special spherical symmetry.The decrease in symmetry affects equations for the gravitational force about an oblate spheroid. The transcendental equations derived by Gauss and summarized by Schmidt [39] were recently recast [45] into a greatly simplified, yet exact, closed analytical form for the special axes:Fobl(r)=−3GMm2a2e2[raeArcSin(aer)−1−a2e2r2−−−−−−−√]; Fobl(z)=−3GMma2e2[1−zaeArcTan(aez)](8)Forces around an oblate body are non-central. Not only does the force not go as 1/r2 from (8), but moreover the lines of force around a flattened oblate only point toward the center along the r and z axes (Figure 9). Behavior of rounder bodies is similar albeit less pronounced [45]. Consequently, stable particle orbits around a spheroidal mass distribution are either polar ellipses or equatorial circles: These limitations underlie the restriction of the orbits of dwarf galaxies around the proximal Milky Way and Andromeda to certain planes [45], and of the orbital patterns inside spiral galaxies to a very few types: normal, counter-rotating, or polar rings.📷Figure 9. Lines of force (red) superimposed on ellipsoidal contours of constant potential (purple or blue): (a) Sphere (black circle) with c/a = 1, where the lines of force are perfectly radial and the orthogonal equipotential surfaces are spherical; (b) Oblate body (black ellipse) with c/a = 0.1, as in spiral galaxies, where the lines of force are not radial and are only perpendicular to the body for the special axial directions. White arrow = spin axis. Equipotential contours are ellipses which become rounder with distance. Contours near the oblate are too finely spaced to be shown.where the RHS gives the limit as e→1. Please note that c/a~0.1 gives e ~0.995.Balancing forces for a test particle or ring in the equatorial plane of the oblate object gives:Min=v23Gr2a2e2[raeArcSin(aer)−1−a2e2r2−−−−−−−√]; Min(e→1)=2a2v23Gr1[raArcSin(ar)−1−a2r2−−−−−√](9)Newton discovered that any given homeoid in an oblate object only experiences a net gravitational attraction to matter in its interior. Hence, a requirement for forward models of RC is that Min grows with r up to radius a of the oblate body, whereupon growth stops. Consequently, flatter oblates have overall flatter rotation curves, whereas moderately round to nearly circular oblate spheroids (e→0), have essentially Keplerian behavior for r > a (Figure 8). A peak exists in RC for all ellipticities.Physically realistic gravitational shapes produce orbital patterns with common features because all members of this family of objects behave similarly, as was deduced long ago by Newton and Maclaurin. Crucially, the gravitational force on objects in the equatorial plane does not follow 1/r2 until great distance is attained, per (8), leading to a much different formulation for the dynamical mass from (9), see Section 3.1.4. Basically, transformation of a sphere into an oblate shape causes proportionally more material to lie near the equatorial zones. The strong forces in the equatorial plane cause rotational motions associated with axial symmetry to lie in the equatorial plane. Comparing Equations (1) and (9) shows that k is close to 2/3, not ~1 as previously assumed. The excessive mass in NOMs stems from assuming central forces that decrease inversely with r2 plus using an overly large moment of inertia, that of a test point or ring. 3.1.3. Approximate Formulae for Rings and Disks Forces around the equator of a thin disk have been incorrectly represented as central in NOMs approach. For transparency, we provide some simple formulae, using an approach similar to that of Kellogg [62] who considered the bar. Upper and lower limits to the force in the plane of an ultrathin ring are:GMringmr2−23/b2<Fring<GMringmr2−b2(10)where r > b, r is the distance of the test mass from the center, and b is the radius of the ring. Integration provides upper and lower limits to the exterior force around a homogeneous ultrathin disk of radius a. Numerical analysis (Figure 10a) of forces encircling a homogeneous ultrathin disk closely match our analytical upper limit of:Fdisk,ext(r)≃GMdiskma2ln[r2r2−a2]; Fdisk,ext(z)=−2GMdiskma2[zz2+a2−−−−−−√−1](11)📷Figure 10. Energetics of ultrathin disks: (a) Numerical calculation of work (dots) done by a test particle (the ant) as it crawls across a disk surface with constant density. Nearly 106 point masses were used in our integration. Light gray curve and inset shows a two-parameter fit. Dotted curve shows a rough power law. Dark gray curve shows a numerical calculation for the work done at r > a by an external test particle (ant in spacesuit); (b) Numerical calculations of velocity inside and outside an ultrathin disk for homogeneous density (solid) and for an exponentially decaying density but with the same mass (dots), showing a discontinuity at r = a. Keplerian orbits (dot-dashes) match these curves at several body radii, as expected. Weakening the exponential decline (moving mass outwards) would provide an RC less steep at the center, but steeper on the outside, moving the peak at r = a closer to the RC for constant density. Strengthening the exponential (concentrating mass inwards) would provide an RC that is steeper at the center and flatter in the middle, such that the peak at r = a is weaker, while remaining higher than the Keplerian orbit al constraint for r > a.Force along the special z-axis, obtained earlier by two different integrations [45,63], is exact. Our results are consistent with exact formulae for the bar [62]. The force experienced when approaching the bar from its top (the z-direction) or the lateral side (the y direction) is well behaved, but the force along x becomes infinite at the tip of the bar.Regarding the interior of a disk, no theorem of Newton exists to guide us. Hence, we provide a numerical model of the work for an ant crossing a constant density disk (Figure 10a), which gives the force inside the disk, via work = force × distance. Fits to interior work are shown in Figure 10a. At the edge, a break in slope occurs, which is consistent with the singularity in our analytical approximation.The interior force in a homogeneous disk is closely represented by our approximation:Fdisk,int(r)≃GMdiskma2(1−r3a)ln[1−ra] for homogeneous density(12)Non-centrality of the gravitational forces for a disk, as recognized by Feng and Gallo [64], is obvious in Equation (12). For the surface modeled here, there is no interior force along the z-direction.Centripetal forces in the equatorial plane remain central, as in (6). Force balance gives the rotation curves. Numerical results are shown in Figure 10b for both constant and exponentially declining density, both inside and outside the ultrathin disk. At the center, the disk spins like a record. This is required under axial symmetry since no net force exists exactly at the center, regardless of the density distribution.The infinite velocity predicted for r = a under constant density was not reproduced due to the finite size of the numerical steps. At the edge, a discontinuity in dv/dr exists for the exponential case. Figure 10b shows a moderately strong decline, and describes the effect of changing the attenuation, considering the limiting cases. The center of a finite sized body always spins like a record and a discontinuity at the edge of the disk will accompany any feasible density distribution for an ultrathin disk with a definite edge, because this is not a gravitational shape. The results converge to the Keplerian approximation at very large r.The mass for a constant density ultrathin disk (from exterior orbits) is approximately:Mdisk≃a2v2rG1ln[r2r2−a2] for r>a(13) 3.1.4. Geometry, Stability, and the Dynamical Mass Currently, the dynamical mass is computed from (1) with k = 1 (e.g., [61]). The implicit, but seemingly ignored, assumption in using Mdyn = rv2/G is that r >> a. For this reason, Burbidge’s [4] evaluation of dwarf satellite galaxy orbits using (1) yielded a reasonable mass for the Milky