Geosynchronous station keeping

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In summary, the person is having difficulty figuring out geosynchronous station keeping requirements due to triaxiality. They found the equilibrium points by finding where angular acceleration equaled zero and then used an equation from Fundamentals of Astrodynamics and Applications by Vallado to find the acceleration necessary to stay in the same place for other longitudes. However, the given equation doesn't produce a realistic maximum, and another perspective from Wertz and Larsen suggests that the change in velocity is actually: \dot V=r\dot\omega + \dot{r} \omega.
  • #1
BobG
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I'm having problems figuring out geosynchronous station keeping requirements due to triaxiality.

So far, I've gotten as far as finding the equilibrium points at about 75 degrees and 255 degrees, but I can't get from there to the station keeping requirements for satellites located at other longitudes. The angular, or longitudinal, accleration should equal:

[tex]\ddot\lambda = -\left(\omega^2_E \left(\frac{R_E}{a}\right)^2\right) \left(-18J_{22} sin(2(\lambda - \lambda_{22}))\right)[/tex]

[tex]\lambda[/tex] is longitude with [tex]\lambda_{22}[/tex] being a constant that goes along with [tex]J_{22}[/tex] for one of the Earth's spherical harmonics due to triaxiality.
[tex]\omega_E[/tex] is the rotation rate of the Earth, [tex]R_E[/tex] is the radius of the Earth, and a is the orbit's semi-major axis.

There's another variation of this I found in NASA's TM-2001-210854, Integrated Orbit, Attitude, and Structural Control Systems Design for Space Solar Power Satellites, but it yields the same results. They just merged the mean motion into the rest of the equation. Since the whole purpose of geosynchronous satellites is for the orbit's mean motion to match the Earth's rotation rate, I like the version where its separate, better.

I found the equilibrium points by finding where angular acceleration equaled zero. If I convert this to linear acceleration, I think I should get the acceleration necessary to stay in the same place for other longitudes. If projected over the course of one year, I get a maximum of around 5.203 meters/second/year, which is about 3 times too big.

Wertz and Larsen's Space Mission Analysis and Design just use the equation:

[tex]\deltaV = 1.715 sin(2(\lambda - \lambda_s))[/tex]

Their equation does produce a realistic maximum. Unfortunately, 1.715 doesn't tell me anything.

Anyone know the missing link, here?
 
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  • #2
Bob, I wish I could offer more help, but when this topic was covered in my space nav class, it went right over my head.

IIRC, Fundamentals of Astrodynamics and Applications by Vallado had a brief section on this. If they've got it in the library, it may be worth a look.
 
  • #3
Something like this:

The potential of the '22' part of the gravitational potential is written out as:

[tex]U_{22}=\frac{GM}{r} \frac{Re^2}{r^2} P_{22} ( \sin \phi) J_{22} \cos 2 ( \lamda - \lambda_{22} )[/tex]

with the associated Legendre polynomial:

[tex]P_{22} ( \sin \phi ) = 3 \cos^2 \phi [/tex]
equals 3 for equatorial orbit.

Getting the acceleration in lambda-direction (alongtrack) means differentiating once wrt r and once wrt lambda:

[tex]a_{\lambda} = \frac{\partial^2 U}{\partial r \partial \lambda} = -3 \left( \frac{GM}{r} \frac{Re}{r^3} \right) \cdot 3 J_{22} \left( -2 \sin 2 ( \lambda - \lambda_{22} ) \right) [/tex]

equating:

[tex] a_{\lambda} = r \ddot{\lambda} [/tex]

filling in [tex]r=a[/tex]

and noting that [tex]\omega_e = \sqrt{ \frac{GM}{a^3} } [/tex] by definition of the geostationary orbit (orbital rotational velocity equals Earth rotational velocity) gives the given equation.

edit - I'm not sure about this (the exact reason concerning the double differentiation and the definition of the acceleration). Have to look it up. But at least this gives the given equation :)
 
Last edited:
  • #4
This gets me to my original equation, but reading another perspective gives me a lot of help. I was kind of looking at it as if, by constantly accelerating (or decelerating) to compensate for the longitudinal drift, that I would hold the radius constant. True enough, but that doesn't mean that the acceleration doesn't have a radial component.

In other words, the change in velocity is actually:

[tex]\dot V=r\dot\omega + \dot{r} \omega[/tex]

And [tex]\dot\omega[/tex] and [tex]\dot{r}[/tex] are always changing in opposite directions. I'll have to play with it a little and see if it works.

(I need to get a better reference for this, as well. The ones that offer a good explanation came up with same numbers I did and the ones with realistic numbers offer no explanation at all.)
 

1. What is geosynchronous station keeping?

Geosynchronous station keeping is a process used to maintain the position of a satellite in geostationary orbit. This involves making small adjustments to the satellite's orbit to counteract the effects of gravity and other forces that can cause it to drift out of position.

2. Why is geosynchronous station keeping necessary?

Geosynchronous station keeping is necessary to ensure that the satellite remains in its designated orbital slot. This is important for maintaining communication and navigation services, as well as for scientific observations and other functions.

3. How does geosynchronous station keeping work?

Geosynchronous station keeping typically involves the use of thrusters or reaction wheels on the satellite to make small adjustments to its orbit. These adjustments are calculated based on precise measurements of the satellite's position and the forces acting on it.

4. What challenges are involved in geosynchronous station keeping?

One of the main challenges in geosynchronous station keeping is the limited amount of fuel that can be carried by the satellite. This means that careful planning and efficient use of fuel are essential to ensure that the satellite can maintain its orbit for as long as possible.

5. How often does geosynchronous station keeping need to be done?

The frequency of geosynchronous station keeping depends on the specific satellite and its orbital characteristics. In general, adjustments need to be made every few days or weeks to counteract the effects of atmospheric drag and other forces that can cause the satellite to drift out of position. However, some satellites may require more frequent adjustments due to their orbit or mission requirements.

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