Implement spherical harmonic representation of planetary gravity

Environment/Gravity

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.

**Units**Specifies the parameter and output units:

Units

Height

`Metric (MKS)`

Meters

`English`

Feet

**Degree**Specify the degree of harmonic model. Recommended degrees are:

Planet Model Degree `EGM2008`

120

`EGM96`

70

`LP100K`

60

`LP165P`

60

`GMM2B`

60

`EIGENGL04C`

70

**Action for out of range input**Specify if out-of-range input invokes a warning, error, or no action.

**Planet model**Specify the planetary model. From the list, select:

Planet Model Notes `EGM2008`

Earth — Is the latest Earth spherical harmonic gravitational model from National Geospatial-Intelligence Agency (NGA). This block provides the WGS-84 version of this gravitational model. You can use the EGM96 planetary model if you need to use the older standard for Earth.

`EGM96`

Earth `LP100K`

Moon — Is best for lunar orbit determination based upon computational time required to compute orbits. This planet model was created in approximately the same year as LP165P with similar data.

`LP165P`

Moon — Is best for extended lunar mission orbit accuracy. This planet model was created in approximately the same year as LP165P with similar data.

`GMM2B`

Mars

`Custom`

Enables you to specify your own planetary model. This option enables the

**Planet mat-file**parameter.`EIGENGL04C`

Earth — Supports the gravity field model, EIGEN-GL04C (

`http://icgem.gfz-potsdam.de/ICGEM/`

). This model is an upgrade to EIGEN-CG03C.When defining your own planetary model, the

**Degree**parameter is limited to the maximum value for int16. When inputting a large degree, you might receive an out-of-memory error. For more information about avoiding out-of-memory errors in the MATLAB^{®}environment, see Memory Usage.**Planet mat-file**Specify a MAT-file that contains definitions for a custom planetary model. The

`aerogmm2b.mat`

file in the Aerospace Toolbox is the default MAT-file for a custom planetary model.This file must contain:

Variable Description *Re*Scalar of planet equatorial radius in meters (m).

*GM*Scalar of planetary gravitational parameter in meters cubed per second squared (m

^{3}/s^{2})*degree*Scalar of maximum degree.

*C*(

*degree*+1)-by-(*degree*+1) matrix containing normalized spherical harmonic coefficients matrix,*C*.*S*(

*degree*+1)-by-(*degree*+1) matrix containing normalized spherical harmonic coefficients matrix,*S*.When using a large value for

**Degree**, you might receive an out-of-memory error. For more information about avoiding out-of-memory errors in the MATLAB environment, see Memory Usage.

[1] Gottlieb, R. G., "Fast Gravity, Gravity Partials,
Normalized Gravity, Gravity Gradient Torque and Magnetic Field: Derivation,
Code and Data," *Technical Report NASA Contractor
Report 188243*, NASA Lyndon B. Johnson Space Center, Houston,
Texas, February 1993.

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

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

[4] Konopliv, A. S., S. W. Asmar, E. Carranza, W. L. Sjogen, D. N. Yuan., "Recent Gravity Models as a Result of the Lunar Prospector Mission, Icarus", Vol. 150, no. 1, pp 1–18, 2001.

[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*, Vol. 106, No. E10, pp 23359-23376,
October 25, 2001.

[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, Texas, 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, F., Schmidt, R., König,
R., Meyer, U., Stubenvoll, R., Rothacher, M., Barthelmes, F., Neumayer,
H., Biancale, R., Bruinsma, S., Lemoine, J.M., Loyer, S., "A
Mean Global Gravity Field Model From the Combination of Satellite
Mission and Altimetry/Gravmetry Surface Data - EIGEN-GL04C", *Geophysical
Research Abstracts*, Vol. 8, 03462, 2006.

[10] Hill, K. A. (2007). Autonomous Navigation in Libration Point Orbits. Doctoral dissertation, University of Colorado, Boulder. http://ccar.colorado.edu/geryon/papers/Misc/Hill_thesis.pdf.

[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", Nederlands Geodetic Commission, vol. 7 No. 3, Delft, The Nederlands, 1984., Reports of the department of Geodetic Science, Report No. 310, The Ohio State University, Columbus, OH., March 1981.

[13] Jones, Brandon A. (2010). Efficient Models for the Evaluation and Estimation of the Gravity Field. Doctoral dissertation, University of Colorado, Boulder. http://ccar.colorado.edu/geryon/papers/Misc/bajones_phd.pdf.

Was this topic helpful?