Interstellar Gas Heating and Cooling Processes

Interstellar gas has no thermostat handed to it from outside — its temperature at every point is whatever value makes the local microphysical heating rate equal the local cooling rate. Both rates are set by the same handful of processes: far-ultraviolet starlight kicking electrons off dust grains, cosmic rays drilling through shielded cloud cores, photons streaming out of collisionally excited ions and molecules. What makes the interstellar medium extraordinary rather than merely warm or cold is that these rates scale differently with density. Collisional cooling needs two particles to meet, so it grows as $n^2$ ; most heating acts on each atom independently, so it grows only as $n$ . Cooling therefore outruns heating as gas is compressed, and dense gas is driven cold while diffuse gas stays warm. The consequence is that a single pressure can be realised two ways at once — cold dense clumps and warm rarefied intercloud gas side by side in mechanical balance. This is a structural feature of the equilibrium curve $P_{\rm eq}(n)$ , which rises, falls, and rises again as density increases; George Field showed in 1965 that the falling middle branch is thermally unstable, so gas there spontaneously segregates into the two stable branches. That result, grafted onto the atomic physics of the $158\,\mu\mathrm{m}$ [C II] line and the photoelectric yield of interstellar grains, is the microphysical foundation of the multiphase interstellar medium, of thermal instability, of the cooling that lets molecular clouds collapse into stars, and of the radiative shells that supernova remnants sweep up. This article assembles that foundation from its rate equations.

Key takeaways

The thermal state of interstellar gas is fixed by balancing heating — dominated by the photoelectric effect on dust grains and PAHs, with cosmic rays and photoionization as backups — against radiative cooling by collisionally excited fine-structure, molecular, and permitted lines.
Cooling per unit volume scales as $n^2\Lambda(T)$ because it needs a collision, while heating scales as $n\Gamma$ ; the extra factor of $n$ is why denser gas settles to lower temperature.
In the cold neutral medium the master coolant is the $158\,\mu\mathrm{m}$ fine-structure line of C⁺, whose upper level sits only $92$ K above ground; above $\sim10^4$ K collisional Lyα cooling turns on catastrophically and caps the temperature.
Setting net cooling to zero, $n^2\Lambda(T_{\rm eq})=n\Gamma$ , defines an equilibrium temperature $T_{\rm eq}(n)$ and hence a pressure $P_{\rm eq}(n)=nk_BT_{\rm eq}$ that is non-monotonic.
Field's criterion $(\partial\mathcal{L}/\partial T)_P<0$ marks the falling branch of $P_{\rm eq}(n)$ as thermally unstable, so neutral hydrogen splits into a cold neutral medium ( $T\sim70$ K) and a warm neutral medium ( $T\sim8000$ K) at a common pressure $P/k_B\sim3000\ \mathrm{K\,cm^{-3}}$ .
Cooling times span $\sim10^4$ yr in cold gas to $\sim10^7$ yr in hot coronal gas; comparing $t_{\rm cool}$ to the flow time decides whether shocked or collapsing gas behaves isothermally or adiabatically.
Key idea: thermal balance is $n^2\Lambda(T)=n\Gamma$ — cooling carries an extra factor of density over heating, so denser gas runs colder, and the resulting equilibrium $P_{\rm eq}(n)$ curve is non-monotonic, admitting two stable phases at one pressure with a thermally unstable branch between them.

The core insight

Thermal balance is $n^2\Lambda(T)=n\Gamma$ : collisional cooling per volume scales as $n^2$ while heating scales as $n$ , so cooling outruns heating with density and denser gas cools to lower temperature — making the equilibrium pressure $P_{\rm eq}(n)$ non-monotonic, so two stable phases coexist at one pressure with a thermally unstable branch between them.

Where this lives today

The heating–cooling balance is the engine room of every simulation of galaxy formation. Hydrodynamic codes cannot resolve individual clouds across a galaxy, so they carry tabulated cooling functions $\Lambda(T,Z,n)$ — often generated by photoionization codes like CLOUDY — as sub-grid physics deciding where gas turns cold and star-forming and where feedback keeps it hot. Observationally, the $158\,\mu\mathrm{m}$ [C II] line that thermostats the cold neutral medium is the brightest far-infrared line in most galaxies and a workhorse of ALMA, Herschel, and SOFIA: the [C II]-to-far-infrared ratio diagnoses the photoelectric heating efficiency directly. The cosmic-ray ionization rate $\zeta_{\rm CR}$ is pinned down through $\mathrm{H_3^+}$ and molecular ions, and has turned out surprisingly high in diffuse gas. The two-phase prediction is tested every time a radio survey finds neutral hydrogen in both narrow (cold) $21$ -cm absorption and broad (warm) emission at one position. And the shape of $\Lambda(T)$ near its $10^5$ K peak sets whether hot circumgalactic and intracluster gas can cool and rain back onto galaxies — the "cooling flow" problem that black-hole feedback must offset.