Way.Equations (9) and (13) for flat shapes yield different formulae for Mdyn in the limit of r = a:Mdyn,obl=23av2Ge2[1eArcSin(e)−1−e2−−−−−√] for r=a; vs. Mdyn,disk→0 for r→a(14)For e = 0, (14) or (9) reduce to the result for the sphere (Mdyn,sphere = av2/G). Considering e→1 underscores limitations of the thin disk geometry. From Maclaurin and Todhunter’s [40] equation describing the connection of ellipticity and angular velocity in a self-gravitating oblates, no rotation in expected. Simply put, the limiting case (e→1, which is a plane) cannot rotate because the mass would be infinite per Equation (14), and the energy would be infinite since a is infinite.A singularity in velocity exists at r = a in our analytical approximations (Figure 10b). The same holds for the tip of a flat bar [62]. This singularity makes it difficult to deduce the total mass from close orbits: i.e., approaching the disk from the outside yields a null mass (14). The limit r >> a yields the point mass formula, Equation (1), as it must, and so the total mass of a finite disk can be estimated from some large r > a. 3.1.5. Geometry, Coordinate Systems, the Theorem of Gauss, and Logarithmic Potentials Many proposed solutions to the galactic rotation problem involve logarithmic potentials. One of two long lists of Evans and Bowden [65] consists of ad hoc modifications of a logarithmic potential, which itself is baseless, as follows: The simple logarithmic form is identical to the textbook solution for an electrical field around a long wire [49]. This familiar form for a Coulombic force is derived by applying Gauss’ theorem to an imaginary cylinder enclosing a wire (Figure 11). No algebraic manipulation can possibly transform the simple mathematical solution for a line of mass along the z-axis into something relevant to spiral galaxies, which are highly flattened in the z-direction.📷Figure 11. Lines of force in cylindrical geometry, showing how these depend on the aspect ratio of the central body (dark gray). The light gray cylinders are imaginary shapes constructed to evaluate the divergence of the lines of force, per Gauss’ method: (a) Line source, where force lines need only be computed in the radial direction; (b) Stubby body, showing edge effects and bending force lines, which are only perfectly straight along the z and r axes; (c) Thin disk. Away from the edges, flux is essentially vertical, and analytic (Equation (11)), with no discontinuities.Using the theorem of Gauss instead of Poisson’s equation requires only a single integration, rather than two, thereby avoiding ambiguity and greatly simplifying visualization. Also, specifying the origin and the coordinate system makes the limits of applicability clear [48].For a long and thin vertical source, equipotential surfaces are concentric cylinders, and the attractive force is purely radial, with no component through the top and bottom of any enveloping cylinder (Figure 11a). For a stubby object (i.e., the vertical and radial length scales are similar, as in Figure 11b), the flux out of the ends of an imagined cylinder becomes increasingly important. For these stubby cases, the radial solution pertains only on the equatorial plane; plus, edge effects become increasingly important. Lines of force are straight only for the special direction (as in the oblate of Figure 9). For a disk, flux along z dominates (Figure 11c) because this is the largest and important surface. The force from a disk along z is a simple analytic solution that is not singular [49], but this is not the case along r: singularities exist, and no simple analytical solution apparently exists.The other long list of [65] consists of ad hoc substitutions of some function of r, z, and a for the spherical radius in Newton’s gravitational potential of a sphere or point. Although dimensionally correct, none of these substitutions provide the prefactor of GmMin/a2 for the force associated with the exact result for an oblate spheroid (8) or with the asymptotic brackets on force for a disk (11). This prefactor must accompany all approximations for the potential of a disk, since it exists in the exact formulae for the special z-axis of the disk (12), as well as in F(z) for the oblate (8). The prefactor of GmMin/a2 originates in the symmetry breaking operation of spin which transforms spherical into cylindrical coordinates.All substitutions listed or cited in [65] rest on the misconception that s in spherical coordinates is interchangeable with r in cylindrical coordinates. In comparing the Laplacian operator in the two distinct coordinate systems (i.e., Poisson’s equation), we neglect the angular variables to focus on the symmetry characteristics of the length coordinates:−4πGρ(r,z)=1r∂∂r(r∂ψ∂r)+∂2ψ∂z2 vs. −4πGρ(s)=1s2∂∂s(s2∂Ψ∂s)(15)Solutions for ψ(s) take on a different mathematical form than solutions for ψ(r) because the mathematical forms for the volume elements in the cylindrical and spherical coordinate systems differ (rdθdzdr vs. r2cosθdθdφdr). Considering Gauss’ theorem leads to the same conclusion [48]. 3.1.6. Toomre’s Mathematically Invalid Analysis of the Disk whereas his 3rd equation is Poisson’s equation, i.e., (15), LHS. Both equations are correct.Toomre’s [46] complicated potential function for the disk promoted use of this shape in galactic models. His derivation incorporates several mathematical errors, each of which invalidates his results. Due to the entrenchment of his study in galactic modeling, a detailed analysis is needed. For reference, Toomre’s 1st equation balances forces in the equatorial plane, as in all forward models:v2r=−∂ψ∂r=−Fr(r)(16) When separation of variables is invoked for a disk of finite thickness, density off the plane at any radius equals the density at z = 0 and that particular r, and thus coaxial cylinders are being considered (Section 3.1.5; also see Section 4). As explicitly stated by Toomre below his 4th equation, no mass can exist off the plane. From (17), no mass off the plane means either H = 0, or the z-dependence of ρ entails a delta function, which amounts to the same thing while requiring separation of variables for ρ (i.e., Perek’s [47] specious declaration, see [14,48]). To provide his 4th equation which contains only surface density, Toomre [46] divided Poisson’s equation by H. Division by 0 invalidates Toomre’s 4th equation, and all equations thereafter, which specify the potential without a z-dependence. It is immaterial whether he set H = 0 before or after division. Additional problems with separation of variables are discussed below.Flat disks (Figure 2f,g; Figure 11c) require a simple relationship between surface mass density (σ), full thickness (H), and thermodynamic density (ρ):σ(r)=∫H/2−H/2ρ(r,z)dz; σ(r)=Hρ(r), if density along z and r involves separable functions(17) Poisson’s equation cannot be applied to a surface, as its use requires volume elements, per the discussion of Garland [66] (see his appendix), who based his work on Kellogg’s [62] 1925 edition. Similarly, MacMillan ([63] p. 124) states that the surface under consideration must be closed, which condition cannot be met by a plane. The above division by H = 0 is a simple explanation for application of (15) to a plane being a faux pas. Proof that an enclosed volume is required for (15) is straightforward per the theorem of Gauss [48]. Toomre’s 2nd equation proposes a solution to (15) which includes an exponential function of the from exp(−k|z|) where k is a dummy index that is used subsequently in integration. In Toomre’s 4th equation and thereafter he sets z = 0. Obviously, his analysis is limited to the plane, which is invalid, as noted above. In his exponential function exp(−k|z|), k must be inversely proportional to some scale length, in accord with dimensional analysis and to provide a dimensionless argument kz. Because the relevant scale length along z is H and H = 0, k must equal some constant divided by H, and so k is infinite. Hence, k does not vary and cannot