Skip to content

Polyhedral Gravity

Polyhedral gravity model using the Tsoulis (2012) line-integral method.

Computes gravitational potential, acceleration, and gravity gradient tensor for irregularly-shaped bodies represented as triangulated surface meshes. This is the standard approach for modelling the gravitational field of asteroids and small bodies using shape models such as those from the DAMIT database.

The algorithm follows the formulation in:

D. Tsoulis, "Analytical computation of the full gravity tensor of a
homogeneous arbitrarily shaped polyhedral source using line integrals,"
*Geophysics*, vol. 77, no. 2, pp. F1-F11, 2012.

with singularity-handling refinements from:

D. Tsoulis and K. Petrovic, "On the singularities of the gravity field
of a homogeneous polyhedral body," *Geophysics*, vol. 66, no. 2,
pp. 535-539, 2001.

All functions use JAX primitives and are compatible with jax.jit, jax.vmap, and jax.grad.

accel_polyhedral_gravity(r_point, r_body, R_body_to_inertial, vertices, faces, density)

Gravitational acceleration from a polyhedral body in an inertial frame.

Transforms the computation point into the body-fixed frame, evaluates the polyhedral gravity acceleration there, and rotates the result back to the inertial frame.

This function is designed to integrate with :func:~astrojax.datasets.damit_spin_to_rotation and :func:~astrojax.datasets.scale_shape_vertices.

Parameters:

Name Type Description Default
r_point ArrayLike

Position of the computation point in the inertial frame [m], shape (3,).

 required
r_body ArrayLike

Position of the body centre in the inertial frame [m], shape (3,).

 required
R_body_to_inertial ArrayLike

Rotation matrix from body-fixed to inertial frame, shape (3, 3).

 required
vertices ArrayLike

Mesh vertex coordinates in the body frame [m], shape (N, 3).

 required
faces ArrayLike

Triangle face indices (0-indexed), shape (M, 3) of int.

 required
density ArrayLike

Bulk density [kg/m^3], scalar.

 required

Returns:

Type Description
Array

Gravitational acceleration in the inertial frame [m/s^2],
Array

shape (3,).

Examples:

import jax.numpy as jnp
from astrojax.orbit_dynamics import accel_polyhedral_gravity

# Body at origin, identity rotation (body = inertial)
vertices = jnp.array([
    [-1, -1, -1], [ 1, -1, -1], [ 1,  1, -1], [-1,  1, -1],
    [-1, -1,  1], [ 1, -1,  1], [ 1,  1,  1], [-1,  1,  1],
], dtype=jnp.float64)
faces = jnp.array([
    [1,3,2],[0,3,1],[0,1,5],[0,5,4],[0,7,3],[0,4,7],
    [1,2,6],[1,6,5],[2,3,6],[3,7,6],[4,5,6],[4,6,7],
])

r_point = jnp.array([5.0, 5.0, 5.0])
r_body = jnp.zeros(3)
R = jnp.eye(3)
a = accel_polyhedral_gravity(r_point, r_body, R, vertices, faces, 1.0)

polyhedral_gravity(r_body_frame, vertices, faces, density)

Gravitational potential, acceleration, and tensor from a polyhedral body.

Evaluates the full gravity field of a homogeneous polyhedron at a single computation point using the Tsoulis (2012) line-integral method. All quantities are computed in the body-fixed frame.

Parameters:

Name Type Description Default
r_body_frame ArrayLike

Computation point in body frame [m], shape (3,).

 required
vertices ArrayLike

Mesh vertex coordinates [m], shape (N, 3).

 required
faces ArrayLike

Triangle face indices (0-indexed), shape (M, 3) of int.

 required
density ArrayLike

Bulk density of the body [kg/m^3], scalar.

 required

Returns:

Type Description
Array

Tuple of:
Array
  • potential — Gravitational potential [m2/s2], scalar.
Array
  • acceleration — Gravitational acceleration [m/s^2], shape (3,). The vector points toward the body (attractive).
tuple[Array, Array, Array]
  • tensor — Gravity gradient tensor [1/s^2], shape (6,) as [Vxx, Vyy, Vzz, Vxy, Vxz, Vyz].

Examples:

import jax.numpy as jnp
from astrojax.orbit_dynamics import polyhedral_gravity

# Unit cube: 8 vertices, 12 triangular faces
vertices = jnp.array([
    [-1, -1, -1], [ 1, -1, -1], [ 1,  1, -1], [-1,  1, -1],
    [-1, -1,  1], [ 1, -1,  1], [ 1,  1,  1], [-1,  1,  1],
], dtype=jnp.float64)
faces = jnp.array([
    [1,3,2],[0,3,1],[0,1,5],[0,5,4],[0,7,3],[0,4,7],
    [1,2,6],[1,6,5],[2,3,6],[3,7,6],[4,5,6],[4,6,7],
])

r = jnp.array([5.0, 5.0, 5.0])
potential, accel, tensor = polyhedral_gravity(r, vertices, faces, 1.0)