
Michael EfroimskyUnited States Naval Observatory | USNO
Michael Efroimsky
Ph. D. (Oxford 1995)
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Publications (113)
The paper addresses the possibility of a young Mars having had a massive moon, which synchronized the rotation of Mars, and gave Mars an initial asymmetric triaxiality to be later boosted by geological processes. It turns out that a moon of less than a third of the lunar mass was capable of producing a sufficient initial triaxiality. The asymmetry...
The paper addresses the possibility of a young Mars having had a massive moon, which synchronised the rotation of Mars, and gave Mars an initial asymmetric triaxiality to be later boosted by geological processes. It turns out that a moon of less than a third of the lunar mass was capable of producing a sufficient initial triaxiality. The asymmetry...
The gas giant Kepler-1658b has been inferred to be spiralling into its sub-giant F-type host star Kepler-1658a (KOI-4). The measured rate of change of its orbital period is $\stackrel{\bf \centerdot }{\textstyle {P}}_{\rm orb}\, =\, -\, 131^{+20}_{-22}\,\,\mbox{ms/yr}\hspace{1.1pt}$, which can be explained by tidal dissipation in the star if its mo...
Parameterised by the Love number k2 and the tidal quality factor Q, and inferred from lunar laser ranging (LLR), tidal dissipation in the Moon follows an unexpected frequency dependence often interpreted as evidence for a highly dissipative, melt‐bearing layer encompassing the core‐mantle boundary. Within this, more or less standard interpretation,...
Parameterised by the Love number k2 and the tidal quality factor Q, and inferred from lunar laser ranging (LLR), tidal dissipation in the Moon follows an unexpected frequency dependence often interpreted as evidence for a highly dissipative, melt-bearing layer encompassing the core-mantle boundary. Within this, more or less standard interpretation,...
Context. It is conceivable that a few thousand confirmed exoplanets initially harboured satellites similar to the moons of the Solar System or larger. We ask the question of whether some of them have survived over the æons of dynamical evolution to the present day. The dynamical conditions are harsh for exomoons in such systems because of the great...
It is conceivable that a few thousand confirmed exoplanets initially harboured satellites similar to the moons of the Solar system or larger. Could some of them have survived over the aeons of dynamical evolution to the present day? The dynamical conditions are harsh for exomoons in such systems because of the greater influence of the host star and...
Interpretation of the data obtained throughout the period of more than 40 years by Lunar Laser Ranging yields an interesting observation: the tidal quality factor of the Moon, which determines the magnitude of ongoing energy dissipation, follows a different frequency dependence than is measured for rocks in laboratory conditions. When the self-grav...
Interpretation of the data obtained by Lunar Laser Ranging provides an interesting observation: the tidal quality factor of the Moon, which determines the magnitude of ongoing energy dissipation, follows a different frequency dependence than is measured for rocks in laboratory conditions (e.g., [1]). When the self-gravity of the lunar body is taken...
Parameterised by the Love number $k_2$ and the tidal quality factor $Q$, and inferred from lunar laser ranging (LLR), tidal dissipation in the Moon follows an unexpected frequency dependence often interpreted as evidence for a highly dissipative, melt-bearing layer encompassing the core-mantle boundary. Within this, more or less standard interpreta...
Interpretation of the data obtained throughout the period of more than 40 years by Lunar Laser Ranging yields an interesting observation: the tidal quality factor of the Moon, which determines the magnitude of ongoing energy dissipation, follows a different frequency dependence than is measured for rocks in laboratory conditions. When the self-grav...
Bodily tides provide key information on the interior structure, evolution, and origin of the planetary bodies. Our Solar system harbors a very diverse population of planetary bodies, including those composed of rock, ice, gas, or a mixture of all. While a rich arsenal of geophysical methods has been developed over several years to infer knowledge a...
Solid body tides provide key information on the interior structure, evolution, and origin of the planetary bodies. Our Solar system harbours a very diverse population of planetary bodies, including those composed of rock, ice, gas, or a mixture of all. While a rich arsenal of geophysical methods has been developed over several years to infer knowle...
WASP-12b stands out among the planets of its class of hot Jupiters because of the observed fast orbital decay attributed to tidal dissipation. The measured rate of the orbital period is $\stackrel{\bf\centerdot}{\textstyle{P}}_{\rm orb}\,=\,-\,29\pm3\;\mbox{ms/yr}\;$=$\;(9.2\pm1.0)\times10^{-10}\;\mbox{s/s}$. In the literature heretofore, all attem...
