Gravitational Collapse and Fragmentation

A giant molecular cloud that crosses its own Jeans threshold does not settle gracefully into one enormous star. It shatters. The reason is a single scaling law wearing a decisive sign: the Jeans mass, the smallest lump self-gravity can hold together against pressure, obeys $M_J\propto T^{3/2}\rho^{-1/2}$ , and during the collapse the density $\rho$ climbs by many orders of magnitude while the temperature $T$ barely moves — dust grains and molecular lines radiate the heat of compression away as fast as gravity delivers it, pinning the gas near $10\,\mathrm{K}$ . With $T$ frozen, every hundredfold rise in density drops $M_J$ tenfold, so sub-regions that were far too small to be self-gravitating at the cloud's original density suddenly exceed the new, smaller Jeans mass and begin to collapse on their own. One cloud becomes many; each fragment can fragment again. This runaway subdivision — gravitational fragmentation — is the physical bridge between diffuse interstellar gas and a bound cluster of stars. The cascade is not endless. As a fragment grows dense it eventually becomes opaque to its own thermal radiation; the heat of compression can no longer escape, the gas turns adiabatic, and now $M_J$ rises with density instead of falling. Fragmentation stalls, and the smallest surviving fragment — the opacity limit, of order $0.01$ to $0.1\,M_\odot$ — sets the lower edge of the stellar and brown-dwarf mass range. This article takes the Jeans mass, free-fall time, and cooling balance as its starting point and asks what they predict when a cloud actually collapses: why stars are born in clusters, why so many are binaries, and where the initial mass function gets its low-mass turnover. Rotation, magnetic fields, and turbulence complicate every step.

Key takeaways

A collapsing cloud fragments because the Jeans mass $M_J\propto T^{3/2}\rho^{-1/2}$ falls as density rises, so ever-smaller pieces become self-gravitating.
The cascade needs near-isothermal collapse: dust and molecular-line cooling dumps the compressional heat into infrared photons, holding $T\approx10\,\mathrm{K}$ while $\rho$ climbs.
A worked cascade at $T=10\,\mathrm{K}$ : $M_J\approx55\,M_\odot$ at $n\approx10^2\,\mathrm{cm^{-3}}$ becomes $0.55\,M_\odot$ at $10^6\,\mathrm{cm^{-3}}$ — factor $100$ in mass per $10^4$ in density.
Denser fragments have shorter free-fall times $t_{\rm ff}=\sqrt{3\pi/32G\rho}\propto\rho^{-1/2}$ , so they collapse faster and decouple.
Fragmentation stops at the opacity limit, $M_{\rm min}\sim0.01$ – $0.1\,M_\odot$ , where the gas turns adiabatic and $M_J$ rises again ( $\gamma_{\rm crit}=4/3$ ); magnetic support, rotation, and turbulence regulate the rest.
Key idea: during near-isothermal collapse the Jeans mass falls as $\rho^{-1/2}$ , so a collapsing cloud spontaneously subdivides into smaller self-gravitating fragments until rising opacity traps the heat and halts the cascade at a minimum mass of $\sim0.01$ – $0.1\,M_\odot$ .

Where this lives today

Fragmentation theory is now tested in the machinery of star formation rather than argued on paper. ALMA resolves prestellar cores and disks at the scales where the Jeans mass predicts fragments should appear, and routinely catches disks fragmenting into companion stars and giant planets — the rotation channel for binaries made visible. Herschel far-infrared surveys found cores piling up at a characteristic mass a few times the thermal Jeans mass, tying the observed core mass function to the stellar initial mass function. On the theory side, radiation-magnetohydrodynamic simulations now follow gravity, turbulence, magnetic fields, and radiative feedback together, and show that heating from the first protostar can suppress further fragmentation nearby, blunting the pure isothermal cascade. The opacity limit remains the anchor of the substellar mass floor, and the origin of the lowest-mass brown dwarfs and free-floating planetary-mass objects — failed fragments or ejected disk material — is an open problem JWST is now attacking directly in nearby clouds.