be used as the variable of integration, which invalidates Toomre’s analysis [63]. Use of an integral formula for the potential is invalid independent of all other mathematical errors. Because all integrals can be recast as summations, the potential Toomre provided is a summation of simpler component potentials. However, Poisson’s equation is non-homogeneous. From Pinsky [67] (Chapter 1), solutions to non-homogeneous differential equations cannot be summed (i.e., superimposed), as in homogeneous equations such as that of Laplace, where ρ = 0 One can arrive at the finding that Toomre’s 2nd equation cannot solve Poisson’s equation from another perspective: Obviously, the exponential dependence on z in Toomre’s 2nd equation involves separation of the potential into a some function of r multiplied by another function of z. Whereas separation of variables is commonly used to solve homogenous differential equations, it is not possible to solve an inhomogeneous differential equation in this way, e.g., [67]. Solving (15) using separation of variables is impossible, as revealed by inspection. Separation of variables for the potential requires that density also be a multiplication of two distinct functions, one of z and another of r. For this representation, the RHS of (17) holds at any given radius, and so the density does not depend on z. For this case, the potential cannot depend on z either. Toomre addressed this problem by setting z = 0, which prohibits solving Poisson’s equation. From another perspective, in “dropping” the z-dependence of the potential in Toomre’s 2nd equation from his 4th equation and beyond, Toomre assumed that density is independent of z (via Equations (15) and (17)), i.e., he actually assumed that density is constant along the z-axis. Zero is a constant. Coaxial cylinders (Figure 2) is actually the geometry described in [46]. Due to the properties of the exponential function, Toomre’s component of the potential along z cannot reduce to the exact result for the special axis of a disk, which was known circa 1930 [63]:ψdisk,ext,axial(z)=−2GMdiskma2[z2+a2−−−−−−√−z](18)Equations (11) and (18) respectively reduce to the correct, inverse square dependence of force with distance at great distance, and of potential with inverse distance, if their limits as a/z approach zero are properly evaluated. The exponential function does not reduce to this required functional dependence.In summary, gravitation is a 3-dimensional phenomenon, as illustrated in Figure 11. Toomre’s incorrect analysis does not describe gravitation of a disk. Force summations are needed (Section 4). 3.1.7. Fundamental Mathematic Problems in Many Post-1998 NOMs Models Simulations of RC in many papers involve summing individual contributions from different geometric constituents such as the bulge, disk, halo, and/or black hole. Most studies after ~1998 use Dehnen and Binneys’ [68] problematic modification of Poisson’s equation:−4πG∑jρj(r)=∇2ψ(19)In these forward models, Equation (16) is used to fit RC, and so centripetal vs. gravitational forces are balanced in the equatorial plane, as in the early approaches.The modification (19) and implementation via (16) are unjustifiable, even if the proper divergence of the gradient (LHS of (15)) is used [14,48]:Densities do not sum, as discussed in numerous books on thermodynamics. Importantly, addition of densities in Equation (17) is equivalent to summing solutions of individual differential equations. Use of linear superposition is indeed described in RC literature [58,69]. Again, Poisson’s equation is a non-homogeneous partial differential equation: it is well-known that solutions to such equations cannot be summed (e.g., [67]). It is immaterial what component is being summed: velocities, masses, densities (e.g., [30]), or v2, which is generally the case [29,58]. All are equally problematic. All such summations amount to linear superposition, which is allowable only for homogeneous differential equations, i.e., when ρ = 0 everywhere.The underlying problem with post-1998 approach is that forces due to different geometrical shapes have different radial dependences except for the case of nested spheres. Hence, one exception exists regarding summation: If all components in the galaxy are distributed spherically, then summing v2 is permitted, since each of the different mass distributions can be represented by some effective point mass at the center. This approach stems from the force balance of Equation (1), and is a reasonable approximation for nearly round elliptical galaxies. Spirals cannot be treated in this way because the forces for this axially symmetric shape are not central and do not vary inversely with r2, as discussed in Section 3.1.2 and Section 3.1.3 and shown in Figure 9.Deductions that NBDM haloes exist around spirals rest on summations that involve at least one non-spherical component. Proposals of haloes are underlain by improper mathematical analyses. 3.1.8. Relativistic Orbital Models General relativity models are complex, so simplifications are required to construct forward models of RC. We discuss three studies by different groups.Brownstein and Moffat [7] assume a symmetrical distribution of mass M over a sphere of radius s, which permits using the volumetric density. The centripetal force of (6) remains, but force on a test particle (mass m) in an orbit is modified toF(s)=−GmM(s)s2{1+M0M−−−−√[1−(1+ss0)exp(−ss0)]}(20)Acceleration (a = F/m) is considered, since m cancels during force balance. The relativistic acceleration of (20) is non-central, and involves three fitting parameters: an acceleration (a0), a reference mass (M0), and a reference radius (r0). The number of free parameters is reduced to two by the Newtonian orbital result of a0 = GM0/r02. Reference values of mass and radius are linked via the formulation for density. For this reason, Brownstein and Moffat [7] consider their fits to involve one parameter. However, the function assumed for the density is also a constraint, and masses for stars and gas were modeled separately.Lin et al. [33] describe relativistic acceleration as an asymptotic function, which makes their model quite similar to MOND (Section 3.1.9). Otherwise, both studies follow the same steps to simulate RC: (1) It is assumed that that s can be replaced by r in the non-central attractive force to represent the equatorial plane. (2) Different mass distributions are assumed for two components (e.g., H gas and other unconsolidated matter vs. stars). (3) Velocities or v2 for these two different mass distributions are summed and fit to RC.Scelza and Stabile [34] base their analysis on (17). Potentials for a spherical bulge and a flat disk are summed. The bulge potential is a function of s only, but this is not so for the disk, where s2 = r2 + z2. As discussed above, velocities can sum only if the masses are spherically distributed and have central forces (i.e., going as 1/s2 where the vectors point to the center and the velocities are tangential to the spherical shells. However, because substitution of r for s is not justified in Newtonian physics (Section 3.1.5), this substitution is equally questionable in general relativity.Other relativistic calculations exist, and are similar to those described above. 3.1.9. Modified Newtonian Orbital Dynamics where a0 = GM0/r02, as above. Because acceleration decreases as r increases, forces in MOND are non-central.MOND is a phenomenological model. The initial proposal of Milgrom [5], used subsequently by Sanders and McGaugh [70] and others, assumes that Newtonian orbital acceleration (F/m) governs the central regions of a galaxy, whereas a smaller acceleration governs the outside. A critical acceleration divides the two regimes. Hence, an interpolation function is used to describe the variation in the acceleration across the galaxy. A popular form is:μ(x)=x1+x2−−−−−√ where x=aa0(21)As in the relativistic calculations, MOND models use two free parameters for the attractive force, assume circular orbits, and sum v2 when more than one mass type is considered. For such a summation to be valid, requires central forces (spherical distributions) for all components. 