Zonal Harmonic Gravity Model

blockType: SubSystem

Path in the library:

/Aerospace/Environment/Gravity/Zonal Harmonic Gravity Model

Description

Block Zonal Harmonic Gravity Model Calculates the zonal harmonic representation of planetary gravity at a specific point based on the planetary gravitational potential. This block allows you to conveniently describe the gravitational field of a planet outside its surface. By default, the block uses the fourth-order zonal coefficient for the Earth to calculate the zonal harmonic gravity. You can also specify a zone coefficient of the second or third order. This block is implemented using the usage of the following values of planetary parameters for each planet:

The planet	Equatorial radius ( ), m	Gravitational parameter ( ), m3/s2	Zonal harmonic coefficients ( )
The Earth	6 378 136.3	3.986004415 × 10¹⁴	[1.0826269 × 10⁻³, -2.5323 × 10⁻⁶, -1.6204 × 10⁻⁶]
Jupiter	71 492 000	1.268 × 10¹⁷	[1.475 × 10⁻², 0, -5.8 × 10⁻⁴]
Mars	3 397 200	4.305 × 10¹³	[1.964 × 10⁻³, 3.6 × 10⁻⁵]
Mercury	2 439 000	2.2032 × 10¹³	6.0 × 10⁻⁵
The moon	1 738 000	4.902799 × 10⁹	2.027 × 10⁻⁴
Neptune	24 764 000	6.809 × 10¹⁵	4.0 × 10⁻³
Saturn	60 268 000	3.794 × 10¹⁶	[1.645 × 10⁻², 0, -1.0 × 10⁻³]
Uranium	25 559 000	5.794 × 10¹⁵	1.2 × 10⁻²
Venus	6 052 000	3.257 × 10¹⁴	2.7 × 10⁻⁵

The planet

Equatorial radius ( ), m

Gravitational parameter ( ), m3/s2

Zonal harmonic coefficients ( )

The Earth

6 378 136.3

3.986004415 × 10¹⁴

[1.0826269 × 10⁻³, -2.5323 × 10⁻⁶, -1.6204 × 10⁻⁶]

Jupiter

71 492 000

1.268 × 10¹⁷

[1.475 × 10⁻², 0, -5.8 × 10⁻⁴]

Mars

3 397 200

4.305 × 10¹³

[1.964 × 10⁻³, 3.6 × 10⁻⁵]

Mercury

2 439 000

2.2032 × 10¹³

6.0 × 10⁻⁵

The moon

1 738 000

4.902799 × 10⁹

2.027 × 10⁻⁴

Neptune

24 764 000

6.809 × 10¹⁵

4.0 × 10⁻³

Saturn

60 268 000

3.794 × 10¹⁶

[1.645 × 10⁻², 0, -1.0 × 10⁻³]

Uranium

25 559 000

5.794 × 10¹⁵

1.2 × 10⁻²

Venus

6 052 000

3.257 × 10¹⁴

2.7 × 10⁻⁵

The block does not take into account the influence of centrifugal forces arising from the rotation of the planet, as well as the influence of the precession of the reference frame.

Ports

Input

# x_ecef (m) — coordinates in the Greenwich rectangular coordinate system
`matrix 3 by m

Details

Coordinates in the Greenwich rectangular coordinate system.

Data types	`Float64`.
Complex numbers support	No

Output

# g_ecef (m/s²) — gravity values
`matrix 3 by m

Details

Gravity values returned in the Greenwich rectangular coordinate system.

Data types	`Float64`.
Complex numbers support	No

Parameters

Main

# Degree — harmonic order
4 | 3 | 2

Details

The order of the harmonic model:

2 is the second order, , the most significant and weighty term of the spherical harmonic, which takes into account the flattening of the planet;
3 - third order, ;
4 is the fourth order, .

Values	`4` \| `3` \| `2`
Default value	`4`
Program usage name	`degree_str`
Tunable	No
Evaluatable	Yes

Planet

Details

Planetary model.

Values	`Mercury` \| `Venus` \| `Earth` \| `Moon` \| `Mars` \| `Jupiter` \| `Saturn` \| `Uranus` \| `Neptune` \| `Custom`
Default value	`Earth`
Program usage name	`planet_model`
Tunable	No
Evaluatable	Yes

# Equatorial radius — equatorial radius

Details

The equatorial radius of a planet.

Dependencies

To use this parameter, set the parameter Planet model to Custom.

Default value	`6.3781363e6`
Program usage name	`equatorial_radius`
Tunable	No
Evaluatable	Yes

# Gravitational parameter — gravity parameter

Details

Planetary gravitational parameters.

Dependencies

To use this parameter, set the parameter Planet model to Custom.

Default value	`398600441500000`
Program usage name	`gm`
Tunable	No
Evaluatable	Yes

# J values — zonal harmonic coefficients

Details

Zone harmonic coefficients.

Dependencies

To use this parameter, set the Planet model parameters to Custom.

Default value	`[0.0010826269 -2.5323e-6 -1.6204e-6]`
Program usage name	`j_values`
Tunable	No
Evaluatable	Yes

Literature

Vallado, David, Fundamentals of Astrodynamics and Applications. New York: McGraw-Hill, 1997.
Fortescue, P., J. Stark, G. Swinerd, eds. Spacecraft Systems Engineering, 3d ed. West Sussex: Wiley & Sons, 2003.
Tewari, A. Atmospheric and Space Flight Dynamics Modeling and Simulation with MATLAB and Simulink. Boston: Birkhäuser, 2007.