The physical picture

Start with a cloud that has just tipped past its Jeans mass and begun to fall inward. As it contracts, the density everywhere climbs and the Jeans mass shrinks. A clump buried inside that was comfortably sub-Jeans — too little mass for its own gravity to grip — is suddenly super-Jeans against the new, lower threshold, and peels away to collapse independently. Repeat at every scale and the single cloud dissolves into a swarm of fragments. The first figure follows one branch of this tree quantitatively: three snapshots of the same collapsing gas, with $M_J$ and $t_{\rm ff}$ recomputed at each density.

Figure

The fragmentation cascade at T = 10 K. As density climbs from n ≈ 10² to 10⁶ cm⁻³ the Jeans mass falls from 55 to 0.55 M⊙ because M_J ∝ ρ^−1/2, and the free-fall time shortens from 3.4 Myr to 34 kyr, so denser fragments detach and collapse first

Why does the temperature stay so low? Because molecular clouds are superb refrigerators. Dust grains absorb the compressional (PdV) heating and re-radiate it as far-infrared continuum, while at lower densities CO and other lines carry heat off directly; both channels stay optically thin over a huge density range, so photons escape almost as fast as gravity supplies energy. The gas rides an isotherm, not an adiabat, and this near-constant $T$ keeps $M_J$ marching downward — until cooling can no longer keep pace. The second figure is the punchline: the run of Jeans mass against density, isothermal on the way down and adiabatic on the way back up, with the opacity-limited minimum at the knee.

Figure

The Jeans mass versus density on log axes at T = 10 K. On the isothermal branch M_J ∝ ρ^−1/2 falls through 55, 5.5, 0.55 M⊙; near n ≈ 10¹⁰ cm⁻³ the gas becomes opaque, turns adiabatic with M_J ∝ ρ^+1/2, and fragmentation halts at M_min ≈ 0.01 M⊙

The descending blue branch is where fragmentation lives; the ascending red branch is where it dies. Because $t_{\rm ff}\propto\rho^{-1/2}$ , the densest sub-clumps collapse in a fraction of the envelope's free-fall time and detach as distinct objects rather than smearing back into one body.

The core equations

The whole subject rests on the Jeans mass — the critical mass above which self-gravity overwhelms thermal pressure for a uniform sphere. It follows from virial balance of thermal and gravitational energy; the prefactor depends on convention, but the scaling is universal.

M_J = \left(\frac{5 k_B T}{G\,\mu m_H}\right)^{3/2}\left(\frac{3}{4\pi\rho}\right)^{1/2} \qquad (1)

where: $M_J$ — Jeans mass, the minimum mass for instability; $T$ — gas temperature; $\rho$ — mass density; $\mu$ — mean molecular weight (about $2.3$ for molecular gas with helium); $m_H$ — hydrogen mass; $k_B,G$ — Boltzmann and gravitational constants. A hotter or more diffuse cloud needs more mass to collapse; a colder, denser one needs less. Strip equation (1) to the two quantities that change during collapse and the engine of fragmentation is laid bare — everything hinges on the opposite signs of the two exponents.

M_J \propto T^{3/2}\,\rho^{-1/2} \qquad (2)

where: $M_J$ — Jeans mass; $T$ — temperature, held nearly constant while cooling is efficient; $\rho$ — density, which rises steeply during collapse. With $T$ pinned, the $\rho^{-1/2}$ term alone survives: as density climbs, $M_J$ falls and smaller pieces become unstable. The natural clock of the collapse is the free-fall time — how long a pressureless uniform sphere takes to collapse to a point. Remarkably it depends only on the density, not the size or mass of the region.

t_{\rm ff} = \sqrt{\frac{3\pi}{32\,G\rho}} \qquad (3)

where: $t_{\rm ff}$ — free-fall (collapse) time; $\rho$ — mean density; $G$ — gravitational constant. Because $t_{\rm ff}\propto\rho^{-1/2}$ , a region a hundred times denser collapses ten times faster — so over-dense sub-clumps run ahead of their surroundings. The instability also has a length scale — the Jeans length, the shortest wavelength self-gravity can destabilize against pressure. Perturbations larger than $\lambda_J$ grow; smaller ones oscillate as sound waves.