Bodily tides provide key information on the interior structure, evolution, and origin of the planetary bodies. Our Solar system harbors a very diverse population of planetary bodies, including those composed of rock, ice, gas, or a mixture of all. While a rich arsenal of geophysical methods has been developed over years to infer knowledge about the...
The parameterised post-Newtonian (PPN) formalism is a weak-field slow-motion approximation for both GR and some of its generalisations. It permits various parameterisations of the motion, among which are the Lagrange-type and Gauss-type orbital equations. Often, these equations are developed under the Lagrange constraint, which makes the evolving o...
The parameterised post-Newtonian (PPN) formalism is a weak-field and slow-motion approximation for both General Relativity (GR) and some of its viable generalisations. Within this formalism, the motion can be approached using various parameterisations, among which are the Lagrange-type and Gauss-type orbital equations. Often, these equations are de...
The origin of the Martian moons, Phobos and Deimos, remains elusive. While the morphology and their cratered surfaces suggest an asteroidal origin1,2,3, capture has been questioned because of potential dynamical difficulties in achieving the current near-circular, near-equatorial orbits4,5. To circumvent this, in situ formation models have been pro...
Observational evidence of white dwarf planetary systems is dominated by the remains of exo-asteroids through accreted metals, debris discs, and orbiting planetesimals. However, exo-planets in these systems play crucial roles as perturbing agents, and can themselves be perturbed close to the white dwarf Roche radius. Here, we illustrate a procedure...
We address the expressions for the rates of the Keplerian orbital elements within a two-body problem perturbed by the tides in both partners. Formulae for these rates appeared in the literature in various forms, at times with errors. We reconsider, from scratch, the derivation of these rates and arrive at the Lagrange-type equations which, in some...
Observational evidence of white dwarf planetary systems is dominated by the remains of exo-asteroids through accreted metals, debris discs, and orbiting planetesimals. However, exo-planets in these systems play crucial roles as perturbing agents, and can themselves be perturbed close to the white dwarf Roche radius. Here, we illustrate a procedure...
We address the expressions for the rates of the Keplerian orbital elements within a two-body problem perturbed by the tides in both partners. The formulae for these rates have appeared in the literature in various forms, at times with errors. We reconsider, from scratch, the derivation of these rates and arrive at the Lagrange-type equations which,...
The geophysics of extrasolar planets is a scientific topic often regarded as standing largely beyond the reach of near-term observations. This reality in no way diminishes the central role of geophysical phenomena in shaping planetary outcomes, from formation, to thermal and chemical evolution, to numerous issues of surface and near-surface habitab...
Highly volcanic exoplanets, which can be variously characterized as 'lava worlds', 'magma ocean worlds', or 'super-Ios' are high priority targets for investigation. The term 'lava world' may refer to any planet with extensive surface lava lakes, while the term 'magma ocean world' refers to planets with global or hemispherical magma oceans at their...
We perform numerical simulations of the TRAPPIST-1 system of seven exoplanets orbiting a nearby M dwarf, starting with a previously suggested stable configuration. The long-term stability of this configuration is confirmed, but the motion of planets is found to be chaotic. The eccentricity values are found to vary within finite ranges. The rates of...
We perform numerical simulations of the TRAPPIST-1 system of seven exoplanets orbiting a nearby M dwarf, starting with a previously suggested stable configuration. The long-term stability of this configuration is confirmed, but the motion of planets is found to be chaotic. The eccentricity values are found to vary within finite ranges. The rates of...
The Darwin-Kaula theory of bodily tides is intended for celestial bodies rotating without libration. We demonstrate that this theory, in its customary form, is inapplicable to a librating body. Specifically, in the presence of libration in longitude, the actual spectrum of Fourier tidal modes differs from the conventional spectrum rendered by the D...
In the preceding paper (Efroimsky 2017), we demonstrated that under weak libration in longitude most dissipation is due to the gravitational tides (including the additional tides generated by libration). The other three sources of dissipation$~-~$which are the alternating parts of the centripetal, toroidal and purely radial deformations$~-~$are les...
Dissipation in a tidally perturbed librating body differs from that in a steadily spinning rotator. First, libration changes the spectral distribution of tidal damping across the tidal modes, as compared to the steady-spin case. This adds both to the dissipation rate and the tidal torque. Second, while a non-librating rotator experiences alternatin...
