Dense Cores and Prestellar Cores

A giant molecular cloud does not collapse as a whole into one enormous star; it fragments, and fragments again, until star formation localizes into discrete, gravitationally bound knots called dense cores. These are the terminal links of the interstellar hierarchy — giant molecular cloud → clump → dense core → protostar — the smallest structures in which an individual star's mass is first assembled. A typical core is cold and quiet: size of order $0.1\ \mathrm{pc}$ , density $n_{\mathrm{H_2}}\sim10^4$ – $10^6\ \mathrm{cm^{-3}}$ , temperature near $10\ \mathrm{K}$ , mass from a fraction of a solar mass to a few $M_\odot$ . Its linewidths are subsonic or transonic — the signature of a region where turbulence has decayed and thermal pressure competes with gravity on equal footing. Cores announce themselves where the ordinary tracer fails. CO, the workhorse of molecular astronomy, freezes onto grain mantles at the density and temperature of a core interior, lighting the envelope but going dark in the nucleus; the dense gas is read instead in ammonia ( $\mathrm{NH_3}$ ) and diazenylium ( $\mathrm{N_2H^+}$ ), whose abundances survive or peak in the frozen interior, and in the cold submillimetre dust continuum. The physics that decides a core's fate is compact and classical: an isothermal, self-gravitating, pressure-confined sphere has a maximum stable mass, the critical Bonnor–Ebert mass. A lighter core rests in equilibrium; a heavier one has no equilibrium at all and must collapse. Because the observed population piles up at that threshold, and its mass spectrum echoes the stellar initial mass function, prestellar cores are read as the direct blueprints of individual stars.

Key takeaways

A dense core is the smallest, most centrally condensed unit of a molecular cloud: $R\sim0.05$ – $0.3\ \mathrm{pc}$ , $n_{\mathrm{H_2}}\gtrsim10^4\ \mathrm{cm^{-3}}$ , $T\approx10\ \mathrm{K}$ , $M\sim0.5$ –a few $M_\odot$ .
Cores are traced by high-density lines ( $\mathrm{NH_3}$ , $\mathrm{N_2H^+}$ ) and cold dust continuum; CO freezes onto grains above $\sim10^4\ \mathrm{cm^{-3}}$ and fails inside.
A prestellar core is a dense core that is additionally gravitationally bound but still starless — the precursor of one protostar, distinct from unbound starless cores that re-disperse.
Its structure is a Bonnor–Ebert sphere: a flat inner plateau and an $r^{-2}$ envelope, with a maximum stable mass $M_{\rm BE}\approx1.18\,c_s^4/\sqrt{G^3P_{\rm ext}}$ that turns over at density contrast $14.1$ .
Thermal, turbulent, and magnetic support sit near equipartition with gravity, so cores cluster at the critical/virial threshold — and the core mass function resembles the IMF shifted by efficiency $\sim1/3$ .
Collapse follows when a core exceeds its critical mass: promptly if magnetically supercritical, or after ambipolar diffusion removes flux from subcritical ones, feeding a protostar by inside-out infall.
Key idea: a prestellar core is a marginally critical Bonnor–Ebert / near-virial sphere — a cold, thermally supported, gravitationally bound clump poised on the edge of runaway collapse — whose critical mass of a few $M_\odot$ at $10\ \mathrm{K}$ sets the scale of an individual star.

The core insight

A prestellar core is a marginally critical Bonnor–Ebert / near-virial sphere: a centrally condensed, thermally supported, gravitationally bound clump of cold molecular gas sitting at the threshold between stable equilibrium and runaway collapse, whose critical mass — a few $M_\odot$ at $T\sim10\ \mathrm{K}$ — directly sets the scale of individual stars.

Where this lives today

Dense and prestellar cores are the front line of modern star-formation science. ALMA now resolves the flat-plus- $r^{-2}$ Bonnor–Ebert structure of individual cores at few-hundred-au scale and maps the blue-asymmetric infall signature in optically thick $\mathrm{HCO^+}$ and $\mathrm{CS}$ — the double-peaked profile, brighter on the blueshifted side with a thin $\mathrm{N_2H^+}$ peak between, that fingerprints genuine collapse rather than rotation. Herschel's Gould Belt surveys delivered the core mass function for thousands of cores and its Salpeter-like tail, offset from the IMF by the $\sim1/3$ efficiency that makes cores the presumptive units of star formation. On the magnetic gate, Zeeman splitting of $\mathrm{OH}$ and $\mathrm{CN}$ pins down line-of-sight fields while polarized dust from JCMT/BISTRO and SOFIA/HAWC+ maps the plane-of-sky field, together yielding mass-to-flux ratios mostly marginally supercritical ( $\mu_\Phi\gtrsim1$ ). The open questions are sharp: is the core mass function truly the origin of the IMF, or reshaped by fragmentation and competitive accretion? Is collapse triggered promptly by turbulence or paced by ambipolar diffusion?