\lambda_J = \left(\frac{\pi c_s^2}{G\rho}\right)^{1/2},\qquad c_s=\sqrt{\frac{k_B T}{\mu m_H}} \qquad (4)

where: $\lambda_J$ — Jeans length, the minimum unstable wavelength; $c_s$ — isothermal sound speed (about $0.19\,\mathrm{km\,s^{-1}}$ at $10\,\mathrm{K}$ ); $\rho$ — density. The Jeans mass is essentially the mass in a Jeans-length sphere, $M_J\sim\rho\lambda_J^3$ , so length and mass criteria are one. Fragmentation lives or dies by the thermodynamics of the collapse. Generalize equation (2) to a polytrope $P\propto\rho^\gamma$ (so $T\propto\rho^{\gamma-1}$ ) and the density dependence of $M_J$ carries one controlling exponent.

M_J\propto\rho^{(3\gamma-4)/2}\quad\Rightarrow\quad \gamma_{\rm crit}=\tfrac{4}{3} \qquad (5)

where: $\gamma$ — effective adiabatic index of the collapsing gas; $\gamma_{\rm crit}=4/3$ — the dividing line. For $\gamma<4/3$ (isothermal is $\gamma=1$ ), $M_J$ falls as density rises and fragmentation proceeds; for $\gamma>4/3$ (opaque, $\gamma\to5/3$ ), $M_J$ rises and fragmentation is suppressed. Whether a magnetized core can collapse is fixed not by mass alone but by its ratio to the threading magnetic flux — a critical mass-to-flux ratio below which magnetic support wins.

\left(\frac{M}{\Phi}\right)_{\rm crit}=\frac{\zeta}{3\pi}\left(\frac{5}{G}\right)^{1/2}\approx\frac{0.13}{\sqrt{G}} \qquad (6)

where: $M$ — core mass; $\Phi$ — magnetic flux threading it; $\zeta\approx0.53$ — a geometry factor of order unity (Mouschovias and Spitzer). A subcritical core cannot collapse until ambipolar diffusion leaks flux out of the neutral gas; a supercritical core collapses as if the field were absent.

Derivation AWhy the Jeans mass falls during isothermal collapse

The goal is to show explicitly that near-isothermal collapse forces the Jeans mass downward as the density rises, and to turn that into a quantitative fragmentation ratio linking the mass scale at any two epochs of the collapse. We begin from the general scaling, impose the condition that efficient cooling holds the temperature fixed, and read off both the mass ratio and its consistency with the free-fall time — the two facts that together guarantee a self-perpetuating cascade rather than a one-off split.

Step 1. Take the scaling $M_J\propto T^{3/2}\rho^{-1/2}$ from equation (2). During collapse, dust and line cooling radiate the compressional heating away on a timescale short compared to $t_{\rm ff}$ , so the gas is very nearly isothermal and $T$ can be treated as constant. Why: freezing the temperature factor lets the density term act alone — the whole content of "isothermal".

Step 2. Form the ratio of the Jeans mass at two densities $\rho_1<\rho_2$ reached at successive moments of the collapse. The temperature factors cancel, leaving $M_J(\rho_2)/M_J(\rho_1)=(\rho_1/\rho_2)^{1/2}$ . Why: cancelling $T^{3/2}$ turns a scaling into a clean prediction — the threshold at the denser epoch is smaller by the square root of the density contrast.

