# Change in energy during orbital decay

Suppose we have a planet orbiting a star and the planet radiates away some amount of energy $\Delta E$. I want to find by how much the orbit decays (i.e the change in orbital radius).

One way to do this is to find the total energy as a function of radius $E(r)$ (it's equal to half the gravitational potential energy $U(r)$ for circular orbits) and set $E(r_f)-E(r_0)=\Delta E$ and solve for $r_f$ in terms of $r_0$ to find $\Delta r$.

I'm not sure if this is the best way possible (but I think at least it's valid). I also want to know if I can do it without knowing the total energy $E$ (that is, just by using the potential energy).

## 1 Answer

By talking about orbital radius you're assuming the orbit is circular, but that may not be the case. In general you need both energy and angular momentum to specify your orbit, and to solve completely you'd need to know the specific mechanism of energy loss to find out what happens to the angular momentum.

You can get around this if you're willing to use the semi-major axis $a$ instead of the radius. If you know the energy $E$ you can relate them by $$a = \frac{GMm}{2|E|}$$

Differentiating we find that

$$da = \frac{GMm}{2E^2}dE = \frac{2a^2}{GMm}dE$$

Which gives the change in semi-major axis for small changes in energy.

I don't know what you mean by doing this only knowing the potential energy. The solution to the Kepler problem is known, so you don't have to work out all the formulas. You only care about the total energy radiated away.