# Spherical Harmonic Gravity Model

Implement spherical harmonic representation of planetary gravity

**Libraries:**

Aerospace Blockset /
Environment /
Gravity

## Description

The Spherical Harmonic Gravity Model block implements the mathematical representation of spherical harmonic planetary gravity based on planetary gravitational potential. It provides a convenient way to describe a planet gravitational field outside of its surface in spherical harmonic expansion.

You can use spherical harmonics to modify the magnitude and
direction of spherical gravity (-GM/r^{2}). The most significant or largest spherical harmonic term is the second degree zonal harmonic, J2, which accounts for oblateness of a planet.

Use this block if you want more accurate gravity values than spherical gravity models. For example, nonatmospheric flight applications might require higher accuracy.

## Limitations

The block excludes the centrifugal effects of planetary rotation, and the effects of a precessing reference frame.

Spherical harmonic gravity model is valid for radial positions greater than the planet equatorial radius. Minor errors might occur for radial positions near or at the planetary surface. The spherical harmonic gravity model is not valid for radial positions less than the planetary surface.

## Ports

### Input

### Output

## Parameters

## Algorithms

The Spherical Harmonic Gravity block works in the fixed-frame coordinate system for the central bodies:

Earth — The fixed-frame coordinate system is the Earth-centered Earth-fixed (ECEF) coordinate system.

Moon — The fixed-frame coordinate system is the Principal Axis system (PA), the orientation specified by JPL planetary ephemeris DE403.

Mars — The fixed-frame coordinate system is defined by the directions of the poles of rotation and prime meridians defined in [14].

## References

[1] Gottlieb, Robert G., "Fast Gravity, Gravity Partials, Normalized Gravity, Gravity Gradient Torque and Magnetic Field: Derivation, Code and Data." NASA-CR-188243. Houston, TX: NASA Lyndon B. Johnson Space Center, February 1993.

[2] Vallado, David.
*Fundamentals of Astrodynamics and Applications*. New York:
McGraw-Hill, 1997.

[3] "Department of Defense World Geodetic System 1984, Its Definition and Relationship with Local Geodetic Systems." NIMA TR8350.2.

[4] Konopliv, A.S., W. Asmar, E.
Carranza, W.L. Sjogren, and D.N. Yuan. "Recent Gravity Models as a Result of the Lunar
Prospector Mission," *Icarus*, 150, no. 1 (2001):
1–18.

[5] Lemoine, F. G., D. E. Smith, D.D.
Rowlands, M.T. Zuber, G. A. Neumann, and D. S. Chinn. "An Improved Solution of the
Gravity Field of Mars (GMM-2B) from Mars Global Surveyor". *Journal Of
Geophysical Research* 106, np E10 (October 25, 2001): pp
23359-23376.

[6] Kenyon S., J. Factor, N. Pavlis, and S. Holmes. "Towards the Next Earth Gravitational Model." Society of Exploration Geophysicists 77th Annual Meeting, San Antonio, TX, September 23–28, 2007.

[7] Pavlis, N.K., S.A. Holmes, S.C. Kenyon, and J.K. Factor, "An Earth Gravitational Model to Degree 2160: EGM2008." Presented at the 2008 General Assembly of the European Geosciences Union, Vienna, Austria, April 13–18, 2008.

[8] Grueber, T., and A. Köhl. "Validation of the EGM2008 Gravity Field with GPS-Leveling and Oceanographic Analyses." Presented at the IAG International Symposium on Gravity, Geoid & Earth Observation 2008, Chania, Greece, June 23–27, 2008.

[9] Förste, C., Flechtner et al, "A
Mean Global Gravity Field Model From the Combination of Satellite Mission and
Altimetry/Gravmetry Surface Data - EIGEN-GL04C." *Geophysical Research
Abstracts* 8, 03462, 2006.

[10] Hill, K. A. "Autonomous Navigation in Libration Point Orbits." Doctoral dissertation, University of Colorado, Boulder. 2007.

[11] Colombo, Oscar L. "Numerical Methods for Harmonic Analysis on the Sphere." Reports of the Department of Geodetic Science, Report No. 310, The Ohio State University, Columbus, OH., March 1981.

[12] Colombo, Oscar L. "The Global Mapping of Gravity with Two Satellites." Netherlands Geodetic Commission 7, no 3, Delft, The Netherlands, 1984., Reports of the Department of Geodetic Science. Report No. 310. Columbus: Ohio State University, March 1981.

[13] Jones, Brandon A. "Efficient Models for the Evaluation and Estimation of the Gravity Field." Doctoral dissertation, University of Colorado, Boulder. 2010.

[14] Report of the IAU/IAG Working Group on cartographic coordinates and rotational elements: 1991.

## Extended Capabilities

## Version History

**Introduced in R2010a**