Internal dissipation in a tidally perturbed librating body differs from the tidal dissipation in a steadily spinning rotator. First, libration changes the spectral distribution of tidal damping across the tidal modes, as compared to the case of steady spin. This changes both the tidal heating rate and the tidal torque. Second, while a non-librating...
In the preceding paper (Efroimsky 2017), we derived an expression for the tidal dissipation rate in a homogeneous near-spherical Maxwell body librating in longitude. Now, by equating this expression to the outgoing energy flux due to the vapour plumes, we estimate the mean tidal viscosity of Enceladus, under the assumption that the Enceladean mantl...
The Darwin-Kaula theory of bodily tides is intended for celestial bodies rotating without libration. We demonstrate that this theory, in its customary form, is inapplicable to a librating body. Specifically, in the presence of libration in longitude, the actual spectrum of Fourier tidal modes differs from the conventional spectrum rendered by the D...
Various perturbations (collisions, close encounters, YORP) destabilise the rotation of a small body, leaving it in a non-principal spin state. Then the body experiences alternating stresses generated by the inertial forces. The ensuing inelastic dissipation reduces the kinetic energy, without influencing the angular momentum. This yields nutation r...
We use a damped mass-spring model within an N-body code to simulate the tidal evolution of the spin and orbit of a self-gravitating
viscoelastic spherical body moving around a point-mass perturber. The damped mass-spring model represents a Kelvin-Voigt viscoelastic
solid. We measure the tidal quality function (the dynamical Love number k2 divided b...
In the Abstract, in Footnote 2, and in the end of Section 6, we erroneously stated that some terms in the expression (31) from Peale & Cassen (1978) are not positive definite. This misunderstanding on our part emerged from a nonconventional definition of the quality factors Q lmpq in Peale & Cassen (1978) where these factors were implied to incorpo...
We use a damped mass-spring model within an N-body code to simulate the tidal evolution of the spin and orbit of a self-gravitating viscoelastic spherical body moving around a point-mass perturber. The damped mass-spring model represents a Kelvin-Voigt viscoelastic solid. We measure the tidal quality function (the dynamical Love number $\,k_2\,$ di...
The rotation of Mercury is a unique case in the Solar System since this planet is locked into a 3:2 spin-orbit resonance. We here simulate the despinning of Mercury, with or without a fluid core, and with a frequency-dependent tidal model employed. The tidal model incorporates the viscoelastic (Maxwell) rebound at low forcing frequencies and a pred...
The rate of tidal evolution of asteroidal binaries is defined by the
dynamical Love numbers divided by quality factors. Common is the (often
illegitimate) approximation of the dynamical Love numbers with their static
counterparts. As the static Love numbers are, approximately, proportional to
the inverse rigidity, this renders a popular fallacy tha...
In Efroimsky & Makarov (2014), we derived from the first principles a formula
for the tidal heating rate in a tidally perturbed homogeneous sphere. We
compared it with the formulae used in the literature, and pointed out the
differences. Using this result, we now present three case studies - Mercury,
Kepler-10b, and a triaxial Io. A very sharp freq...
A formula for the tidal dissipation rate in a spherical body is derived from
first principles, to correct some mathematical inaccuracies found in the
literature. The development is combined with the Darwin-Kaula formalism for
tides. Our intermediate results are compared with those by Zschau (1978) and
Platzman (1984). When restricted to the special...
Mercury's spin state is peculiar, in that it is locked into the 3:2
spin-orbit resonance. Its rotation period, 58 days, is exactly two thirds of
its orbital period. It is accepted that the eccentricity of Mercury (0.206)
favours the trapping into this resonance. More controversial is how the capture
took place. A recent study by Makarov has shown t...
Spin-orbit coupling is often described in the "MacDonald torque" approach
which has become the textbook standard. Within this method, a concise
expression for the additional tidal potential, derived by MacDonald (1964; Rev.
Geophys. 2, 467), is combined with an assumption that the Q factor is
frequency-independent (i.e., that the geometric lag angl...
Tidal torques play a key role in rotational dynamics of celestial bodies.
They govern these bodies' tidal despinning, and also participate in the subtle
process of entrapment of these bodies into spin-orbit resonances. This makes
tidal torques directly relevant to the studies of habitability of planets and
their moons.
