Gravity

Point-mass gravity from every body in the scenario, with the J2, J3 and J4 zonal terms for the Earth.

Status: point-mass gravity and J2 are Validated; J3 and J4 are Implemented.

Point-mass gravity

Every gravity source in the scenario attracts every body. No body is treated as the single fixed centre, so the same model serves an Earth orbit and the Earth and Moon system. For a source with gravitational parameter μ at distance r:

a = μ / r² · r̂

Zonal harmonics

The Earth is not a perfect sphere. Its flattening is described by the zonal harmonic terms, of which J2 is by far the largest. KnownX MPT models degrees 2, 3 and 4 about the body’s spin axis. With k the unit spin axis, ξ = r̂ · k, Rₑ the equatorial radius and Pₗ the Legendre polynomial of degree l, each degree adds:

aₗ = (μ Jₗ Rₑˡ / rˡ⁺²) · [ ((l+1) Pₗ(ξ) + ξ Pₗ′(ξ)) r̂ − Pₗ′(ξ) k ]

The main visible effect of J2 is a slow turning of the orbital plane (nodal regression) and of the orbit within its plane. J3 and J4 are several hundred times smaller.

Constants

Quantity	Value
Earth gravitational parameter	3.986004418 × 10¹⁴ m³/s²
Earth equatorial radius	6,378,137 m
J2	1.08262668 × 10⁻³
J3	−2.5327 × 10⁻⁶
J4	−1.6196 × 10⁻⁶

In the application

Press J to switch the Earth’s zonal terms on or off. The run restarts from the same initial state, so the two cases can be compared.
The Inspector lists the models active for the selected body.

Scope

The spin axis and coefficients are fixed in time. Terms above degree 4, tesseral and sectoral terms, and the gravity of the Sun and Moon on spacecraft are not in this release.

Validation

J2 is checked against the analytic rate of nodal regression. See the technical note on propagation accuracy.