Step 3. Read the numbers. A factor $100$ rise in density lowers $M_J$ by a factor $10$ ; a cloud starting at $M_J\sim55\,M_\odot$ can, after its density has risen $10^4$ -fold, support fragments of $\sim0.55\,M_\odot$ , each of which repeats the trick. Why: the same drop applies at every stage, so the process is a cascade generating a hierarchy of ever-smaller clumps.

Step 4. Confirm the pieces actually separate. Because $t_{\rm ff}\propto\rho^{-1/2}$ , a sub-region denser by the same factor collapses faster by its square root, so it runs ahead of the envelope and detaches as a dynamically distinct object. Why: a falling $M_J$ only licenses fragmentation; the shorter free-fall time is what physically decouples the fragment. *The $M_J$ values $55,5.5,0.55,0.055\,M_\odot$ at $n=10^2,10^4,10^6,10^8\,\mathrm{cm^{-3}}$ follow directly from equation (1) at $T=10\,\mathrm{K}$ , $\mu=2.3$ .*

Go deeper · Derivation B · 1 minThe opacity limit that stops the cascade

Derivation BThe opacity limit that stops the cascade

The goal is to locate where fragmentation must end by asking when a collapsing fragment can no longer radiate the heat of compression away fast enough to stay isothermal. We compare the rate at which gravity liberates energy to the maximum blackbody radiation rate, set them equal to mark the transition, and insert fragment conditions to recover the minimum fragment mass — the origin of the smallest stellar and substellar seeds.

Step 1. Estimate the heating rate. A fragment of mass $M$ and radius $R$ liberates gravitational energy $\sim GM^2/R$ over roughly a free-fall time, so $L_{\rm grav}\sim GM^2/(R\,t_{\rm ff})$ . Why: this is the power that must be radiated away for the temperature to hold steady — the demand side of the balance.

Step 2. Estimate the maximum cooling. No opaque body can radiate faster than a blackbody at its surface temperature, $L_{\rm max}=4\pi R^2\sigma T^4$ — the ceiling on how fast an optically thick fragment can shed heat. Why: the blackbody limit is the supply side — the largest luminosity physics allows.

Step 3. Set $L_{\rm grav}\simeq L_{\rm max}$ . While $L_{\rm grav}<L_{\rm max}$ the gas radiates freely and stays isothermal; when $L_{\rm grav}$ overtakes $L_{\rm max}$ the fragment can no longer dump its heat, the temperature rises, and the collapse turns adiabatic. Why: this crossing is the knee in the second figure — the condition that flips the $M_J$ – $\rho$ trend from falling to rising.

Step 4. Insert conditions appropriate to a collapsing fragment, $T\sim10$ to a few hundred kelvin as the gas begins to heat, and the balance yields $M_{\rm min}\sim0.01$ – $0.1\,M_\odot$ , depending only weakly on $T$ and the radiative efficiency. Why: this is the observed lower edge of the stellar and brown-dwarf range, near the hydrogen-burning limit. *The transition exponent follows from $\gamma_{\rm crit}=4/3$ in equation (5).*

Go deeper · Derivation C · 1 minFree-fall time from the collapse equation

Derivation CFree-fall time from the collapse equation

The goal is to derive the free-fall time of a pressureless uniform sphere from Newton's second law and to show it depends only on the initial density, not on the size of the region — the property that makes $t_{\rm ff}$ a clean universal clock for collapse and fragmentation. We follow a shell at the sphere's edge, integrate its equation of motion once for velocity as a function of radius, and perform the standard trigonometric substitution that carries the shell to the center.

Step 1. Write Newton's law for a shell at the edge, $\mathrm{d}^2r/\mathrm{d}t^2=-GM/r^2$ , where $M=\tfrac{4}{3}\pi\rho_0 r_0^3$ is the enclosed mass. Since interior mass is conserved in a pressureless collapse, $M$ is a constant of the motion. Why: fixing $M$ makes this a one-body Kepler-like fall — the edge shell feels only the constant interior mass beneath it.