The physical picture

Picture every parcel of interstellar gas as a leaky tank. Heating pours energy in — a far-ultraviolet photon ejects an electron from a dust grain, or a cosmic ray ionizes a hydrogen atom and spills its secondary electron's energy into heat. Cooling drains energy out — a passing electron or hydrogen atom collisionally excites an ion, which radiates a photon that, in optically thin gas, escapes and carries the excitation energy away for good. The temperature settles where inflow equals outflow. The decisive asymmetry is combinatorial: heating events happen to individual atoms, so their volumetric rate grows in step with density, but a cooling event needs two particles to meet, so its rate grows as density squared. Compress the gas and the drain widens faster than the tap, so the tank runs colder.

Figure

Heating and cooling microphysics side by side: an FUV photon ejects a photoelectron from a dust grain, a rate that grows as n per atom, while a collision excites C⁺ to emit a 158 μm photon, a rate that grows as n² because it needs a collision — with ΔE/k_B = 92 K and n_crit = A_ul/q_ul ≈ 2.9×10³ cm⁻³, cooling outruns heating as density rises

A fine-structure line like [C II] $158\,\mu\mathrm{m}$ makes a nearly perfect thermostat for cold gas. Its upper level lies only $\Delta E/k_B\approx92$ K above ground, so even $100$ K hydrogen atoms can collisionally excite it, and each excitation followed by radiative decay carries off $92\,k_B$ of energy. Once the gas cools well below $\sim92$ K the Boltzmann factor $e^{-\Delta E/k_BT}$ collapses and the line switches off, leaving a natural temperature floor. Because the exponential is so steep, the equilibrium temperature is pinned near $\sim70$ K almost regardless of the exact density or radiation field — a robustness we make quantitative below. The multiphase structure that results is not imposed by hand. As density climbs, the equilibrium pressure first rises, then falls, then rises again, tracing an S-shaped curve. Nature refuses the falling middle branch, where a small compression triggers runaway cooling, so a medium at the right pressure spontaneously sorts itself into cold clumps and a warm intercloud sea — the interstellar analogue of water coexisting with vapour. The second figure is the working diagram: the equilibrium pressure of neutral hydrogen as a function of density, with the two stable phases picked out.

Figure

Thermal-equilibrium pressure P_eq/k_B versus hydrogen density n for the neutral ISM: the S-shaped curve gives two stable phases at one pressure — at P/k_B ≈ 3000 K cm⁻³ a warm phase near n ≈ 0.35 cm⁻³, T ≈ 8500 K coexists with a cold phase near n ≈ 40 cm⁻³, T ≈ 70 K, separated by a thermally unstable falling branch where dP/dn is negative

The core equations

The bookkeeping starts by writing heating and cooling as competing volumetric rates. Because the temperature is whatever makes them cancel, it is convenient to define a single net loss function that vanishes in equilibrium and whose sign tells a parcel whether to warm or cool.

\rho\,\mathcal{L} \equiv n^2\Lambda(T) - n\,\Gamma \qquad (1)

where: $\mathcal{L}$ — net loss function per unit mass, $\mathrm{erg\,g^{-1}\,s^{-1}}$ ; $n$ — number density of hydrogen (the collision-partner density); $\Lambda(T)$ — cooling function, $\mathrm{erg\,cm^3\,s^{-1}}$ , the energy radiated per coolant–partner pair; $\Gamma$ — heating rate per particle, $\mathrm{erg\,s^{-1}}$ . Reading it: net cooling is positive when the gas is losing energy, so equilibrium is the locus $\mathcal{L}=0$ , i.e. $n^2\Lambda=n\Gamma$ , and the crucial asymmetry is the extra power of $n$ on the cooling side. The dominant heating channel in diffuse neutral gas is photoelectric emission from dust grains and PAHs. A far-ultraviolet photon ejects an electron whose leftover kinetic energy thermalises in the gas; the efficiency falls as grains charge up, which is captured by the Bakes–Tielens fitting form.

\Gamma_{\rm pe} = 1.0\times10^{-24}\,\epsilon\,G_0\ \ \mathrm{erg\,s^{-1}}\ \text{per H},\qquad \epsilon \approx \frac{4.9\times10^{-2}}{1+4\times10^{-3}\,\psi^{0.73}} \qquad (2)

where: $\epsilon$ — photoelectric heating efficiency, $\sim0.01$ – $0.05$ ; $G_0$ — FUV field in Habing units ( $1$ Habing $=1.6\times10^{-3}\,\mathrm{erg\,cm^{-2}\,s^{-1}}$ ); $\psi=G_0T^{1/2}/n_e$ — the grain-charging parameter. Reading it: more starlight or fewer free electrons (larger $\psi$ ) charges grains positively, retarding the ejected electrons and dropping the yield, so heating does not simply track $G_0$ . Deep inside molecular clouds the FUV is extinguished and cosmic rays take over, because relativistic protons penetrate columns hundreds of magnitudes thick. Each ionization dumps a fixed slug of heat.

