= Solution
{c}
Put $r=r_0+\eta$, where $|\eta|\ll r_0$. The <Taylor expansion> at a <circular orbit> is
$$
V_{\mathrm{eff}}(r_0+\eta)=V_{\mathrm{eff}}(r_0)
+\frac12V_{\mathrm{eff}}''(r_0)\eta^2+O(\eta^3).
$$
The constant $V_{\mathrm{eff}}(r_0)$ has no effect on the <force>. The paper's displayed <harmonic potential> should be read as this expansion after subtracting that constant and neglecting higher orders; it is not an equality for the absolute <effective potential> in general. For example, when $m=J=\beta=1$ and $k=2$, $r_0=2^{-1/4}$ and $V_{\mathrm{eff}}(r_0)=\sqrt2$, whereas the printed quadratic expression is zero at $r_0$.
Differentiating again and then using $\beta k r_0^{k-2}=J^2/(mr_0^4)$ gives
$$
\begin{aligned}
V_{\mathrm{eff}}''(r_0)&=\beta k(k-1)r_0^{k-2}+\frac{3J^2}{mr_0^4}\\
&=(k+2)\frac{J^2}{mr_0^4}.
\end{aligned}
$$
The linearized radial <equation of motion> is $m\ddot\eta=-V_{\mathrm{eff}}''(r_0)\eta$, so the <radial frequency in a power-law central potential> is
$$
\boxed{\omega_r=\sqrt{k+2}\,\frac{|J|}{mr_0^2},\qquad k>-2.}
$$
For $k<-2$ the <circular orbit> is unstable and radial disturbances grow exponentially, with growth rate $\sqrt{-(k+2)}|J|/(mr_0^2)$. At $k=-2$ the linear restoring <force> vanishes, as described in part (c). Thus an attractive <central force> alone does not guarantee <small oscillations>.
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