Step 2. Integrate once using $\dot r=0$ at $r=r_0$ to obtain the energy integral $\tfrac12\dot r^2=GM(1/r-1/r_0)$ , giving the infall speed as a function of radius. Why: the first integral converts the second-order equation into a separable first-order one.

Step 3. Substitute $r=r_0\cos^2\xi$ , which maps the radial integral into $\int\cos^2\xi\,\mathrm{d}\xi$ ; as the shell falls from $r_0$ to $0$ , the angle $\xi$ runs from $0$ to $\pi/2$ . Why: the substitution linearizes the awkward $\sqrt{1/r-1/r_0}$ dependence into an elementary cosine-squared integral.

Step 4. Evaluate from $\xi=0$ to $\pi/2$ and eliminate $r_0$ using $M=\tfrac{4}{3}\pi\rho r_0^3$ . The initial radius cancels completely, leaving $t_{\rm ff}=\sqrt{3\pi/32G\rho}$ . Why: $r_0$ cancels, so every shell reaches the center at the same instant — the sphere collapses homologously and $t_{\rm ff}$ depends on density alone. *At $n=10^2\,\mathrm{cm^{-3}}$ ( $\rho=3.85\times10^{-19}\,\mathrm{kg\,m^{-3}}$ ) equation (3) gives $t_{\rm ff}=3.4\,\mathrm{Myr}$ .*

Limiting cases

The behaviour of the Jeans mass, and of fragmentation, is fixed almost entirely by the thermodynamics of the collapse and whether magnetic support is present:

Case	Result	Meaning
Isothermal collapse ( $\gamma=1$ )	$M_J\propto\rho^{-1/2}$	Efficient cooling pins $T$ ; $M_J$ falls as density rises, so fragmentation proceeds. This is the molecular-cloud regime.
General polytrope	$M_J\propto\rho^{(3\gamma-4)/2}$	The exponent changes sign at $\gamma=4/3$ : below it fragmentation runs, above it stops.
Adiabatic / opaque ( $\gamma=5/3$ )	$M_J\propto\rho^{+1/2}$	Heat is trapped, $T$ rises with $\rho$ , $M_J$ climbs — the opacity limit, fragmentation suppressed.
Opacity-limited minimum	$M_{\rm min}\sim0.01$ – $0.1\,M_\odot$	Smallest fragment where the gas turns adiabatic; the low-mass end of the stellar and brown-dwarf range.
Magnetic support (subcritical)	$M/\Phi<(M/\Phi)_{\rm crit}$	Magnetic pressure prevents collapse until ambipolar diffusion removes flux from the neutral gas.

Common trapThe Jeans criterion is an exact stability boundary for a collapsing cloud

It is not. The classical Jeans analysis linearizes small perturbations about a static, infinite, uniform medium, discarding the gravity of the unperturbed background — the notorious "Jeans swindle", since a uniform infinite self-gravitating medium cannot actually be in equilibrium. A real collapsing cloud is contracting and manifestly out of equilibrium, so $M_J(\rho,T)$ is a running order-of-magnitude threshold, not a sharp on-off switch: expect fragmentation when a region exceeds a few times its local Jeans mass, not the instant it crosses it. Two further cautions. First, fragmentation demands near-isothermality; if the gas heats appreciably ( $\gamma>4/3$ ), $M_J$ rises and the cascade is quenched — the opacity limit is exactly this failure. Second, real clouds are turbulent and magnetized, so the effective Jeans mass includes turbulent and magnetic support, not thermal pressure alone; the purely thermal $M_J$ can underestimate the collapse mass by a large factor.

Worked examples

1 · Jeans mass, length, and free-fall time of a cloud clump (confidence builder)

A molecular clump has $T=10\,\mathrm{K}$ , $\mu=2.3$ , and $n=10^2\,\mathrm{cm^{-3}}$ (so $\rho=\mu m_H n=3.85\times10^{-19}\,\mathrm{kg\,m^{-3}}$ ). Find its sound speed, Jeans length, Jeans mass, and free-fall time.