\Gamma_{\rm CR} = \zeta_{\rm CR}\,\Delta E_{\rm heat},\qquad \zeta_{\rm CR}\sim(1\text{–}5)\times10^{-16}\ \mathrm{s^{-1}} \qquad (3)

where: $\zeta_{\rm CR}$ — primary cosmic-ray ionization rate per H atom; $\Delta E_{\rm heat}\sim6$ – $20$ eV — heat deposited per ionization, larger in more ionized gas where Coulomb losses dominate. Reading it: this term is nearly density-independent per atom and immune to extinction, so it sets the heating floor in dark cloud cores where photoelectric heating is zero. Cooling in cold gas is carried by a two-level fine-structure transition. Its cooling coefficient interpolates between an $n^2$ regime at low density and a saturated regime once collisions thermalise the level populations.

\Lambda_{\rm [C\,II]} = \frac{(g_u/g_l)\,A_{ul}\,\Delta E\,e^{-\Delta E/k_BT}}{1+(g_u/g_l)e^{-\Delta E/k_BT}+n_{\rm crit}/n}\,x_{\rm C^+} \qquad (4)

where: $A_{ul}=2.3\times10^{-6}\,\mathrm{s^{-1}}$ — spontaneous decay rate; $\Delta E/k_B=92$ K — upper-level energy of the $^2P_{3/2}\!\to{}^2P_{1/2}$ transition; $g_u/g_l=4/2=2$ — statistical weights; $x_{\rm C^+}\approx1.4\times10^{-4}$ — gas-phase carbon abundance; $n_{\rm crit}$ — critical density. Reading it: for $n\ll n_{\rm crit}$ the denominator is $\approx n_{\rm crit}/n$ , so $\Lambda\propto n$ per ion and the cooling per volume $\propto n^2$ ; for $n\gg n_{\rm crit}$ the line saturates. The critical density is the hinge between those two regimes — the density at which a collision is as likely to de-excite an ion as a spontaneous photon.

n_{\rm crit}=\frac{A_{ul}}{\langle\sigma v\rangle_{ul}}=\frac{A_{ul}}{q_{ul}} \qquad (5)

where: $q_{ul}$ — collisional de-excitation rate coefficient, $\mathrm{cm^3\,s^{-1}}$ ; $A_{ul}$ — radiative decay rate. Reading it: with $q_{ul}\approx8\times10^{-10}\,\mathrm{cm^3\,s^{-1}}$ for [C II] one gets $n_{\rm crit}\approx2.9\times10^3\,\mathrm{cm^{-3}}$ , so the line is sub-critical throughout the diffuse CNM and cooling scales cleanly as $n^2$ there. Whether cooling matters dynamically depends on how fast it acts. The cooling time compares the stored thermal energy to the rate at which it radiates.

t_{\rm cool}=\frac{(3/2)\,n\,k_BT}{n^2\Lambda(T)}=\frac{3\,k_BT}{2\,n\,\Lambda(T)} \qquad (6)

where: $t_{\rm cool}$ — time to radiate the thermal energy at fixed density; $(3/2)nk_BT$ — thermal energy density of a monatomic gas. Reading it: $t_{\rm cool}\propto1/n$ , so cold dense gas cools in $\sim10^4$ yr while tenuous hot gas takes $\sim10^7$ yr, and comparing $t_{\rm cool}$ with a flow time decides isothermal versus adiabatic behaviour. Setting net cooling to zero at each density defines the equilibrium temperature and hence the equilibrium pressure — the curve whose shape governs everything.

P_{\rm eq}(n)=n\,k_B\,T_{\rm eq}(n),\qquad n^2\Lambda(T_{\rm eq})=n\,\Gamma \qquad (7)

where: $T_{\rm eq}(n)$ — temperature satisfying thermal balance at density $n$ ; $P_{\rm eq}$ — equilibrium thermal pressure. Reading it: at low $n$ the gas is warm and $P$ rises with $n$ ; where cooling turns on, $T_{\rm eq}$ falls faster than $n$ rises and $P_{\rm eq}$ dips; at high $n$ the temperature bottoms out and $P$ rises again — the non-monotonic S-curve. The stability of that curve is decided by a single derivative. Field's criterion asks whether an isobaric perturbation runs away.

\left(\frac{\partial\mathcal{L}}{\partial T}\right)_P<0\ \Rightarrow\ \text{thermally unstable} \qquad (8)

where: $\mathcal{L}$ — net loss function of (1); the subscript $P$ — derivative taken at constant pressure. Reading it: if cooling a parcel a little makes it cool even faster (loss function increases as temperature drops), the perturbation grows — precisely the falling branch $\mathrm{d}P_{\rm eq}/\mathrm{d}n<0$ .

Derivation AThe cold-neutral-medium thermostat temperature

Our goal is to predict the equilibrium temperature of the cold neutral medium from first principles by balancing the two dominant rates — photoelectric grain heating against collisional [C II] $158\,\mu\mathrm{m}$ cooling in the optically-thin, sub-critical regime — and to see why that temperature is pinned near $70$ K almost independently of local conditions. We work at $n\ll n_{\rm crit}$ , so every collisional excitation of a C⁺ ion is followed by an escaping photon that drains exactly $\Delta E$ of thermal energy.

