Jeans instability dispersion relation

What you are seeing: the Jeans dispersion relation $\omega^2 = c_s^2 k^2 - 4 \pi G \rho$ for a uniform self-gravitating medium. Modes with $\omega^2 \lt 0$ grow exponentially (gravitational collapse seeds star formation); modes with $\omega^2 \gt 0$ oscillate as sound waves. The crossover wavelength is the Jeans length $\lambda_J = \sqrt{\pi c_s^2 / G \rho}$. Sliders set temperature $T$ (which sets $c_s = \sqrt{k_B T / m_p}$ for isothermal hydrogen) and number density $n$ (which sets $\rho = n m_p$).

For a cold dense molecular cloud ($T = 10$ K, $n = 10^3$ cm$^{-3}$), $\lambda_J \sim 1.5$ pc and $M_J \sim 50\,M_\odot$: cores larger than this collapse to form stars. The plot is on a log-log axis; the unstable band ($k \lt k_J$) is shaded.