= Circular orbits in an exponential central potential
{title2=$l^2=r^3e^{-r}$}
For unit mass with specific <angular momentum> $l$ in $V(r)=-e^{-r}$, the <effective potential> is $-e^{-r}+l^2/(2r^2)$. A positive-radius <circular orbit> satisfies the displayed equation. Since $r^3e^{-r}$ has its unique maximum $27/e^3$ at $r=3$, exactly two radii occur for $0<l^2<27/e^3$. Their radial stability follows from $V_{\rm eff}''=e^{-r}(3/r-1)$: the inner orbit is stable at fixed <angular momentum>, the outer unstable. The case $l=0$ has no finite-radius <circular orbit> despite satisfying the upper bound.
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