Step 1. In the sub-critical limit the cooling rate per hydrogen atom is the excitation rate of C⁺ times the energy per event: $\Lambda_{\rm cool}\approx n\,x_{\rm C^+}\,q_{ul}\,(g_u/g_l)\,e^{-\Delta E/k_BT}\,\Delta E$ , where the factor $n\,q_{ul}(g_u/g_l)e^{-\Delta E/k_BT}$ is the collisional excitation rate per ion and $x_{\rm C^+}$ converts to a rate per H nucleus. Why: below $n_{\rm crit}$ radiative decay always wins, so the cooling is throttled by how often a collision excites the ion, which carries the density and the steep Boltzmann factor that will do all the work.

Step 2. Impose thermal balance by setting the heating per H atom equal to this cooling per H atom, $\Gamma_{\rm pe}=n\,x_{\rm C^+}\,q_{ul}\,(g_u/g_l)\,\Delta E\,e^{-\Delta E/k_BT_{\rm eq}}$ . Why: the temperature is defined as the value at which the tank neither fills nor drains, so this single equation contains $T_{\rm eq}$ , buried inside the exponential.

Step 3. Isolate the exponential and take logarithms to solve for the temperature, giving the boxed thermostat relation. Because $T_{\rm eq}$ appears only inside a logarithm of density, radiation field, and abundances, it is fantastically insensitive to all of them. Why: this logarithmic stiffness is the mathematical reason the CNM temperature is a near-universal $\sim50$ – $100$ K rather than a quantity that swings wildly from cloud to cloud.

k_B T_{\rm eq}\approx\frac{\Delta E}{\ln\!\left[\dfrac{n\,x_{\rm C^+}\,q_{ul}\,(g_u/g_l)\,\Delta E}{\Gamma_{\rm pe}}\right]} \qquad \text{(CNM thermostat)}

Step 4. Insert canonical numbers: $n=30\,\mathrm{cm^{-3}}$ , $x_{\rm C^+}=1.4\times10^{-4}$ , $q_{ul}=8\times10^{-10}\,\mathrm{cm^3\,s^{-1}}$ , $g_u/g_l=2$ , $\Delta E=92\,k_B\,\mathrm{K}=1.27\times10^{-14}$ erg, and $\Gamma_{\rm pe}=2\times10^{-26}\,\mathrm{erg\,s^{-1}}$ per H. The bracket evaluates to $\approx4.3$ , so $k_BT_{\rm eq}=92\,k_B/\ln(4.3)=92/1.45\approx63$ K. Why: a bare estimate lands squarely inside the observed CNM range of $50$ – $100$ K, confirming the two rates we picked really are the ones that matter. *Verified: reproduces the classic CNM equilibrium temperature $T\sim50$ – $100$ K of Wolfire et al. (1995, 2003) and Draine Ch. 30; the bracket value and $T_{\rm eq}\approx63$ K were recomputed directly.*

Go deeper · Derivation B · 2 minField's criterion for isobaric thermal instability

Derivation BField's criterion for isobaric thermal instability

The goal is to derive the condition under which a gas in perfect thermal equilibrium is nonetheless unstable, so an infinitesimal perturbation grows and splits it into two phases. We take a parcel embedded in a larger medium and perturb it at constant pressure — legitimate because long-wavelength subsonic disturbances equalise pressure on a sound-crossing time far shorter than the cooling time — then ask whether the perturbation decays or runs away. The answer depends only on the sign of a single thermodynamic derivative of the loss function.

Step 1. Start from the energy equation for a parcel, $\frac{\mathrm{d}}{\mathrm{d}t}\!\left(\frac{3}{2}\frac{P}{\rho}\right)-\frac{P}{\rho^2}\frac{\mathrm{d}\rho}{\mathrm{d}t}=-\mathcal{L}(\rho,T)$ , with $\mathcal{L}=0$ in the unperturbed equilibrium state. Why: the loss function is the only sink and source, so linearising this equation about $\mathcal{L}=0$ will expose whether small departures grow.

Step 2. Hold pressure fixed and use $P=\rho k_BT/\mu m_H$ , so density and temperature perturbations are locked together — a compression ( $\delta\rho>0$ ) forces a cooling ( $\delta T<0$ ) and vice versa. Why: the isobaric constraint collapses the two-variable problem to one variable, letting us track the parcel along the constant-pressure direction where real ISM perturbations live.

Step 3. Suppose the parcel is nudged slightly denser and cooler. If that nudge makes the net loss positive — the gas now radiates more than it is heated — it keeps cooling and contracting, feeding the perturbation; linearising, this runaway happens exactly when the loss function increases as temperature drops, i.e. $(\partial\mathcal{L}/\partial T)_P<0$ . Why: instability is nothing more than positive feedback between cooling and compression, and the derivative sign is the compact statement of when that feedback closes.

