Estimating asteroid density distributions from shape and gravity information

https://doi.org/10.1016/S0032-0633(00)00064-7Get rights and content

Abstract

A least-squares approach for estimating the internal density distribution of an asteroid is presented and applied to a simple polyhedron asteroid shape. The method assumes that the asteroid gravity field is measured to a specified degree and order and that a polyhedral model of the asteroid is available and has been discretized into a finite number of constant density polyhedra. The approach is derived using several basic properties of spherical harmonic gravitational expansions and can explicitly accommodate a fully correlated covariance matrix for the estimated gravity field. For an asteroid shape discretized into M constant density polyhedra and a gravity field measured to degree and order N, the least-squares problem is under-determined if M>(N+1)2 and is over-determined if M<(N+1)2. For both cases a singular-value decomposition (SVD) approach will yield solutions. We apply our approach to a number of ideal test situations using an asteroid shape consisting of 508 tetrahedra. We show that the under-determined case is sensitive to non-uniform density distributions. The over-determined case shows very good performance independent of the initial density distribution guess.

Introduction

Two of the main products of the Near Earth Asteroid Rendezvous (NEAR) mission to asteroid 433 Eros (Farquhar, 1995) will be the gravity field, measured at least to degree and order 12 (Yeomans et al., 1997), and the shape, measured with both the imager and lidar instruments (Veverka et al., 1997; Zuber et al., 1997). By combining the total mass and the asteroid volume it is, of course, possible to extract the bulk density of the body. But to determine the distribution of that density is a non-trivial problem which cannot be uniquely determined in general. In this paper we present a method for estimating the internal density distribution of an asteroid given its gravity field and shape. This approach relies on the discretization of the asteroid shape into a finite number of constituent polyhedra, and then uses a least-squares approach to assign densitites to these polyhedra. When the number of polyhedra is less than (N+1)2, where N is the degree of the gravity field, this assignment procedure is over-determined and the resulting density assignment is unique (assuming a given set of weights). Thus, using such a procedure reduces the problem of finding the density distribution to finding the regions of different density distributions. Using this approach it should be possible to gain additional insight into the internal structure of asteroids, comets, or any irregularly shaped body for which we have a gravity field and a shape. This approach makes no distinction between the computation of density variations or the identification of voids within the asteroid.

There is an important additional application that becomes available once the internal density distribution of an asteroid is approximated — the ability to evaluate the gravity field close to the surface of the body. For a general body which has a major to minor axis ratio greater than 2 (which includes almost all asteroid shapes measured to date) the spherical harmonic gravity field description diverges once one moves within the circumscribing sphere about that body. It can be shown that this divergence cannot be remedied by discarding higher-order coefficients, as the coefficients themselves at all orders are no longer properly defined. The polyhedron-based gravity field that we derive, however, is free of this divergence and can be used to model the gravity field close to and on the surface, enabling both studies of surface ejecta and analysis of spacecraft trajectories that closely approach or land on the asteroid's surface.

Section snippets

Spherical harmonic gravity field description

The usual form for a spherical harmonic gravity potential is (Kaula, 1966)U=GMr1+l=1Nm=0lRrlPlm(sinθ)(Clmcosmλ+Slmsinmλ),where G is the universal gravitation constant, M is the total asteroid mass, l is the degree, m is the order of the gravity field, N is the maximum degree of the expansion, Plm is the associated Legendre polynomial, R is an arbitrary radius scaling factor used in conjunction with the gravity coefficients, r is the distance of the test particle from the center of the

Measured data

We assume as input the measured spherical harmonic coefficients of the asteroid up to degree and order N and the correlated covariance matrix of these coefficients. Also, we assume that a polyhedral model of the asteroid shape exists, which is comprised of an arbitrary number of tetrahedra, NT. These tetrahedra can be grouped into arbitrary collections of polyhedra. In general the groupings of these polyhedra will either be based on some physical reasoning or to ensure uniform tiling of the

Results

The following results are tests of our basic approach using a 508 tetrahedra model of Eros based on radar data (Mitchell et al., 1998), shown in Fig. 4. In these tests we assemble the tetrahedra into a number of polyhedra and assign densities to each polyhedron to compute the “true” gravity field potential. Then we use our algorithm on the same shape discretization in order to verify that we recover the proper densities for each body. We also investigate the limitations of this method when the

Conclusions

This paper presents a mathematical framework for estimating the internal density distribution of an arbitrarily shaped body, assuming that a measured gravity field and shape for that body exist. The algorithm is explicitly developed and tested with a 508 tetrahedra shape model based on radar measurements of the asteroid 433 Eros (Mitchell et al., 1998). The technique will be especially useful for any and all missions to asteroids and comets, where the determination of internal density

Acknowledgements

This work was supported by a Grant from NASA's Planetary Geology and Geophysics Program.

References (10)

There are more references available in the full text version of this article.

Cited by (43)

  • The geophysical environment of Bennu

    2016, Icarus
    Citation Excerpt :

    One item of significant interest, but with no current information, is the internal density distribution of this body, which is encoded in the body’s gravity field. While constant density gravity field coefficients can be computed, it is only once the gravity field of Bennu is measured that explicit comparisons can be made in order to detect internal density non-uniformities (Scheeres et al., 2000; Takahashi and Scheeres, 2014a). In this paper the constant density gravity field is either computed directly from the shape model using the technique by (Werner and Scheeres, 1997) or with spherical harmonics using gravity field coefficients computed directly from the shape model with a constant density distribution assumption (Werner, 1997), a severe limitation which will be corrected once the actual gravity field is estimated.

  • Gravity and Topography of the Terrestrial Planets

    2015, Treatise on Geophysics: Second Edition
  • Morphology driven density distribution estimation for small bodies

    2014, Icarus
    Citation Excerpt :

    The discrepancies in the spherical harmonic coefficients between the two models form a measurement that can be used to detect inhomogeneity. Also, they discussed the density estimation technique in an informal conference report (Takahashi and Scheeres, 2013) by leveraging the results of Scheeres et al. (2000), where the density distribution was estimated from the spherical harmonic coefficients determined from OD. Zuber et al. (2000) showed the center-of-mass (COM) and center-of-figure (COF) offset of 433 Eros can be explained by small variations in Eros’ internal mechanical structure, given the global homogeneity in surface composition.

View all citing articles on Scopus

Poster 24.18-P, presented at the Asteroids, Comets, Meteors Conference, Cornell, 1999.

View full text