Jeans mass ($M_J$)

The Jeans mass is the characteristic gas mass above which self-gravity overcomes pressure support. Up to the convention used to define the collapsing region,for an ideal gas of fixed composition.

Replacing the sound speed in a Jeans mass estimate by the Alfvén speed $B/\sqrt{{\mu }_0\rho }$ gives the displayed magnetic-support mass scale. A sphere whose diameter is the corresponding wavelength gives $C={\pi }^{5/2}/6$; the crossing-time cloud support mass instead gives $C=(4\pi /3)(3\pi /32{)}^{3/2}$. The coefficient is model dependent, and a real magnetic field supplies anisotropic stresses rather than isotropic gas pressure.

During isotropic magnetic flux freezing, $B\propto {\rho }^{2/3}$, so $M_B$ and the number of such masses in a fixed material cloud remain constant. This is another expression of a conserved mass-to-flux ratio. It contrasts with the decreasing thermal Jeans mass of isothermal fragmentation, and does not rule out motion or fragmentation along field lines.

Equate the free-fall time of a uniform sphere $\sqrt{3\pi /(32G\rho )}$ to a radius-based sound-crossing time $R/c_s$, then substitute $R$ in $M=4\pi \rho R^3/3$. This gives the displayed mass. It has the scaling of a Jeans mass, but its coefficient differs from the convention using a sphere of diameter equal to the Jeans length. Equality of these two times is an approximate support comparison, not the exact marginal mode of an infinite-medium Jeans instability.

For a cloud supported by an effective speed of sound $v$ and confined by a surface pressure $P$, dimensional analysis fixes the gravitational instability mass scale to $v^4/(G^{3/2}{P}^{1/2})$. The dimensionless coefficient depends on the equilibrium model and the convention used for the instability threshold.

For a uniform ideal gas sphere, set $\rho =mn$, $p=n{k}_{B}T$ and $M=4\pi \rho R^3/3$. Building the sphere by concentric shells gives gravitational potential energy $W=-3G{M}^2/(5R)$. During a small homologous contraction, released gravitational energy balances the work against gas pressure when $3G{M}^2|dR|/(5R^2)=4\pi R^{2}p|dR|$, or $GMm/R=5k_{B}T$. Eliminating $R$ gives the displayed mass. At the same initial mass density and temperature, larger masses have stronger gravitational contraction than pressure support in this estimate. Its numerical coefficient belongs to the uniform-sphere pressure-work model; the infinite-medium Jeans instability and a pressure-confined isothermal sphere use different equilibrium and boundary assumptions.

During isothermal collapse, $T$ is constant and the Jeans mass decreases as ${\rho }^{-1/2}$. Successively denser regions can therefore become independently unstable and fragment to smaller masses.

For adiabatic monatomic gas, $T\propto {\rho }^{2/3}$ and the Jeans mass increases as ${\rho }^{1/2}$. Once cooling fails and collapse becomes adiabatic, further fragmentation is suppressed.