3.2. Comparison of Orbital Forward Models Forward models simulate RC. The accuracy of a fit does not validate any particular model because certain factors supersede: these include the nature of the assumptions and the number of free parameters, most of which are used to describe the density as a function of radius for each galactic component invoked. For each component, the minimum number of parameters is two: one for constant density and one for its shape, whether the choice is explicit or not (Table 2). Additional parameters are required for more complicated density functions. The function for density also involves some choices, although this can be constrained from the luminosity with additional assumptions. Some force laws involve a free parameter. Only the force constant G of Newtonian gravitational force is experimentally constrained.Table 2. Forward Models Applied to Rotation Curves.📷Table 2 summarizes forward models. Problems with some approaches are discussed in Section 3.1. Uncertainties and systematic errors in RC (Section 2.2) are neglected in the discussion to follow. 3.2.1. Allowable Number of Free Parameters The number of free parameters in a robust physical model should not exceed the number of dependent, observational variables it proposes to explain. For example, one can weigh a colorless gem on a scale, measure its volume, and then ascertain whether this is a diamond or glass from the density. The problem is not constrained if additional free parameters or unknowns exist. If two gemstones are placed on the scale, one cannot tell which one is the fake, without making a second measurement. Elementary physics books present many such exercises. The precepts of linear algebra are quite clear: one equation (or observation) with one unknown can be solved; two equations (or observations) with two unknowns has a solution; three equations (or observations) are needed to constrain three unknowns, etc.Images of galaxies constrain their shape and size. This information (Section 1 and Section 2) underlies the mathematical construction, as indicated in Table 2 and discussed in Section 3.1.A rotation curve constrains one variable, the tangential orbital velocity (v) as a function of the radius (r) in the equatorial plane. The function v(r) may be complex, but for gravitational shapes, the velocity at any given radius is related to mass inside the orbit (Min), also as a function of r. In forward models, the assumed mass distribution is the input, although this can be cast as density for a specific shape. In general, terms the forward computational approach is (after Groetsch [27]):Min(r)⇒equation for force law⇒ v(r)(22)Only one quantity, v(r), is known, so one free parameter is permissibly calculable at any given radius, for any given force law. This parameter is the mass or density function. Hence, summing components does not provide a definitive answer, even if summing the solutions for each component shape were mathematically permissible, which it is not unless all mass components were spherical (Section 3.1.6).Ambiguity that results from multiple free parameters is discussed in cosmology [71], and has been recognized by users of the NOMs approach (e.g., [36,69,72,73]). Commonly, the NBDM halo component is minimized during fitting [35]. Even with such biasing, available estimates of DM proportions are huge, ~75% for the Milky Way [51,52] and >99% for some dwarf galaxies [19]. This situation occurs because a single mass distribution controls rotation via (22).Please note that some authors pursuing relativistic or MOND models state that one free parameter was used. This count refers to the acceleration or reference parameter only, and does not take into account that the form used for density also involves parameter(s). If more than one mass is considered (e.g., stars and gas), more than one parameter was used for each component.Thus, multicomponent models for RC are inconclusive, independent of their mathematical shortcomings. Haloes inferred in the NOMs approach are without basis, simply given the excessive number of free parameters, which is ~8 due to considering 4 shapes. Feng [20] provides further discussion.The subsubsections below focus on models in Table 2 with a single geometry, none of which require NBDM. 3.2.2. Ambiguities in Force Laws When the force law is not known, the RC forward problem cannot be solved unambiguously. MOND and relativistic models both use the Newtonian constraint for the central region [7], but model acceleration at great distance in different ways. For each of MOND and relativistic models, uniform acceleration parameters are obtained for many galaxies. This uniformity of parameters results from the following combination of factors: (1) use of non-central forces in these non-Newtonian models; (2) ellipticity in spirals being similar, so that the Newtonian force law of Equation (8) is similar for all galaxies; and 3) rotation curves have similar mathematical forms (Figure 7b).Newtonian force laws depend on shape. If the shape is known, this is not a free parameter, but a constraint, because “G” in Newton’s law is experimentally constrained (see Section 3.1). 3.2.3. Density Formulations for Disk Models Based on Central Forces All simulations of RC either assume a density structure or component mass, or construct a model for the mass in stars based on luminosity measurements. For the latter case, some approximations are needed to estimate the mass of unconsolidated material, which is not unreasonable but introduces uncertainties. Most of the functions used (e.g., the popular exponential function [74,75]) give high density near the center, consistent with central regions being brighter. Numerical calculations for the interior of a disk (Figure 10a) show that a simple exponential function provides tangential velocities that first increase with radius, and then flatten, in accord with one of the generic types of RC (Figure 7b).Density functions are also restricted by the central limit of r→0, as follows: Because M is proportional to ρr3 for the oblate (or ρr2 for a disk), the existence of finite mass at the center requires that ρ increases at least as strongly as 1/r3 at the center of an oblate (1/r2 for a disk).The log-normal function meets neither of these criteria. Yet, Marr [12,13] obtained reasonable fits to RC by employing a central force law, as in the NOMs approach. RC are replicated because the log-normal function compensates for the inappropriate use of a central force law (F~1/r2) for flattened shapes (Section 3.1). The apparent realism of the computed RC shape underscores the uncertainties inherent to forward modeling. 3.2.4. Approximate Analytical Models of the Disk with a Single Density Our numerical calculations and approximate formulae for the disk yield a velocity that initially increases at small radius for both constant density and exponentially declining density (Figure 10b). The singularity in the force at the edge of the disk yields a discontinuity in v at r = a. Notably, both forces and velocities around either a point or a wire with finite mass become infinite as r→0. Neither the point mass nor the infinitely thin wire are real entities: these are mathematical conveniences. In applying these formulae, the test particle must be outside the central mass. The same holds for the disk, as the numerical calculations show that the entire disk contributes to the force on any given point: this was recognized in the inverse models of disks previously ([8,9,10,11,76], Section 4).For a rotating thin disk of constant density, the gravitational pull of ~r1.06 per unit mass would be balanced by the centrifugal force ~v2/r, also per mass. The velocity inside such a disk would increase greatly outwards, stronger than r, as shown in our numerical calculation (Figure 10b). Hence, a hypothetical thin galaxy of constant density would spin with increasingly faster velocities on the outside than would a solid record with interior cohesive forces. Any concentration of mass towards the center would slow down the spin of the outer rings. Thus, rotation of the thin disk with some density function which decreases with r could, in principle, provide the basic shape of RC whereby velocities first increase with r, flatten, and then sometimes decrease roughly as 1/r. Figure 10b compares the numerical results for velocities in a thin disk with density that decreases exponentially outwards (as is commonly assumed) to v of a constant density disk and of a point source, all with the same total mass. Declining density provides an RC similar to those observed for galaxies, but with an undesirable edge effect. 