The physical picture

Picture a cold, nearly isothermal ball of molecular gas embedded in the warmer, thinner cloud that presses on it from every side. Thermal pressure holds it up, but gravity bites hardest at the centre, so the equilibrium is not a uniform sphere but a flat-topped plateau wrapped in an $r^{-2}$ halo. Add mass, or squeeze harder, and the centre grows more concentrated — until a critical concentration beyond which no pressure structure balances its own weight. That turnover is the whole story: on one side a core is a placid, long-lived equilibrium; a hair past it, the centre runs away. The first figure is the anatomy of a core, the second the stability curve whose peak sets the critical Bonnor–Ebert mass.

Figure

Cross-section of a dense core: a CO-bright envelope at n(H₂) ~ 10⁴ cm⁻³ surrounds a CO-depleted shell and a cold N₂H⁺ / NH₃ nucleus reaching n_c ≳ 10⁵–10⁶ cm⁻³ at T ≈ 10 K, and the radial density is flat inside r₀ then falls as ρ ∝ r⁻² through the envelope

The tracers behave like filters tuned to density. CO glows in the diffuse envelope but vanishes in the frozen nucleus; ammonia and $\mathrm{N_2H^+}$ need densities above $\sim10^4\ \mathrm{cm^{-3}}$ to be excited or to survive, so they switch on where CO switches off. ( $\mathrm{N_2H^+}$ is destroyed by gas-phase CO, so its abundance rises as CO freezes out — a near-perfect flashlight for the collapsing centre.) And the near-coincidence of thermal, turbulent, and magnetic energy with gravity is not accidental: over-supported clumps re-expand and under-supported ones have already collapsed, so the catalogued population is the marginally critical one — which is why measured cores pile up at the peak of the stability curve below.

Figure

The Bonnor–Ebert stability curve: dimensionless bounding pressure versus central-to-edge density contrast ρc / ρedge. Equilibria on the rising green branch are stable, the curve peaks at contrast 14.1 (dimensionless radius ξmax = 6.45) — the critical sphere of mass M_BE ≈ 1.18 cs⁴ / √(G³ Pext) — and beyond it no pressure balances gravity so the core collapses

The resemblance between the core mass function and the stellar IMF closes the argument: a star largely remembers the mass of the core it condensed from. Fragmentation, outflows, and accretion modulate the map, but to leading order the interstellar medium pre-sorts gas into a mass spectrum before a nuclear fire is lit.

The core equations

A dense core is defined observationally by a compact bundle of numbers — the density, temperature, size, and mass that place it below "clump" and above "protostar". Everything analytic follows.

n_{\mathrm{H_2}}\sim10^4\text{–}10^6\ \mathrm{cm^{-3}},\quad T\approx10\ \mathrm{K},\quad R\sim0.05\text{–}0.3\ \mathrm{pc},\quad M\sim0.5\text{–}\text{few}\ M_\odot \qquad (1)

where: $n_{\mathrm{H_2}}$ — molecular hydrogen number density; $T$ — gas and dust kinetic temperature (coupled here); $R$ — core radius; $M$ — core mass. Reading: cold, dense, sub-parsec, roughly stellar in mass — which is why one core makes one star. Support against gravity is thermal, and because cooling pins $T$ during contraction the relevant speed is the isothermal sound speed, not the adiabatic one — it sets every scale below.

c_s=\sqrt{\frac{k_B T}{\mu m_H}}\approx0.19\ \mathrm{km\,s^{-1}}\left(\frac{T}{10\ \mathrm{K}}\right)^{1/2} \qquad (2)

where: $c_s$ — isothermal sound speed; $k_B$ — Boltzmann constant; $\mu\approx2.33$ — mean weight per particle for $\mathrm{H_2}$ + He; $m_H$ — hydrogen mass. Reading: at $10\ \mathrm{K}$ the thermal speed is a fifth of a km/s — any linewidth much broader than this is turbulence, not heat. The equilibrium structure is a Bonnor–Ebert sphere — isothermal, self-gravitating gas bounded by external pressure — nearly flat inside a scale radius $r_0$ and falling as $r^{-2}$ outside.

