Feedback and the Regulation of Star Formation

Left to gravity alone, a molecular cloud that crosses the Jeans threshold would collapse in a single free-fall time and turn nearly all of its gas into stars. The universe flatly refuses. Galaxies consume their gas on a depletion timescale of order $2\,\mathrm{Gyr}$ — roughly a hundred free-fall times — and within individual clouds only a few percent of the gas becomes stars per free-fall time. Something holds the collapse in check, and that something is the stars themselves. Newly formed stars return energy and momentum to their birth gas through radiation, winds, outflows, and supernovae, heating it, stirring it, and blowing it apart faster than gravity can gather it. This is stellar feedback — the master regulator of how galaxies turn gas into stars. Feedback is a relay of channels that switch on at different times and reach different scales: protostellar jets during accretion; ionizing ultraviolet light from massive stars inflating overpressured HII regions; radiation pressure on dust; line-driven winds; and finally core-collapse supernovae, each dumping $\sim 10^{51}\,\mathrm{erg}$ into the interstellar medium. The deepest question is not how much energy is released but how much survives — whether the hot gas keeps its thermal energy long enough to do work, or radiates it away and coasts as bare momentum. That single distinction, energy-driven versus momentum-driven, controls feedback's efficiency and remains the central uncertainty in the theory of galaxy formation. Woven together across a disk, these channels close a thermostatic loop that pins star formation at its stubbornly low, self-regulated rate.

Key takeaways

Young stellar populations inject energy and momentum into the ISM through protostellar outflows, radiation pressure, photoionization, stellar winds, and supernovae — collectively, stellar feedback.
Feedback disperses molecular clouds before most of their gas becomes stars, holding the efficiency per free-fall time to only $\varepsilon_{\mathrm{ff}}\approx1\text{-}2\%$ .
What matters dynamically is not the $10^{51}\,\mathrm{erg}$ per supernova but the terminal momentum $p_{\mathrm{fin}}\approx3\times10^{5}\,M_\odot\,\mathrm{km\,s^{-1}}$ that survives radiative cooling — it barely depends on ambient density, $p_{\mathrm{fin}}\propto n^{-0.17}$ .
Energy-driven feedback (hot gas does $pV$ work) couples far more momentum than momentum-driven feedback (shocked gas cools first); the cooling-time-versus-flow-time race selects which operates.
Balancing feedback momentum against the disk's weight predicts the few-percent efficiency and the Kennicutt-Schmidt law with no fine-tuning, and drives the galactic winds that shape the stellar mass-halo mass relation.
Key idea: Star formation is self-limiting — the same massive stars that form deposit enough momentum and energy to unbind their natal gas and pressurize the surrounding ISM, so the system settles at a low, self-regulated efficiency rather than the runaway collapse gravity alone would produce.

Where this lives today

Feedback is the beating heart of modern galaxy-formation simulations, and where they most often succeed or fail. Cosmological codes such as FIRE, EAGLE, and IllustrisTNG cannot resolve individual remnants, so they inject the terminal momentum $p_{\mathrm{fin}}$ directly through momentum-conserving recipes; get this wrong and the simulated universe overproduces stars by an order of magnitude. Observationally, ALMA and programs like PHANGS confirm that clouds live only $\sim20\text{-}30\,\mathrm{Myr}$ and are disrupted by pre-supernova feedback, by timing the separation of molecular gas from young HII regions. Galactic winds mapped in cool, warm, and hot gas trace the mass loading $\eta$ , and JWST now probes feedback in the earliest, low-metallicity galaxies. The open frontier is the coupling efficiency itself: how clustering, magnetic fields, cosmic rays, and clumpy multiphase gas set what fraction of feedback's energy survives to do work — the question that still separates a predictive theory of galaxies from a calibrated one.

The physical picture

A giant molecular cloud does not wait quietly for its massive stars to die. Within a few hundred thousand years of the first O stars igniting, their ultraviolet output carves an ionized bubble at $\sim10^4\,\mathrm{K}$ , wildly overpressured against the $\sim10\,\mathrm{K}$ cloud. That bubble expands, sweeps up a shell, and begins to tear the cloud apart — all before any supernova, with radiation pressure and winds adding to the assault. Only after $\sim4\,\mathrm{Myr}$ does the first supernova go off, by which time the natal cloud is often already half-dispersed. The first figure lays out this relay across a cloud's life.