3.2.5. Forward Models of the Spinning, Oblate Shape Orbits are governed by the potential exterior to an object, whereas axial spin is controlled by the interior potential of the object itself. These two gravitational potentials are mathematically distinct but match at the surface. Section 4 focusses on the organized motions of the interior. Here we follow a forward modeling approach that is equivalent to the NOMs models, i.e., an orbiting test particle is considered but the interior is taken to be an oblate body composed of nested, equipotential homeoids.Applying the Virial theorem while assuming the homeoids have the same ellipticity e as the surface of the oblate gives:v2homeoid=3GMin2rarcsin(e)e(23)[15]. The interior mass is obtained from:Min=4π1−e2−−−−−√∫r0ρ(q)q2dq(24)Figure 12 shows results for two different power law formulations for varying internal volumetric density which grades into the surroundings. These density distributions are similar to forms considered by Binney and Tremaine [43]. Figure 12a depicts the case ρ(r) = brn, where b was chosen for each value of n to provide a value for Min of 1011 solar masses at an equatorial radius of 18 kpc, which approximates the Milky Way. For n = 0, velocity v depends linearly on r, because density is the same for all shells; this result is the same as the interior of a rigidly rotating sphere or spheroid of Figure 8. Interestingly, for a power law with n = −2, the equatorial velocity does not vary with horizontal distance, but instead remains constant at [6πGb ArcSin(e)/e]½. Figure 12b was similarly constructed.📷Figure 12. Simulated RC for oblate bodies with varying internal density, calculated for c/a = 0.1 = (1-e2)½ and Min = 1011Msun at 18 kpc. An edge is not assumed: (a) Power law; (b) Formulae resembling that of Binney and Tremaine [43], which these authors used to cancel terms and make the integral tractable. The case of n = −3/2 has the same shape as the result provided by Figure 2.13 in Ref. [43]), which to our knowledge is the only analytical solution heretofore available for an oblate spheroid.Another possible density function is Emden’s polytrope [77], which is associated with adiabatic (actually isentropic) compression of an ideal gas. Presumably, galaxies form when an immense gas cloud collapses under its own weight, suggesting that the present density distribution could be derived from that of the earlier state. Polytropes represent termination of mass at a finite radius (Figure 13a), and thus this function describes spiral interiors only, e.g., up to the edge in visual images of galaxies or to a fall-off in density to IGM values. Galaxies do not extend to infinity as does the popular exponential decay in density, although these are enveloped in a gaseous H2, H and He atmosphere (Figure 1).📷Figure 13. Polytropic model for galaxies: (a) Density for polytropes (black patterns, with indices labeled, which are normalized to the central density as a function of scaled radius normalized to unity at the center and zero at the surface (r = a). Index n = 0 for constant density is not shown. Gray shows linear and exponential densities for comparison. Reproduction of Figure 4b from Hofmeister and Criss [14], which is open access under a Creative Commons Attribution 4.0 International License: (b) Rotation curves for differentially rotating homeoids inside an oblate with the same ellipticity. During the normalization factors involve the ellipticity cancel.The shape of the polytrope function depends on an index, n, such that n = 1.5 corresponds to monatomic gas (e.g., H or He) while n = 2.5 corresponds to a diatomic gas such as H2 (e.g., [78]). Indices > 1 give model RC declining at large distance (Figure 13b) that are similar to many measurements. Indices near 2 or 3 are promising because these have flat, concentrated density near the center (Figure 13a), and so resemble a galaxy with a bulge.Many of the curves shown in Figure 12 and Figure 13 resemble generic RC of Figure 7b and actual RC (Figure 3 and Figure 4). This exercise indicates that RC variations result from density variations. 3.2.6. Summary In summary, forward fitting models are inherently ambiguous because density is assumed and its radial dependence in a real galaxy need not be a simple, smooth function. The beautiful patterns of rings and spiral arms show that complexity exists whereby perfect radial symmetry is an approximation, and that density cannot vary simply with radius. A forward approach only allows evaluation of whether an assumed density (or mass distribution) is possible or plausible. Additional constraints are important, such as the luminosity and information on the proportions of stars to unconsolidated matter, both as a function of radius.We have shown from elementary physics that the rotational velocity inside a self-gravitating body must increase with radius, irrespective of material properties. This increase exists for self-gravitating oblate spheroids and the limiting case of the sphere. The point mass is not an appropriate analogy for a flat spiral galaxy. Yet, this analogy underlies all proposals of non-baryonic haloes. The vertical line of infinite mass, which underlies the popular logarithmic potential [65] is likewise inapplicable.These problematic analogies stem from the focus on orbits in astronomy, and from Gauss providing transcendental equations for the gravitational force around oblate spheroids [39], which is the most appropriate shape. More recent emphasis on the disk rests on Perek’s [47] unsubstantiated and incorrect declaration regarding density, and on Toomre’s [46] misapplication of Poisson’s equation to a plane (Section 3.1.6). Focus of the galactic community on the disk for > 50 years, which is not a gravitational shape, greatly limited progress.Importantly, forces in and around a disk or an oblate spheroid are non-central, i.e., the dependence is not 1/r2 and the lines of force are only parallel to the special directions along the special directions (Figure 9 and Figure 10, [45]). Assuming central forces in NOMs models contributed to overestimating galactic mass. Forward models which are non-central, even with formulae that are unverified with experiments, provide reasonable masses and do not require NBDM haloes.Despite the limitations of an orbital model, it is possible to fit the rotation curves with certain assumed density distributions. This has been demonstrated for ellipticals (e.g., Romanowsky et al. [79]) with a density distribution similar to that in Figure 12b. The particular distribution (n = −3/2) cancels terms in the relevant integral which permitted Binney and Tremaine [43] to provide a solution to the density with radius for an oblate body. Using equations describing nested homeoids in an oblate spheroid with diverse density formulations provides many solutions (Figure 12 and Figure 13) that resemble smooth, generic rotation curves (Figure 7b).This exercise underscores that many simple density distributions are compatible with smooth, generic RC, and that the rotation of all spiral galaxies are “dynamically supported.” Literally, NBDM haloes are