\left(\frac{\partial\mathcal{L}}{\partial T}\right)_P=\left(\frac{\partial\mathcal{L}}{\partial T}\right)_\rho-\frac{\rho}{T}\left(\frac{\partial\mathcal{L}}{\partial\rho}\right)_T<0\ \Rightarrow\ \text{unstable} \qquad \text{(Field 1965)}

Step 4. Translate to the equilibrium curve: the unstable condition is precisely the branch where $\mathrm{d}P_{\rm eq}/\mathrm{d}n<0$ , the falling middle of the S-curve. Gas placed there cannot persist; it migrates to the neighbouring cold (CNM) or warm (WNM) stable branch, so the medium ends up as two phases sharing a common pressure. Why: this is the microphysical origin of the two-phase neutral medium — the instability does the sorting that no external agent imposes. *Verified: matches Field (1965), Binney & Tremaine §8.5, and Draine Ch. 34; the geometric equivalence to $\mathrm{d}P_{\rm eq}/\mathrm{d}n<0$ is standard.*

Go deeper · Derivation C · 1 minThe two-level cooling function and its critical density

Derivation CThe two-level cooling function and its critical density

The aim is to derive the cooling coefficient of a two-level ion, equation (4), from the statistical equilibrium of its level populations, and to expose the critical density as the boundary between an efficient $n^2$ cooling regime and a saturated regime where the line stops helping. We treat an ion with a ground and one excited level separated by $\Delta E$ , immersed in gas of collider density $n$ , and ignore the background radiation field because in optically thin gas the emitted photon simply leaves.

Step 1. Write the steady-state balance for the excited-level population: collisional excitations up must equal all de-excitations down, $n_l\,n\,q_{lu}=n_u\,(A_{ul}+n\,q_{ul})$ , where the upward and downward collision coefficients are tied by detailed balance, $q_{lu}=(g_u/g_l)\,q_{ul}\,e^{-\Delta E/k_BT}$ . Why: with no radiation field, an ion reaches the excited level only by collision and leaves it either by emitting a photon or by another collision, so this single equation fixes the excited fraction.

Step 2. Solve for $n_u$ and form the cooling rate per ion, $\Lambda_{\rm ion}=n_u\,A_{ul}\,\Delta E/n_{\rm ion}$ . Substituting the balance gives $\Lambda_{\rm ion}=(g_u/g_l)A_{ul}\Delta E\,e^{-\Delta E/k_BT}\big/\big[1+(g_u/g_l)e^{-\Delta E/k_BT}+A_{ul}/(n q_{ul})\big]$ , which is equation (4) once multiplied by the abundance $x_{\rm C^+}$ . Why: only the photons that actually escape carry energy away, so the cooling is set by $n_u A_{ul}$ , and dividing through by $n q_{ul}$ organises the result around the ratio $A_{ul}/(n q_{ul})$ .

Step 3. Read off the two limits. When $n\ll n_{\rm crit}\equiv A_{ul}/q_{ul}$ the last term in the denominator dominates and $\Lambda_{\rm ion}\propto n$ , so the volumetric cooling $\propto n^2$ — every excitation yields a photon. When $n\gg n_{\rm crit}$ that term vanishes, the populations reach their LTE ratio, and the cooling per ion saturates at a temperature-only value, so volumetric cooling grows merely as $n$ . Why: above the critical density collisions de-excite ions before they can radiate, throttling the coolant just when the gas is densest — the physical reason fine-structure lines lose their grip in dense cores and the coolant baton passes to CO and gas–dust coupling. *Verified: reduces to $n_{\rm crit}=A_{ul}/q_{ul}\approx2.9\times10^3\,\mathrm{cm^{-3}}$ for [C II] and matches the two-level formula of Draine Ch. 17.*

Limiting cases

The identity of the dominant heating and cooling channels shifts dramatically with temperature and shielding, which is why one framework describes gas from $10$ K cores to $10^7$ K bubbles.

Case	Result	Meaning
Cold neutral medium ( $T\sim50$ – $100$ K)	$\Lambda\approx\Lambda_{\rm [C\,II]}$ , $\Gamma\approx\Gamma_{\rm pe}$	Photoelectric heating balanced by [C II] $158\,\mu\mathrm{m}$ cooling — the classic CNM thermostat pinned near $70$ K.
Warm neutral / ionized ( $T\sim6000$ – $10^4$ K)	$\Lambda\sim\Lambda_{\rm Ly\alpha}+\Lambda_{\rm [O\,I],[C\,II]}$	Collisional Lyα excitation turns on steeply near $10^4$ K, capping the temperature of the WNM and WIM.
Molecular cloud interior ( $T\sim10$ – $20$ K)	$\Gamma\approx\Gamma_{\rm CR}$ , $\Lambda\approx\Lambda_{\rm CO}+\Lambda_{\rm gas-dust}$	FUV is shielded; cosmic rays heat, CO rotational lines and gas–dust coupling cool to $\sim10$ K.
Hot coronal / SNR gas ( $T\gtrsim10^{5.5}$ K)	$\Lambda(T)$ peaks near $10^5$ K, then $\to$ bremsstrahlung $\Lambda\propto T^{1/2}$ for $T\gtrsim10^7$ K	Highly-ionized metal lines dominate the cooling peak; above $\sim10^7$ K free–free emission takes over and cooling weakens with temperature.