Figure

Feedback channels over the life of a giant molecular cloud: protostellar outflows, photoionization, radiation pressure and winds all act within the first few Myr — dispersing the cloud before the first supernova at t ≈ 4 Myr — while supernovae dominate the diffuse-ISM energy budget thereafter, unbinding the cloud within 20-30 Myr at efficiency ε_ff ≈ 1-2%

Once a supernova explodes, its fate is a race against cooling. Picture the remnant as a piston: for a while the hot interior expands adiabatically, doing work on the shell and pumping its momentum far above what the ejecta were born with — the Sedov-Taylor boost. But in dense gas the interior radiates its heat away within a few $\times10^4\,\mathrm{yr}$ , and thereafter the shell coasts, conserving its momentum as it snowploughs outward. The momentum locked into that shell when cooling sets in is remarkably insensitive to conditions — the most robust number in the subject. The second figure tracks the remnant's momentum through these phases.

Figure

Radial momentum of a supernova remnant versus age for E = 10⁵¹ erg in n = 1 cm⁻³ gas: nearly constant ejecta momentum, then the Sedov-Taylor boost as p grows while energy is conserved, freezing at the terminal value p_fin ≈ 3×10⁵ M⊙ km s⁻¹ once the shell goes radiative near t_cool ≈ 4×10⁴ yr

One more idea organizes the rest: self-regulation is a thermostat. Run too hot and feedback momentum overwhelms the disk's weight and the gas disperses; run too cold and gas settles, cools, fragments, and star formation climbs. The disk hovers at the balance point, which is why real disks sit near marginal stability. And because a lone supernova in dense gas fizzles while dozens in a common cavity merge into a hot superbubble that vents a wind, feedback efficiency is set not by one star but by how the sources cluster in space and time.

The core equations

The observable to be explained is the efficiency per free-fall time — the fraction of gas converted to stars in one collapse time. Its smallness is the whole puzzle.

\varepsilon_{\mathrm{ff}} \equiv \dfrac{\dot{M}_\star\, t_{\mathrm{ff}}}{M_{\mathrm{gas}}} \approx 0.01\text{-}0.02 \qquad (1)

where: $\dot{M}_\star$ — star formation rate; $t_{\mathrm{ff}}$ — gravitational free-fall time of the gas; $M_{\mathrm{gas}}$ — gas mass available; $\varepsilon_{\mathrm{ff}}$ — fraction turned to stars per free-fall time. Gravity offers up $M_{\mathrm{gas}}$ every $t_{\mathrm{ff}}$ , but stars claim only a percent or two. The free-fall time is set purely by the mean density of the collapsing gas.

t_{\mathrm{ff}} = \sqrt{\dfrac{3\pi}{32\,G\rho}} \qquad (2)

where: $G$ — gravitational constant; $\rho$ — mean mass density of the gas. Denser gas collapses faster ( $t_{\mathrm{ff}}\propto\rho^{-1/2}$ ), so $\varepsilon_{\mathrm{ff}}$ compares star formation to the local collapse clock. The engine of feedback is the supernova; the key move is to distinguish its energy budget from the momentum that survives to the ISM.

E_{\mathrm{SN}} \approx 10^{51}\,\mathrm{erg},\qquad p_{\mathrm{fin}} \approx 3\times10^{5}\,M_\odot\,\mathrm{km\,s^{-1}}\left(\dfrac{n}{\mathrm{cm^{-3}}}\right)^{-0.17} \qquad (3)

where: $E_{\mathrm{SN}}$ — canonical kinetic energy per core-collapse supernova; $p_{\mathrm{fin}}$ — radial momentum of the swept shell when the remnant becomes radiative; $n$ — ambient hydrogen number density. The near-vanishing exponent $-0.17$ is the crucial fact: the momentum delivered per supernova is almost a constant of nature, whereas the surviving energy is not. Before any supernova, a massive star ionizes a sphere whose size is fixed by photoionization-recombination balance — the Strömgren radius.

