= Solution
For a <test particle> outside a spherical central <mass>, the gravitational <force> is central, so <angular momentum> is conserved and the orbit is planar. With <specific angular momentum> $h=r^2\dot\theta$ and <specific orbital energy> $\epsilon=\dot r^2/2+h^2/(2r^2)-GM/r$, the radial <effective potential> combines attraction with the <centrifugal barrier>. Its nonradial solutions are <conic sections>:
$$
\boxed{r(\theta)=\frac{p}{1+e\cos(\theta-\theta_0)},\qquad p=\frac{h^2}{GM},\quad e^2=1+\frac{2\epsilon h^2}{(GM)^2}.}
$$
For nonzero $h$, negative <energy> gives an <ellipse> with $0\le e<1$, zero <energy> a <parabolic Kepler orbit> and positive <energy> a <hyperbolic Kepler orbit>. The <circle> is the special case $e=0$. Exactly radial motion is a separate zero-angular-momentum case and may hit the central body; a mathematical point-mass trajectory is not automatically a physically noncolliding orbit.
Conservation of $h$ gives constant <areal velocity> $h/2$, so equal <areas> are swept in equal times. For a bound <ellipse> of <semimajor axis> $a$, $\epsilon=-GM/(2a)$ and $T^2=4\pi^2a^3/(GM)$. A <circular orbit> has <speed> $\sqrt{GM/r}$, while the minimum local <escape speed> is $\sqrt{2GM/r}$, obtained by setting the total <energy> to zero. These are the test-mass two-body results: a comparable second <mass> requires the relative-coordinate <gravitational parameter> $G(M+m)$, and nonspherical structure or drag can perturb the conic and its conserved quantities.
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