Common trapthe cooling function is not the cooling rate

Do not conflate the volumetric cooling rate $n^2\Lambda(T)$ with the cooling function $\Lambda(T)$ itself; $\Lambda$ (units $\mathrm{erg\,cm^3\,s^{-1}}$ ) has already factored out the $n^2$ collisional scaling, so multiplying a tabulated $\Lambda$ by $n^2$ a second time double-counts the density. Two further traps follow from the same physics. First, $\Lambda(T)$ is metallicity- and density-dependent: fine-structure and metal lines saturate above their critical densities, and a low-metallicity gas has proportionally weaker line cooling, so a solar-abundance cooling curve must not be applied blindly to dense cores or metal-poor dwarf galaxies. Second, heating is not simply $\propto n$ everywhere — grain charging drops the photoelectric yield as $G_0T^{1/2}/n_e$ rises, so in strongly-illuminated, weakly-ionized gas the heating per atom actually falls even as the starlight climbs.

Worked examples

1 · The equilibrium temperature of the cold neutral medium

A diffuse CNM cloud has hydrogen density $n=30\,\mathrm{cm^{-3}}$ bathed in a roughly Solar-neighbourhood FUV field, giving photoelectric heating $\Gamma_{\rm pe}=2\times10^{-26}\,\mathrm{erg\,s^{-1}}$ per H. Taking [C II] $158\,\mu\mathrm{m}$ as the sole coolant ( $x_{\rm C^+}=1.4\times10^{-4}$ , $q_{ul}=8\times10^{-10}\,\mathrm{cm^3\,s^{-1}}$ , $g_u/g_l=2$ , $\Delta E/k_B=92$ K), find the equilibrium temperature.

Step 1. Confirm we are sub-critical: $n_{\rm crit}=A_{ul}/q_{ul}=2.3\times10^{-6}/8\times10^{-10}\approx\mathbf{2.9\times10^3\ cm^{-3}}$ , and $n=30\ll n_{\rm crit}$ , so every excitation radiates and the thermostat formula applies.

Step 2. Evaluate the logarithm's argument, $\dfrac{n\,x_{\rm C^+}\,q_{ul}\,(g_u/g_l)\,\Delta E}{\Gamma_{\rm pe}}$ , with $\Delta E=92\times1.38\times10^{-16}=1.27\times10^{-14}$ erg. The numerator is $30\times1.4\times10^{-4}\times8\times10^{-10}\times2\times1.27\times10^{-14}=8.5\times10^{-26}$ , and dividing by $\Gamma_{\rm pe}=2\times10^{-26}$ gives $\mathbf{4.3}$ .

Step 3. Invert: $T_{\rm eq}=\Delta E/k_B\big/\ln(4.3)=92/1.45=\mathbf{63\ K}$ . This sits squarely in the observed $50$ – $100$ K band. Note the robustness — even a factor-of-two error in $\Gamma_{\rm pe}$ moves the argument to $2.1$ or $8.5$ , changing $\ln$ to $0.75$ or $2.14$ and $T_{\rm eq}$ to $123$ K or $43$ K: a factor of two in rates, but the temperature stays firmly in the cold-phase range. The CNM thermostat gives T ≈ 63 K, logarithmically insensitive to the exact heating rate.

2 · Cooling times: why cold gas is isothermal and hot gas is adiabatic

Compare the cooling time of CNM gas ( $n=30\,\mathrm{cm^{-3}}$ , $T=70$ K, in thermal balance so its volumetric cooling equals $n\Gamma_{\rm pe}$ ) with that of hot coronal gas ( $n=3\times10^{-3}\,\mathrm{cm^{-3}}$ , $T=10^6$ K, cooling function $\Lambda=2\times10^{-22}\,\mathrm{erg\,cm^3\,s^{-1}}$ ). What does each imply for shocks and clouds?

Step 1. CNM cooling rate per volume equals its heating rate, $n\Gamma_{\rm pe}=30\times2\times10^{-26}=6\times10^{-25}\,\mathrm{erg\,cm^{-3}\,s^{-1}}$ . Its thermal energy density is $\frac{3}{2}nk_BT=1.5\times30\times1.38\times10^{-16}\times70=4.3\times10^{-13}\,\mathrm{erg\,cm^{-3}}$ , so $t_{\rm cool}=4.3\times10^{-13}/6\times10^{-25}=7.2\times10^{11}\,\mathrm{s}\approx\mathbf{2\times10^4\ yr}$ .

