The representation of the force functions of the Earth's and Moon's gravitation in ellipsoidal coordinates

This paper derives asymptotic expansions of ellipsoidal coordinates in Cartesian coordinates and an expansion in spherical harmonics of the dominant term for the solution of Laplace's equation corresponding to the gravitational force function for a two-dimensional finite body.On comparing the expansion of the dominant term derived here with known expansions of the force functions of the Earth's and Moon's gravitation the author obtains values for the semimajor axes and eccentricities of the singular ellipses of these bodies in terms of the second degree harmonic coefficientsc ₂₀ andc ₂₂.     

The exact transformation from spherical harmonic to ellipsoidal harmonic coefficients for gravitational field modeling

The spherical and ellipsoidal harmonic series of the external gravitational potential for a given mass distribution are equivalent in their mutual region of uniform convergence. In an instructive case, the equality of the two series on the common coordinate surface of an infinitely large sphere reveals the exact correspondence between the spherical and ellipsoidal harmonic coefficients. The transformation between the two sets of coefficients can be accomplished via the numerical methods by Walter (Celest Mech 2:389–397, 1970) and Dechambre and Scheeres (Astron Astrophys 387:1114–1122, 2002), respectively. On the other hand, the harmonic coefficients are defined by the integrals of mass density moments in terms of the respective solid harmonics. This paper presents general algebraic formulas for expressing the solid ellipsoidal harmonics as a linear combination of the corresponding solid spherical harmonics. An exact transformation from spherical to ellipsoidal harmonic coefficients is found by incorporating these connecting expressions into the density integral. A computational procedure is proposed for the transformation. Numerical results based on the nearly ellipsoidal Martian moon, Phobos, are presented for validation of the method.     

Collisional Spin Up of Nonspherical Asteroids

We investigate the spin up of nonspherical asteroids by successive random collisions with other smaller asteroids. The spin-up rates are calculated for ellipsoidal and spherical bodies of the same mass. Then, we derive the relative spin-up efficiency, that is, the ratio of the spin-up rate of the ellipsoidal body to that of the spherical body. We suppose a collisional process in our model different from that adopted in the previous works. The results show that ellipsoidal bodies can spin up more rapidly than spherical bodies.     

Small body surface gravity fields via spherical harmonic expansions

Conventional gravity field expressions are derived from Laplace’s equation, the result being the spherical harmonic gravity field. This gravity field is said to be the exterior spherical harmonic gravity field, as its convergence region is outside the Brillouin (i.e., circumscribing) sphere of the body. In contrast, there exists its counterpart called the interior spherical harmonic gravity field for which the convergence region lies within the interior Brillouin sphere that is not the same as the exterior Brillouin sphere. Thus, the exterior spherical harmonic gravity field cannot model the gravitation within the exterior Brillouin sphere except in some special cases, and the interior spherical harmonic gravity field cannot model the gravitation outside the interior Brillouin sphere. In this paper, we will discuss two types of other spherical harmonic gravity fields that bridge the null space of the exterior/interior gravity field expressions by solving Poisson’s equation. These two gravity fields are obtained by assuming the form of Helmholtz’s equation to Poisson’s equation. This method renders the gravitational potentials as functions of spherical Bessel functions and spherical harmonic coefficients. We refer to these gravity fields as the interior/exterior spherical Bessel gravity fields and study their characteristics. The interior spherical Bessel gravity field is investigated in detail for proximity operation purposes around small primitive bodies. Particularly, we apply the theory to asteroids Bennu (formerly 1999 RQ36) and Castalia to quantify its performance around both nearly spheroidal and contact-binary asteroids, respectively. Furthermore, comparisons between the exterior gravity field, interior gravity field, interior spherical Bessel gravity field, and polyhedral gravity field are made and recommendations are given in order to aid planning of proximity operations for future small body missions.     

Ellipsoidal geometry in asteroid thermal models: The standard radiometric model

This paper reports results of the incorporation of ellipsoidal geometry into the standard radiometric model for asteroids. For small departures from spherical shape the standard model using spherical geometry predicts fluxes in good agreement with ellipsoidal models. Large departures from spherical shape, however, can produce substantial differences in the calculated flux depending on the subsolar temperature and the wavelength of interest. The results derived here suggest that radiometric measurements of highly nonspherical, low-obliquity asteroids interpreted with spherical models result in systematically smaller diameter and higher albedos. In addition, non-spherical shape can also result in a systematic difference in the diameter of a particular asteroid derived from separate 10- and 20-μm flux measurements interpreted with spherical models. Thermal-infrared diurnal lightcurves calculated for ellipsoids have amplitudes that depend on wavelength as well as projected area, and phase curves calculated for ellipsoids are indistinguishable from those calculated for spheres.     