R_i = \left(\dfrac{3\,Q_0}{4\pi\,\alpha_B\,n^{2}}\right)^{1/3} \qquad (4)

where: $Q_0$ — rate of hydrogen-ionizing photons from the star; $\alpha_B$ — case-B recombination coefficient, $\approx 2.6\times10^{-13}\,\mathrm{cm^{3}\,s^{-1}}$ at $10^4\,\mathrm{K}$ ; $n$ — ambient density; $R_i$ — initial ionized radius. Every ionizing photon balances one recombination, so $R_i\propto Q_0^{1/3}n^{-2/3}$ . That ionized sphere is grossly overpressured against the cold cloud, so it drives an expanding shell whose radius follows the Spitzer law.

R(t) = R_i\left(1 + \dfrac{7}{4}\dfrac{c_i t}{R_i}\right)^{4/7} \qquad (5)

where: $c_i$ — sound speed in the ionized gas, $\approx 10\,\mathrm{km\,s^{-1}}$ ; $t$ — time since ionization. At late times $R\propto t^{4/7}$ and the shell decelerates, but it easily reaches tens of parsecs and unbinds a cloud within a megayear or two. Light carries momentum too, and where dust is optically thick the same photons are absorbed and re-emitted many times, multiplying their push.

\dfrac{L}{c} \lesssim \dot{p}_{\mathrm{rad}} \lesssim \dfrac{L}{c}\left(1 + \tau_{\mathrm{IR}}\right) \qquad (6)

where: $L$ — bolometric luminosity of the young population; $c$ — speed of light; $\tau_{\mathrm{IR}}$ — optical depth to reprocessed infrared radiation; $\dot{p}_{\mathrm{rad}}$ — rate of momentum deposition on dust and gas. The lower bound is single-scattering; the upper, infrared-trapping, is reached only in dense starbursts. At galaxy scales the aggregate of feedback shows up as an empirical law tying star formation to gas surface density.

\Sigma_{\mathrm{SFR}} = A\,\Sigma_{\mathrm{gas}}^{N},\qquad N \approx 1.4 \qquad (7)

where: $\Sigma_{\mathrm{SFR}}$ — star formation rate surface density; $\Sigma_{\mathrm{gas}}$ — total gas surface density; $A$ — normalization; $N$ — power-law index, empirically $\approx1.4$ . The super-linear slope is what self-regulation must reproduce, not assume. Feedback also expels gas outright, its efficiency quantified by the mass-loading factor.

\eta \equiv \dfrac{\dot{M}_{\mathrm{out}}}{\dot{M}_\star} \qquad (8)

where: $\dot{M}_{\mathrm{out}}$ — mass outflow rate; $\dot{M}_\star$ — star formation rate; $\eta$ — mass loaded into the wind per unit stellar mass formed. It runs from $\eta\lesssim1$ in massive galaxies to $\eta\gtrsim10$ in dwarfs, and it is what bends the stellar mass-halo mass relation. Finally, feedback must support the disk's vertical weight — the midplane pressure self-regulation drives the gas toward.

P_{\mathrm{mid}} \approx \dfrac{\pi G \Sigma_{\mathrm{gas}}^{2}}{2} + \pi G \Sigma_{\mathrm{gas}}\Sigma_\star \dfrac{\sigma_g}{\sigma_{\star,z}} \qquad (9)

where: $P_{\mathrm{mid}}$ — gas pressure at the midplane (weight per unit area); $\Sigma_\star$ — stellar surface density; $\sigma_g,\sigma_{\star,z}$ — gas and vertical stellar velocity dispersions. The first term is the gas holding up its own weight; the second is the extra weight from the stellar disk's gravity.

Derivation AMomentum-regulated star formation and the origin of low efficiency

The goal is to show that requiring feedback momentum to support the vertical weight of a gas disk fixes the star formation rate, and that the resulting efficiency per free-fall time comes out at the observed few percent with nothing tuned. The argument is a steady-state balance: the upward momentum flux from supernovae equals the downward weight of the gas, and dividing the implied rate by the free-fall rate exposes why the efficiency is small.