\rho(r)\to\rho_c\ \ (r\ll r_0),\qquad \rho(r)\propto r^{-2}\ \ (r\gg r_0),\qquad r_0=\frac{c_s}{\sqrt{4\pi G\rho_c}} \qquad (3)

where: $\rho_c$ — central density; $r_0$ — flattening scale, the isothermal Jeans length at the centre. Reading: the flat core plus $r^{-2}$ envelope is the fingerprint of a pressure-bounded equilibrium, not a collapsing singular sphere. The central result of the topic is that this equilibrium has a maximum mass. Confined at pressure $P_{\rm ext}$ , an isothermal sphere holds no more than the critical Bonnor–Ebert mass; a heavier core has no static solution and collapses.

M_{\rm BE}=\frac{m_1\,c_s^4}{\sqrt{G^3P_{\rm ext}}},\qquad m_1\approx1.18 \qquad (4)

where: $M_{\rm BE}$ — maximum stable mass of the confined sphere; $P_{\rm ext}$ — external pressure; $G$ — gravitational constant; $m_1\approx1.18$ — constant from the Lane–Emden turnover. Reading: $M_{\rm BE}\propto c_s^4/\sqrt{P_{\rm ext}}$ , so colder or lower-pressure gas has a larger critical mass; at $10\ \mathrm{K}$ it is of order one solar mass — the stellar mass scale. A complementary, model-independent test is the virial theorem: comparing kinetic-plus-thermal energy against self-gravity yields a virial mass and a parameter for whether a core is bound.

M_{\rm vir}=\frac{5\,\sigma^2 R}{G},\qquad \alpha_{\rm vir}\equiv\frac{M_{\rm vir}}{M}=\frac{5\sigma^2 R}{GM} \qquad (5)

where: $\sigma^2=c_s^2+\sigma_{\rm NT}^2$ — one-dimensional velocity dispersion (thermal + non-thermal); $M_{\rm vir}$ — mass balancing gravity against internal motions for a uniform sphere. Reading: $\alpha_{\rm vir}\lesssim1$ – $2$ is a bound, near-virial core; $\alpha_{\rm vir}\gg1$ a transient fluctuation. Once bound and supercritical, a core collapses on essentially the free-fall time — the clock of star formation, set by density.

t_{\rm ff}=\sqrt{\frac{3\pi}{32\,G\rho}}\approx1.0\times10^5\ \mathrm{yr}\left(\frac{n_{\mathrm{H_2}}}{10^5\ \mathrm{cm^{-3}}}\right)^{-1/2} \qquad (6)

where: $t_{\rm ff}$ — free-fall time; $\rho=\mu_c m_H n_{\mathrm{H_2}}$ with $\mu_c\approx2.8$ (mass per $\mathrm{H_2}$ including helium). Reading: the free-fall clock is $\sim10^5\ \mathrm{yr}$ at core densities; that cores live several times longer signals near-critical support. Whether collapse begins promptly also depends on the field, through the mass-to-flux ratio: only a supercritical core, whose gravity beats its magnetic tension, contracts without first shedding flux.

\mu_\Phi\equiv\frac{M/\Phi}{(M/\Phi)_{\rm crit}},\qquad \left(\frac{M}{\Phi}\right)_{\rm crit}=\frac{c_\Phi}{\sqrt{G}},\quad c_\Phi\approx0.13 \qquad (7)

where: $\Phi$ — magnetic flux threading the core; $\mu_\Phi>1$ supercritical, $\mu_\Phi<1$ subcritical (supported); $c_\Phi\approx0.13$ for a uniform sphere. Reading: a subcritical core must lose flux by ambipolar diffusion first, delaying collapse by megayears. Finally, the mass itself is most often measured from cold dust: in the optically thin limit the submillimetre flux is proportional to mass, the standard observational estimator.

M=\frac{S_\nu\,d^2}{\kappa_\nu\,B_\nu(T_d)},\qquad I_\nu=B_\nu(T_d)\,(1-e^{-\tau_\nu})\approx B_\nu(T_d)\,\kappa_\nu\Sigma \qquad (8)

where: $S_\nu$ — flux density; $d$ — distance; $\kappa_\nu$ — dust opacity (including gas-to-dust $\sim100$ ); $B_\nu(T_d)$ — Planck function; $\Sigma$ — column density; $\tau_\nu$ — dust optical depth. Reading: mass scales as $1/\kappa_\nu$ , whose factor-of-few uncertainty dominates every core-mass catalogue.

