gravityzonal

Implement zonal harmonic representation of planetary gravity

Description

Default Degree of Harmonic and Planetary Model

example

[gravityXcoord gravityYcoord,gravityZcoord] = gravityzonal(planetCoord) implements the mathematical representation of zonal harmonic planetary gravity based on planetary gravitational potential. The function takes an m-by-3 matrix that contains planet-centered planet-fixed coordinates from the center of the planet in meters. This function calculates the arrays of m gravity values in the x-, y-, and z-axes of the planet-centered planet-fixed coordinates.

This function does not include the potential due planet rotation, which excludes the centrifugal effects of planetary rotation and the effects of a precessing reference frame.

[gravityXcoord,gravityYcoord,gravityZcoord] = gravityzonal(planetCoord,action) specifies the action for out-of-range input.

Degree of Harmonic Model and Planetary Model

example

[gravityXcoord,gravityYcoord,gravityZcoord] = gravityzonal(planetCoord,degreeGravityModel) uses the degree of harmonic model.

[gravityXcoord,gravityYcoord,gravityZcoord] = gravityzonal(planetCoord,planetModel) uses the planetary model.

[gravityXcoord,gravityYcoord,gravityZcoord] = gravityzonal(planetCoord,planetModel,degreeGravityModel) uses the degree of harmonic model and planetary model.

[gravityXcoord,gravityYcoord,gravityZcoord] = gravityzonal(planetCoord,planetModel,degreeGravityModel,action) specifies the action for out-of-range input.

Custom Planetary Model

example

[gravityXcoord,gravityYcoord,gravityZcoord] = gravityzonal(planetCoord,'Custom',Re,planetaryGravitional,zonalHarmonicCoeff) uses the equatorial radius, planetary gravitational parameter, and zonal harmonic coefficients for the custom planetary model.

[gravityXcoord,gravityYcoord,gravityZcoord] = gravityzonal(planetCoord,'Custom',Re,planetaryGravitional,zonalHarmonicCoeff,action) specifies the action for out-of-range input.

Examples

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Calculate the gravity in x-axis at the equator on surface of Earth using the fourth degree model with no warning actions.

gx = gravityzonal([-6378.1363e3 0 0])
gx =
9.8142

Calculate the gravity using the close approximation method at 100 m over the geographic South Pole of Earth with error actions.

[gx, gy, gz] = gravityzonal([0 0 -6356.851e3],'Error')
gx =
0

gy =
0

gz =
9.8317

Calculate the gravity at 15,000 m over the equator and 11,000 m over the geographic North Pole using a second order Mars model with warning actions.

p = [2412.648e3 -2412.648e3 0; 0 0 3376.2e3];
[gx, gy, gz] = gravityzonal(p,'Mars',2,'Warning')
gx =
-2.6224
0

gy =
2.6224
0

gz =
0
-3.7542

Calculate the gravity at 15,000 m over the equator and 11,000 m over the geographic North Pole using a custom planetary model with no actions.

p= [2412.648e3 -2412.648e3 0; 0 0 3376e3];
GM = 42828.371901e9;  % m^3/s^2
Re = 3397e3;          % m
Jvalues = [1.95545367944545e-3 3.14498094262035e-5 ...
-1.53773961526397e-5];
[gx, gy, gz] = gravityzonal(p,'custom',Re,GM, ...
Jvalues,'None')
gx =
-2.6090
0

gy =
2.6090
0

gz =
0.0002
-3.7352

Input Arguments

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Planet-centered planet-fixed coordinates from center of planet, specified as an m-by-3 matrix in meters. If planetModel has a value of 'Earth', this matrix contains Earth-centered Earth-fixed (ECEF) coordinates.

Data Types: double

Planetary model, specified as:

• 'Mercury'

• 'Venus'

• 'Earth'

• 'Moon'

• 'Mars'

• 'Jupiter'

• 'Saturn'

• 'Uranus'

• 'Neptune'

• 'Custom'

'Custom' requires you to specify your own planetary model using the equatorialRadius, planetaryGravitional, and zonalHarmonicCoeff parameters.

Data Types: double

Degree of harmonic model, specified as a scalar of one of these values.

DegreeDescriptionDefault WhenplanetModel Is
4Fourth degree, J4
• 'Earth

• 'Jupiter'

• 'Saturn'

• 'Custom'

2Second degree, J2
• 'Mercury'

• 'Venus'

• 'Moon'

• 'Uranus'

• 'Neptune'

3Third degree, J3'Mars'

Data Types: double

Equatorial radius, specified as a scalar in meters.

Data Types: double

Planetary gravitational parameter, specified as a scalar in meters cubed per second squared.

Data Types: double

Zonal harmonic coefficients to calculate zonal harmonics planetary gravity, specified as a 3-element array.

Data Types: double

Action for out-of-range input, specified as:

• 'Error' — Displays warning and indicates that the input is out of range.

• 'Warning' — Displays error and indicates that the input is out of range.

• 'None' — Does not display warning or error.

Data Types: char | string

Output Arguments

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Gravity values in the x-axis, returned as an array of m gravity values of the planet-centered planet-fixed coordinates in meters per second squared.

Gravity values in the y-axis, returned as an array of m gravity values of the planet-centered planet-fixed coordinates in meters per second squared.

Gravity values in the z-axis, returned as an array of m gravity values of the planet-centered planet-fixed coordinates in meters per second squared.

Algorithms

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

PlanetEquatorial Radius (Re) in MetersGravitational Parameter (GM) in m3/s2 Zonal 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

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

 Fortescue, Peter, Graham Swinerd, and John Stark, eds. Spacecraft Systems Engineering, 3rd ed. West Sussex: Wiley & Sons, 2003.

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

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