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I am considering the motion of two satellites around a Protostar. I need an equation to define the change in orbital radius given a situation where angular momentum is conserved but energy is lost.

My Understanding: the changes will have opposite signs

My Idea

I define the velocity and the total energy of a satellite of mass $m$ in a near-circular orbit of radius $r$ around a Protostar of mass $M$ are

$$v = \sqrt{\frac{GM}{r}}$$ and $$E = -G\frac{mM}{2r}$$

I beleive the total angular momentum of the system would be $$L = m\sqrt{GM}\left(\sqrt{r_{1}}+\sqrt{r_{2}}\right)$$

And the total energy of the system would be

$$E = -G\frac{mM}{2}\left(\frac{1}{\sqrt{r_1}}+\frac{1}{\sqrt{r_2}}\right)$$

By losing energy does $E\rightarrow 0$?

Edit $r_1$ is the radius of satellite $1$, $r_2$ is the radius of satellite $2$,

David
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  • Do you realize that the momentum is a vector and energy is a scalar? Both energy and angular momentum are always conserved. See https://physics.stackexchange.com/questions/92051/how-can-momentum-but-not-energy-be-conserved-in-an-inelastic-collision – Amirhossein Rezaei Jan 15 '21 at 18:21
  • In this case the energy of the rotating system is said to be lost – David Jan 15 '21 at 20:05

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