Derivation ABonnor-Ebert critical mass from hydrostatic equilibrium

The goal is to show from first principles that a pressure-confined isothermal self-gravitating sphere has a maximum mass, and to obtain its numerical coefficient and its value for a real $10\ \mathrm{K}$ core. We start from hydrostatic balance, impose the isothermal equation of state and Poisson's equation, recast into the dimensionless isothermal Lane–Emden form, locate the turnover of the mass–pressure relation, and read off the critical mass — the result that underlies the entire notion of a well-defined stellar mass scale.

Step 1. Balance the pressure gradient against gravity, $\dfrac{\mathrm{d}P}{\mathrm{d}r}=-\dfrac{GM(r)\rho}{r^2}$ , with $\mathrm{d}M/\mathrm{d}r=4\pi r^2\rho$ . Why: hydrostatic equilibrium says the core is neither expanding nor contracting — the condition whose breakdown is collapse.

Step 2. Impose the isothermal equation of state $P=\rho c_s^2$ with $c_s$ constant, so $\mathrm{d}P=c_s^2\,\mathrm{d}\rho$ . Why: cooling keeps $T$ — hence $c_s$ — fixed during contraction, so the isothermal (not adiabatic) closure is correct for a $10\ \mathrm{K}$ core.

Step 3. Combine into one second-order equation, $\dfrac{c_s^2}{\rho}\dfrac{\mathrm{d}}{\mathrm{d}r}\!\left(\dfrac{r^2}{\rho}\dfrac{\mathrm{d}\rho}{\mathrm{d}r}\right)=-4\pi G r^2$ . Why: eliminating $M(r)$ turns two coupled relations into one equation for the density profile alone.

Step 4. Non-dimensionalize with $\rho=\rho_c e^{-\psi}$ , $\xi=r/r_0$ , $r_0=c_s/\sqrt{4\pi G\rho_c}$ , giving the isothermal Lane–Emden equation $\dfrac{1}{\xi^2}\dfrac{\mathrm{d}}{\mathrm{d}\xi}\!\left(\xi^2\dfrac{\mathrm{d}\psi}{\mathrm{d}\xi}\right)=e^{-\psi}$ , integrated outward from $\psi(0)=\psi'(0)=0$ . Why: rescaling collapses every core onto one universal solution, so stability becomes a single number, not a family of models.

Step 5. Truncate at radius $R$ where the medium sets $P_{\rm ext}=\rho(R)c_s^2$ , then find the peak of $P_{\rm ext}(M)$ at fixed mass. The series turns over at $\xi_{\max}\simeq6.45$ , where the contrast is $\rho_c/\rho_{\rm edge}\approx14.1$ . Why: beyond this concentration a small compression raises gravity faster than pressure responds, so no stable solution exists — the origin of the critical mass.

Step 6. Evaluate the mass at the turnover: $M_{\rm BE}=1.18\,c_s^4/\sqrt{G^3P_{\rm ext}}\approx1.15\,c_s^3/\sqrt{G^3\rho_c}$ . For $T=10\ \mathrm{K}$ ( $c_s=0.19\ \mathrm{km\,s^{-1}}$ ) and $P_{\rm ext}/k_B\sim10^5\ \mathrm{K\,cm^{-3}}$ this gives $M_{\rm BE}\approx1.2\,M_\odot$ . Why: this coefficient and near-solar mass make the Bonnor–Ebert sphere a quantitative theory of the stellar mass scale. *The values $\xi_{\max}\simeq6.45$ , contrast $\simeq14.1$ , and $m_1\simeq1.18$ are the standard Bonnor–Ebert constants; at $10\ \mathrm{K}$ and $P/k_B=10^5\ \mathrm{K\,cm^{-3}}$ they yield $M_{\rm BE}=1.2\,M_\odot$ .*

Go deeper · Derivation B · 1 minVirial condition for a bound prestellar core

Derivation BVirial condition for a bound prestellar core

The goal is to derive, independently of any assumed density profile, the mass at which self-gravity overcomes the combined thermal, turbulent, and magnetic support, and to produce the virial parameter used to classify cores as bound or transient. We take the scalar virial theorem for a magnetized cloud, drop the surface and time-derivative terms, insert the uniform-sphere energies, solve for the critical mass, then restore the magnetic term to recover the mass-to-flux criterion.

Step 1. Write the scalar virial theorem for a static core with negligible surface pressure, $2(K+U_{\rm th})+W+\mathcal{M}=0$ ( $W<0$ gravity, $\mathcal{M}\ge0$ magnetic). Why: the virial theorem is an energy budget holding for any equilibrium regardless of internal structure, so it tests boundedness without a model.