Our work begins with an expla...
We reexamine the popular belief that a telluric planet or satellite on an
eccentric orbit can, outside a spin-orbit resonance, be captured in a
quasi-static tidal equilibrium called pseudosynchronous rotation. The existence
of such configurations was deduced from oversimplified tidal models assuming
either a constant tidal torque or a torque linear...
It was demonstrated back in 2001 that fitting of the LLR data results in
the quality factor Q of the Moon scaling as the frequency ξ to a
negative power [8]: Q ˜ ξp , where p = -0.19 . (1) At the same
time, numerous measurements by various seismological teams agree on the
exponent being positive, not negative [4]. The positive sign of the
exponent...
GJ 581d is a potentially habitable super-Earth in the multiple system of
exoplanets orbiting a nearby M dwarf. We investigate this planet's long-term
dynamics, with an emphasis on its probable final rotation states acquired via
tidal interaction with the host.
The published radial velocities for the star are re-analysed with a benchmark
planet dete...
Mathematical modeling of bodily tides can be carried out in various ways.
Most straightforward is the method of complex amplitudes, which is often used
in the planetary science. Another method, employed both in planetary science
and astrophysics, is based on decomposition of each harmonic of the tide into
two bulges oriented orthogonally to one ano...
While the seismic quality factor and phase lag are defined solely by the bulk properties of the mantle, their tidal counterparts are determined by both the bulk properties and the size effect (self-gravitation of a body as a whole). For a qualitative estimate, we model the body with a homogeneous sphere, and express the tidal phase lag through the...
We study the tidal history of an icy moon, basing our approach on a dissipation model, which combines viscoelasticity with anelasticity and takes into account the microphysics of attenuation. We apply this approach to Iapetus, the most remote large icy moon in the Saturnian system. Different authors provide very different estimates for Iapetus's de...
This authoritative book presents the theoretical development of gravitational physics as it applies to the dynamics of celestial bodies and the analysis of precise astronomical observations. In so doing, it fills the need for a textbook that teaches modern dynamical astronomy with a strong emphasis on the relativistic aspects of the subject produce...
Text of IAU Resolutions of 1997 Adopted at the XXIIIrd General Assembly, KyotoText of IAU Resolutions of 2000 Adopted at the XXIVth General Assembly, ManchesterText of IAU Resolutions of 2006 Adopted at the XXVIth General Assembly, PragueText of IAU Resolutions of 2009 Adopted at the XXVIIth General Assembly, Rio de Janeiro
Post-Newtonian Equations of Orbital MotionRotational Equations of Motion of Extended BodiesMotion of Spherically-Symmetric and Rigidly-Rotating BodiesPost-Newtonian Two-Body ProblemReferences
Prolegomena – Classical Mechanics in a NutshellThe N-body ProblemThe Reduced Two-Body ProblemA Perturbed Two-Body ProblemRe-examining the ObviousEpilogue to the ChapterReferences
This appendix contains sections titled:
This appendix contains sections titled:
Spin-orbit coupling can be described in two approaches. The method known as
"the MacDonald torque" is often combined with an assumption that the quality
factor Q is frequency-independent. This makes the method inconsistent, because
the MacDonald theory tacitly fixes the rheology by making Q scale as the
inverse tidal frequency.
Spin-orbit coupling...
While currently Iapetus is locked into a 1:1 spin-orbit resonance with Saturn, its original state was likely one of rapid rotation. Simulations (Aleshkina 2009) have demonstrated that: (1) with a high arbitrariness in the initial conditions, Iapetus had passed without delay through the 5:2, 2:1, and 3:2 spin-orbit commensurabilities, and (2) its ti...
Given the impressive investment by the nation in observational Astronomy and Astrophysics facilities coming on line now and in the near future, we advocate for an increased investment in applied and fundamental research on Astrophysical and Celestial Dynamics (ACD). Specifically we call for a) continued and expanded support for applied research in...
This paper deals with the quantitative analysis of a recently introduced mechanism of mechanical alignment of suprathermally rotating cosmic-dust grains. These grains, whose rapid rotation is mostly due to H2 formation, are insensitive to stochastic torques arising from gas-grain collisions and can be aligned either by gaseous torques during short...
We have designed and constructed a "dispersed Fourier transform spectrometer" (dFTS), consisting of a conventional FTS followed by a grating spectrometer. By combining these two devices, we negate a substantial fraction of the sensitivity disadvantage of a conventional FTS for high-resolution, broadband, optical spectroscopy, while preserving many...
