Applying the <Poisson equation for Newtonian gravity> to a <flattened power-law gravitational potential> yields
$$
\rho=-\frac{\beta\Phi_0R_0^\beta}{4\pi G}\frac{(\lambda^2-\beta)R^2+\lambda^2[2-(\beta+1)\lambda^2]z^2}{(R^2+\lambda^2z^2)^{(\beta+4)/2}}.
$$
For $0<\beta<1$, nonnegative <mass density> everywhere outside the origin requires and is ensured by $\beta\leq\lambda^2\leq2/(\beta+1)$. The central density singularity is locally integrable and carries no point mass, although the total mass diverges at large radius in this scale-free model.

Density positivity for a flattened power-law potential

{title2=$\alpha_{\rm crit}=\arctan(|\lambda|/\sqrt{\beta+2})$}

In a <flattened power-law gravitational potential> $\Phi=\Phi_0R_0^\beta(R^2+\lambda^2z^2)^{-\beta/2}$, a magnetic surface corotating with its disk footpoint has an <effective potential> with meridional <Hessian matrix> $\Omega_f^2\operatorname{diag}[-(\beta+2),\lambda^2]$ there. A cold outward displacement along a field inclined by $\alpha$ to the vertical is downhill when $\tan^2\alpha>\lambda^2/(\beta+2)$. Equality is marginal and can depend on higher derivatives and field-line curvature; this is a local launching criterion, not a guarantee of a global escaping wind.

Magnetocentrifugal launching criterion in a flattened power-law potential

Set $A=\Phi_0R_0^\beta<0$, $q=\lambda^2$ and $s=R^2+qz^2$. The <flattened power-law gravitational potential> is $\Phi=As^{-\beta/2}$. In axisymmetric <cylindrical coordinates>, the <Poisson equation for Newtonian gravity> gives
$$
4\pi G\rho=\frac1R\partial_R(R\partial_R\Phi)+\partial_z^2\Phi.
$$
The first derivatives are $\partial_R\Phi=-\beta ARs^{-(\beta+2)/2}$ and $\partial_z\Phi=-\beta Aqzs^{-(\beta+2)/2}$. Differentiating again and collecting powers gives
$$
\nabla^2\Phi=-\beta As^{-(\beta+4)/2}\left[(2+q)s-(\beta+2)(R^2+q^2z^2)\right].
$$
Therefore the required <mass density> is
$$
\boxed{\rho(R,z)=-\frac{\beta\Phi_0R_0^\beta}{4\pi G}\frac{(\lambda^2-\beta)R^2+\lambda^2[2-(\beta+1)\lambda^2]z^2}{(R^2+\lambda^2z^2)^{(\beta+4)/2}}.}
$$
This is a formal density for arbitrary $\lambda$, but a physical <dark matter> distribution must be nonnegative. The <density positivity for a flattened power-law potential> condition is
$$
\boxed{\beta\leq\lambda^2\leq\frac2{\beta+1}.}
$$
Necessity follows by evaluating on the midplane and symmetry axis; sufficiency follows because both numerator coefficients are then nonnegative. In this range the origin is a locally integrable density cusp: the mass enclosed near radius $r$ scales as $r^{1-\beta}$ and tends to zero, so no point mass needs adding. The scale-free distribution has infinite total mass at large radius and represents an idealized background, not a finite isolated halo.

For the cold, non-self-gravitating disk, radial balance of a <circular orbit> is $R\Omega^2=\partial_R\Phi(R,0)$. Thus
$$
\boxed{\Omega^2(R)=-\beta\Phi_0R_0^\beta R^{-\beta-2},\qquad\Omega(R)=\frac{\sqrt{-\beta\Phi_0}}{R_0}\left(\frac R{R_0}\right)^{-(\beta+2)/2}.}
$$
The displayed positive root chooses the rotation orientation; the opposite orientation has the negative of this angular frequency.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 314 / 3 / a / Solution

{title2=$\Phi=\Phi_0[R_0/(R^2+\lambda^2z^2)^{1/2}]^\beta$}

A flattened power-law <Newtonian gravitational potential> has equipotential spheroids $R^2+\lambda^2z^2=\text{constant}$. For $\Phi_0<0$ and $0<\beta<1$, its midplane <circular orbit> angular frequency satisfies $\Omega^2=-\beta\Phi_0R_0^\beta R^{-\beta-2}$.

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 Incoming links: Flattened power-law gravitational potential