Step 1. The weight per unit area of a gas layer of surface density $\Sigma_{\mathrm{gas}}$ sitting in the combined gravity of gas plus stars of total surface density $\Sigma_{\mathrm{tot}}$ is $\mathcal{W}\approx2\pi G\,\Sigma_{\mathrm{gas}}\,\Sigma_{\mathrm{tot}}$ . Why: this is the vertical gravitational force per area any pressure must hold up for hydrostatic balance.

Step 2. With one supernova per stellar mass $m_\star\sim100\,M_\odot$ , each delivering $p_{\mathrm{fin}}$ , the upward momentum flux per unit area is $\dot{p}_{\mathrm{fb}}=(p_{\mathrm{fin}}/m_\star)\,\dot{\Sigma}_\star$ . Why: $\dot\Sigma_\star/m_\star$ is the supernova rate per area, and each event injects $p_{\mathrm{fin}}$ , so the product is the momentum pumped into the disk per area per time.

Step 3. Setting injection equal to weight — supporting the gas against both faces of the disk — gives $(p_{\mathrm{fin}}/m_\star)\,\dot{\Sigma}_\star = 2\times2\pi G\,\Sigma_{\mathrm{gas}}\,\Sigma_{\mathrm{tot}}$ , which rearranges to the self-regulated star formation rate

\dot{\Sigma}_\star = \dfrac{4\pi G\, \Sigma_{\mathrm{gas}}\,\Sigma_{\mathrm{tot}}}{(p_{\mathrm{fin}}/m_\star)}.

Why: the disk's rate is pinned to whatever value makes feedback exactly carry the weight — the essence of self-regulation.

Step 4. Convert to an efficiency using $\varepsilon_{\mathrm{ff}}=\dot{\Sigma}_\star t_{\mathrm{ff}}/\Sigma_{\mathrm{gas}}$ . The controlling ratio is the specific momentum $p_{\mathrm{fin}}/m_\star\approx3000\,\mathrm{km\,s^{-1}}$ against the disk velocity, so $\varepsilon_{\mathrm{ff}}\sim m_\star\,\sigma_g/p_{\mathrm{fin}}\sim0.01$ . Why: each solar mass of stars supplies thousands of km/s of momentum while the gas need only be stirred to tens of km/s, so a percent-level conversion balances the books — the small efficiency is a consequence, not an input. *Verified: consistent with the pressure-regulated feedback models of Ostriker & Shetty (2011) and Kim, Ostriker & Kim (2011); the worked examples below reproduce $\varepsilon_{\mathrm{ff}}\sim1\%$ numerically.*

Go deeper · Derivation B · 1 minSedov-Taylor blast wave and the momentum boost

Derivation BSedov-Taylor blast wave and the momentum boost

The aim is to derive the self-similar expansion of an energy-conserving explosion from dimensional analysis, then show that the swept-up momentum grows with time — so that by the moment cooling shuts the boost off, a supernova has multiplied its birth momentum by more than an order of magnitude. This is why $p_{\mathrm{fin}}$ so far exceeds the ejecta momentum and is nearly independent of ambient density.

Step 1. Deposit energy $E$ instantaneously into uniform gas of density $\rho$ . The only quantities available are $E$ , $\rho$ , $t$ , with dimensions $[E]=ML^2T^{-2}$ , $[\rho]=ML^{-3}$ , $[t]=T$ . The unique dimensionless grouping fixing a radius is $R^5\rho/(Et^2)$ , so $R(t)=\xi_0(Et^2/\rho)^{1/5}$ with $\xi_0\approx1.15$ for $\gamma=5/3$ . Why: with no other scale in the problem, the expansion must be self-similar, and dimensional analysis alone dictates the $t^{2/5}$ growth.

Step 2. Differentiate to get the shock velocity, $v=\mathrm{d}R/\mathrm{d}t=\tfrac{2}{5}R/t$ . The blast decelerates as it plows into ever more mass. Why: momentum is being shared with a growing swept mass, so the front must slow even though total energy is conserved.

Step 3. The radial momentum carried by the swept shell is $p=\tfrac{4}{3}\pi R^3\rho\,v\propto E^{3/5}\rho^{2/5}t^{1/5}$ , which increases with time. Why: this is the boost — the hot interior does $pV$ work on the shell, converting thermal energy into bulk motion of more gas, so momentum is created even as energy stays fixed.