Step 2. Insert the uniform-sphere energies $K+U_{\rm th}=\tfrac32M\sigma^2$ and $W=-\tfrac35\,GM^2/R$ , with $\sigma^2=c_s^2+\sigma_{\rm NT}^2$ . Why: one dispersion $\sigma$ lets a single linewidth stand in for all thermal and turbulent support.

Step 3. Set $\mathcal{M}=0$ to isolate the hydrodynamic balance, $3M\sigma^2=\tfrac35\,GM^2/R$ , and solve for $M_{\rm vir}=5\sigma^2 R/G$ ; equivalently $\alpha_{\rm vir}=5\sigma^2 R/(GM)$ . Why: $\alpha_{\rm vir}$ compresses the energy budget into one number — below about unity the core is bound and can collapse, well above it it is unbound and transient.

Step 4. Restore the field: magnetic energy adds a mass floor $M_\Phi=c_\Phi\Phi/\sqrt{G}$ that gravity must also beat, so collapse requires both $\alpha_{\rm vir}\lesssim1$ and $M>M_\Phi$ ( $\mu_\Phi>1$ ). Why: a core can be bound yet magnetically supported, so the mass-to-flux ratio is a second, independent gate on collapse.

Step 5. Combine the gates: a prestellar core is simultaneously starless, bound ( $\alpha_{\rm vir}\lesssim1$ ), super-BE, and supercritical — the intersection selecting the objects destined to form stars. Why: this is the precise definition separating "prestellar" from the broader, mostly transient "starless" class. *The coefficient $5$ in $M_{\rm vir}$ follows directly from the uniform-sphere gravitational energy $W=-\tfrac{3}{5}GM^2/R$ .*

Go deeper · Derivation C · 1 minCore mass from optically thin dust continuum

Derivation CCore mass from optically thin dust continuum

The goal is to justify the workhorse mass estimator $M=S_\nu d^2/[\kappa_\nu B_\nu(T_d)]$ , which turns a single submillimetre flux into a core mass, and to expose the assumptions it rests on. We take the transfer equation in the optically thin limit, integrate intensity over the solid angle, convert to physical area at the distance, invert for the mass, and flag the dominant systematic.

Step 1. In the optically thin limit $\tau_\nu\ll1$ the transfer solution $I_\nu=B_\nu(T_d)(1-e^{-\tau_\nu})$ linearizes to $I_\nu\approx B_\nu(T_d)\,\kappa_\nu\Sigma$ . Why: cold cores are genuinely thin at submillimetre wavelengths, so emission tracks column density rather than saturating at the Planck value.

Step 2. Integrate over the beam, $S_\nu\approx B_\nu(T_d)\,\kappa_\nu\!\int\Sigma\,\mathrm{d}\Omega$ , and use $\mathrm{d}\Omega=\mathrm{d}A/d^2$ to write $\int\Sigma\,\mathrm{d}\Omega=M/d^2$ . Why: the solid-angle integral of the column is just the total mass over distance squared — geometry does the bookkeeping.

Step 3. Combine to $S_\nu=\kappa_\nu B_\nu(T_d)\,M/d^2$ and invert for $M=S_\nu d^2/[\kappa_\nu B_\nu(T_d)]$ . Why: this is the formula behind every submillimetre core-mass catalogue.

Step 4. Flag the systematics: $M\propto\kappa_\nu^{-1}$ with $\kappa_\nu$ uncertain by a factor of a few, while $T_d\sim10\ \mathrm{K}$ must be fixed independently from the SED peak. Why: these two inputs, not the flux, set the true error bar on a core mass — and it propagates into the core mass function. *This result is the standard submillimetre mass formula; the Rayleigh–Jeans and full-Planck evaluations of $B_\nu(T_d)$ give consistent masses at $10\ \mathrm{K}$ .*

Limiting cases

The character of a core — and whether it will ever form a star — is set by which support dominates and by whether the core has crossed its critical mass.

Case	Result	Meaning
Thermally dominated (subsonic)	$\sigma_{\rm NT}\ll c_s,\ M\lesssim M_{\rm BE}$	A quiescent, near-hydrostatic Bonnor–Ebert sphere supported by thermal pressure; typical of isolated cores in low-mass clouds like Taurus.
Turbulence supported	$\sigma_{\rm NT}\gtrsim c_s$	Supersonic non-thermal linewidths; turbulent (often plus magnetic) support dominates, common in massive cluster-forming clumps.
Magnetically subcritical	$M<M_\Phi=c_\Phi\Phi/\sqrt{G}$	Magnetic flux prevents collapse; the core must first lose flux by ambipolar diffusion over several Myr before it can contract.
Supercritical / prestellar	$M>M_{\rm BE}$ and $\mu_\Phi>1$	Bound and unstable: gravity beats thermal + magnetic support, triggering inside-out collapse toward a protostar.
Just below critical	$M\to M_{\rm BE}^-$	Marginally stable near contrast $14.1$ ; long-lived — the state most catalogued cores are caught in.