unsupported. Although much can be learned from a proper mathematical analysis of RC curves via forward models, the solutions are not unique, cf. Figure 12 and Figure 13. Extracting density as a function of radius from RC requires an inverse model, covered next. 4. Inverse Models of Galactic Rotation The inverse approach is in the “reverse” direction of Equation (22), i.e., the character of the source is inferred from its effect [27]:v(r)⇒equation for force law⇒ Min(r)(25)Ideally, no additional assumptions are needed, and the solution is unique. However, Newton’s law is 3-dimensional, whereas our observational view of galaxies is 2-dimensional. Hence, galaxies appear rounder than they actually are, motions are presumed to be circular, and Doppler measurements are presumed to represent the equatorial plane. Insufficient information requires some assumptions.All the available inverse models for galaxies are grounded in Newtonian physics, but consider different shapes and use different computational approaches (Table 3). Each inverse model is unique, so we discuss these approaches individually. Early efforts on the oblate shape are covered in Section 4.3.Table 3. Newtonian Inverse Models Applied to Rotation Curves.📷Oblate models provide volumetric density. Plane and disk models provide surface density (σ) which is actually a mass cross section, Equation (17). One can imagine a surface perpendicular to z, but above the galaxy: σ represents the total mass situated below the imaginary surface. The physical meaning of σ lies in its connection with the galactic luminosity, but adjustment must be made for internal attenuation of light, component luminosity, and for the estimated galactic tilt relative to the LOS. 4.1. Numerical Disk-Ring Models Gallo and Feng [76] and Feng and Gallo [8,64,80] were the first to invert detailed measured RC. The geometry considered is a ring-disk model, similar to that of Bergeman [50] (Section 2.2.1). Feng [9] summarizes these efforts. A finite height, H, is assumed for the coin-like shape. 4.1.1. Mathematical Construct Feng and Gallo’s [8,64,80] analysis is based on a force balance that sums a series of concentric rings (actually hollow cylinders). The dimensional equivalent of their key equation is:Amtest[v(r)]2r=GHmtest∫a0[∫2π0(qcosθ−r)dθ(r2+q2−2rqcosθ)3/2]ρ(q)qdq(26)where q is a dummy radial variable, the integral is over the disk to its full radius, a, and A = 1 in this dimensionalized form. Also assumed is that the moment of inertia = mtestr2, which is valid for a thin ring or hollow cylinder. The total mass M of the galaxy is related to the volumetric density (ρ) or the surface density (σ) by the following:M=2πH∫a0ρ(q)qdq=2π∫a0σ(q)qdq(27)Volumetric density cannot be constrained without additional assumptions regarding H.For numerical evaluation, (26) is non-dimensionalized, using A = avc2/(MG), a characteristic velocity (vc) defined by the RC, and two elliptical integrals (K and E):A2v2(r)=∫10[K(m)q−r−E(m)q+r]σ(q)qdq(28)This is Equation (6) in Feng and Gallo [64], who state that evaluating the integral of (25) for the case of finite ρ at the center gives the null value in the limit of r→0. Velocity at the center would also approach 0. This is also true for some singularities in density, as can be confirmed by setting r = 0 in the integral of (26). Cancelling the test mass, and multiplying both sides by r gives:v|r→0→rGH[∫2π0cosθdθ]{∫a0ρ(q)qdq}→rGH[0]∫a0ρ(q)qdq→rGH[0]ρ|r→0=rG[0]σ|r→0(29)Because the integral over the angle approaches 0 at small r, velocity will increase linearly with radius at the center as long as σ climbs to infinity no faster than 1/r. Feng and Gallo reached this conclusion by considering properties of elliptical integrals in (28).Mathematically, Feng and Gallo’s inverse model is sound, but the assumptions need to be specified to interpret their results. Values for H are arbitrary, as stated by the authors, who assumed H = 0.01a. In this formulation, using σ(r) rather than ρ(r) eliminates the free parameter, H from (26). However, this simplification stems from assuming that density in (24) is independent of z, i.e., ρ(z,r) = ρ(0,r). Hence, the model actually describes rotating coaxial cylinders. As sketched in Figure 1 and Figure 2f, these can be tall, since H is unconstrained. The limit of q = a was applied in evaluating the integral of (26). This step assumes that material near the edge of the disk affects motions of material near the center. This behavior is unlike Newton’s analysis of self-gravitating spherical and spinning oblate bodies, which shows that only mass internal to the test mass controls its orbit. Evaluation over the entire disk or cylinder is needed because these shapes are not gravitationally stable (Figure 2e). 4.1.2. Results of Ring-Disk Models Feng and Gallo explored both idealized and measured RC. In all cases, their model produces singularities both at the center (r→0) and the edge (r→a). The central singularity is consistent with a coaxial cylinder geometry, which becomes the line source as r→0 (Figure 11). The additional singularity at the edge is inherent to a finite size disk (Section 3.1.3).For an example of their extraction, Figure 14 shows RC and results for a moderate size (R)SA(r) counter-rotating galaxy, which motions require axial symmetry. NGC 4736 exhibits a ring and its velocity decreases with radius, which also indicate simple axial spin. Its rotation curve was determined out to the visible edge [53] and so depicts the dense interior.📷Figure 14. Inverse analysis of NGC 4736 of Feng and Gallo [64], based on measured RC of De Blok et al. [53], shown as a black dotted line: (a) Results for σ from numerical evaluation of (24) by [64]. Dashed line = least squares fit below 1 kpc, which equals 1/r within the uncertainty of the selected cutoff; (b) Mass obtained form (25), which integration smooths σ. Black dashed line = a third order polynomial obtained from a least squares fit. Heavy lines in the lower right corner show mass computed for a thin, constant density disk from (13), using two approximations for RC. Average v = 159 km s−1. Thin dash-dotted line shows the extrapolation used. Equation (13) has a singularity in v at r = a, but merges with the point mass approximation at very large r.Calculated surface density below 1 kpc (Figure 14a) goes as 1/r, which is quantitatively consistent with connection of a coaxial cylinder geometry to a line source. The surface density from ~1 to 9 kpc, which avoids the singularities, is well-described by an exponential function, as discussed by the Feng and Gallo [64].Mass in Figure 14b was obtained by integrating σ, to eliminate specifying a numerical value for H. The form of the volume element in (25) leads to M = 0 at r = 0, despite the 1/r singularity in density. Hence, the mass is 4.45 × 1011 solar masses out of 10 kpc for NGC 4736. Luminosity in the visible is much lower, ~1 × 1010 solar [42] for this nearly face-on spiral. The relatively large computed mass is connected with the assumption of disk geometry, whereby the thickness H is uniform from r = 0 to r = a, and in addition, ρ is assumed to the constant in the z-direction. For comparison, mass inside r = a, calculated for an ultrathin disk from the approximate formula of (13), is ~4 × 1010 solar masses, see Figure 14b. This rough value confirms that the additional density (mass) above and below the equatorial plane influences the inversion.We emphasize that the large mass results from assuming a coin-shaped geometry. Despite its popularity in galactic astronomy, the disk is not a stable gravitational shape and so cannot quantitatively represent rotation of spiral galaxies. 4.2. Numerical Mass Summations in the Equatorial Plane Sipols and Pavlovich [11] improved the matrix inversion model of Pavolich et al. [10], and explore the connection of mass with luminosity in detail. Our review is limited to their extractions of σ and mass from RC data. The geometry considered is the equatorial plane with a finite radius, a. Hence, negligible thickness was assumed. 