Gravity / Spherical Harmonics

The gravity force model accounts for the gravitational acceleration experienced by a spacecraft due to the non-uniform mass distribution of the central body (typically Earth). While a point-mass gravity model assumes the central body is perfectly spherical and uniform, real celestial bodies have irregular mass distributions that create additional gravitational perturbations.

Physical Description

The Earth's gravitational field deviates from that of a perfect sphere due to:

• Oblateness: The Earth is flattened at the poles due to rotation
• Mass irregularities: Variations in density and topography
• Tidal deformation: Deformation due to external gravitational forces

These irregularities are mathematically represented using spherical harmonics, which decompose the gravitational potential into a series expansion. The most significant terms are the zonal harmonics (particularly J₂), which account for the Earth's oblateness.

The gravitational acceleration including harmonics is:

a_grav = -∇U = -∇(μ/r + ΔU)

Where:

• μ is the gravitational parameter
• r is the distance from the center of mass
• ΔU represents the harmonic corrections to the potential

Harmonic Coefficients

The spherical harmonic expansion uses coefficients Cₙₘ and Sₙₘ where:

• n is the degree (order of the harmonic)
• m is the order (number of nodal lines)
• Zonal harmonics (m=0): Symmetric about the rotation axis
• Tesseral harmonics (m≠0): Asymmetric terms

Components

The gravity force model in AstroForceModels provides two main implementations:

GravityHarmonicsAstroModel

A comprehensive model that includes spherical harmonic terms:

• gravity_model: Contains the harmonic coefficients and reference data
• eop_data: Earth orientation parameters for coordinate transformations
• order: Maximum order of harmonics to compute (-1 for maximum available)
• degree: Maximum degree of harmonics to compute (-1 for maximum available)

KeplerianGravityAstroModel

A simplified point-mass gravity model for comparison or computational efficiency:

• Assumes perfectly spherical, uniform central body
• Only includes the μ/r² term
• Useful for initial orbit determination or when high precision isn't required

Usage Example

using AstroForceModels
using SatelliteToolboxGravityModels
using SatelliteToolboxBase

# Load a gravity model (e.g., EGM96)
grav_coeffs = GravityModels.load(IcgemFile, fetch_icgem_file(:EGM96))

# Load Earth orientation parameters
eop_data = fetch_iers_eop()

# Create high-fidelity gravity model
gravity_model = GravityHarmonicsAstroModel(
    gravity_model = grav_coeffs,
    eop_data = eop_data,
    order = 20,    # Use up to degree/order 20
    degree = 20
)

# Compute acceleration (typically called within integrator)
acceleration(state, parameters, time, gravity_model)

Gravity Models Available

Common gravity models supported:

• EGM96: Earth Gravitational Model 1996 (360×360)
• EGM2008: High-resolution model (2190×2190)
• GGM03C: GRACE-based model
• JGM-3: Joint Gravity Model 3
• WGS84: World Geodetic System 1984 reference

Any ICGEM format is accepted, see SatelliteToolboxGravityModels.jl for details.

Implementation Details

The harmonic gravity computation involves:

1. Coordinate Transformation: Converting position to body-fixed coordinates
2. Legendre Polynomials: Computing associated Legendre functions
3. Harmonic Summation: Evaluating the spherical harmonic series
4. Gradient Calculation: Computing spatial derivatives for acceleration
5. Coordinate Transformation: Converting back to inertial frame

References

[1] Vallado, David A. "Fundamentals of Astrodynamics and Applications." 4th ed., 2013. [2] Montenbruck, Oliver, and Eberhard Gill. "Satellite Orbits: Models, Methods and Applications." Springer, 2000. [3] Hofmann-Wellenhof, B., and H. Moritz. "Physical Geodesy." 2nd ed., Springer, 2006. [4] Lemoine, F. G., et al. "The Development of the Joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) Geopotential Model EGM96." NASA Technical Publication, 1998.