gravitysphericalharmonic

Implement spherical harmonic representation of planetary gravity

Syntax

[gx gy gz] = gravitysphericalharmonic(planet_coordinates)
[gx gy gz] = gravitysphericalharmonic(planet_coordinates, model)
[gx gy gz] = gravitysphericalharmonic(planet_coordinates, degree)
[gx gy gz] = gravitysphericalharmonic(planet_coordinates, model, degree)
[gx gy gz] = gravitysphericalharmonic(planet_coordinates, model, degree, action)
[gx gy gz] = gravitysphericalharmonic(planet_coordinates, 'Custom', degree, {datafile dfreader}, action)


Description

[gx gy gz] = gravitysphericalharmonic(planet_coordinates) implements the mathematical representation of spherical harmonic planetary gravity based on planetary gravitational potential. This function calculates arrays of N gravity values in the x-axis, y-axis, and z-axis of the Planet-Centered Planet-Fixed coordinates for the planet. It performs these calculations using planet_coordinates, an M-by-3 array of Planet-Centered Planet-Fixed coordinates. By default, this function assumes 120th degree and order spherical coefficients for the 'EGM2008' (Earth) planetary model.

[gx gy gz] = gravitysphericalharmonic(planet_coordinates, model) implements the mathematical representation for the planetary model, model.

[gx gy gz] = gravitysphericalharmonic(planet_coordinates, degree) uses the degree and order that degree specifies.

[gx gy gz] = gravitysphericalharmonic(planet_coordinates, model, degree) uses the degree and order that degree specifies. model specifies the planetary model.

[gx gy gz] = gravitysphericalharmonic(planet_coordinates, model, degree, action) uses the specified action when input is out of range.

[gx gy gz] = gravitysphericalharmonic(planet_coordinates, 'Custom', degree, {datafile dfreader}, action) implements the mathematical representation for a custom model planet. datafile defines the planetary model. dfreader specifies the reader for datafile.

This function has the following limitations:

• The function 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 planetary surface.

Input Arguments

planet_coordinates	M-by-3 array of Planet-Centered Planet-Fixed coordinates in meters. The z-axis is positive toward the North Pole. If model is 'EGM2008' or 'EGM96' (Earth), the planet coordinates are ECEF coordinates.
model	Planetary model. Default is 'EGM2008'. Specify one: Planetary Model Planet 'EGM2008' Earth Gravitational Model 2008. Planet coordinates are ECEF (WGS84). 'EGM96' Earth Gravitational Model 1996. Planet coordinates are ECEF (WGS84). 'LP100K' 100th degree Moon model. 'LP165P' 165th degree Moon model. 'GMM2B' Goddard Mars model 2B. 'Custom' Custom planetary model that you define in datafile. Note To deploy a custom planetary model, explicitly include the custom data and reader files to the MATLAB® Compiler™ (mcc) command at compilation. For example: mcc -m mycustomsphericalgravityfunction... -a customDataFile -a customReaderFile For other planetary models, use the MATLAB Compiler as usual. 'EIGENGL04C' Combined Earth gravity field model EIGEN-GL04C. When inputting a large PCPF array and a high degree value, you might receive an out-of-memory error. For more information about avoiding out-of-memory errors in the MATLAB environment, see Resolve “Out of Memory” Errors. When inputting a large PCPF array, you might receive a maximum matrix size limitation. To determine the largest matrix or array that you can create in the MATLAB environment for your platform, see Performance and Memory.
degree	Scalar value that specifies the degree and order of the harmonic gravity model. Planetary Model Degree and Order 'EGM2008' Maximum degree and order is 2159. Default degree and order are 120. 'EGM96' Maximum degree and order is 360. Default degree and order are 70. 'LP100K' Maximum degree and order is 100. Default degree and order are 60. 'LP165P' Maximum degree and order is 165. Default degree and order are 60. 'GMM2B' Maximum degree and order is 80. Default degree and order are 60. 'Custom' Maximum degree is default degree and order. 'EIGENGL04C' Maximum degree and order is 360. Default degree and order are 70. When inputting a large PCPF array and a high degree value, you might receive an out-of-memory error. For more information about avoiding out-of-memory errors in the MATLAB environment, see Performance and Memory. When inputting a large PCPF array, you might receive a maximum matrix size limitation. To determine the largest matrix or array that you can create in the MATLAB environment for your platform, see Performance and Memory.
action	Action for out-of-range input. Specify one: 'Error' 'Warning' (default) 'None'
'Custom'	Character vector or string that specifies that datafile contains definitions for a custom planetary model.
datafile	File that contains definitions for a custom planetary model. For an example of file content, see aerogmm2b.mat. This file must contain the following variables. Variable Description Re Scalar of planet equatorial radius in meters (m) GM Scalar of planetary gravitational parameter in meters cubed per second squared (m3/s2) 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 This parameter requires that you specify a program in the dfreader parameter to read the data file.
dfreader	Specify a MATLAB function to read datafile. The reader file that you specify depends on the file type of datafile. Data File Type Description MATLAB file Specify the MATLAB load function, for example, @load. Other file type Specify a custom MATLAB reader function. For examples of custom reader functions, see astReadSHAFile.m and astReadEGMFile.m. Note the output variable order in these files.

Output Arguments

gx	Array of N gravity values in the x-axis of the Planet-Centered Planet-Fixed coordinates in meters per second squared (m/s2).
gy	Array of N gravity values in the y-axis of the Planet-Centered Planet-Fixed coordinates in meters per second squared (m/s2).
gz	Array of N gravity values in the z-axis of the Planet-Centered Planet-Fixed coordinates in meters per second squared (m/s2).

Examples

Calculate the gravity in the x-axis at the equator on the surface of Earth. This example uses the default 120 degree model of EGM2008 with default warning actions:

gx = gravitysphericalharmonic( [-6378.1363e3 0 0] )

Calculate the gravity at 25000 m over the south pole of Earth. This example uses the 70 degree model of EGM96 with error actions:

[gx, gy, gz] = gravitysphericalharmonic( [0 0 -6381.751e3], 'EGM96', 'Error' )

Calculate the gravity at 15000 m over the equator and 11000 m over the North Pole. This example uses a 30th order GMM2B Mars model with warning actions:

p  = [2412.648e3 -2412.648e3 0; 0 0 3397.2e3]
[gx, gy, gz] = gravitysphericalharmonic( p, 'GMM2B', 30, 'Warning' )

Calculate the gravity at 15000 m over the equator and 11000 m over the North Pole. This example uses a 60th degree custom planetary model with no actions:

p       = [2412.648e3 -2412.648e3 0; 0 0 3397e3]
[gx, gy, gz] = gravitysphericalharmonic( p, 'custom', 60, ...
{'GMM2BC80_SHA.txt' @astReadSHAFile}, 'None' )

Calculate the gravity at 25000 meters over the south pole of Earth using a 120th order EIGEN-GL04C Earth model with warning actions:

p  = [0 0 -6381.751e3]
[gx, gy, gz] = gravitysphericalharmonic( p, 'EIGENGL04C', ...
120, 'Warning' )   

Tips

• When inputting a large PCPF array and a high degree value, you might receive an out-of-memory error. For more information about avoiding out-of-memory errors in the MATLAB environment, see Performance and Memory.

• When inputting a large PCPF array, you might receive a maximum matrix size limitation. To determine the largest matrix or array that you can create in the MATLAB environment for your platform, see Performance and Memory.

References

[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.

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[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.

[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.