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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:

Planet	Equatorial Radius (Re) in Meters	Gravitational Parameter (GM) in m3/s2	Zonal Harmonic Coefficients (J Values)
Earth	6378.1363e3	3.986004415e14	[ 0.0010826269 -0.0000025323 -0.0000016204 ]
Jupiter	71492.e3	1.268e17	[0.01475 0 -0.00058]
Mars	3397.2e3	4.305e13	[ 0.001964 0.000036 ]
Mercury	2439.0e3	2.2032e13	0.00006
Moon	1738.0e3	4902.799e9	0.0002027
Neptune	24764e3	6.809e15	0.004
Saturn	60268.e3	3.794e16	[0.01645 0 -0.001]
Uranus	25559.e3	5.794e15	0.012
Venus	6052.0e3	3.257e14	0.000027

Dialog Box

Units

Specify the input units:

Units	Position	Equatorial Radius	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.

Equatorial radius

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).

Input	Dimension Type	Description
First	m-by-3 matrix	Contains 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.

Output	Dimension Type	Description
First	m-by-3 array	Contains 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.

World Magnetic Model 2005

Aerospace Units

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