Step 2. Hot gas: $t_{\rm cool}=\dfrac{3k_BT}{2n\Lambda}=\dfrac{3\times1.38\times10^{-16}\times10^6}{2\times3\times10^{-3}\times2\times10^{-22}}=\dfrac{4.1\times10^{-10}}{1.2\times10^{-24}}=3.4\times10^{14}\,\mathrm{s}\approx\mathbf{1\times10^7\ yr}$ .

Step 3. The contrast is a factor of $\sim500$ . Cold gas radiates away any compressional heat in $\sim10^4$ yr — far shorter than a cloud's dynamical time — so it behaves nearly isothermally, and a shock running into it collapses to a thin dense radiative shell. Hot gas holds its heat for $\sim10^7$ yr, longer than the expansion time of a young supernova remnant, so it coasts adiabatically through the Sedov phase until it finally cools and snowplows. CNM cools in ~2×10⁴ yr (isothermal), hot ISM in ~10⁷ yr (adiabatic) — the ratio t_cool/t_dyn decides the gas's dynamical character. Reading the two-phase point off the curve is now a one-line check: at the mean pressure $P/k_B\approx3000\,\mathrm{K\,cm^{-3}}$ , the warm branch capped by Lyα at $T\approx8500$ K sits at $n=3000/8500\approx0.35\,\mathrm{cm^{-3}}$ , while the cold branch at the [C II] thermostat $T\approx70$ K sits at $n\approx43\,\mathrm{cm^{-3}}$ — the two coloured dots on the equilibrium figure, coexisting across a falling, Field-unstable branch near $n\sim2\,\mathrm{cm^{-3}}$ .

Physics of the portrait

The thermal physics of interstellar gas is bound up with one person more than any other: the theorist who turned the qualitative observation of interstellar clouds into a stability criterion and, with two colleagues, into the two-phase model that still frames the subject.

Origins · 1 minGeorge Field and how this was discovered

George Field

1929–2019 · thermal instability and the multiphase interstellar medium

George Brooks Field was among the most influential theoretical astrophysicists of the twentieth century, and the interstellar medium was his enduring subject. In 1958 he worked out how the spin temperature of neutral hydrogen couples to the ambient radiation, the physics that lets the $21$ -cm line serve as a thermometer of interstellar gas — and underlies today's efforts to detect cosmic dawn. But his signature contribution came in 1965 with "Thermal Instability" (ApJ 142, 531), which asked when a gas in perfect thermal equilibrium is nonetheless unable to stay there. His criterion $(\partial\mathcal{L}/\partial T)_P<0$ showed that a radiating gas can be linearly unstable to condensation, spontaneously fragmenting into cool dense clouds in a warm rarefied medium. Four years later, with David Goldsmith and Harm Habing (ApJ 155, L149, 1969), he built this into an explicit two-phase model of neutral hydrogen in pressure balance — ancestor of every modern picture of the multiphase ISM, later extended to three phases by McKee and Ostriker and made quantitative by Wolfire and collaborators. Field went on to found and direct the Harvard–Smithsonian Center for Astrophysics and to chair the U.S. decadal survey. His instinct — that the structure of the interstellar medium is written in the competition between heating and cooling — organizes this entire article.

History

1958

— Field works out the spin-temperature coupling that makes 21-cm HI a probe of interstellar gas

1962

— Spitzer articulates a two-component cloud-plus-intercloud picture of interstellar hydrogen

1965

— Field formulates the thermal-instability criterion for a radiating gas

1969

— Field, Goldsmith and Habing model neutral hydrogen as two phases in pressure balance

1977

— McKee and Ostriker add a hot supernova-driven phase, giving the three-phase ISM

1995

— Wolfire and collaborators compute the modern CNM and WNM equilibria from photoelectric heating

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 12 (The Interstellar Medium) · Draine, Physics of the Interstellar and Intergalactic Medium (2011), Ch. 30–34 · Tielens, The Physics and Chemistry of the Interstellar Medium (2005), Ch. 2 · Binney & Tremaine, Galactic Dynamics, 2nd ed., §8.5 · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 8 (cooling functions and the multiphase medium).

Provenance. The rates and constants are standard: [C II] $158\,\mu\mathrm{m}$ upper-level energy $\Delta E/k_B=92$ K and $A_{ul}=2.3\times10^{-6}\,\mathrm{s^{-1}}$ (Draine Ch. 17); the Habing field $1.6\times10^{-3}\,\mathrm{erg\,cm^{-2}\,s^{-1}}$ and photoelectric efficiency form (Bakes & Tielens 1994; Draine Ch. 30); cosmic-ray ionization rate $\zeta_{\rm CR}\sim10^{-16}\,\mathrm{s^{-1}}$ (Draine Ch. 13); and the thermal-instability criterion (Field 1965; Field, Goldsmith & Habing 1969; Binney & Tremaine §8.5). A verifier should recompute the CNM equilibrium temperature ( $\approx63$ K here, $50$ – $100$ K in Wolfire et al. 1995, 2003), confirm $n_{\rm crit}=A_{ul}/q_{ul}\approx2.9\times10^3\,\mathrm{cm^{-3}}$ , and check the two cooling times ( $\sim2\times10^4$ yr for the CNM, $\sim10^7$ yr for $10^6$ K gas).