Dynamic expansion points: an extension to Hadjidemetriou’s mapping method

Series expansions are widely used objects in perturbation theory in Celestial Mechanics and Physics in general. Their application nevertheless is limited due to the fact of convergence problems of the series on the one hand and constricted to regions in phase space, where small (expansion) parameters remain small on the other hand. In the mapping case, to overcome the latter problem, e.g., different expansion points are used to cover the whole phase space, resulting in a set of dynamical mappings for one dynamical system. In addition, the accuracy of such expansions depend not only on the order of truncation but also on the definition of the grid of the expansion points in phase space. A simple modification of the usual approach allows to increase the accuracy of the expanded mappings and to cover the whole phase space, where the series converge. Convergence problems due to the nonintegrability of the system can never be ruled out of the system, but the convergence of the series expansions in mapping models, which are convergent can be improved. The underlying idea is based on dynamic expansion points, which are the main subject of this article. As I will show it is possible to derive unique linear mappings, based on dynamically expanded generating functions, for the 3:1 resonance and the coupled standard map, which are valid in their whole phase spaces.     

The dependence of convection in planetary interiors upon the magnitude of the rayleigh number

The conservation equations of mass, energy and momentum when applied to the problem of convection produced by isothermal compression as well as by thermal expansion, in self-gravitating, homogeneous, non rotating spheres with uniform viscosity and under steady flow conditions, lead to a relationship between two characteristic numbers. In the present paper these characteristic numbers are evaluated, for both rigid and free surfaces, for spherical harmonic disturbances of orders 16.     

Source flow expansion of monatomic gas into vacuum under external field of forces

The steady flow expansions of monatomic gases consisting of Maxwellia molecules into a vacuum moving under external forces with a source of spherical symmetry are investigated. The analysis is based on the B-G-K model of the Boltzmann equation with the approximation in hypersonic limit. The kinetic equation is solved by using the moment method. Analytical forms for the density and temperature are obtained for small and large distances from the source. The results show that the temperature in free expansion is less than that in the case of the expansion under the influence of external field of forces.     

Association of spherical and ellipsoidal gravity coefficients of the Earth's potential

A comparison is drawn between the expansion of the potential in spherical harmonics on the one hand and in ellipsoidal harmonics on the other, with the objective of associating the spherical and ellipsoidal gravity coefficients of the Earth's potential.For this purpose the properties of orthogonality of the Lamé functions of the first kind have been tailored to this subject of investigation and become instrumental in establishing the mathematical expressions which relate the two classes of gravity coefficients to each other. In deriving the elements of the transition matrices elliptic integrals have been encountered whose reduction to the three kinds of canonical elliptic integrals is discussed.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969.     

Spherical needlets for cosmic microwave background data analysis

We discuss spherical needlets and their properties. Needlets are a form of spherical wavelets which do not rely on any kind of tangent plane approximation and enjoy good localization properties in both pixel and harmonic space; moreover needlet coefficients are asymptotically uncorrelated at any fixed angular distance, which makes their use in statistical procedures very promising. In view of these properties, we believe needlets may turn out to be especially useful in the analysis of cosmic microwave background (CMB) data on the incomplete sky, as well as of other cosmological observations. As a final advantage, we stress that the implementation of needlets is computationally very convenient and may rely completely on standard data analysis packages such as healp ix.     

Short-axis-mode rotation of a free rigid body by perturbation series

A simple rearrangement of the torque free motion Hamiltonian shapes it as a perturbation problem for bodies rotating close to the principal axis of maximum inertia, independently of their triaxiality. The complete reduction of the main part of this Hamiltonian via the Hamilton–Jacobi equation provides the action-angle variables that ease the construction of a perturbation solution by Lie transforms. The lowest orders of the transformation equations of the perturbation solution are checked to agree with Kinoshita’s corresponding expansions for the exact solution of the free rigid body problem. For approximately axisymmetric bodies rotating close to the principal axis of maximum inertia, the common case of major solar system bodies, the new approach is advantageous over classical expansions based on a small triaxiality parameter.     

