Engee documentation

Zonal Harmonic Gravity Model

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blockType: SubSystem

zonal harmonic gravity model

Path in the library:

/Aerospace/Environment/Gravity/Zonal Harmonic Gravity Model

Description

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

Planet Equatorial radius ( ), m Gravitational parameters ( ), m³/s² Zonal harmonic coefficients ( )

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

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]

Uranus

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 effect of centrifugal forces arising from the rotation of the planet, as well as the effect of the precession of the reference frame.

Ports

Input

# xecef (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

# gecef (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

# Planet model — planet
Mercury | Venus | Earth | Moon | Mars | Jupiter | Saturn | Uranus | Neptune | Custom

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

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

  2. Fortescue, P., J. Stark, G. Swinerd, eds. Spacecraft Systems Engineering, 3d ed. West Sussex: Wiley & Sons, 2003.

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