Gravity Field Model

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

A gravity field model is a computational model that approximates a celestial body's gravitational potential using a set of numerical parameters, most commonly a truncated spherical harmonic expansion:

U(r,\phi,\lambda) = \frac{\mu}{r} \left[ 1 + \sum_{\ell=2}^{N} \sum_{m=0}^{\ell} \left(\frac{R}{r}\right)^{\!\ell} \bar{P}_{\ell m}(\sin\phi) \bigl( \bar{C}_{\ell m} \cos m\lambda + \bar{S}_{\ell m} \sin m\lambda \bigr) \right]

where $N$ is the truncation degree and $\bar{C}_{\ell m}$ , $\bar{S}_{\ell m}$ is a published set of Stokes coefficients, with $R$ being the reference radius. The coefficients are typically recovered from satellite tracking data, GNSS, satellite altimetry, lunar laser ranging, or dedicated gravity missions (GRACE, GOCE, GRAIL) (Yin et al. 2024). Once published, users simply read the coefficients to recover the corresponding gravitational acceleration in an orbit integrator.

A gravity field model can be understood as the discrete, engineering-ready implementation of the mathematical concept of "gravitational potential." Multiple model versions exist for the same body, differing primarily in observational data, solution strategies, and truncation degree.

Common Earth Gravity Field Models

Model	Data Source	Typical Degree	Primary Use
EGM2008	Ground gravity, satellite altimetry, GRACE	Full 2190; missions commonly use 50×50 or 70×70 truncation	Global static high-resolution Earth gravity field, precision orbit determination, geoid
GGM05C	GRACE and GOCE combined	180 (commonly truncated to 20×20 in missions)	Medium/low Earth orbit satellite prediction
WGS84 EGM96/EGM2008 companion	GPS tracking and gravity data	Commonly 8×8, 36×36 truncations	Navigation, general orbital mechanics

EGM2008 is one of the most widely used static Earth gravity field models, with its full version containing 2190-degree spherical harmonic coefficients. Engineering missions rarely use the full degree: for Earth-Moon transfers and near-Earth segments where only $\sim 10^{-3}\ \text{m/s}^2$ orbital perturbations matter, 50×50 or 70×70 truncation usually suffices; higher degrees are required for low-Earth precision orbit determination.

Common Lunar Gravity Field Models

Model	Data Source	Typical Degree	Primary Use
GRGM660PRIM	GRAIL primary mission data	660	Current highest-accuracy static lunar gravity model, high-fidelity orbital dynamics
GL0660B	GRAIL	660	One GRAIL-derived version; engineering commonly uses lower truncation
LP150Q	Lunar Prospector	150	Lunar vicinity orbit analysis and transfer design
LP165P	Lunar Prospector	165	Lunar orbit computation
LP100K	Lunar Prospector	100	Low lunar orbit segment prediction

The lunar gravity field is more irregular than Earth's: the presence of significant mascons on the lunar near side causes high-degree harmonic coefficients to decay more slowly. Degree selection for lunar models is more sensitive than for Earth — for low lunar orbits (50–100 km altitude), insufficient truncation degree will underestimate the long-term drift and decay of the orbital plane (Trofimov et al. 2020).

Engineering Principles for Degree Selection

A higher truncation degree $N$ is not always better. Selection criteria typically include (Vallado 2022):

Orbit altitude and perturbation scale. Near-surface or low lunar orbits are more affected by high-degree terms, requiring higher $N$ ; farther from the central body, the $(R/r)^\ell$ factor decays more rapidly, so lower-degree models suffice.
Mission accuracy requirements. Rough transfer design may use 8×8; precision orbit determination and low-energy Earth-Moon transfers commonly require 50×50 to 150×150; only global geodesy needs $N > 200$ .
Computational cost and real-time constraints. For each increase in degree, the Legendre function computation per integration step grows as $O(N^2)$ . On-board autonomous navigation and Monte Carlo campaigns must balance accuracy with efficiency.
Model consistency. Different mission segments (e.g., geocentric, lunar capture, libration point) may use different degrees or even different celestial body models; consistency of reference frame, reference radius, and normalization convention must be ensured during transitions.

Relationship to Other Models

Simplified force model: Retains only central-body gravity $-\mu/r^2$ and optionally $J_2$ , used for preliminary analytical analysis and rapid search.
High-fidelity force model: In addition to high-degree gravity, superimposes third-body perturbations, solar radiation pressure, atmospheric drag, etc.
Polyhedral / mascon methods: When the target body is highly irregular (asteroids, comets), spherical harmonic expansion may diverge near the body surface; in such cases, polyhedral or mascon models are used instead (Yin et al. 2024).

References

Vallado, D. A., 2022, Fundamentals of Astrodynamics and Applications, 5th ed., Microcosm Press. Ch. 8.6.1 — Mathematical foundation of gravity field modeling and normalization.
Yin, Z., Zhang, K., Duan, Y., Liu, J., Mu, Q., 2024, Theoretical research progress of gravitational field modeling in Earth science and deep-space exploration, Reviews of Geophysics and Planetary Physics, 55(5): 501–512. — Systematic review of Earth and deep-space gravity field modeling theory and methods.
Pavlis, N. K. et al., 2012, The development and evaluation of the Earth Gravitational Model 2008 (EGM2008), Journal of Geophysical Research: Solid Earth, 117(B4). — Source and accuracy of EGM2008.
Konopliv, A. S. et al., 2013, The JPL lunar gravity field to spherical harmonic degree 660 from the GRAIL Primary Mission, Journal of Geophysical Research: Planets, 118(7): 1415–1434. — Theoretical basis of GRGM660PRIM and GL0660B.
Trofimov, S. et al., 2020, Transfers from NRHOs to low-perilune orbits, Acta Astronautica, 167: 260–271. — Impact of irregular lunar gravity on low-perilune orbits.
Zuber, M. T. et al., 2013, Gravity field of the Moon from the Gravity Recovery and Interior Laboratory (GRAIL) mission, Science, 339(6120): 668–671. — GRAIL mission and lunar gravity field models.