Figure–figure interaction between bodies having arbitrary shapes and mass distributions: a power series expansion approach

We derive an expression for the mutual gravitational force and torque of two bodies having arbitrary shapes and mass distributions, as an expansion in power series of their products of inertia and of the relative coordinates of their centres of mass. The absolute convergence of all the power series developed is rigorously demonstrated. The absence of transcendental functions makes this formalism suitable for fast numerical applications. The products of inertia used here are directly related to the spherical harmonics coefficients, and we provide a detailed analysis of this relationship.     

The equilibrium of rubble-pile satellites: The Darwin and Roche ellipsoids for gravitationally held granular aggregates

Many new small moons of the giant planets have been discovered recently. In parallel, satellites of several asteroids, e.g., Ida, have been found. Strikingly, a majority of these new-found planetary moons are estimated to have very low densities, which, along with their hypothesized accretionary origins, suggests a rubble internal structure. This, coupled to the fact that many asteroids are also thought to be particle aggregates held together principally by self-gravity, motivates the present investigation into the possible ellipsoidal shapes that a rubble-pile satellite may achieve as it orbits an aspherical primary. Conversely, knowledge of the shape will constrain the granular aggregate's orbit—the closer it gets to a primary, both primary's tidal effect and the satellite's spin are greater. We will assume that the primary body is sufficiently massive so as not to be influenced by the satellite. However, we will incorporate the primary's possible ellipsoidal shape, e.g., flattening at its poles in the case of a planet, and the proloidal shape of asteroids. In this, the present investigation is an extension of the first classical Darwin problem to granular aggregates. General equations defining an ellipsoidal rubble pile's equilibrium about an ellipsoidal primary are developed. They are then utilized to scrutinize the possible granular nature of small inner moons of the giant planets. It is found that most satellites satisfy constraints necessary to exist as equilibrated granular aggregates. Objects like Naiad, Metis and Adrastea appear to violate these limits, but in doing so, provide clues to their internal density and/or structure. We also recover the Roche limit for a granular satellite of a spherical primary, and employ it to study the martian satellites, Phobos and Deimos, as well as to make contact with earlier work of Davidsson [Davidsson, B., 2001. Icarus 149, 375–383]. The satellite's interior will be modeled as a rigid-plastic, cohesion-less material with a Drucker–Prager yield criterion. This rheology is a reasonable first model for rubble piles. We will employ an approximate volume-averaging procedure that is based on the classical method of moments, and is an extension of the virial method [Chandrasekhar, S., 1969. Ellipsoidal Figures of Equilibrium. Yale Univ. Press, New Haven] to granular solid bodies.     

East-west faults due to planetary contraction

Contraction, expansion and despinning have been common in the past evolution of Solar System bodies. These processes deform the lithosphere until it breaks along faults. Their characteristic tectonic patterns have thus been sought for on all planets and large satellites with an ancient surface. While the search for despinning tectonics has not been conclusive, there is good observational evidence on several bodies for the global faulting pattern associated with contraction or expansion, though the pattern is seldom isotropic as predicted. The cause of the non-random orientation of the faults has been attributed either to regional stresses or to the combined action of contraction/expansion with another deformation (despinning, tidal deformation, reorientation). Another cause of the mismatch may be the neglect of the lithospheric thinning at the equator or at the poles due either to latitudinal variation in solar insolation or to localized tidal dissipation. Using thin elastic shells with variable thickness, I show that the equatorial thinning of the lithosphere transforms the homogeneous and isotropic fault pattern caused by contraction/expansion into a pattern of faults striking east-west, preferably formed in the equatorial region. By contrast, lithospheric thickness variations only weakly affect the despinning faulting pattern consisting of equatorial strike-slip faults and polar normal faults. If contraction is added to despinning, the despinning pattern first shifts to thrust faults striking north-south and then to thrust faults striking east-west. If the lithosphere is thinner at the poles, the tectonic pattern caused by contraction/expansion consists of faults striking north/south. I start by predicting the main characteristics of the stress pattern with symmetry arguments. I further prove that the solutions for contraction and despinning are dual if the inverse elastic thickness is limited to harmonic degree two, making it easy to determine fault orientation for combined contraction and despinning. I give two methods for solving the equations of elasticity, one numerical and the other semi-analytical. The latter method yields explicit formulas for stresses as expansions in Legendre polynomials about the solution for constant shell thickness. Though I only discuss the cases of a lithosphere thinner at the equator or at the poles, the method is applicable for any latitudinal variation of the lithospheric thickness. On Iapetus, contraction or expansion on a lithosphere thinner at the equator explains the location and orientation of the equatorial ridge. On Mercury, the combination of contraction and despinning makes possible the existence of zonal provinces of thrust faults differing in orientation (north-south or east-west), which may be relevant to the orientation of lobate scarps.     