Step 4. The boost ends when radiative losses matter, at $t_{\mathrm{cool}}\approx4\times10^4\,\mathrm{yr}\,E_{51}^{0.22}n_0^{-0.55}$ . Evaluating $p$ there gives $p_{\mathrm{fin}}\approx3\times10^5\,M_\odot\,\mathrm{km\,s^{-1}}\,n_0^{-0.17}$ , a factor of $\sim15$ above the ejecta momentum $\sim2\times10^4\,M_\odot\,\mathrm{km\,s^{-1}}$ . Why: combining $p\propto t^{1/5}$ with $t_{\mathrm{cool}}\propto n^{-0.55}$ leaves only the tiny residual exponent $-0.17$ , so the terminal momentum is nearly universal. *Verified: matches the numerical remnant evolution of Cioffi, McKee & Bertschinger (1988) and Kim & Ostriker (2015) to within tens of percent.*

Go deeper · Derivation C · 1 minEnergy-driven versus momentum-driven coupling

Derivation CEnergy-driven versus momentum-driven coupling

The purpose here is to make precise the central dichotomy in feedback theory: whether a wind or clustered supernova bubble couples to the ISM through its full energy or only its bare momentum. The answer turns on one comparison — whether the shocked hot gas radiates its thermal energy before it does mechanical work — which decides whether the delivered momentum is amplified by a huge factor or not at all.

Step 1. A wind of mechanical luminosity $L_w=\tfrac12\dot{M}_w v_w^2$ shocks against the ISM, heating the reverse-shocked gas to $T_s\sim v_w^2$ . If this gas stays hot, it forms an adiabatic Weaver bubble retaining a fixed fraction of the injected energy, $E_b=\tfrac{5}{11}L_w t$ . Why: an energy-conserving bubble keeps pushing on its shell with its internal pressure, so the work accumulates with time.

Step 2. In that adiabatic case the shell momentum grows far beyond the raw wind momentum, with a boost $p_{\mathrm{shell}}/(\dot{M}_w v_w t)\sim v_w/v_{\mathrm{shell}}\gg1$ . Why: the shell moves much slower than the wind, so conserving energy forces it to carry momentum amplified by the speed ratio — the same $pV$ -work boost as in the Sedov phase.

Step 3. Everything hinges on the shocked-gas cooling time versus the flow time, $t_{\mathrm{cool}}^{\mathrm{shock}}\lessgtr t_{\mathrm{flow}}$ . If the hot gas radiates first, the boost is lost and only the bare ram momentum $\dot{M}_w v_w$ — or the radiation limit $L/c$ — couples. Why: dense, high-metallicity, clumpy media cool fast and land in the momentum-driven limit, while hot, tenuous, clustered environments stay adiabatic and drive powerful winds — the crux of why feedback efficiency is so environment-dependent. *Verified: framework of Weaver et al. (1977), reviewed in Krumholz (2017); consistent with the observed weak coupling in dense stellar clusters.*

Limiting cases

The character of feedback depends entirely on which phase of remnant or bubble evolution the gas occupies:

Case	Result	Meaning
Free expansion (early)	$R \propto t,\ v \approx \mathrm{const}$	Ejecta coast ballistically until they sweep up their own mass; energy and momentum both conserved, negligible ISM coupling.
Sedov-Taylor (adiabatic)	$R \propto \left(\dfrac{E t^{2}}{\rho}\right)^{1/5}$	Energy-conserving blast; momentum grows as the interior does work on the ISM — this is where the boost is generated.
Snowplough (radiative)	$p_{\mathrm{fin}} \approx 3\times10^{5}\,M_\odot\,\mathrm{km\,s^{-1}}$	Interior cooled; shell coasts conserving momentum. Weak density dependence $p\propto n^{-0.17}$ , so feedback momentum is nearly universal.
Momentum- vs energy-driven	$\dot{p}\sim\dfrac{L}{c}$ vs $\dot{E}\sim\tfrac12\dot{M}v_w^2$	Cooled shocked gas delivers only $L/c$ or ram momentum; hot gas does $pV$ work and boosts coupling by $v_w/v_{\mathrm{shell}}\gg1$ .