Step 1. Sound speed from equation (4): $c_s=\sqrt{k_B T/\mu m_H}=\mathbf{1.9\times10^2\ m\,s^{-1}}$ , about $0.19\,\mathrm{km\,s^{-1}}$ . That is far below the few- $\mathrm{km\,s^{-1}}$ turbulent linewidths observed — a first hint that turbulence, not thermal pressure, dominates the real support.

Step 2. Jeans length: $\lambda_J=\sqrt{\pi c_s^2/G\rho}=6.6\times10^{16}\,\mathrm{m}=\mathbf{2.1\ pc}$ . Any enhancement larger than a couple of parsecs at this density is unstable.

Step 3. Jeans mass from equation (1): the first factor is $5k_BT/(G\mu m_H)=2.69\times10^{15}\,\mathrm{kg\,m^{-1}}$ and the second $(3/4\pi\rho)^{1/2}=7.9\times10^{8}$ , so $M_J=1.1\times10^{32}\,\mathrm{kg}=\mathbf{55\ M_\odot}$ — a clump must gather about fifty solar masses before gravity wins here.

Step 4. Free-fall time from equation (3): $t_{\rm ff}=\sqrt{3\pi/32G\rho}=1.1\times10^{14}\,\mathrm{s}=\mathbf{3.4\ Myr}$ . So at $T=10\,\mathrm{K}$ , $n=10^2\,\mathrm{cm^{-3}}$ : $c_s\approx0.19\,\mathrm{km\,s^{-1}}$ , $\lambda_J\approx2.1\,\mathrm{pc}$ , $M_J\approx55\,M_\odot$ , and $t_{\rm ff}\approx3.4\,\mathrm{Myr}$ .

2 · Following the cascade down to the opacity limit

Keep $T=10\,\mathrm{K}$ fixed and track the Jeans mass and free-fall time as the collapse drives the density from $n=10^2$ up to $10^8\,\mathrm{cm^{-3}}$ . Where does the near-isothermal cascade run out?

Step 1. No constants need re-evaluating: because $M_J\propto\rho^{-1/2}$ , each factor $10^2$ in density lowers $M_J$ by exactly $10$ . Starting from $55\,M_\odot$ , the sequence is $\mathbf{55\to5.5\to0.55\to0.055\ M_\odot}$ at $n=10^2,10^4,10^6,10^8\,\mathrm{cm^{-3}}$ .

Step 2. The free-fall time, $t_{\rm ff}\propto\rho^{-1/2}$ , drops in lockstep: $\mathbf{3.4\ Myr\to340\ kyr\to34\ kyr\to3.4\ kyr}$ . The densest fragments finish long before the envelope — the decoupling that makes them separate protostars.

Step 3. Locate the end. Fragmentation needs the gas isothermal, i.e. $L_{\rm grav}\lesssim L_{\rm max}$ . This fails around $n\sim10^{10}$ – $10^{11}\,\mathrm{cm^{-3}}$ , where the fragment becomes optically thick to its own dust emission; below that scale $M_J$ turns around and climbs as $\rho^{+1/2}$ .

Step 4. The turnover sets the minimum fragment mass. Extrapolating the isothermal branch to the opacity density gives a few $\times10^{-2}\,M_\odot$ , matching the radiative estimate $M_{\rm min}\sim0.01$ – $0.1\,M_\odot$ . The cascade converts one $\sim55\,M_\odot$ clump into many fragments and bottoms out near $M_{\rm min}\approx0.01\,M_\odot$ — the brown-dwarf boundary — not because gravity weakens but because the gas stops cooling.