4.2.1. Mathematical Construct The key equation is (26), where surface mass is considered and A = 1; the mass defined in (26) also holds. Sipols and Pavlovich [11] approximated Equation (26) as a summation over a radial distribution of point masses, designated as mq for each radius (q) of interest:V2rr=G∑q[mq∑θr−qcosθ(r2−2rqcosθ+q2)3/2](30)As in Section 4.1, the entire disk contributes to motions of particles in the interior.Rotation curves consist of N datapoints, which represent N orbits, and thus Equation (30) describes each of these orbits. The system is linear, and so inverting the above matrix [10,11] gives a unique solution for mr. Since these point masses are spaced in area elements, σ vs. r is provided, and the result is unique. To obtain the proper spacing of point masses, i.e., mass per unit area, the authors accounted for circumference being proportional to the radius. Pavlovich et al. [10] provide a detailed diagram of their geometrical construction.Mathematically, the inverse model of Sipols and Pavlovich is sound: here we summarize the assumptions for comparison with other models:An equatorial plane of finite radius a is modeled: so no mass exists above or below z = 0. Each orbit is affected by all the mass points in this plane: As discussed in Section 4.1.1, no theorem of Newton exists to guide evaluation of an integral (or summation) in a disk geometry. 4.2.2. Results of Mass Summations in the Equatorial Plane Sipols and Pavlovich [11] explored RC measured for a wide range of galaxies. To exemplify their extraction, Figure 15 shows RC and results for a moderate size, counter-rotating type (R)SAB(s)a galaxy, which motions require axial symmetry. Velocity of NGC 1808 decreases with radius, consistent with simple axial spin. Its rotation curves were determined out to the visible edge [81] and so depict the starry interior.📷Figure 15. Inverse analysis of NGC 1808 by Sipols and Pavlovich [11], based on measured RC of Sofue et al. [81], shown as a heavy dotted line: (a) Results for σ from matrix inversion of (29) by [11]. Dashed line = least squares fit below 0.8 kpc. Thin line = fit from 2 to 12 kpc; (b) Mass obtained form (26), which integration smooths σ. Black dashed line = the least squares fit listed in the box. For comparison, mass was computed for a thin, constant density disk from (13) using extrapolated velocity (thin dash-dotted line). Equation (13) has a singularity in v at r = a, while merging with the point mass approximation at large r.Calculated surface density at r = 0 is finite. Near the origin, σ declines exponentially whereas a weaker exponential decline describes most of the spiral galaxy (Figure 15a). The upturn in σ as r→a is consistent with the singularity inferred analytically, but not seen numerically due to finite step size in the forward modeling (Figure 7b).Mass in Figure 15b was obtained by integrating σ via (27). Their calculated mass is 4.4 × 1010 solar masses at 14.2 kpc for NGC 1808. Luminosity in the visible is lower, 5.9 × 109 solar [42], in part due to the tilted presentation, where the major axis appears to be 2.1× the minor axis. Because the area of an ellipse is πab, but the area of circle is πa2, the luminosity of a relevant, face-on presentation would be 1.2 × 1010 solar. NGC 1808 is actively forming stars and so likely has substantial unconsolidated gas. For comparison, we calculated mass outside r = a, for an ultrathin disk from the approximate formula of (14) as ~4 × 1010 solar masses, which is in good agreement with the matrix inversion of [11], see Figure 15b.Results for the plane from [11] are reasonable. However, mass exists above and below the plane in spiral galaxies (Figure 1), and affects the rotation curves. 4.3. Analytical Model of the Oblate Results of Newton, Maclaurin, Gauss, Todhunter, and Clausius were combined by Criss and Hofmeister [15,44] to provide an analytical inverse model of the volumetric density, ρ(r,z), from data on v(r) and ellipticity of differentially spinning galaxies. Our model assumes quasi-steady-state conditions, which underlie interpretation of galaxy Doppler patterns as circular motions. This assumption is consistent with the oblate spheroid shapes of spiral galaxies documented in images (NED [42]) and intensity contours (e.g., Figure 1), because this shape is the hallmark of a gravitationally stable, spinning entity. The preferred axis of rotation and the organized motions of galaxy constituents show that the phenomenon is spin, not orbits.Here we note that previous efforts, predominantly of Burbidge [4] and Brandt [82], used the exterior potential and orbits. This approach does not permit inversion of RC and investigation of the interior, particularly as Brandt divided by zero, see [15]. These efforts are more closely related to our forward models, but do not account for Newton’s homeoid theorem. After the long hiatus in considering the oblate shape, we started from the beginning, considering galaxies as differentially rotating homeoidal shells. 4.3.1. Mathematical Construct Each homeoidal shell (Figure 2d) is equipotential, which requires constant density, and provides stability, per Newton. As the galaxy is a bound state (restricted in both r and z), Clausius’ Virial theorem holds, which relates kinetic to potential energy. Conservation laws are independent, additional restrictions [83]. The gravitational self-potential was provided by Todhunter [40], based on the historic geometrical constructions of Maclaurin. Differentiating this well-known result gives the gravitational self-potential for a homeoid of mass m = ρdV = 3ρVdr/r that surrounds an interior mass Min. A similar approach gives its moment of inertia:−Ug,homeoid=GMinmrarcsin(e)e ; Ihomeoid=23mr2(31)Applying the Virial theorem or balancing forces gives the same result:v2homeoid=3GMin2rarcsin(e)e=v2tan(32)Differentiating and some algebra [15,44] yields the inverse solution as:ρ(r)=16πG(2vr∂v∂r+v2r2)e1−e2−−−−−√arcsin(e)(33)Only the r dependence of density need be specified because the z-dependence is defined by the ellipsoidal shape of the homeoids (Section 2.2.1). Homeoids are nested: the same ellipticity as the oblate surface is used for the interior of a galaxy.Equation (33) is analytic and exact, has no free parameters, and allows direct and unambiguous extraction of density and mass profiles from RC. If e is not known, c/a = 0.1 is used, as indicated by light contours of edge-on galaxies. Ellipticity does not have a large effect on velocity (32) because the geometrical factor arcsin(e)/e ranges only from 1 to ~1.57, in covering shapes from a sphere to the infinite plane (e = 0 to 1). 4.3.2. Results for Differential Spin Equation (33) was applied to 36 spiral galaxies, and 15 additional types (spiral, lenticular, irregular, spheroidal, elliptical, and polar ring) using a commercial spreadsheet [15].Excluding the bright centers and diminishing furthest reaches, density approximately follows r−1.8. For all galaxies examined, ρ at the visual edge does not vary much, such that the average value of 1.1 × 10−21 kg m−3 matches ISM density. The visual edge, being an isophote, is defined by a certain concentration of luminous matter in the galaxy. Association of the visual edge with a certain value of density indicates that total mass correlates with star mass. Consistency exists: luminosity of the 51 galaxies studied linearly depends on the radius associated with the visible edge (Figure 14 in [15]). Further out, the density decreases to roughly IGM values. Near the centers of most galaxies, ρ is like that independently determined for molecular cloud cores, such that larger galaxies, are more concentrated at their centers.Galaxies are density stratified just like any other self-gravitating body. Likewise, galaxies have a visible edge with some surrounding gas (H, H2) whose density becomes very low at great distance. Galaxies thus have “atmospheres.”Figure 16 shows two examples, those analyzed using disk models of Figure 14 and Figure 15. NGC 4736 was analyzed earlier by us, but using lower resolution data of Sofue et al. [81]. Results are nearly the same for mass. Larger NGC 1808 is more massive, consistent with its size. Also indicated in Figure 16 is that its density is higher overall. The high-resolution curve of NGC 4736 shows internal structure: rings of rarefied regions are prominent, which is consistent with its morphology [42].