No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
A Guided Tour of Planetary Interiors
Abstract
We explore the gravitational dynamics of falling through planetary interiors.
Two trajectory classes are considered: a straight cord between two surface
points, and the brachistochrone path that minimizes the falling time between
two points. The times taken to fall along these paths, and the shapes of the
brachistochrone paths, are examined for the Moon, Mars, Earth, Saturn, and the
Sun, based on models of their interiors. A toy model of the internal structure,
a power-law gravitational field, characterizes the dynamics with one parameter,
the exponent of the power-law, with values from -2 for a point-mass to +1 for a
uniform sphere. Smaller celestial bodies behave like a uniform sphere, while
larger bodies begin to approximate point-masses, consistent with an effective
exponent describing their interior gravity.
... In the terrestrial brachistochrone problem [65,67], tunnels of various curved shape described by functions r(θ) in polar coordinates, are dug out in the uniform Earth of radius R, and one looks for the shape that minimizes the transit time between two given points. This problem has seen much renewed attention recently, mostly in the pedagogical literature [72][73][74][75][76][77][78][79][80][81][82][83][84][85][86] but not only [87][88][89]. ...
... The problem of the terrestrial brachistochrone, which has seen renewed attention recently [70,[72][73][74][75][76][77][78][79][80][81][83][84][85][86][87][88][89], provides an explicit example of a universe dominated by a phantom fluid with non-linear equation of state, which can be solved explicitly and exhibits a finite future singularity at a finite value of the scale factor, where the Hubble function, Ricci scalar, energy density, and pressure all diverge. Finite time singularities have been the subject of much literature in the past decade [91,[98][99][100][101][102][103][104][105][106] hence, in this problem, the mechanical side of the analogy helps the cosmology side in the sense that the known exact solution for the terrestrial brachistochrone problem can be immediately translated into an analytical solution of the corresponding cosmology with complicated (non-linear) equation of state. ...
... Expanding to first order in δa/a min 1, using Equation (72), and keeping only the zero order and the linear terms, one obtains δä C 2 a min 2a 2 0 − a 2 min + 2C 2 a min 2a 2 0 − a 2 min + 1 + 1 a min + 1 δa + a min . ...
Several classic one-dimensional problems of variational calculus originating in non-relativistic particle mechanics have solutions that are analogues of spatially homogeneous and isotropic universes. They are ruled by an equation which is formally a Friedmann equation for a suitable cosmic fluid. These problems are revisited and their cosmic analogues are pointed out. Some correspond to the main solutions of cosmology, while others are analogous to exotic cosmologies with phantom fluids and finite future singularities.
We describe how orbital tunnels could be used to transport payloads through the Earth. If you use a brachistochrone for the tunnel, the body forces in the tunnel become overwhelmingly large for small angular distances traveled. Projectiles move along an orbital tunnel faster than they would along a brachistochrone connecting the same points but the body force components cancel. We describe how parabolic Keplerian orbits outside the object merge onto quasi-Keplerian orbits inside the object. We use models of the interior of the Earth with three values of the polytropic index (n) to calculate interior orbits that travel between surface points. The n = 3 results are also scaled to the Sun. Numerical integrations of the equations describing polytropes were used to generate the initial models. Numerical integration of the equations of motion are then used to calculate the angular distance you can travel along the surface and the traversal time as a function of the parabolic periapsis distance for each model. Trajectories through objects of low central condensation show a focussing effect that decreases as the central condensation increases. Analytic solutions for the trajectories in a homogeneous sphere are derived and compared to the numeric results. The results can be scaled to other planets, stars, or even globular clusters.
Dropping objects into a tunnel bored through Earth has been used to visualize simple harmonic motion for many years, and even imagined for use as rapid transport systems. Unlike previous studies that assumed a constant density Earth, here we calculate the fall-through time of polytropes, models of Earth's interior where the pressure varies as a power of the density. This means the fall-through time can be calculated as the central condensation varies from one to large within the family of polytropes. Having a family of models, rather than a single model, helps to explore the properties of planets and stars. Comparing the family of phase space solutions shows that the fall-through time and velocity approach the limit of radial free-fall onto a point mass as the central condensation increases. More condensed models give higher maximum velocities but do not have the right global properties for Earth. The angular distance one can travel along the surface is calculated as a brachistochrone (path of least time) tunnel that is a function of the depth to which the tunnel is bored. We also show that completely degenerate objects, simple models of white dwarf stars supported by completely degenerate electrons, have sizes similar to Earth but their much higher masses mean a much larger gravitational strength and a shorter fall-through time. Numerical integrations of the equations describing polytropes and completely degenerate objects are used to generate the initial models. Analytic solutions and numerical integration of the equations of motion are used to calculate the fall-through time for each model, and numerical integrations with analytic approximations at the boundaries are used to calculate the brachistochrones in the polytropes. Scaling relationships are provided to help use these results in other planets and stars.
Data from the Global Oscillation Network Group (GONG) project and other helioseismic experiments provide a test for models
of stellar interiors and for the thermodynamic and radiative properties, on which the models depend, of matter under the extreme
conditions found in the sun. Current models are in agreement with the helioseismic inferences, which suggests, for example,
that the disagreement between the predicted and observed fluxes of neutrinos from the sun is not caused by errors in the models.
