Non-Spherical Gravity Perturbation

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

Definition

Non-spherical gravity perturbation (also called aspherical gravity perturbation) is the additional gravitational acceleration caused by a celestial body's actual gravity field deviating from a perfect uniform sphere. A uniform sphere's gravity can be represented as a point-mass force (the Newton two-body problem). Real celestial bodies, however, are neither geometrically nor materially spherically symmetric—due to rotation, internal mass distributions, and surface topography—and the gravitational differences that result constitute non-spherical perturbations.

This is a conservative perturbation, expressible as the gradient of a gravity potential function $\mathbf{a} = \nabla U$ , modelled by a spherical-harmonic expansion (Vallado 2022, Ch. 8.6.1):

U(r, \phi, \lambda) = \frac{\mu}{r} \sum_{\ell=0}^{\infty} \sum_{m=0}^{\ell} \left(\frac{R_e}{r}\right)^{\ell} P_{\ell m}(\sin\phi) \left[ C_{\ell m} \cos m\lambda + S_{\ell m} \sin m\lambda \right]

where $r$ is the distance to the body's centre of mass, $\phi$ is the geocentric latitude, $\lambda$ is the longitude, $R_e$ is the reference equatorial radius, $P_{\ell m}$ are the associated Legendre polynomials, and $C_{\ell m}$ , $S_{\ell m}$ are dimensionless spherical-harmonic coefficients. The $\ell = 0$ term is the two-body gravity; the $\ell \geq 1$ terms constitute the non-spherical (perturbing) part. By convention, the $\ell = 2, m = 0$ term is written as $J_2 = -C_{2,0}$ , with a positive sign indicating oblateness.

Classification of Spherical-Harmonic Terms

According to the $(\ell, m)$ combination, the terms are classified into three types (Vallado 2022, Ch. 8.6.1):

Zonal harmonics ( $m = 0$ ): depend only on latitude, not longitude—on the sphere they form "zones of latitude." $J_2$ is the leading zonal harmonic, representing the equatorial bulge (oblateness); $J_3$ describes north-south asymmetry (the "pear shape"); $J_4, J_5, \dots$ describe higher-order latitudinal structure.
Tesseral harmonics ( $m \neq 0$ and $\ell \neq m$ ): form a "checkerboard" pattern on the sphere, varying with both latitude and longitude. Higher-order tesseral terms in Earth's field reflect large-scale mass anomalies such as continents and oceans.
Sectoral harmonics ( $\ell = m$ ): depend only on longitude, forming "longitude sectors" on the sphere.

The J2 Term — The Dominant Oblateness Perturbation

$J_2$ ( $\approx 1.0826 \times 10^{-3}$ ) is the largest term in the non-spherical field, describing Earth's equatorial bulge (the polar radius is about 21 km shorter than the equatorial radius). It produces two key secular effects on near-Earth orbits (Vallado 2022, Ch. 9.6.1):

\dot{\Omega} = -\frac{3}{2} J_2 \frac{R_e^2}{a^2(1-e^2)^2} n \cos i \quad \text{(RAAN regression)}

\dot{\omega} = \frac{3}{2} J_2 \frac{R_e^2}{a^2(1-e^2)^2} n \left(2 - \frac{5}{2}\sin^2 i\right) \quad \text{(argument-of-perigee precession)}

Engineering applications:

Sun-Synchronous Orbit (SSO): choose an inclination (approximately 98°) such that $\dot{\Omega} \approx 0.9856^{\circ}/\text{day}$ (Earth's orbital angular rate), keeping the orbital plane at a fixed orientation relative to the Sun.
Critical Inclination Orbit: choose $i = 63.4^{\circ}$ or $116.6^{\circ}$ so that $\dot{\omega} = 0$ , holding the perigee in a fixed direction (the Molniya orbit exploits this principle).

