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 Quote by twofish-quant It has *everything* to do with thermodynamics and negative heat capacities.
If internal energy U of the satellite is held constant and only the mechanical energy (K+V) is varied, then you do not need thermodynamics, but only mechanics. Indeed, the laws of mechanics alone explains, perfectly, why the satteliite modifies its orbit and falles when you take away kinetic energy.

It is evident that you do not need thermodynamics for this. Gibbs and heat capacities are not even mentioned.

 Quote by twofish-quant Once you have enough satellites, it becomes a gas, and once you have a gas, you can do thermodynamics on it. The system of earth and satellites is a closed thermodynamic system and you can define temperatures and heat capacities.
Taking an enough collection of satellites does not magically convert a mechanical problem into a thermodynamic one. Moreover, even assuming that the system of satellites can be considered a thermodynamic system, still there is none need to confound internal energy U with total energy E (including potential energy due to Earth) as done in previous posts. Similar remarks about stars.

The equation

$$\frac{\partial u}{\partial t} = c_V \frac{\partial T}{\partial t} + u_k \frac{\partial n_k}{\partial t}$$

in presence of field with potential $\psi$ changes to

$$\frac{\partial u}{\partial t} = c_V \frac{\partial T}{\partial t} + (u_k + \tau_k \psi) \frac{\partial n_k}{\partial t}$$

with $\tau_k$ the coupling factor. The heat capacity $c_V$ is the same, in presence or absence of field, this is well-know, at least for thermodynamicians.

Of course if you want to confound rest energy with internal energy U and with total energy E, if you pretend that a black hole is a closed thermodynamic system, etc. then you can obtain anything that you want, but that is 'astrophysics', not thermodynamics.