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# Zonal Harmonic Gravity Model

Calculate zonal harmonic representation of planetary gravity

## Library

Environment/Gravity

## Description

The Zonal Harmonic Gravity Model block calculates the zonal harmonic representation of planetary gravity at a specific location based on planetary gravitational potential. This block provides a convenient way to describe the gravitational field of a planet outside its surface.

By default, the block uses the fourth order zonal coefficient for Earth to calculate the zonal harmonic gravity. It also allows you to specify the second or third zonal coefficient.

gravityzonal is implemented using the following planetary parameter values for each planet:

PlanetEquatorial Radius (Re) in MetersGravitational Parameter (GM) in m3/s2Zonal Harmonic Coefficients
(J Values)
Earth6378.1363e33.986004415e14[ 0.0010826269 -0.0000025323 -0.0000016204 ]
Jupiter71492.e31.268e17[0.01475 0 -0.00058]
Mars3397.2e34.305e13 [ 0.001964 0.000036 ]
Mercury2439.0e3 2.2032e130.00006
Moon1738.0e3 4902.799e90.0002027
Neptune24764e3 6.809e150.004
Saturn60268.e3 3.794e16 [0.01645 0 -0.001]
Uranus25559.e35.794e15 0.012
Venus6052.0e3 3.257e140.000027

## Dialog Box

Units

Specify the input units:

Units

Position

Gravitational Parameter

Metric (MKS)

Meters

Meters

Meters cubed per second squared

English

Feet

Feet

Feet cubed per second squared

Degree

Specify the degree of harmonic model.

• 2 — Second degree, J2. Most significant or largest spherical harmonic term, which accounts for the oblateness of a planet.

• 3 — Third degree, J3.

• 4 — Fourth degree, J4 (default).

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 Mercury, Venus, Earth, Moon, Mars, Jupiter, Saturn, Uranus, Neptune, or Custom.

Selecting Custom enables you to specify your own planetary model. This option enables the Equatorial radius, Gravitational parameter, and J values parameters.

Selecting Mercury, Venus, Moon, Uranus, or Neptune limits the degree to 2.

Selecting Mars limits the degree to 3.

Specify the planetary equatorial radius in the length units that the Units parameter defines.

Gravitational parameter

Specify the planetary gravitational parameter in the length units cubed per second squared that the Units parameter defines.

J values

Specify a 3-element array that defines the zonal harmonic coefficients.

## Inputs and Outputs

This block accepts only scalar inputs (m=1).

InputDimension TypeDescription

First

m-by-3 matrixContains planet-centered planet-fixed coordinates from the center of the planet in the selected length units. If Planet model has a value of Earth, this matrix contains Earth-centered Earth-fixed (ECEF) coordinates.

OutputDimension TypeDescription

First

m-by-3 arrayContains gravity values in the x-axis, y-axis and z-axis of the planet-centered planet-fixed coordinates in the selected length units per second squared.

## References

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

Fortescue, P., J. Stark, G. Swinerd, (Eds.). Spacecraft Systems Engineering, Third Edition, Wiley & Sons, West Sussex, 2003.

Tewari, A., Atmospheric and Space Flight Dynamics Modeling and Simulation with MATLAB and Simulink, Birkhäuser, Boston, 2007.