= Solution
For a test <mass> moving around a fixed central <mass> $M$, <gravity> is central and conservative. <Specific angular momentum> $h=|\mathbf r\times\mathbf v|$ and <specific orbital energy> $\epsilon=v^2/2-GM/r$ are constant. The motion is planar and sweeps equal areas in equal times, $dA/dt=h/2$.
For a nonradial orbit, the inverse-square <force> gives a <conic section>,
$$
\boxed{r=\frac{h^2/(GM)}{1+e\cos\theta},\qquad
e^2=1+\frac{2\epsilon h^2}{(GM)^2}.}
$$
Negative <energy> gives an <ellipse>, zero <energy> a <parabola> and positive <energy> a <hyperbola>. For an <ellipse> of <semimajor axis> $a$,
$$
\boxed{\epsilon=-\frac{GM}{2a},\qquad
P^2=\frac{4\pi^2a^3}{GM},\qquad
v^2=GM\left(\frac2r-\frac1a\right).}
$$
A <circular orbit> is the $e=0$ case, with $v=\sqrt{GM/r}$. Escape from radius $r$ requires $v\ge\sqrt{2GM/r}$ in the absence of further <forces>. For two comparable <masses>, apply the relative two-body equation with $GM$ replaced by $G(M+m)$, rather than treating one body as fixed.
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