We point out that the MacDonald formula for body-tide torques is valid only
in the zeroth order of e/Q, while its time-average is valid in the first order.
So the formula cannot be used for analysis in higher orders of e/Q. This
necessitates corrections in the theory of tidal despinning and libration
damping.
We prove that when the inclination is l...
This chapter discusses the gauge freedom in astrodynamics. Both orbital and attitude dynamics employ the method of variation of parameters. In a non-perturbed setting, the coordinates get expressed as functions of the time and six adjustable constants called elements. The perturbed velocity consists of a partial time derivative and a convective ter...
In geophysics and seismology, it is a common knowledge that the quality
factors Q of the mantle and crust materials scale as the tidal frequency to a
positive fractional power (Karato 2007, Efroimsky and Lainey 2007). In
astronomy, there exists an equally common belief that such rheological models
introduce discontinuities into the equations and th...
Any model of tides is based on a specific hypothesis of how lagging depends
on the tidal-flexure frequency. For example, Gerstenkorn (1955), MacDonald
(1964), and Kaula (1964) assumed constancy of the geometric lag angle, while
Singer (1968) and Mignard (1979, 1980) asserted constancy of the time lag.
Thus, each of these two models was based on a c...
The Cassini orbiter is equipped with the capability to measure the
gravity field of Saturn and its satellites. Specifically, the gravity
science experiment on Titan aims to determine Titan's dynamic potential
Love number k2. We show that this parameter alone does not allow an
unambiguous detection of a deep ocean inside Titan. This is due to the
fa...
Different theories of bodily tides assume different forms of dependence of the angular lag δ upon the tidal frequency χ. In the old theory (Gerstenkorn 1955, MacDonald 1964, Kaula 1964) the geometric Iag angle is assumed constant (i.e., δ ∼ χ0), while the new theory (Singer 1968; Mignard 1979, 1980) postulates constancy of the time lag Δt (which is...
We have designed and constructed a ``dispersed Fourier Transform Spectrometer'' (dFTS), consisting of a conventional FTS followed by a grating spectrometer. By combining these two devices, we negate a substantial fraction of the sensitivity disadvantage of a conventional FTS for high resolution, broadband, optical spectroscopy, while preserving man...
Both orbital and attitude dynamics employ the method of variation of parameters. In a non-perturbed setting, the coordinates (or the Euler angles) get expressed as functions of the time and six adjustable constants called elements. Under disturbance, each such expression becomes ansatz, the "constants" being endowed with time dependence. The pertur...
The Origins Billions Star Survey is a mission concept addressing the astrophysics of extrasolar planets, Galactic structure, the Galactic halo and tidal streams, the Local Group and local supercluster of galaxies, dark matter, star formation, open clusters, the solar system, and the celestial reference frame by determining the position, parallax, a...
We continue the study undertaken in Efroimsky (2005a) where we explored the influence of spin-axis variations of an oblate planet on satellite orbits. Near-equatorial satellites had long been believed to keep up with the oblate primary's equator in the cause of its spin-axis variations. As demonstrated by Efroimsky and Goldreich (2004), this opinio...
Construction of an accurate theory of orbits about a precessing and nutating oblate planet, in terms of osculating elements defined in a frame associated with the equator of date, was started in Efroimsky and Goldreich (2004) and Efroimsky (2004, 2005, 2006a, b). Here we continue this line of research by combining that analytical machinery with num...
This paper reviews the Serret-Andoyer (SA) canonical formalism in rigid-body dynamics and presents some new results. As is well known, the problem of unsupported and unperturbed rigid rotator can be reduced. The availability of this reduction is offered by the underlying symmetry, which stems from conservation of the angular momentum and rotational...
Description of tides is based on the form of dependence of the geometric lag
on the tidal frequency. Some authors assume the lag angle to be constant,
others set it to be linear in the frequency. The actual dependence of the lag
on the frequency is complicated and is determined by the planet's rheology. A
particular form of this dependence will fix...
In orbital and attitude dynamics the coordinates and the Euler angles are expressed as functions of the time and six constants called elements. Under disturbance, the constants are endowed with time dependence. The Lagrange constraint is then imposed to guarantee that the functional dependence of the perturbed velocity on the time and constants sta...
