Acceleration of a Satellite in General Relativity

[Matter_Matters]

Hi there guys,

I'm struggling! I've been looking at the International Earth Rotation Services (IERS) "standards" for motion of a satellite in GR. the expression is far from trivial and I'm battling to determine where to even start with this bad boy.

The expression is given by

[tex]
\Delta \ddot{\mathbf{r}} = \frac{GM_E}{c^2r^3} \left\{\left[2(\beta+\gamma)\frac{GM_E}{r} - \gamma \dot{\mathbf{r}} \cdot \dot{\mathbf{r}} \right] \mathbf{r} + 2(1+\gamma)(\mathbf{r}\cdot\dot{\mathbf{r}})\dot{\mathbf{r}} \right\} + (1+\gamma)\frac{GM_E}{c^2r^3} \left[ \frac{3}{r^2}(\mathbf{r}\times\dot{\mathbf{r}})(\mathbf{r}\cdot\mathbf{J})+(\dot{\mathbf{r}}\times \mathbf{J})\right] + \left\{ (1+2\gamma)\left[\dot{\mathbf{R}} \times \left( \frac{-GM_S \mathbf{R}}{c^2R^3} \right) \right] \times\dot{\mathbf{r}} \right\}.
[/tex]

The terms in the expression correspond to the following:
[itex]c = [/itex] speed of light.
[itex] \beta, \gamma [/itex] = PPN (parameterized post-Newtonian) parameters, equal to 1 in General Relativity.
[itex] \mathbf{r} [/itex] is the position of the satellite with respect to the Earth.
[itex] \mathbf{R} [/itex] is the position of the Earth with respect to the Sun.
[itex] J [/itex] is the Earth’s angular momentum per unit mass.
[itex] GM_E [/itex] and [itex] GM_S [/itex] are the gravitational coefficients of the Earth and Sun, respectively.

Now, obviously nobody in their right mind is going to know how to derive this monster off the top of their heads, unless of course you wrote the technical note, BUT, does anyone have experience with Relativistic mechanics of satellite is geocentric reference frame or barycentric for that matter?

 

[pervect]

I don't know if this will help, but the approach that comes to mind is to first find what metric the IERS is using for the Earth-sun-satellite system. The next step is to say that the satellite is following a geodesic of said metric, and write the geodesic equations.

To "simplify" things, it looks like they're using 3-vector notation rather than tensors, as evidenced by the use of cross products, which are only defined in 3 dimensions. These are presumably there to account for the effects of the Earths orbital and rotational angular momentum on the metric.

Presumably they're using some variant of the PPN metric, it rather looks like the same version MTW uses in Gravitation, due to the presence of beta's and gamma's. There are a couple of different versions of the PPN metric out there (at least according to Wikipedia.) It doesn't look (to me) like they're using the same symbolism as the IAU recommends for their solar system metric in the IAU 2000 resolutions, https://syrte.obspm.fr/IAU_resolutions/Resol-UAI.htm, or the 2006 ammendments of the resolutions (which I don't have a link to).

I got a link error when I tried to look at the technical note, I'll try again.

 

[Matter_Matters]

I don't know if this will help, but the approach that comes to mind is to first find what metric the IERS is using for the Earth-sun-satellite system. The next step is to say that the satellite is following a geodesic of said metric, and write the geodesic equations.

So my very limited knowledge is that they are following the Schwarzschild metric including both Lens-Thirring and DeSitter effects. It is quite confusing because the documentation is a derivative with respect to coordinate time also and not proper time, like one would expect in relativity.

To "simplify" things, it looks like they're using 3-vector notation rather than tensors, as evidenced by the use of cross products, which are only defined in 3 dimensions. These are presumably there to account for the effects of the Earths orbital and rotational angular momentum on the metric.

I think you are right as these terms correspond to frame dragging and precession.

Presumably they're using some variant of the PPN metric, it rather looks like the same version MTW uses in Gravitation, due to the presence of beta's and gamma's. There are a couple of different versions of the PPN metric out there (at least according to Wikipedia.) It doesn't look (to me) like they're using the same symbolism as the IAU recommends for their solar system metric in the IAU 2000 resolutions, https://syrte.obspm.fr/IAU_resolutions/Resol-UAI.htm, or the 2006 ammendments of the resolutions (which I don't have a link to).

I got a link error when I tried to look at the technical note, I'll try again.

My intuition is saying that it is some PPN approximation of the Schwarzschild metric. I've updated the link, hopefully it works now. Thanks for the reply.

 

[pervect]

There's more than just the Schwarzschild metric in there, as they have effects from the Earth and the Sun included. The Schwarzschild metric would include effects from only one dominant mass.

The PPN formalsim can handle that (approximately) though.

 

[Matter_Matters]

There's more than just the Schwarzschild metric in there, as they have effects from the Earth and the Sun included. The Schwarzschild metric would include effects from only one dominant mass.

The PPN formalsim can handle that (approximately) though.

I think it's time to start reading up and giving some proper attention to the PPN formalism! Cheers.