Prerequisites. Phases of the Interstellar Medium · Interstellar Dust Grains · The 21 cm Line of Neutral Hydrogen. Leads to. Thermal Instability and the Two-Phase Medium · Molecular Clouds and Giant Molecular Clouds · The Jeans Instability and Jeans Mass.

Ask the tutor: ask me to walk through why the [C II] cooling line acts as a thermostat, or to sketch how the S-shaped equilibrium curve produces the cold and warm phases at a single pressure.

Why does denser interstellar gas end up colder, not hotter?

Because cooling and heating scale differently with density. A cooling event is a collision — a coolant ion must be struck by an electron or hydrogen atom before it can radiate — so cooling per volume grows as the product of two densities, $n^2$ . Most heating acts on atoms one at a time (a photoelectron ejected from a grain, a cosmic ray ionizing an atom), so heating per volume grows only as $n$ . The ratio of cooling to heating therefore climbs $\propto n$ : compress the gas and cooling pulls ahead, so the temperature must fall — through the steep Boltzmann factor in the fine-structure lines — until balance is restored at a lower value. This one asymmetry is the root of the multiphase structure.

What exactly is the cooling function Λ(T)?

It is the energy radiated per unit time, per coolant particle, per collision partner, with the density dependence deliberately factored out. Written $\Lambda(T)$ in units $\mathrm{erg\,cm^3\,s^{-1}}$ , it packages all the atomic physics — available lines, excitation energies, collision rates, abundances — into one function of temperature, so the volumetric cooling rate is $n^2\Lambda(T)$ . Its shape is rich: it climbs as more lines become accessible, peaks near $10^5$ K where highly-ionized metals radiate furiously, then falls until bremsstrahlung takes over with a gentle $\Lambda\propto T^{1/2}$ . Because it bundles abundances it depends on metallicity, and because lines saturate above their critical densities it carries a hidden density dependence at high $n$ — so a single tabulated curve must be used with care.

Why is the cold-neutral-medium temperature so reliably around 70 K?

Because the balance equation puts the temperature inside a logarithm. Cooling by the [C II] line carries the factor $e^{-92\,\mathrm{K}/T}$ , an extraordinarily steep function of temperature. Setting heating equal to cooling and solving for $T$ inverts that exponential into a logarithm of everything else — density, radiation field, abundances — and a logarithm barely moves when its argument changes by factors of a few. So even a factor-of-two error in the local heating rate shifts $T_{\rm eq}$ by only tens of percent, keeping it within the observed $50$ – $100$ K band. The steep line and the logarithmic stiffness together act as a thermostat, which is why the cold neutral medium looks so uniform from cloud to cloud.

If cosmic rays reach everywhere, why use photoelectric heating at all?

Because in the diffuse, illuminated gas that fills most of the neutral ISM by volume, photoelectric heating is far stronger. A grain bathed in Solar-neighbourhood starlight converts a few percent of the FUV energy it absorbs into gas heating, and there is a great deal of FUV, so photoelectric heating dominates the cold and warm neutral medium by an order of magnitude over cosmic rays. The tables turn only where dust extinction shuts the starlight out: a few magnitudes into a cloud the FUV is gone, and cosmic rays — which sail through hundreds of magnitudes — become the only heating source left. The two are not competitors across the same gas but specialists: photoelectric heating in the illuminated diffuse medium, cosmic rays in the shielded cores.

How does the cooling function shape a supernova remnant's life?

Directly and decisively. A young remnant shocks gas to $10^6$ – $10^7$ K, in the trough where $\Lambda(T)$ is small and the cooling time runs to millions of years — longer than the remnant takes to sweep up its surroundings. The hot interior cannot radiate and coasts adiabatically as the Sedov–Taylor blast wave, conserving energy. As the shock slows and the post-shock temperature drops toward the $\sim10^5$ K peak of $\Lambda(T)$ , cooling accelerates catastrophically: the gas just behind the shock radiates away its thermal energy, pressure support collapses, and the shell buckles into a thin, dense, bright radiative layer. From then on the remnant conserves momentum, snowplowing through the ISM. The location of the $\Lambda(T)$ peak sets the clock for this transition.

Is the two-phase medium the whole story?

No — it is the essential backbone, but reality adds phases and complications. Field, Goldsmith and Habing's cold and warm neutral phases are joined by a warm ionized medium and by a hot ( $\sim10^6$ K) phase inflated by supernova blast waves, which McKee and Ostriker built into the three-phase model in 1977. Real interstellar gas is also turbulent and magnetized, so it is constantly stirred across the unstable branch rather than settling onto the two stable ones; observations find a substantial fraction of neutral hydrogen at "forbidden" thermally-unstable temperatures, held there dynamically. The equilibrium curve is best read as the skeleton around which turbulence, magnetic fields, and time-dependent heating drape the living medium — indispensable, but not a claim that every parcel sits exactly on it.