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The optically thin gas cooling time is thermal energy density divided by the radiative loss rate. Write $\chi =n_{\mathrm{tot}}/n_H$ for the number of thermal particles per hydrogen nucleus, $X$ for the hydrogen mass fraction, and $f_g={\rho }_g/{\rho }_{\mathrm{grav}}$ for the gas fraction of the gravitating mass. ThenThe last expression is the free-fall time of a uniform sphere. Equating the two times gives the cooling-to-free-fall equality curveAt fixed temperature, $t_{\mathrm{cool}}/t_{\mathrm{ff}}=\sqrt{n_{H,\mathrm{crit}}/n_H}$: above the curve, $t_{\mathrm{cool}}<t_{\mathrm{ff}}$; below it, cooling is slower. The temperature dependence is approximately $n_{H,\mathrm{crit}}\propto T^2/{\Lambda }^2$. Strong line cooling therefore creates low-density troughs, whereas the fully ionized bremsstrahlung tail gives $n_{H,\mathrm{crit}}\propto T$.

For the diagram choose a self-gravitating uniform gas cloud, $f_g=1$, with $X=0.76$, proton mass $m_p\simeq 1.67\times 10^{-24}\mathrm{g}$, and the fully ionized particle ratio $\chi =2+3(1-X)/(4X)\simeq 2.24$. Using this fixed ratio is a convenient sketch normalization; the varying low-temperature ion fraction changes the coefficient by a factor of order unity. The supplied cooling values then give $n_{H,\mathrm{crit}}\sim 10^{-9}\mathrm{c}{\mathrm{m}}^{-3}$ at $1.5\times 10^4\mathrm{K}$ and $n_{H,\mathrm{crit}}\sim 0.1\mathrm{c}{\mathrm{m}}^{-3}$ at $10^7\mathrm{K}$. A dark-matter-dominated potential has $f_g<1$, shortening the free-fall time and moving the equality curve upward by $1/f_g$.

To relate the plane to mass, use the virial theorem for a uniform self-gravitating sphere. Its gravitational potential energy is $-3G{M}^2/(5R)$, so $k_B{T}_{\mathrm{vir}}=GM\mu m_p/(5R)$, where $\mu =1/(X\chi )$. With $M=(4\pi /3)\rho R^3$, the virial mass contours in a cooling diagram satisfyThe coefficient $5$ is for this uniform-sphere convention; other halo profiles change it. The horizontal mean-density label is $n_{H,0}\simeq 2\times 10^{-7}\mathrm{c}{\mathrm{m}}^{-3}$. Real collapsing clouds are overdense, and their characteristic densities rise at earlier epochs.

The cooling criterion for galaxy formation requires a cloud to lose shock-generated virial heat rapidly enough to contract and fragment. At higher virial masses and temperatures, hydrogen-helium line cooling ceases to be efficient, so a growing bound aggregate can remain a hot, pressure-supported atmosphere rather than condense as one giant luminous galaxy. Combined with virial mass contours and the densities at which clouds assemble, the diagram gives a characteristic upper galaxy-scale cooling mass, conventionally of order $10^{12}{M}_{\odot }$ in the simple baryonic-cloud argument. Larger aggregates are predominantly groups or clusters containing smaller galaxies and hot gas.

This upper galaxy mass from gas cooling is a physical scale, not an exact universal mass cutoff derivable from the two cooling labels alone. Composition, formation density, geometry, metals and a dark-matter potential shift it. Cooling that is slower than free fall but faster than the available cosmic time can still yield gradual central condensation. Mergers of existing stellar galaxies can also assemble a larger stellar system without rapidly cooling the entire gas mass of its host halo.