For cislunar libration-point orbits (halo, NRHO, DRO, etc.) the $J_2$ effect is comparatively minor—these orbits are dominated by three-body gravitation, and the relative-motion deviation from Earth oblateness is typically at the millimetre level, negligible in most analyses.

Earth's Non-Spherical Gravity Field

Earth gravity-field models (EGM96, EGM2008, JGM-3, GGM series) are solved jointly from satellite altimetry, SLR, and gravimetry (GOCE, GRACE), with maximum expansion to $\ell_{max} = 2190$ . For orbital mechanics, 70×70 (LEO high-precision) or 5×5 (cislunar) is usually sufficient (Vallado 2022, Ch. 8.6.1; Framework paper, 2023).

Reference ellipsoid parameters (WGS-84):

Equatorial radius $R_e = 6\,378\,137$ m;
Flattening reciprocal $1/f = 298.257\,223\,563$ ;
$J_2 = 1.082\,63 \times 10^{-3}$ ;
Leading zonal coefficients: $J_3 \approx -2.532 \times 10^{-6}$ , $J_4 \approx -1.620 \times 10^{-6}$ .

The Moon's Non-Spherical Gravity Field

The Moon's gravity field differs significantly from Earth's (Vallado 2022, Ch. 8.6.1; Folta et al. 2010; LP-150Q model):

Far-side unknown: Due to tidal locking, the Moon always shows the same face to Earth; the far-side field was, for a long time, inferred only indirectly from Apollo residuals. The GRAIL mission (2012) dramatically improved knowledge.
Mascons (mass concentrations): large positive gravity anomalies beneath major impact basins (Mare Imbrium, Mare Serenitatis, Mare Crisium, etc.) create significant "gravity wells" that perturb low-orbiting satellites. Post-GRAIL models (GRGM-1200A, etc.) expand to degree 1200.
Basin positive anomalies, crater negative anomalies: basins host positive mascons, while smaller craters often exhibit a "central peak plus negative anomaly ring" structure.
Overall higher roughness: without atmosphere, oceans, or plate tectonics to "round off" the field, the Moon's gravity is inherently rougher.

Engineering impact: the orbital lifetime of low lunar orbit (LLO) satellites is strongly affected by mascons; many LLO orbits decay within days due to mascon perturbations—this is one of the key reasons DROs and NRHOs are chosen as long-term orbits.

Model Selection (Cislunar)

Recommended model orders by orbital regime (Vallado 2022, Ch. 8; Framework paper, 2023):

Orbit	Earth Gravity Degree/Order	Lunar Gravity Degree/Order
LEO high-precision OD	70×70 or higher	N/A
GEO / MEO	8×8–20×20	N/A
Earth-Moon transfer (high fidelity)	8×8–20×20	20×20–50×50
Cislunar debris OD	5×5	Optional (point-mass proxy)
Low Lunar Orbit (LLO)	N/A	50×50 or higher
Earth-Moon libration point	4×4	4×4

The spherical-harmonic expansion converges poorly near the lunar far-side and poles; for precise LLO orbit determination, auxiliary point-mass models are often used as a supplement.

References

Vallado, 2022, Fundamentals of Astrodynamics and Applications (Ch. 8.6.1 Gravity Field of a Central Body—derivation of spherical-harmonic expansion, zonal/tesseral/sectoral classification, the J2 term; Ch. 9.6.1—secular J2 effect formulas; Ch. 8.6.3—the Moon's distinctive gravity field).
A model framework for high-accuracy orbit determination and propagation of cislunar space debris, 2023 (cislunar-debris OD: Earth 5×5 harmonics plus lunar point-mass empirical basis).
Peng & Zhang, 2016, Review of Earth-Moon transfer trajectory schemes for manned lunar landing (Earth-Moon transfer mechanical-model configuration).
Folta et al., 2010, Earth-Moon libration point orbit stationkeeping (negligible impact of $J_2$ on Earth-Moon libration-point orbits).