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In a short time $dt$, the radius vector sweeps an oriented area $d\mathcal{A}=\frac{1}{2}{r}^{2}d\theta$. ThusThe constant $h$ is twice the oriented areal velocity, or the angular momentum per unit mass. Its constancy is Kepler's second law: equal areas are swept in equal times.

For a circular orbit of radius $a$, set $\ddot{r}=0$ in the radial equation. The balance isHence a circular orbit exists for any $a>0$ with this nonzero $|h|$. Its period gives Kepler's third law:To check circular-orbit stability for a power-law central force directly, keep $h$ fixed and set $r=a+\rho$. Expandingabout the balanced radius givesThe linear radial perturbation therefore obeysIt remains bounded and oscillates, proving stability under the stipulated small perturbations. Equivalently, the specific effective potential $h^2/(2r^2)-GM/r$ has a strict minimum at $a$.

In a circular orbit, $r=a$ is constant, so the radial acceleration vanishes. The radial equation requiresFor every $a>0$, this equation permits a real nonzero specific angular momentum, $h=\pm \sqrt{k{a}^{n+3}}$, and hence a circular orbit. This chooses $h$ for the desired radius; it does not claim that one prescribed value of $h$ permits every radius.

To test circular-orbit stability for a power-law central force, keep that chosen specific angular momentum fixed and write $r=a+\rho$, with $|\rho |\ll a$. The radial equation becomesExpanding to first order and using $h^2=k{a}^{n+3}$ givesFor $n>-3$, the linearized equation is the harmonic oscillator equation, withThe radial disturbance undergoes bounded small oscillations, rather than growing. Equivalently, the effective potential has a strict local minimum: its second derivative at $a$ is $(n+3)k{a}^{n-1}>0$. By the effective potential stability criterion, sufficiently small radial disturbances remain small, proving stability for $n>-3$. For comparison, $n<-3$ gives exponential growth. At $n=-3$, the balancing value $h^2=k$ makes the radial acceleration identically zero, so a small radial velocity produces drift rather than a restoring oscillation.

Write $r=r_0+\eta$ and keep $L$ fixed. The small radial oscillations in a power-law potential have squared angular frequencyThe circular orbital angular frequency has magnitude $\Omega =|L|/(m{r}_0^2)$, so ${\Omega }^2=(\beta k/m)r_0^{k-2}$. HenceA restoring radial oscillation exists for $k>-2$, subject to the attractive-force condition in part (b). For $k<-2$ the radial stationary point is unstable; $k=-2$ is the degenerate boundary. This agrees with circular-orbit stability for a power-law central force, since the actual force is proportional to $-r^{k-1}$.