Common trapDo not conflate starless, prestellar, and gravitationally bound

These three labels are routinely used as synonyms, and they are not. Every prestellar core is starless — no embedded protostar — but many starless cores are unbound: transient, pressure-confined fluctuations with $\alpha_{\rm vir}\gg1$ that re-disperse without ever forming a star. Only the bound subset ( $\alpha_{\rm vir}\lesssim1$ and $M\gtrsim M_{\rm BE}$ ) earns the name prestellar. The second half of the trap is observational: do not use CO to trace a core interior. CO depletes onto grains above $\sim10^4\ \mathrm{cm^{-3}}$ , so a CO map is blind to the densest gas and underestimates the central density — the region that decides the core's fate. Use $\mathrm{NH_3}$ , $\mathrm{N_2H^+}$ , or dust instead.

Worked examples

1 · Is a Taurus core prestellar? (fully worked)

A dense core in Taurus has $T=10\ \mathrm{K}$ , sits in cloud gas of pressure $P_{\rm ext}/k_B=2\times10^5\ \mathrm{K\,cm^{-3}}$ , and is measured from dust continuum to have $M=1.5\,M_\odot$ , radius $R=0.05\ \mathrm{pc}$ , and one-dimensional velocity dispersion $\sigma\approx0.20\ \mathrm{km\,s^{-1}}$ . Is it prestellar?

Step 1. Sound speed: $c_s=\sqrt{k_BT/(\mu m_H)}$ with $\mu=2.33$ , $T=10\ \mathrm{K}$ gives $c_s=\mathbf{0.19\ km\,s^{-1}}$ . The dispersion $\sigma=0.20\ \mathrm{km\,s^{-1}}$ is barely above $c_s$ , so the core is essentially thermally supported.

Step 2. Critical Bonnor–Ebert mass: with $P_{\rm ext}=2.8\times10^{-12}\ \mathrm{Pa}$ , $M_{\rm BE}=1.18\,c_s^4/\sqrt{G^3P_{\rm ext}}=\mathbf{0.82\,M_\odot}$ . The observed $1.5\,M_\odot$ exceeds this, so no hydrostatic equilibrium exists at this pressure — the sphere is super-critical.

Step 3. Cross-check with the virial parameter $\alpha_{\rm vir}=5\sigma^2R/(GM)$ . With $\sigma=200\ \mathrm{m\,s^{-1}}$ , $R=1.54\times10^{15}\ \mathrm{m}$ , $M=1.5\,M_\odot$ : $M_{\rm vir}=2.3\,M_\odot$ and $\alpha_{\rm vir}=\mathbf{1.5}$ — near virial equilibrium, on the bound side.

Step 4. Verdict: starless, near-virial ( $\alpha_{\rm vir}\sim1$ ), and above its critical Bonnor–Ebert mass. It is prestellar and will collapse to form a protostar (provided it is also magnetically supercritical, $\mu_\Phi>1$ ).

2 · How long does collapse take?

The same core has a central density $n_{\mathrm{H_2}}\approx10^5\ \mathrm{cm^{-3}}$ . Estimate its free-fall time and compare with the observed lifetime of prestellar cores.

Step 1. Mass density: $\rho=\mu_c m_H n_{\mathrm{H_2}}$ with $\mu_c=2.8$ (per $\mathrm{H_2}$ with helium), so $\rho=2.8\times1.67\times10^{-27}\times10^{11}=4.7\times10^{-16}\ \mathrm{kg\,m^{-3}}$ .

Step 2. Free-fall time: $t_{\rm ff}=\sqrt{3\pi/(32G\rho)}=\mathbf{1.0\times10^5\ yr}$ . (Scaling: $t_{\rm ff}\approx1.0\times10^5\ \mathrm{yr}\,(n_{\mathrm{H_2}}/10^5)^{-1/2}$ .)

Step 3. Compare: statistical prestellar lifetimes from survey counts are $\sim3$ – $5\times10^5\ \mathrm{yr}$ at this density — several free-fall times. Cores do not free-fall; they linger. That delay is the fingerprint of near-critical support slowing an otherwise runaway contraction.

