Gravitational Potential

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

Definition

Gravitational potential is a scalar potential function $U(\mathbf{r})$ that describes a celestial body's gravitational field. Its gradient gives the gravitational acceleration vector:

\mathbf{g}(\mathbf{r}) = \nabla U(\mathbf{r})

The gravitational potential provides a more compact mathematical description than the vector field — a single scalar function carries the complete information of the gravitational field (Vallado 2022). A spacecraft's potential energy $V$ relates to the gravitational potential via $U = -V/m$ , with the potential energy referenced to zero at infinity.

Point-mass potential. When the central body is treated as a point mass or a homogeneous sphere, $U = \mu / r$ , where $\mu = GM$ is the gravitational parameter and $r$ is the distance from the body's center of mass. This is the foundation of the two-body problem — applying $\nabla U$ yields Newton's gravitational acceleration $-\mu \,\hat{\mathbf{r}} / r^2$ .

Spherical Harmonic Expansion for Non-Spherical Potentials

Real celestial bodies are not homogeneous spheres; their uneven mass distribution and non-spherical shape cause actual gravity to deviate from the point-mass model. For a point $P(r, \phi, \lambda)$ in the exterior space ( $r$ is distance from the body's center, $\phi$ is latitude, $\lambda$ is longitude), the gravitational potential can be expanded as a spherical harmonic series (Vallado 2022, Ch. 8.6; Yin et al. 2024):

U(r,\phi,\lambda) = \frac{\mu}{r} \left[ 1 + \sum_{\ell=2}^{\infty} \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 $R$ is the reference radius (for Earth, the equatorial radius $a_e$ ), $\bar{P}_{\ell m}$ are normalized associated Legendre functions, and $\bar{C}_{\ell m}$ , $\bar{S}_{\ell m}$ are normalized spherical harmonic coefficients (Stokes coefficients). The coefficients encode the body's entire mass distribution — low-degree terms describe large-scale shape (oblateness, etc.), while high-degree terms capture local details. The expansion starts from $\ell = 2$ because $\ell = 0$ recovers the point-mass potential $\mu/r$ , and $\ell = 1$ terms vanish when the coordinate origin is placed at the body's center of mass.

Three Types of Spherical Harmonic Terms

Based on the relationship between order $m$ and degree $\ell$ , spherical harmonic terms fall into three categories (Vallado 2022; Yin et al. 2024):

Type	Condition	Geometric Characteristic	Physical Meaning
Zonal harmonics	$m = 0$	Vary only with latitude, symmetric about the polar axis	Equatorial bulge ( $J_2$ ), pear shape ( $J_3$ ), and other global shapes
Sectoral harmonics	$\ell = m$	Depend only on longitude, appear as "orange slices"	Longitudinal mass concentrations
Tesseral harmonics	$\ell \neq m \neq 0$	Vary with both latitude and longitude, "checkerboard" pattern	Regional mass anomalies

J notation convention. Zonal harmonics commonly use the notation $J_\ell$ instead of $C_{\ell,0}$ , with the convention $J_\ell = -C_{\ell,0}$ (Vallado 2022, Eq. 8-20). Earth's $J_2 \approx 1.0826 \times 10^{-3}$ is the largest coefficient in absolute value — roughly 1000 times larger than the next largest coefficient $J_3$ — and dominates the non-spherical gravitational perturbation (causing orbital plane precession, perigee rotation, etc.).

Normalization. The raw coefficients $C_{\ell m}$ and $S_{\ell m}$ become extremely small as $\ell$ and $m$ increase, introducing round-off errors. Therefore, published models use normalized coefficients $\bar{C}_{\ell m}$ , $\bar{S}_{\ell m}$ :

\bar{C}_{\ell m} = \sqrt{\frac{(\ell+m)!}{(\ell-m)!\,k\,(2\ell+1)}} \; C_{\ell m}, \qquad k = \begin{cases} 1 & m=0 \\ 2 & m \neq 0 \end{cases}

The corresponding Legendre functions must be inversely normalized to maintain the product $\bar{C}_{\ell m} \bar{P}_{\ell m} = C_{\ell m} P_{\ell m}$ (Vallado 2022, Eq. 8-22).

Normal and Disturbing Potential

In geodesy, the gravitational potential is decomposed into a reference part and a deviation part (Vallado 2022):

U = V + T

Normal potential $V$ : Gravitational potential produced by a rotationally symmetric ellipsoid (normal Earth), containing only even-degree zonal harmonics.
Disturbing potential $T$ : Difference between the true and normal potentials, $T = U - V$ . The disturbing gravitational acceleration $\delta\mathbf{g} = \nabla T$ is on the order of $200\ \text{mgal}$ ( $10^{-5}\ \text{m/s}^2$ ) and cannot be neglected in precision orbit determination or altimetry.

Earth vs. Moon Comparison

Property	Earth	Moon
$J_2$ magnitude	$\sim 1.08 \times 10^{-3}$	$\sim 2.03 \times 10^{-4}$
High-degree field irregularity	Relatively smooth	Significantly more irregular — near-side mascons, slower decay of high-degree coefficients
Impact on low orbits	$J_2$ dominates perturbations	High-degree terms ( $\ell$ up to tens) still significantly affect orbits $<100\ \text{km}$
Representative model	EGM2008 ( degree 2190)	GRGM660PRIM ( degree 660)

The irregularity of the lunar gravity field is a critical constraint for cislunar mission design — below approximately 100 km altitude, gravitational irregularities make it difficult to maintain stable orbits with single impulses (Trofimov et al. 2020).

References

Vallado, D. A., 2022, Fundamentals of Astrodynamics and Applications, 5th ed., Microcosm Press. Ch. 8.6.1 — Complete derivation of spherical harmonic expansion, normalization conventions, J notation, and three-way harmonic classification.
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 gravitational field definition, spherical harmonic expansion, and multipole expansion theory.
Chao, B. F. & Shih, S. A., 2021, Multimultipole expansion: Unifying formalism for Earth and planetary gravitational dynamics, Surveys in Geophysics, 42: 803–838. — Complex-variable spherical harmonic representation.
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.