= Solution
Let $X$ be the <hydrogen mass fraction>, $f_g=\rho_g/\rho_{\rm tot}$ the gravitating gas fraction, and $\chi=n_{\rm tot}/n_H$ the total-particle-to-hydrogen ratio. The <optically thin gas cooling time> and <free-fall time of a uniform sphere> are
$$
t_{\rm cool}=\frac{(3/2)n_{\rm tot}k_BT}{n_H^2\Lambda}=\frac{3\chi k_BT}{2n_H\Lambda},\qquad t_{\rm ff}=\sqrt{\frac{3\pi Xf_g}{32Gm_pn_H}},
$$
where $\rho_{\rm tot}=m_pn_H/(Xf_g)$. Equating them gives the <cooling-to-free-fall equality curve>:
$$
\boxed{n_{H,{\rm crit}}(T)=\frac{32Gm_p}{3\pi Xf_g}\left(\frac{3\chi k_BT}{2\Lambda(T)}\right)^2.}
$$
At fixed composition, its shape is $n_{\rm crit}\propto T^2/\Lambda^2$. \b[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 $n_{\rm crit}\propto T$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-60-cooling-diagram.png]
{title=Cooling versus free-fall diagram with uniform-cloud virial mass contours and illustrative virial densities}
The displayed uniform self-gravitating-cloud example uses $f_g=1$, $X=0.76$, $\mu\simeq0.59$ and $\chi\simeq2.24$. 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 $n_{\rm crit}(1.5\times10^4\,\mathrm K)\simeq9.6\times10^{-10}\,\mathrm{cm^{-3}}$ and $n_{\rm crit}(10^7\,\mathrm K)\simeq0.107\,\mathrm{cm^{-3}}$. 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 $-3GM^2/(5R)$ and its thermal <kinetic energy> is $(3/2)Mk_BT/(\mu m_p)$. The <virial theorem> gives $k_BT_{\rm vir}=\mu m_pGM/(5R)$, while $M=(4\pi/3)R^3\rho_{\rm tot}$. Eliminating $R$ gives
$$
\boxed{n_H=\frac{375Xf_gk_B^3T_{\rm vir}^3}{4\pi G^3\mu^3m_p^4M^2}.}
$$
Thus 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 $k_BT_{\rm vir}=\mu m_pGM/(2R)$ shifts the mass contours but preserves their slopes. The horizontal illustrative formation-density lines are $n_{H,{\rm vir}}\simeq\Delta_v(2\times10^{-7})(1+z)^3\,\mathrm{cm^{-3}}$, 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 $10^8M_\odot$ 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 $t_{\rm cool}/t_{\rm ff}\propto T^{1/2}\propto M^{1/3}$ 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.
\b[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.
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