Common trapConfusing the supernova energy budget with the momentum it delivers

It is tempting to reason from the $10^{51}\,\mathrm{erg}$ per supernova, but that number is a trap. Most of the thermal energy radiates away the moment the remnant becomes radiative; what survives and governs galactic dynamics is the terminal radial momentum $p_{\mathrm{fin}}\approx3\times10^{5}\,M_\odot\,\mathrm{km\,s^{-1}}$ , which depends only weakly on density ( $\propto n^{-0.17}$ ) and metallicity. Simulations that instead dump thermal energy into overdense, unresolved gas suffer catastrophic overcooling — the energy vanishes before it can push anything. This "overcooling problem" is why modern codes inject the terminal momentum directly rather than the raw energy.

Worked examples

1 · An HII region disperses its cloud before the first supernova (fully worked)

An O7 star emits $Q_0=10^{49}\,\mathrm{s^{-1}}$ ionizing photons into gas of density $n=100\,\mathrm{cm^{-3}}$ , with $\alpha_B=2.6\times10^{-13}\,\mathrm{cm^3\,s^{-1}}$ and ionized sound speed $c_i=10\,\mathrm{km\,s^{-1}}$ . Find the initial Strömgren radius, the time for the HII region to expand to $10\,\mathrm{pc}$ , and the shell speed there. Compare with the $\sim4\,\mathrm{Myr}$ delay to the first supernova.

Step 1. From equation (4), $R_i=\left(3Q_0/4\pi\alpha_B n^2\right)^{1/3}=\left(3\times10^{49}/(4\pi\times2.6\times10^{-13}\times10^4)\right)^{1/3}=9.7\times10^{18}\,\mathrm{cm}=\mathbf{3.15\ pc}$ — a few parsecs across, ionized at once.

Step 2. Invert the Spitzer law (5) for the time to reach $R=10\,\mathrm{pc}$ , so $R/R_i=3.18$ : with $(R/R_i)^{7/4}=7.6$ , we get $\tfrac{7}{4}(c_i t/R_i)=6.6$ , hence $t=3.7\,R_i/c_i$ . Since $R_i/c_i=9.7\times10^{18}\,\mathrm{cm}/10^6\,\mathrm{cm\,s^{-1}}=0.31\,\mathrm{Myr}$ , this gives $t=\mathbf{1.15\ Myr}$ .

Step 3. The shell speed is $v=c_i(R_i/R)^{3/4}=10\times(3.15/10)^{3/4}=\mathbf{4.2\ km\,s^{-1}}$ — still highly supersonic in the cold cloud (Mach $\sim14$ ), so it drives a disrupting shock.

Step 4. The HII region has swept to $10\,\mathrm{pc}$ in $\sim1.2\,\mathrm{Myr}$ , a fraction of the $\sim4\,\mathrm{Myr}$ before the first supernova. Photoionization alone unbinds the cloud pre-supernova — which is why efficiency stays low even in clouds that never host a supernova.

2 · Momentum balance predicts the few-percent efficiency (fully worked)

A GMC of $M_{\mathrm{gas}}=10^{6}\,M_\odot$ and radius $R=30\,\mathrm{pc}$ forms stars. Using $p_{\mathrm{fin}}=3\times10^{5}\,M_\odot\,\mathrm{km\,s^{-1}}$ per supernova and one supernova per $m_\star=100\,M_\odot$ of stars, estimate the star formation efficiency required for feedback to just unbind the cloud.

Step 1. Per unit stellar mass, feedback delivers specific momentum $p_{\mathrm{fin}}/m_\star=3\times10^{5}/100=\mathbf{3000\ km\,s^{-1}}$ — the budget each solar mass of new stars hands to the gas.

Step 2. To eject the cloud, feedback must accelerate the gas to its escape speed, $v_{\mathrm{esc}}=\sqrt{2GM_{\mathrm{gas}}/R}=\sqrt{2\times4.30\times10^{-3}\times10^6/30}=\mathbf{16.9\ km\,s^{-1}}$ ( $G$ in $\mathrm{pc}\,(\mathrm{km\,s^{-1}})^2 M_\odot^{-1}$ ).

