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Implement spherical harmonic representation of planetary 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
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Texas, February 1993.

[2] Vallado, D. A., *Fundamentals of Astrodynamics
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[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.

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[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,
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H., Biancale, R., Bruinsma, S., Lemoine, J.M., Loyer, S., "A
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