In orbital and attitude dynamics the coordinates and the Euler angles are expressed as functions of the time and six constants called elements. Under disturbance, the constants are endowed with time dependence. The Lagrange constraint is then imposed to guarantee that the functional dependence of the perturbed velocity on the time and constants sta...
We present initial results from the dispersed Fourier Transform Spectrometer. This new device couples a conventional FTS to a grating spectrometer to achieve a sensitivity increase of a factor of several thousand as compared to a conventional FTS. A laser-based metrology system provides precise PSF, wavelength and velocity calibration. The bandpass...
Construction of an accurate theory of orbits about a precessing and nutating oblate planet, in terms of osculating elements defined in a frame associated with the equator of date, was started in Efroimsky & Goldreich (2004) and Efroimsky (2005a,b). Here we continue this line of research by combining those analytics with numerical tools. This semian...
When the dynamical equations, written in terms of variable "constants," are demanded to be symplectic, these "constants" make conjugated pairs and are called Delaunay elements, in the orbital case, or Serret-Andoyer elements, in the rotational case. These sets of elements share a feature not readily apparent: in certain cases, the standard equation...
The Hamiltonian theory of Earth rotation, known as the Kinoshita-Souchay theory, operates with nonosculating Andoyer elements. This situation parallels a similar phenomenon that often happens (but seldom gets noticed) in orbital dynamics, when the standard Lagrange-type or Delaunay-type planetary equations unexpectedly render nonosculating orbital...
Both orbital and rotational dynamics employ the method of variation of parameters. We express, in a non-perturbed setting, the coordinates (Cartesian, in the orbital case, or Eulerian in the rotation case) via the time and six adjustable constants called elements (orbital elements or rotational elements). If, under disturbance, we use this expressi...
It was believed until very recently that a near-equatorial satellite would always keep up with the planet's equator (with oscillations in inclination, but without a secular drift). As explained in Efroimsky and Goldreich (2004), this opinion originated from a wrong interpretation of a (mathematically correct) result obtained in terms of non-osculat...
The goal of this paper is to demonstrate how the internal symmetry of the N-body celestial-mechanics problem can be exploited in orbit calculation.
We start with summarising research reported in (Efroimsky [CITE], [CITE]; Newman & Efroimsky [CITE]; Efroimsky & Goldreich [CITE]) and develop its application to planetary equations in non-inertial fra...
The method of variation of constants is an important tool used to solve systems of ordinary differential equations, and was invented by Euler and Lagrange to solve a problem in orbital mechanics. This methodology assumes that certain "constants" associated with a homogeneous problem will vary in time in response to an external force. It also introd...
In most books the Delaunay and Lagrange equations for the orbital elements are derived by the Hamilton-Jacobi method: one begins with the 2-body Hamilton equations, performs a canonical transformation to the orbital elements, and obtains the Delaunay system. A standard trick is then used to generalise the approach to the N-body case. We re-examine...
We revisit the Lagrange and Delaunay systems of equations for the orbital elements, and point out a previously neglected aspect of these equations: in both cases the orbit resides on a certain 9-dimensional submanifold of the 12-dimensional space spanned by the orbital elements and their time derivatives. We demonstrate that there exists a vast fre...
Neutron stars, asteroids, comets, cosmic-dust granules, spacecraft, as well as whatever other freely spinning body dissipate energy when they rotate about any axis different from principal. We discuss the internal-dissipation-caused relaxation of a freely precessing rotator towards its minimal-energy mode (mode that corresponds to the spin about th...
We develop a comprehensive quantitative description of the cross-section mechanism discovered several years ago by Lazarian. This is one of the processes that determine grain orientation in clouds of suprathermal cosmic dust. The cross-section mechanism manifests itself when an ensemble of suprathermal paramagnetic granules is placed in a magnetic...
We revisit the Lagrange and Delaunay systems of equations of celestial mechanics, and point out a previously neglected aspect of these equations: in both cases the orbit resides on a certain 9(N-1)-dimensional submanifold of the 12(N-1)-dimensional space spanned by the orbital elements and their time derivatives. We demonstrate that there exists a...
Whenever a freely spinning body is found in a complex rotational state, this means that either the body is a recent victim of an impact or a tidal interaction, or is a fragment of a recently disrupted progenitor. Another factor (relevant for comets) is outgassing. Due to impacts, tidal forces and outgassing, the asteroidal and cometary precession m...
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