Existence of particular solutions in the generalised restricted problem of three bodies with variable masses

The restricted problem of three bodies with variable masses is considered. It is assumed that the infinitesimal body is axisymmetric with constant mass and the finite bodies are spherical with variable masses such that the ratio of their masses remains constant. The motion of the finite bodies are determined by the Gyldén-Meshcherskii problem. It is seen that the collinear, triangular, and coplanar solutions not exist, but these solutions exist when the infinitesimal body be a spherical.     

On some group-theoretical aspects of the study of non-radial oscillations

Group-theoretical techniques are used for the analysis of non-radial oscillations of spherical stars, for the classification and the study of the splitting of the modes under the influence of rotation, tides and magnetic fields, as well as for ellipsoidal configurations. The general conclusions are compared with the analytical and numerical results which have up to now been worked out only for artificial models.     

Gravitational instability of spheroidal expansions: A cosmogonic fragmentation mechanism

Expanding flows of self-gravitating gas are common in astronomy and their stability therefore of some concern. Recently, gravitational instability in anisotropic expansions was proposed as an effective mechanism for the fragmentation called for in various cosmogonic hypotheses. (The anisotropy permits strong linear instabilities, while expansion avoids some nonlinear problems.) Here we analyze in detail the gravitational stability of spheroidal expansions of rotating gas masses. This extends Jeans' static results, and includes the well-known special cases of nonrotating spherical expansion and nonexpanding rotation. We obtain generalized Jeans criteria for instability, and in particular, a criterion for the instability to grow exponentially is derived. For an important class of asymptotic flows, we explicitly evaluate the instability strength and show that it is enough to initiate a fragmentation process. From the instability criteria we estimate a lower bound to the possible fragment masses. The Appendix discusses another asymptotic flow, modeling a high energy jet which is highly unstable.     

The dependence of convection in planetary mantles upon the magnitude of the Rayleigh number

The conservation equations of mass, energy and momentum when applied to the problem of convection produced by isothermal compression as well as by thermal expansion in a self-gravitating sphere of uniform density, consisting of a core extending to a fraction of the radius of the sphere and with a viscous mantle overlying the core, lead to a relationship between two characteristic numbers. In the present paper these characteristic numbers are evaluated, for the case when both boundaries of the mantle are free, for spherical harmonic disturbances of orders 1 6. It is found that the ease with which different patterns of convection can be excited changes as the characteristic numbers are varied.     

Analytical development of the lunisolar disturbing function and the critical inclination secular resonance

We provide a detailed derivation of the analytical expansion of the lunar and solar disturbing functions. Although there exist several papers on this topic, many derivations contain mistakes in the final expansion or rather (just) in the proof, thereby necessitating a recasting and correction of the original derivation. In this work, we provide a self-consistent and definite form of the lunisolar expansion. We start with Kaula’s expansion of the disturbing function in terms of the equatorial elements of both the perturbed and perturbing bodies. Then we give a detailed proof of Lane’s expansion, in which the elements of the Moon are referred to the ecliptic plane. Using this approach the inclination of the Moon becomes nearly constant, while the argument of perihelion, the longitude of the ascending node, and the mean anomaly vary linearly with time. We make a comparison between the different expansions and we profit from such discussion to point out some mistakes in the existing literature, which might compromise the correctness of the results. As an application, we analyze the long-term motion of the highly elliptical and critically-inclined Molniya orbits subject to quadrupolar gravitational interactions. The analytical expansions presented herein are very powerful with respect to dynamical studies based on Cartesian equations, because they quickly allow for a more holistic and intuitively understandable picture of the dynamics.