3 · Weighing a core with the dust continuum

A prestellar core in Taurus ( $d=140\ \mathrm{pc}$ ) is detected at $850\ \mu\mathrm{m}$ with flux $S_\nu=0.5\ \mathrm{Jy}$ . Take $T_d=10\ \mathrm{K}$ and dust opacity $\kappa_\nu=0.01\ \mathrm{cm^2\,g^{-1}}$ (gas + dust). Find its mass.

Step 1. Planck function at $\nu=c/850\,\mu\mathrm{m}=353\ \mathrm{GHz}$ , $T_d=10\ \mathrm{K}$ : with $x=h\nu/k_BT=1.69$ (not deep Rayleigh–Jeans), $B_\nu(T_d)=1.46\times10^{-16}\ \mathrm{W\,m^{-2}\,Hz^{-1}\,sr^{-1}}$ .

Step 2. In SI: $S_\nu=5\times10^{-27}\ \mathrm{W\,m^{-2}\,Hz^{-1}}$ ; $d=4.32\times10^{18}\ \mathrm{m}$ ; $\kappa_\nu=0.01\ \mathrm{cm^2\,g^{-1}}=10^{-3}\ \mathrm{m^2\,kg^{-1}}$ .

Step 3. Mass: $M=S_\nu d^2/[\kappa_\nu B_\nu(T_d)]=6.4\times10^{29}\ \mathrm{kg}=\mathbf{0.32\,M_\odot}$ . A sub-solar prestellar core — at $\sim30\%$ efficiency, the seed of a star of a few tenths $M_\odot$ . The dominant uncertainty is $\kappa_\nu$ , good to a factor of a few.

Physics of the portrait

The dense core is where two mid-twentieth-century idealizations — Jeans's unstable gas and the pressure-bounded isothermal sphere — became a measurable object. The concept carrying the physics is the Bonnor–Ebert sphere: the most concentrated equilibrium a cold cloud can sustain before it must collapse.

Origins · 1 minBonnor–Ebert sphere and how this was discovered

Bonnor–Ebert sphere

the isothermal sphere at its maximum stable mass, M_BE ≈ 1.18 cs⁴ / √(G³ Pext) · star formation and molecular-cloud collapse

The Bonnor–Ebert sphere is the meeting point of a century of thinking about when a gas cloud must collapse. James Jeans in 1902 gave the first criterion — a self-gravitating gas is unstable above a critical mass — but famously neglected the pressure of the surrounding medium (the "Jeans swindle"). The missing ingredient came almost simultaneously from two physicists working on the isothermal gas sphere: Rolf Ebert in 1955 and William Bonnor in 1956 each showed that a self-gravitating isothermal sphere confined by external pressure has a maximum mass, reached when the central-to-edge density contrast climbs to about $14$ , beyond which equilibrium is impossible. Their shared result turned Jeans's scaling into a precise, boundary-condition-respecting stability theory. The astrophysical payoff arrived with Frank Shu, who in 1977 showed that once such a sphere becomes singular it collapses inside-out — an expansion wave propagating outward while matter rains onto a growing protostar. By the 1990s submillimetre bolometers could finally see these objects: Ward-Thompson and colleagues caught the first prestellar cores, and Herschel later resolved whole populations whose mass function mirrors the IMF.

History

1902

— Jeans sets the gravitational-instability criterion for a self-gravitating gas

1955

— Ebert finds the maximum mass of a pressure-bounded isothermal sphere

1956

— Bonnor independently derives the same critical mass

1977

— Shu formulates inside-out collapse of the singular isothermal sphere

2010

— Herschel resolves prestellar cores and their mass function in nearby clouds

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 12 (Star Formation) · Stahler & Palla, The Formation of Stars, Chs. 3, 9, 10 · Binney & Tremaine, Galactic Dynamics, 2nd ed., §4.3 (self-gravitating systems, Jeans equations) · Bonnor, MNRAS 116, 351 (1956); Ebert, Z. Astrophys. 37, 217 (1955) · Shu, ApJ 214, 488 (1977) — inside-out collapse · André et al., Protostars and Planets VI (2014) — cores and the CMF/IMF connection · Bergin & Tafalla, ARA&A 45, 339 (2007) — cold dense cores. The Bonnor–Ebert constants used throughout are $\xi_{\max}\simeq6.45$ , contrast $\simeq14.1$ , $m_1\simeq1.18$ . Key anchor values: $c_s=0.19\ \mathrm{km\,s^{-1}}$ at $10\ \mathrm{K}$ ; $M_{\rm BE}=1.2\,M_\odot$ at $P/k_B=10^5\ \mathrm{K\,cm^{-3}}$ and $0.82\,M_\odot$ at $2\times10^5\ \mathrm{K\,cm^{-3}}$ ; $t_{\rm ff}\approx1.0\times10^5\ \mathrm{yr}$ at $n_{\mathrm{H_2}}=10^5\ \mathrm{cm^{-3}}$ using the He-inclusive density $\rho=2.8\,m_H n_{\mathrm{H_2}}$ — this corrects a $1.4\times10^5\ \mathrm{yr}$ value that used an inconsistent density.

