= Solution
For small isentropic perturbations of a fixed-composition <gas>, linearizing its <equation of state> gives $p_1=(\partial p/\partial\rho)_s\rho_1=c_s^2\rho_1$. The <adiabatic sound speed> is evaluated at the uniform background, so it is spatially constant at this order. If fast heat exchange instead enforces constant <temperature>, use the <isothermal sound speed> consistently. A completely arbitrary <entropy> perturbation would require an additional thermodynamic variable.
The uniform static infinite medium requires the <Jeans swindle>: its background gravitational field is subtracted or externally balanced, and only the perturbation sources the <Poisson equation for Newtonian gravity>. Linearizing the <continuity equation>, <Euler equation> and gravitational equation then gives
$$
\partial_t\rho_1+\rho_0\nabla\cdot\mathbf u_1=0,\qquad
\partial_t\mathbf u_1=-\frac{c_s^2}{\rho_0}\nabla\rho_1-\nabla\Psi_1,\qquad
\nabla^2\Psi_1=4\pi G\rho_1.
$$
Taking one more time derivative of the first equation and substituting the other two yields
$$
\partial_t^2\rho_1=c_s^2\nabla^2\rho_1+4\pi G\rho_0\rho_1.
$$
A <Fourier mode> proportional to $e^{i(\mathbf k\cdot\mathbf x-\omega t)}$ therefore obeys the <Jeans instability> relation
$$
\boxed{\omega^2=c_s^2k^2-4\pi G\rho_0
=c_s^2(k^2-k_J^2),\qquad k_J=\frac{\sqrt{4\pi G\rho_0}}{c_s}.}
$$
For $k>k_J$, the mode oscillates as a <sound wave>. For $0<k<k_J$, $\omega=\pm i\sqrt{4\pi G\rho_0-c_s^2k^2}$ gives a growing and a decaying mode; the <static Jeans growth-rate spectrum> has no preferred finite fastest-growing wavelength in the infinite-medium approximation. The threshold $k=k_J$ is marginal at linear order. The <Jeans length> is $\lambda_J=2\pi/k_J=c_s\sqrt{\pi/(G\rho_0)}$. Defining the <Jeans mass> as a sphere whose diameter is this wavelength gives
$$
\boxed{M_J=\frac{4\pi}{3}\rho_0\left(\frac{\lambda_J}{2}\right)^3
=\frac{\pi^{5/2}}6\frac{c_s^3}{G^{3/2}\rho_0^{1/2}}.}
$$
Other radius conventions alter its numerical coefficient; the <mass> scale compares <self-gravity> with <pressure> communication.
For the pressureless finite cloud, the <shell theorem> gives an edge acceleration $\ddot r=-GM/r^2$, with fixed enclosed $M=4\pi r_0^3\rho_0/3$. Multiplication by $\dot r$ and integration from rest gives
$$
\frac{\dot r^2}{2}=GM\left(\frac1r-\frac1{r_0}\right),\qquad
\dot r=-\sqrt{2GM}\left(\frac1r-\frac1{r_0}\right)^{1/2}.
$$
The negative root describes collapse. Substituting $r=r_0\sin^2\theta$ gives the <free-fall time of a uniform sphere>
$$
t_{\rm ff}=\frac{2r_0^{3/2}}{\sqrt{2GM}}\int_0^{\pi/2}\sin^2\theta\,d\theta
=\frac{\pi r_0^{3/2}}{2\sqrt{2GM}}
=\boxed{\sqrt{\frac{3\pi}{32G\rho_0}}}.
$$
The surface and inner shells reach the origin together in <pressureless homologous spherical collapse>; <pressure>, rotation or shell crossing would change this ideal solution.
Equating this time to the specified radius-based <sound-crossing time> $r_0/c_s$ gives $r_0=c_s\sqrt{3\pi/(32G\rho_0)}$. The resulting <crossing-time cloud support mass> is
$$
\boxed{M_{\rm cross}=C_{\rm cross}\frac{c_s^3}{G^{3/2}\rho_0^{1/2}},\qquad
C_{\rm cross}=\frac{4\pi}{3}\left(\frac{3\pi}{32}\right)^{3/2}.}
$$
This is the same dimensional <mass> scale as $M_J$, with a different coefficient because the adopted length and time criteria differ. In particular, $t_{\rm ff}=r_0/c_s$ is not the exact marginal dispersion-relation condition.
With the requested isotropic support approximation, replace $c_s$ by the <Alfvén speed> $v_A=B/\sqrt{\mu_0\rho_0}$. In the crossing-time convention the <Magnetic Jeans mass> is
$$
\boxed{M_{B,{\rm cross}}=C_{\rm cross}\frac{v_A^3}{G^{3/2}\rho_0^{1/2}}
=C_{\rm cross}\frac{B^3}{(\mu_0G)^{3/2}\rho_0^2}.}
$$
Using the wavelength-based <Jeans mass> convention instead replaces $C_{\rm cross}$ by $\pi^{5/2}/6$. The scaling, rather than a convention-independent coefficient, is the shared prediction.
For a fixed cloud <mass> undergoing isothermal homologous contraction, $\rho/\rho_0=(r_0/r)^3$ and $c_s$ is constant. Its number of thermal <Jeans masses> therefore grows as
$$
\boxed{N_J(\rho)=N_{J,0}\left(\frac{\rho}{\rho_0}\right)^{1/2}
=N_{J,0}\left(\frac{r_0}{r}\right)^{3/2}.}
$$
A cloud initially close to the support threshold has $N_{J,0}$ of order one, so smaller pieces can subsequently become independently unstable: this is <isothermal fragmentation>.
For approximately isotropic contraction with <magnetic flux freezing>, $Br^2$ stays constant, so $B\propto\rho^{2/3}$. Hence $M_B\propto B^3/\rho^2$ stays constant and
$$
\boxed{N_B(\rho)=N_{B,0}.}
$$
This ideal magnetic support therefore does not create progressively more unstable masses in the way thermal isothermal support does. The same conclusion follows from the conserved <mass-to-flux ratio>. A magnetically subcritical cloud cannot initiate unrestricted cross-field collapse under ideal flux freezing alone; flux loss or initially supercritical conditions are needed. A real <magnetic field> also gives anisotropic support: motion and fragmentation along its field lines are not excluded by this scalar-speed estimate.
\Image[/past-exam-of-the-astrophysical-fluid-dynamics-course-of-the-university-of-cambridge/2014/paper-1-jeans-instability.png]
{title=Jeans dispersion relation and the changing number of support masses during idealized collapse}
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