I'm having problems figuring out geosynchronous station keeping requirements due to triaxiality.(adsbygoogle = window.adsbygoogle || []).push({});

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'sSpace Mission Analysis and Designjust 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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# Geosynchronous station keeping

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