Physics of the portrait

The theory of fragmentation is the bridge from Jeans's equilibrium instability to the modern picture of a hierarchical, radiation-regulated collapse — largely the legacy of one restlessly original mind who also wrote the periodic table of the stars.

Origins · 1 minFred Hoyle and how this was discovered

Fred Hoyle

1915–2001 · the fragmentation cascade and stellar nucleosynthesis

James Jeans supplied the criterion in 1902 — a self-gravitating gas is unstable above a critical wavelength — but Jeans imagined a single condensation, not a shattering. It was Fred Hoyle who, in a landmark 1953 paper, recognized that a collapsing cloud need not condense as a whole: because the Jeans mass falls as the density rises under near-isothermal cooling, the cloud must break repeatedly into ever-smaller pieces, a hierarchical cascade that explains why stars are born in clusters rather than as isolated giants. Hoyle was already the most inventive astrophysicist of his generation. With Bondi and Gold he proposed the steady-state cosmology, and he coined the term "Big Bang", in derision, for the rival theory. His deepest contribution came in 1957, when the B²FH paper — Burbidge, Burbidge, Fowler, and Hoyle — laid out how essentially every element heavier than helium is forged in stars, built on his prediction of a resonance in carbon-12 (the "Hoyle state") deduced purely from the fact that carbon-based life exists. The cascade he launched was made quantitative by others: Larson's 1969 calculations showed the non-homologous, inside-out character of real protostellar collapse, and in 1976 Low and Lynden-Bell, with Rees independently, fixed the lower end at the opacity limit, $\sim0.01\,M_\odot$ . Hoyle stayed gloriously contrarian to the end, but the picture of a cloud dissolving into a spectrum of stars, set in motion by a falling Jeans mass, is his enduring monument.

History

1902

— Jeans publishes the instability criterion for a self-gravitating gas

1928

— Jeans frames gravitational condensation in Astronomy and Cosmogony

1953

— Hoyle proposes the hierarchical fragmentation cascade of a collapsing cloud

1969

— Larson computes the non-homologous collapse of a protostellar cloud

1976

— Low, Lynden-Bell, and Rees fix the opacity-limited minimum fragment mass

Going further

Texts. Carroll and Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 12 (Star Formation) · Stahler and Palla, The Formation of Stars (fragmentation, opacity limit, protostellar collapse) · Binney and Tremaine, Galactic Dynamics, 2nd ed., §5.2 (Jeans instability and free-fall) · Mo, van den Bosch, and White, Galaxy Formation and Evolution, ch. 8 (gravitational instability and cooling) · Dodelson, Modern Cosmology (Jeans length and perturbation growth). Key anchor values: $M_J\propto T^{3/2}\rho^{-1/2}$ from virial balance gives $M_J=55\,M_\odot$ at $T=10\,\mathrm{K}$ , $n=10^2\,\mathrm{cm^{-3}}$ , $\mu=2.3$ ; the free-fall time $t_{\rm ff}=\sqrt{3\pi/32G\rho}=3.4\,\mathrm{Myr}$ at that density follows from the $r=r_0\cos^2\xi$ substitution; $\gamma_{\rm crit}=4/3$ from $(3\gamma-4)/2=0$ ; the opacity-limited minimum is $\sim0.01$ – $0.1\,M_\odot$ ; and the critical mass-to-flux ratio is $\zeta/(3\pi)\sqrt{5/G}\approx0.13/\sqrt{G}$ .

Prerequisites. The Jeans Instability and Jeans Mass · Virial Theorem for Molecular Clouds · Molecular Clouds and Giant Molecular Clouds · Interstellar Gas Heating and Cooling Processes. Leads to. The Stellar Initial Mass Function · Protostellar Disks and Binary Formation · Thermal Dust Emission and Infrared Radiation.

Ask the tutor: Want to see why the Jeans mass turns over at exactly $\gamma=4/3$ , or how ALMA catches a disk fragmenting into a binary — just ask.

Why does a cloud fragment instead of collapsing into one big star?