However, the GONG data reveal subtle errors in the models, such as an excess in sound speed just beneath the convection zone.
These discrepancies indicate effects that have so far not been correctly accounted for; for example, it is plausible that
the sound-speed differences reflect weak mixing in stellar interiors, of potential importance to the overall evolution of
stars and ultimately to estimates of the age of the galaxy based on stellar evolution calculations.
How long does it take to fall down a tunnel through the center of the Earth
to the other side? Assuming a uniformly dense Earth, it would take 42 minutes,
but this assumption has not been validated. This paper examines the gravity
tunnel without this restriction, using the internal structure of the Earth as
ascertained by seismic data, and the dynamics are solved numerically. The time
taken to fall along the diameter is found to be 38 rather than 42 minutes. The
time taken to fall along a straight line between any two points is no longer
independent of distance, but interpolates between 42 minutes for short trips
and 38 minutes for long trips. The brachistochrone path (minimizing the fall
time between any two points) is similar to the uniform density solution, but
tends to reach a greater maximum depth and takes less time to traverse.
Although the assumption of uniform density works well in many cases, the
simpler assumption of a constant gravitational field serves as a better
approximation to the true results.
Few objective constraints exist on radial density variations in
planetary interiors. Even if the mean density and mean moment of inertia
were known with perfect accuracy, they would only provide integral
constraints on the density models. However, among the family of
monotonic radial density models with a given mean density and mean
moment of inertia, the simple two-layer piecewise constant models have
useful extremal properties. The model has only three parameters, an
inner density, an outer density, and a transition radius. Once the
radial moment constraints are applied, there is only one remaining
degree of freedom. If the outer region density is somehow specified,
then the inner region density of the two-layer model provides a firm
lower bound on central density of monotonic models. Likewise, if an
upper bound on central density can be provided, then the two-layer model
provides a lower bound on outer region density. An envelope of
acceptable density models can be generated by scanning the transition
depth from the center to the surface. Any monotonic model with specified
mean density and mean inertial moment must lie within that envelope.
Resulting extremal density envelopes for the Earth, Moon, and Mars are
compared to published radial density profiles. For the Moon, the moment
constraints are quite restrictive. For the Earth, which is more
centrally condensed, the allowed envelope of density profiles is rather
broad. Mars is an intermediate case. Present geodetic and astrometric
observations only constrain the Martian mean moment of inertia to lie
somewhere in the range 0.325<=I/MR2<=0.365. However,
our analysis shows that any value within that range can be accommodated
without invoking geochemically implausible density minima or maxima.
Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies
A large data set consisting of about 1000 normal mode periods, 500 summary travel time observations, 100 normal mode Q values, mass and moment of inertia have been inverted to obtain the radial distribution of elastic properties, Q values and density in the Earth's interior. The data set was supplemented with a special study of 12 years of ISC phase data which yielded an additional 1.75 × 10^6 travel time observations for P and S waves. In order to obtain satisfactory agreement with the entire data set we were required to take into account anelastic dispersion. The introduction of transverse isotropy into the outer 220 km of the mantle was required in order to satisfy the shorter period fundamental toroidal and spheroidal modes. This anisotropy also improved the fit of the larger data set. The horizontal and vertical velocities in the upper mantle differ by 2–4%, both for P and S waves. The mantle below 220 km is not required to be anisotropic. Mantle Rayleigh waves are surprisingly sensitive to compressional velocity in the upper mantle. High S_n velocities, low P_n velocities and a pronounced low-velocity zone are features of most global inversion models that are suppressed when anisotropy is allowed for in the inversion.
The Preliminary Reference Earth Model, PREM, and auxiliary tables showing fits to the data are presented.
Brachistochrones are constructed for attractive central force, with logarithmic potential. Each pair of points (except those with the centre between them) are connected by an unique brachistochrone.
Saturn's internal rotation period is unknown, though it must be less than 10 hours, 39 minutes, and 22 seconds, as derived
from magnetic field plus kilometric radiation data. By using the Cassini spacecraft's gravitational data, along with Pioneer
and Voyager radio occultation and wind data, we obtain a rotation period of 10 hours, 32 minutes, and 35 ± 13 seconds. This
more rapid spin implies slower equatorial wind speeds on Saturn than previously assumed, and the winds at higher latitudes
flow both east and west, as on Jupiter. Our related Saturn interior model has a molecular-to-metallic hydrogen transition
about halfway to the planet's center.
Figure 5: The line fall times as a function of surface angle for the five discussed bodies, comparing the prediction of a fit exponent to that of the density profile. The case for the point mass is shown for comparison
- J Garry
- Tee
Garry J Tee. Isochrones and brachistochrones. Technical report, Department of Mathematics, The University
of Auckland, New Zealand, 1998.
Figure 5: The line fall times as a function of surface angle for the five discussed bodies, comparing the prediction
of a fit exponent to that of the density profile. The case for the point mass is shown for comparison. The y-axis
in each figure is different.