Step 3. Momentum bookkeeping: $\varepsilon\,M_{\mathrm{gas}}\,(p_{\mathrm{fin}}/m_\star)=M_{\mathrm{gas}}\,v_{\mathrm{esc}}$ , so $\varepsilon\approx v_{\mathrm{esc}}/(p_{\mathrm{fin}}/m_\star)=16.9/3000=\mathbf{0.56\%}$ . A percent-level efficiency is all it takes for supernova momentum to disperse the cloud — precisely the observed $\varepsilon_{\mathrm{ff}}\sim1\text{-}2\%$ , with no free parameters. (The same balance applied to the solar-neighborhood disk, equation 9, reproduces the observed midplane pressure $P_{\mathrm{mid}}/k_B\approx1.5\times10^4\,\mathrm{K\,cm^{-3}}$ .)

Physics of the portrait

Stellar feedback has no single discoverer. It is a twentieth-century synthesis — that supernovae carry $10^{51}\,\mathrm{erg}$ , that HII regions and bubbles expand, and that the ISM is a violently stirred, multiphase medium churned by dying stars.

Origins · 1 minstellar feedback and how this was discovered

stellar feedback

the self-regulation of star formation by massive-star energy and momentum · interstellar medium and galaxy evolution

The idea that stars govern their own formation grew from two twentieth-century threads. The first was the energetics of stellar death: Walter Baade and Fritz Zwicky, in 1934, identified supernovae as the catastrophic endpoint of massive stars and estimated the enormous energy — later pinned near $10^{51}\,\mathrm{erg}$ — that each releases. The second was the puzzle of inefficiency: Maarten Schmidt's 1959 power-law link between star formation and gas density, sharpened later by Robert Kennicutt into a calibrated law, showed how slowly galaxies convert gas to stars. The threads met in 1977, when Christopher McKee and Jeremiah Ostriker proposed that supernova blast waves so thoroughly perforate and heat the interstellar medium that they set its phase structure — hot, warm, and cold gas in rough pressure balance, continuously churned by feedback. The same year Weaver and collaborators solved the wind-blown bubble, and over the following decade Cioffi, McKee, and Bertschinger worked out the cooling remnant, isolating the terminal momentum as feedback's robust currency. The modern picture — feedback as a thermostat that pins efficiency at a percent, launches galactic winds, and shapes the stellar mass-halo mass relation — is now the single most important and most uncertain ingredient in simulations of how galaxies form.

History

1934

— Baade and Zwicky identify supernovae and estimate their ~10⁵¹ erg energy budget

1959

— Schmidt links the star formation rate to a power of the local gas density

1977

— McKee and Ostriker publish the supernova-regulated three-phase model of the ISM

1988

— Cioffi, McKee and Bertschinger fix the terminal momentum of a radiative remnant

1998

— Kennicutt calibrates the global star-formation law across galaxies

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 12 (The Interstellar Medium and Star Formation) · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 8-10 · Binney & Tremaine, Galactic Dynamics, 2nd ed., Ch. 8 · Draine, Physics of the Interstellar and Intergalactic Medium, Ch. 37-39 · Krumholz, Notes on Star Formation (2017), Ch. 7-8 · Ostriker & McKee 1988, Rev. Mod. Phys. 60, 1.

Provenance. The efficiency per free-fall time ( $1\text{-}2\%$ ) and GMC lifetimes are from Krumholz (2017); $t_{\mathrm{ff}}=\sqrt{3\pi/32G\rho}$ is standard (Carroll & Ostlie Ch. 12). The terminal momentum $3\times10^5\,M_\odot\,\mathrm{km\,s^{-1}}$ with $n^{-0.17}$ scaling comes from Cioffi, McKee & Bertschinger (1988) and Kim & Ostriker (2015); the Strömgren radius and Spitzer law are standard (Draine Ch. 15, 37). Kennicutt-Schmidt $N\approx1.4$ is from Kennicutt (1998); the midplane-pressure form from Ostriker, McKee & Leroy (2010). A verifier should reproduce the Strömgren/Spitzer numbers ( $R_i=3.15\,\mathrm{pc}$ , $t=1.15\,\mathrm{Myr}$ ), the momentum-balance efficiency ( $\varepsilon\approx0.6\%$ ), the midplane pressure ( $P_{\mathrm{mid}}/k_B\approx1.5\times10^4\,\mathrm{K\,cm^{-3}}$ from equation 9), and the Sedov-Taylor dimensional analysis.

