Gravity from surface triangulation: convergence acceleration with nested grids

Abstract

The determination of the gravitational potential by the polyhedral method is revisited in the case where the surface of a body is composed of triangular facets. Based upon six test-shapes of astrophysical interest (sphere, spheroid, triaxial, lemon-shape, dumbell and torus) projected on nested grids, we verify that the convergence toward reference values is second-order in the step size of the grid, inside the body, at the surface and outside. We then show that the accuracy or computing time can be drastically enhanced by implementing the Repeated Richardson Extrapolation. This technique is especially efficient when the body's surface is smooth enough, and is therefore well adapted to the theory of figures (single and multi-layer fluids) and to dynamical studies (test-particle and mutual interactions), which require a large number of field evaluations. For real objects like asteroids that have very irregular terrains at small scales, the gain is modest. In that context, we estimate the discretization level beyond which the typical error in potential values due to altimetric uncertainties dominates over the contribution of sub-grid cavities and bumps. For bodies close to spherical, the criterion reads $T \gtrsim \frac{64 D}{3 λ},$ where $D$ is the diameter of the body, $λ$ the typical shape error and $T$ the number of triangular facets involved. The case of 433 Eros is considered as an example.

Figures (16)

• Figure 1: Part of a sphere triangulated with $960$ triangles obtained with $N=2M=32$ equally-spaced nodes along the azimuth $\phi$ and $M$ equally-spaced nodes in colatitude $\theta$. Also shown are a particular triangle ( bold line), the local basis vectors ( red arrows), a point P$(x,y,z)$ of space where the potential $\Psi$ is evaluated, the intermediate vertex A defined as the projection of point P in the plane of the actual triangle ( black cross), and the basis vectors of the global reference frame ( black arrows)
• Figure 2: A triangle ${\cal T}_{ijk}=\{$V$_i$,V$_j$,V$_k\}$ as part of the triangulated surface ${\cal S}$ of the body, its plane and the normal vector $\mathbf{n}_{ijk}$, oriented outwards. Point A is the projection of P in this plane following its normal $\mathbf{n}_{ijk}$, leaving $3$ intermediate triangles AV$_i$V$_j$, AV$_j$V$_k$ and AV$_k$V$_i$; see Fig. \ref{['fig:configV2.eps']}
• Figure 3: Local Cartesian coordinates $(X',Y',Z')$ in the plane of the triangle ${\cal T}_{ijk}=\{$V$_i$,V$_j$,V$_k\}$ ( dark gray) and local basis is $(\mathbf{e}_{X'},\mathbf{e}_{Y'},\mathbf{e}_{Z'})$, with $\mathbf{e}_{Z'} \equiv \mathbf{n}_{ijk}$; see Fig. \ref{['fig:config.eps']}. The height of the intermediate triangle AV$_i$V$_j$ ( light gray) is $h$
• Figure 4: Same caption as for Fig. \ref{['fig:sphere.pdf']}, but showing the test-lines defined by Eq. \ref{['eq:kspace']} along which the potential is compared with references values
• Figure 5: Error index $E$ for the gravitational potential computed from surface triangulation by direct summation, i.e., from Eq. \ref{['eq:psitotal']}, for the sphere with $a=1$ ( left), for the spheroid with $a=1$ and $b=3a/4$ ( middle) and for the triaxial ellipsoid with $a=1$, $c=a/2$ and $b=3a/4$ ( right) along the nine test-lines defined by Eq. \ref{['eq:kspace']}, with $0 \le k \le K$ and $K=32$. The surface triangulation is performed with $N=2M=32$; see Fig. \ref{['fig:sphere_testlines.pdf']} for the sphere