Because the Jeans mass falls as the collapse proceeds. Self-gravity beats pressure only above $M_J\propto T^{3/2}\rho^{-1/2}$ , and while the cloud stays cold ( $T$ nearly constant) the rising density drives $M_J$ down. A sub-region that held far less than the original Jeans mass suddenly holds more than the new, smaller one, so it becomes unstable in its own right and collapses independently. The process repeats at every scale, subdividing a single $\sim10^4$ – $10^6\,M_\odot$ cloud into hundreds or thousands of stellar-mass cores — which is exactly why stars form in clusters rather than as isolated monsters.

What keeps the gas isothermal during collapse, and why does it matter?

Molecular clouds are extraordinarily efficient radiators. Dust grains absorb the heat of compression and re-emit it as far-infrared photons that escape freely; at lower densities, molecular lines such as CO carry heat off directly. Both channels stay optically thin over a wide density range, so the gas radiates almost as fast as gravity supplies heat, holding the temperature near $10\,\mathrm{K}$ . This matters because the cascade lives or dies on the temperature staying fixed: only with $T$ constant does $M_J\propto\rho^{-1/2}$ fall as the cloud contracts. If the gas heated adiabatically instead, $M_J$ would rise and the cloud would collapse monolithically — so isothermality is the enabling condition for fragmentation.

What actually stops the fragmentation cascade?

Opacity. As a fragment collapses it becomes optically thick to its own thermal radiation, so the heat of compression can no longer escape: the gas heats, the effective adiabatic index rises above the critical $4/3$ , and the Jeans mass turns around and increases with density. Pressure now wins locally, and the fragment settles into a single pressure-supported protostar rather than splitting again. Balancing gravitational heating against the maximum blackbody cooling gives a minimum fragment mass of order $0.01$ – $0.1\,M_\odot$ — the boundary between the smallest stars and brown dwarfs, roughly the lowest stellar mass we observe.

Why is the free-fall time independent of the cloud's size?

Because a uniform sphere collapses homologously. Every shell feels only the mass interior to it, and for uniform density that enclosed mass scales so that all shells reach the center at the same instant. When you integrate, $r_0$ cancels completely, leaving $t_{\rm ff}=\sqrt{3\pi/32G\rho}$ , a function of density alone. So a diffuse cloud and a dense clump of the same density collapse in the same time — but denser regions collapse faster, since $t_{\rm ff}\propto\rho^{-1/2}$ . That density dependence, not size, is what lets over-dense fragments outrun their surroundings.

How do magnetic fields change the fragmentation story?

They can prevent it. A magnetic field resists compression perpendicular to the field lines, adding support thermal pressure does not. Whether the field wins is set by the mass-to-flux ratio: a subcritical core, with $M/\Phi$ below $\approx0.13/\sqrt{G}$ , cannot collapse while the field is frozen into the gas. Collapse then waits on ambipolar diffusion — the slow drift of neutral gas past the ions tied to the field — which leaks flux out of the core until it becomes supercritical. This regulates the pace of low-mass star formation in magnetically dominated regions, and magnetic braking simultaneously drains angular momentum, shaping whether a core fragments into a binary.

Does fragmentation explain the stellar initial mass function?

It explains the ends of it, and frames the middle. The opacity-limited minimum, $\sim0.01$ – $0.1\,M_\odot$ , naturally accounts for the low-mass turnover of the IMF near the hydrogen-burning limit and the existence of brown dwarfs below it — fragmentation cannot make anything smaller. The subdivision into many cores explains why most of the mass ends up in low-mass stars rather than a few giants. But the detailed shape of the IMF — the high-mass slope, the characteristic mass near a few tenths of a solar mass — needs more than the thermal Jeans argument: it emerges from turbulent density fluctuations, competitive accretion, and radiative feedback, which is why the IMF gets its own article. Fragmentation sets the stage; turbulence and feedback write the script.