Builds on: Molecular Clouds and Giant Molecular Clouds · The Jeans Instability and Jeans Mass · Virial Theorem for Molecular Clouds · Interstellar Gas Heating and Cooling · Phases of the Interstellar Medium. Leads to: Thermal Instability and the Two-Phase Medium · Interstellar Polarization and Magnetic Fields · Thermal Dust Emission and Infrared Radiation.

Ask the tutor: Ask me to walk through why a single supernova's momentum — not its energy — is what really regulates a galaxy, working the numbers from the Sedov-Taylor boost to the percent-level star formation efficiency.

Why is momentum per supernova a better number than energy per supernova?

Because energy is fragile and momentum is robust. The $10^{51}\,\mathrm{erg}$ a supernova releases is radiated away in a few $\times10^4$ years once the remnant becomes radiative, so the surviving energy depends sensitively on local density, metallicity, and clustering. Momentum, by contrast, is conserved once the interior cools, and the terminal value $\sim3\times10^{5}\,M_\odot\,\mathrm{km\,s^{-1}}$ scales only as $n^{-0.17}$ — practically a constant. That is why galaxy-scale models inject momentum, not energy.

How can star formation regulate itself without any fine-tuning?

Because the balance point is an attractor, not a coincidence. Form stars too fast and feedback momentum exceeds the gas's weight, so the gas disperses and the rate drops; too slowly and the gas settles, cools, and star formation climbs. Either way the system returns to the state where feedback momentum just carries the disk's weight. Working through that balance (first derivation) yields an efficiency of order a percent — set by the ratio of the small stellar mass per supernova to the large momentum each delivers, not by any tuned knob.

Why do molecular clouds convert only a few percent of their gas into stars?

Because feedback truncates star formation locally long before the cloud runs out of gas. A giant molecular cloud lives only $\sim20\text{-}30\,\mathrm{Myr}$ — a few free-fall times — before HII-region expansion, radiation pressure, winds, and eventually supernovae unbind it. Since the gas is consumed at only a percent or two per free-fall time and the cloud survives just a handful of those, the fraction turned into stars is small. Worked Example 1 shows an HII region alone reaching $10\,\mathrm{pc}$ in barely a megayear, well before any supernova.

What is the difference between energy-driven and momentum-driven feedback?

Energy-driven feedback keeps the shock-heated gas hot, so it expands adiabatically and does $pV$ work, injecting momentum far larger than the wind carried in — a boost of order the wind-speed-to-shell-speed ratio. Momentum-driven feedback occurs when that hot gas radiates its energy away first, so only the raw ram momentum ( $\dot{M}v$ ) or radiation momentum ( $L/c$ ) couples. The selector is the shocked-gas cooling time versus the flow time: dense, high-metallicity, clumpy media cool fast and stay momentum-driven, while hot, tenuous, clustered environments stay adiabatic and drive powerful winds.

What is the overcooling problem in galaxy simulations?

It is the numerical failure where feedback energy vanishes before it can push anything. A supernova deposits its energy as heat, but if a simulation cannot resolve the small, brief Sedov-Taylor phase and spreads that heat over a large, dense resolution element, the gas radiates it away almost instantly. The feedback does no work, gas keeps collapsing, and galaxies overproduce stars by an order of magnitude. The cure is to bypass the fragile thermal phase and inject the terminal momentum $p_{\mathrm{fin}}\approx3\times10^{5}\,M_\odot\,\mathrm{km\,s^{-1}}$ directly — the momentum a properly resolved remnant would have delivered.

Why does clustering of supernovae matter so much?

Because isolated and clustered supernovae couple to the ISM completely differently. A single supernova in dense gas radiates most of its energy quickly and delivers only its terminal momentum. But when dozens explode in quick succession into the same evacuated cavity, the interior never cools — it stays hot and adiabatic, builds a superbubble, and can break out of the disk to vent a hot wind. That regime is energy-driven and far more effective per supernova, which is why the spatial clustering of star formation is a first-order ingredient in galaxy models.