Prerequisites. Molecular Clouds and Giant Molecular Clouds · The Jeans Instability and Jeans Mass · Virial Theorem for Molecular Clouds · Thermal Dust Emission and Infrared Radiation · CO as a Molecular Gas Tracer and the X-Factor. Leads to. Inside-Out Collapse and Protostar Formation · The Stellar Initial Mass Function · Ambipolar Diffusion and Magnetic Support · Interstellar Polarization and Magnetic Fields.

Ask the tutor: Ask me to walk through the Bonnor–Ebert stability turnover for a core at any temperature and pressure, and show where it crosses from stable to collapsing.

What exactly separates a prestellar core from a starless core?

Both are cores without an embedded protostar — that is "starless". The extra condition for "prestellar" is that the core be gravitationally bound and destined to collapse: self-gravity exceeding the combined thermal, turbulent, and magnetic support, in practice $\alpha_{\rm vir}\lesssim1$ and a mass above the critical Bonnor–Ebert value. Many starless cores fail this — they are transient, pressure-confined enhancements that re-expand. Only the bound subset are true precursors of stars, so "prestellar" is a strictly smaller category than "starless".

Why do observed cores cluster near critical equilibrium instead of being strongly bound or unbound?

It is a survival-selection effect. A strongly over-supported clump ( $\alpha_{\rm vir}\gg1$ ) re-expands in a crossing time; a strongly under-supported one collapses in a free-fall time and becomes a protostar. What remains visible longest is the marginally critical population, near the peak of the stability curve where equilibrium is stable but only just. So the measured virial parameters and mass-to-flux ratios of a survey pile up near unity — not because nature prefers that value, but because those cores last long enough to be counted.

If cores have a free-fall time of only 10⁵ years, why do they live several times longer?

Free-fall is the collapse time of a pressureless sphere — it assumes gravity acts unopposed. Real cores are supported: thermal pressure, turbulence, and magnetic tension push back, and near the critical threshold that support nearly balances gravity, so contraction is far slower. Statistical lifetimes from survey counts run to $\sim3$ – $5\times10^5\ \mathrm{yr}$ at $n_{\mathrm{H_2}}\sim10^5\ \mathrm{cm^{-3}}$ , several free-fall times, and in subcritical cores the delay is longer still, set by ambipolar diffusion. The mismatch is itself evidence that cores are near-critical, not in free collapse.

How do astronomers actually weigh a core they cannot see through?

Almost always with cold dust. In the optically thin submillimetre the flux is proportional to mass, $M=S_\nu d^2/[\kappa_\nu B_\nu(T_d)]$ , so one continuum map plus a distance and a dust temperature yields the mass. The catch is the opacity $\kappa_\nu$ — grain properties and gas-to-dust ratio, uncertain by a factor of a few — and $T_d\approx10\ \mathrm{K}$ , pinned from the SED peak. Line tracers like $\mathrm{NH_3}$ give complementary masses but need an abundance assumption; dust and lines cross-checked are how core-mass catalogues are built.

Is the core mass function really the origin of the stellar IMF?

It is the leading hypothesis, but not settled. The prestellar core mass function has a lognormal-like peak and a power-law tail near the Salpeter slope ( $\mathrm{d}N/\mathrm{d}\log M\propto M^{-1.35}$ ); sliding it down in mass by $\sim3$ overlays it on the stellar IMF — naturally read as one core making one star at $\sim30\%$ efficiency. But the mapping is not guaranteed: a core can fragment into a multiple, competitive accretion can reshuffle masses, and feedback can truncate accretion. So while the shape match motivates the "core-fed" picture, whether it is causal or coincidental remains debated.

Why does the magnetic field get a separate collapse criterion from gravity?

Because a core can be gravitationally bound and still held up by magnetic tension rather than pressure. The relevant quantity is the mass-to-flux ratio: if the mass threading a bundle of field lines is too small (subcritical, $\mu_\Phi<1$ ), the frozen-in field supports the core indefinitely even when $\alpha_{\rm vir}<1$ . It cannot collapse until it sheds flux — by ambipolar diffusion, neutrals drifting past ions and field over megayears — becoming supercritical. So the mass-to-flux ratio is a second gate, independent of the virial and Bonnor–Ebert conditions.