Isotropic stellar mass loss ($\ddot{\mathbf{r}}=-\mu (t)\mathbf{r}/r^3$)

Isotropic stellar mass loss changes the central gravitational parameter $\mu (t)$ while preserving the central direction of the force, provided the escaping material causes no recoil or drag on the orbiting body. The relative equation is $\ddot{\mathbf{r}}=-\mu (t)\mathbf{r}/r^3$. Specific angular momentum remains exactly constant; specific orbital energy obeys $\dot{C}=-\dot{\mu }/r$. Orbital expansion is adiabatic only when the mass-loss law varies slowly on the orbital period.

Suppose an initially circular Kepler orbit of radius $R$ surrounds central mass $M$, and a fraction $0<f<1/2$ is removed quickly compared with the orbital period. Assume no recoil or wind drag, so the satellite's position and velocity are continuous through the change. Its specific angular momentum stays $\sqrt{GMR}$, while its new specific orbital energy is $-GM(1-2f)/(2R)$. The resulting elliptic Kepler orbit has semi-major axis $a=(1-f)R/(1-2f)$, pericentre $r_p=R$, apocentre $r_a=R/(1-2f)$ and orbital eccentricity $e=f/(1-f)$. The limiting loss fraction $f=1/2$ gives a parabolic Kepler orbit; larger losses unbind the initially circular satellite. Slow removal instead produces adiabatic orbital expansion under isotropic mass loss, which preserves circularity to leading adiabatic order.

Time averaging over a Kepler orbit gives $⟨\cos f⟩=-e$, as follows from $dt=(1-e\cos E)dE/n$ and $\cos f=(\cos E-e)/(1-e\cos E)$. Consequently slow isotropic stellar mass loss preserves the secular orbital eccentricity. Conserved specific angular momentum then gives $\mu a=\mathrm{constant}$, so pericentre, apocentre and semimajor axis expand inversely with the remaining gravitational parameter.

The signed instantaneous contribution of apsidal precession to azimuthal motion during isotropic stellar mass loss is $\dot{\varpi }/\dot{\theta }=-(\dot{\mu }/\mu n)(1-e^2{)}^{3/2}\sin f/[e(1+e\cos f{)}^2]$. Its maximum absolute magnitude is small when $n|\mu /\dot{\mu }|\gg F(e)$, where $F(e)\sim 1/e$ near a circular orbit and $F(e)\to 3\sqrt{3}/4$ as $e\to 1$. The low-orbital eccentricity coordinate singularity can be avoided by evolving the eccentricity vector.

For isotropic stellar mass loss the eccentricity vector obeys $\dot{\mathbf{e}}=-(\dot{\mu }/\mu )(\mathbf{e}+\hat{\mathbf{r}})$. With true anomaly $f$ and $e>0$, this gives $\dot{e}=-(\dot{\mu }/\mu )(e+\cos f)$ and $\dot{\varpi }=-(\dot{\mu }/\mu )\sin f/e$. The osculating pericentre satisfies $\dot{q}/q=-(\dot{\mu }/\mu )(1-\cos f)/(1+e)$, hence cannot decrease during mass loss and is instantaneously stationary at pericentre. The vector form is nonsingular at a circular orbit.