📷Figure 16. Inverse analysis within the visible disk of counter-rotating spirals. Arrows point to the relevant y-scale: (a) NCG 4736, using RC data from deBlok et al. [53], shown in Figure 14; (b) NGC 1808, using RC data of Sofue et al. [81], shown in Figure 15. 4.4. Comparison of Inverse Models Our analysis of the oblate shape, which assumes c/a = 0.1 if unknown, provides less mass than the thick disk model, which includes mass in the corners, but more mass than the ultrathin disk model, which does not include material off the equatorial plane. Thus, masses obtained in the three inverse models depend on the volume for the assumed shape in the expected order. 5. Discussion and Conclusions This paper reviews and evaluates two important matters regarding the rotational motions of spiral galaxies. First, we have analyzed the mathematical extractions of rotation curves from Doppler data on spiral galaxies. We have shown above that Doppler data need to be reanalyzed, and that rotations about more than one axis might be important, particularly for the complex motions within large galaxies.Second, we have evaluated and compared the various models applied to RC to infer distributions of mass in galaxies that are associated with their presumably circular rotational motions and presumed gravitational stability. Forward models cannot provide unambiguous results, because the number of free parameters is excessive. Also, smooth variations in density with radius are too simple, given the complexity of visual images. Only inverse models can be analytically unambiguous, but the results depend on the geometry assumed. To us, the oblate spheroid is the most appropriate simple shape for galaxies, as it is supported by thousands of images and consistent with Newtonian physics. As was evident centuries ago, gravitation is a 3-dimensional phenomenon, and the oblate spheroid is the stable shape of a spinning object, large or small.To better understand the present emphasis on the coin-like shape (the disk) in galactic modeling, a quote from Samuel Pierpont Langley [84] is useful. He likened the progress of science to “a pack of hounds, which in the long-run perhaps catches its game, but … where the louder-voiced bring many to follow them nearly as often in a wrong path than in a right one.” Perhaps we can offer additional insights, from the privileged perspective afforded by the currently rapid advances in science and engineering.The focus on the coin shape for galactic models originated with declaration of Perek [47] in 1958 that density in z and r directions are independent. This unsupportable contention is one component of the improper treatment of Poisson’s inhomogeneous equation by Toomre [46]. One could argue that the oblate shape was set aside because Gauss’s transcendental formulation of its potential is not readily tractable [39]. However, Toomre’s mathematics are even more obscure, far less user friendly, and do not reduce to the proper Newtonian limits at large distance. Perhaps the answer lies in the discovery that velocity profiles describing the outskirts of galaxies are rather flat. In inappropriately comparing this observation to the familiar behavior of the few discrete planets in the Solar System, another wrong turn was taken, one that spawned the concept of non-baryonic dark matter surrounding galaxies. Rotation curves at centers were also misanalysed, in part because differences between discrete and effectively continuous distributions of mass were not considered, but also because the disk, unlike the stable oblate spheroid, lacks a theorem of Newton to facilitate the analysis of its interior. It may seem strange, but the sphere is a better approximation to a spiral galaxy than a disk, as shown in figures on the density and mass extracted for the Milky Way [15] and Andromeda [44] for the case of e = 0. This is verifiably true, due to Todhunter’s recasting of Maclaurin’s geometrical constraints into useful formulae for the gravitational self-potential [40]: An oblate body is nothing more than a flattened sphere.A type of aether, NBDM, is supported by cosmological models, which contain many free parameters [71], and cannot be validated, any more than the origin of life can be established. Not all questions can be answered with scientific methods, as evolutionary phenomena hide their beginnings, e.g., via bifurcations, as discussed by Nobel Laureate Ilya Prigogine [85].Over-specialization and compartmentalization may explain the publication of Brandt and Toomre’s papers on galactic rotation, both of which divide by zero. One can argue that the author of any article holds the ultimate responsibility for the correctness of its contents, rather than any reviewer, editor, or publisher. This is true, but critical analytical review, followed by post-publication scrutiny, are of the utmost importance to scientific progress. Experimental evaluation can indeed be difficult in astronomical studies. However, model-to-model comparisons provide no proof that a methodology is valid. Software development and popularity of multicomponent modeling have masked many errors in mathematics and physics incorporated in galactic orbital models.A third problem is the pretense that consensus is the arbiter of correctness (see e.g., [1]). Practically no major advance in human history has ever satisfied this criterion. Among human shortcomings are that no one likes to admit they were wrong. Max Planck provided a lengthy analysis, which Paul A. Samuelsen encapsulated as “Science progresses, funeral by funeral” [86].Most scientific research represents incremental embellishments of standard schemes. At least up until the current time, no progress can be made in galactic dynamics within the current framework of two competing, major camps with minimal mutual interaction. Specifically, real progress has reached a standstill for both the non-baryonic dark matter and non-Newtonian camps. We hope that the reader, in closely examining this paper and others in this special issue, will agree that it is time to pursue alternative lines of investigation. Only new avenues will further understanding of the physics underlying the rotational motions of the immense, self-gravitating, and evolving entities known as galaxies.Author Contributions Both authors contributed equally to all aspects of this research, but A.M.H. was primarily responsible for manuscript preparation. 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I have use HISAT2 for alignment . Now i want to obtain contig sequence from from BAM file in galaxy server?
I have used Samtools View. That is giving me some sequence but I am not sure whether it is contig or not?
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Yuxin Yang please recommend me proper tools present on galaxy server to obtain contig or consensus sequence from a BAM file?
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Just think of a photo sent by the Chandra x-ray telescope clicking a large galaxy
at a distance of 100 million light years and it be true ? aren't we seeing a virtual galaxy
whose light travelled 100 million years in past ? to reach us now
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Light travels at around 300,000 km/s; that's about a foot per nanosecond. Since my television is about 8 feet from my couch, when I watch tv, I'm seeing light that it emitted 8 nanoseconds ago. That doesn't make it a 'virtual television'. It's just that it takes that long for its light to reach me.