Equation to define the change in orbital radius given a situation where angular momentum is conserved but energy is lost

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 $$E = -Gfrac{mM}{2r}$$

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

And the total energy of the system would be

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

By losing energy does $Erightarrow 0$?

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