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Let $X$ be the hydrogen mass fraction, $f_g={\rho }_g/{\rho }_{\mathrm{tot}}$ the gravitating gas fraction, and $\chi =n_{\mathrm{tot}}/n_H$ the total-particle-to-hydrogen ratio. The optically thin gas cooling time and free-fall time of a uniform sphere arewhere ${\rho }_{\mathrm{tot}}=m_p{n}_H/(X{f}_g)$. Equating them gives the cooling-to-free-fall equality curve:At fixed composition, its shape is $n_{\mathrm{crit}}\propto T^2/{\Lambda }^2$. Gas above this curve cools faster than it falls; gas below it cools more slowly. Strong line cooling produces a deep low-density trough. In the high-temperature free-free regime, $\Lambda \propto T^{1/2}$ gives $n_{\mathrm{crit}}\propto T$.

The displayed uniform self-gravitating-cloud example uses $f_g=1$, $X=0.76$, $\mu \simeq 0.59$ and $\chi \simeq 2.24$. Holding these ionized-gas factors fixed makes the diagram transparent, but near the atomic threshold their actual temperature dependence changes numerical normalizations. With these conventions the supplied cooling values imply $n_{\mathrm{crit}}(1.5\times 10^4\mathrm{K})\simeq 9.6\times 10^{-10}\mathrm{c}{\mathrm{m}}^{-3}$ and $n_{\mathrm{crit}}(10^7\mathrm{K})\simeq 0.107\mathrm{c}{\mathrm{m}}^{-3}$. For gas supported in a dark-matter-dominated dark matter halo, take $f_g<1$: at fixed $n_H$ the free-fall time of a uniform sphere is shorter and the critical curve shifts upward by $1/f_g$.

To add virial mass contours in a cooling diagram, use the uniform self-gravitating sphere consistently. Its Newtonian gravitational potential energy is $-3G{M}^2/(5R)$ and its thermal kinetic energy is $(3/2)M{k}_{B}T/(\mu m_p)$. The virial theorem gives $k_B{T}_{\mathrm{vir}}=\mu m_{p}GM/(5R)$, while $M=(4\pi /3)R^3{\rho }_{\mathrm{tot}}$. Eliminating $R$ givesThus constant total gravitating mass gives slope three in the logarithmic density-temperature plane, with larger mass contours lying at smaller density. A different density profile changes the structural coefficient; the customary halo temperature scale $k_B{T}_{\mathrm{vir}}=\mu m_{p}GM/(2R)$ shifts the mass contours but preserves their slopes. The horizontal illustrative formation-density lines are $n_{H,\mathrm{vir}}\simeq {\Delta }_v(2\times 10^{-7})(1+z{)}^3\mathrm{c}{\mathrm{m}}^{-3}$, using ${\Delta }_v=18{\pi }^2$. They show why increasing formation cosmological redshift helps gas reach the rapid-cooling region.

At the low-mass end, a dark matter halo's virial temperature may not reach the atomic threshold. The atomic and molecular cooling thresholds for galaxy formation therefore favor an atomic-cooling mass of order $10^8{M}_{\odot }$ at early formation epochs, with its precise value varying as approximately $(1+z{)}^{-3/2}$. Smaller primordial systems require molecules; photoheating and stellar feedback further suppress their luminous output.

At the high-mass end, the virial temperature at a fixed assembly density scales as $M^{2/3}$. Beyond the line-cooling interval the gas becomes increasingly hot, while thermal bremsstrahlung improves only as $T^{1/2}$. Hence at fixed assembly density $t_{\mathrm{cool}}/t_{\mathrm{ff}}\propto T^{1/2}\propto M^{1/3}$ in that regime. Large systems can remain pressure-supported hot atmospheres for many dynamical times, instead of promptly fragmenting into one stellar object. This is the upper galaxy mass from gas cooling: it associates efficient condensation with roughly galactic, rather than group or cluster, masses and motivates the broad $10^8$–$10^{12}{M}_{\odot }$ interval.

The diagram gives characteristic cooling scales, not exact universal mass bounds. Density profiles, gas fraction, enrichment and formation epoch move the boundaries; the two supplied values alone do not determine a complete numerical cutoff. Some gas can cool in dense cores of larger dark-matter haloes, early smaller progenitors can form stars before merging, and subsequent galaxy mergers can assemble more massive galaxies. Long-lived hot atmospheres also require heating to prevent excessive late-time cooling. These qualifications explain why the cooling criterion for galaxy formation is useful without equating each dark dark matter halo to one galaxy.

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.