The disturbance that grows instead of travelling

A wave equation lets a shape travel because whatever is displaced is pushed back toward where it came from. This essay is about what happens to that equation when the medium also pulls on itself.

One term, subtracted #

Perturb a uniform gas of density $\rho$ and sound speed $c_s$, and the perturbation obeys the ordinary wave equation, whose solutions are $\omega = c_s k$: everything travels, at one speed, whatever its wavelength.

Now let the gas produce the gravitational field it sits in. A density enhancement attracts more gas, which enhances it further, and carrying that through Poisson’s equation adds a constant to the dispersion relation:

$$\omega^2 = c_s^2 k^2 - 4\pi G\rho$$

Nothing else changes. The equation is the same equation; one term has been subtracted from it, and the term does not depend on the wavelength.

That is the whole of the instability. At large $k$ — short wavelengths — the $c_s^2k^2$ term dominates and $\omega^2$ is positive, so $\omega$ is real and the disturbance is a sound wave. At small $k$ the constant wins, $\omega^2$ is negative, $\omega$ is imaginary, and $e^{-i\omega t}$ is no longer an oscillation but a growth.

One term is subtracted, and that is the whole of it. The speed of sound in a gas is a stiffness set against an inertia, and every ordinary wave problem has both terms positive. The self-gravitating case takes that same ratio and subtracts a term that carries no stiffness at all — gravity does not push back — so the sum can change sign. When it does, the quantity that was a squared frequency becomes negative, and what was a wave becomes a growth.

Why it is a wavelength and not a strength #

The usual framing — gravity against pressure — leaves out that neither is a number until a region has been chosen. Comparing two times rather than two forces makes the size dependence obvious.

Gravity collapses a region of density $\rho$ in the free-fall time

$$t_{\text{ff}} = \sqrt{\frac{3\pi}{32G\rho}}$$

which contains no radius at all: a large region and a small one at the same density fall together in the same time, because the extra mass and the extra distance cancel exactly. Pressure, meanwhile, can only respond at the speed of sound, so it takes a time $R/c_s$ to push back on a region of size $R$ — and that grows with the region.

So there is always a size above which pressure is too slow, whatever the pressure is. The changeover is at

$$\lambda_J = c_s\sqrt{\frac{\pi}{G\rho}}$$

and the mass of gas in a sphere of that diameter is the Jeans mass.

Why it is a wavelength rather than a strength is the part worth being careful about. A column of gas held up by pressure against external gravity has no preferred size: the gravity is given, and enlarging the region does not change it. Here the gravity is made by the gas being supported, so enlarging the region strengthens both sides of the comparison at once — and they strengthen at different rates. Gravity grows with the mass enclosed while pressure has to cross the region to respond, so beyond a certain size pressure loses, and that size is the criterion.

The curve has no scale of its own #

The dispersion relation contains $c_s$ and $\rho$ and nothing else, so measuring $k$ in units of $\sqrt{4\pi G\rho}/c_s$ and $\omega^2$ in units of $4\pi G\rho$ should collapse every gas onto one curve. It does, to the last bit of a double.

That is worth stating as a fact about the equation rather than about any object: there is no privileged density, no privileged temperature and no privileged size in it. A laboratory gas would obey it too, and does not collapse only because $4\pi G\rho$ for anything in a laboratory corresponds to a wavelength larger than the Earth.

The two exponents are the content. At fixed temperature the unstable mass falls as $\rho^{-1/2}$, so a denser gas is unstable in smaller pieces; and it rises as $T^{3/2}$, because pressure support goes with the sound speed cubed. Both follow from the expression above with no further physics.

The sound speed is the thermal speed with a numerical factor in front of it, so a gas’s entire ability to resist its own gravity is a statement about how fast its molecules are moving. That speed enters the critical mass cubed, which is why the criterion is so sensitive to temperature — a factor of four in temperature is a factor of eight in the mass that can collapse.

The mean molecular weight enters the same way and is easy to overlook: molecular hydrogen has twice the mass per particle of atomic hydrogen, so the same gas at the same temperature has a sound speed lower by √2 once it is molecular, and an unstable mass lower by a factor of 2.8. Becoming molecular is itself destabilising, before any cooling is counted.

The background the analysis cannot have #

There is an objection to all of this that is usually mentioned in a footnote and deserves better, because it is where the derivation is actually weak.

The calculation perturbs a uniform, static, self-gravitating medium. No such thing exists. A uniform medium with self-gravity is not in equilibrium — it is already collapsing — so the unperturbed state does not satisfy its own equations, and the perturbation is being taken about something that is not a solution.

Jeans knew, and did it anyway; the step is called the Jeans swindle, which is an unusually honest name for a piece of standard physics. What rescues it is that the repair changes the answer very little: redoing the analysis in a uniformly expanding background, where a solution does exist, gives the same criterion with a growth that is a power of time rather than an exponential.

The general shape is worth carrying. A linear stability analysis needs an equilibrium to perturb, and the interesting systems often have none; what is then computed is a growth rate about a state the system is passing through rather than sitting in, and its meaning is that a disturbance grows faster than the background changes, which has to be checked separately.

What the growth rate actually is #

An imaginary frequency is easy to write and worth converting into a time, because the number decides whether the instability matters.

At the longest wavelengths — where the pressure term is negligible — $\omega^2 \to -4\pi G\rho$, so the growth rate is $\sqrt{4\pi G\rho}$ and the e-folding time is within a factor of two of the free-fall time. That is the fastest anything can grow, and it is set by the density alone.

Nearer the threshold the growth is slower and vanishes exactly at $\lambda_J$, where a disturbance neither travels nor grows: it sits. That marginal wavelength is the one a static self-gravitating equilibrium is built from, and it is why the Bonnor–Ebert sphere — the largest isothermal cloud a given external pressure can hold up — has a radius of about that size.

So the instability has no slow regime worth speaking of. Either a disturbance is short enough to oscillate, or it grows on the free-fall time — and the free-fall time for interstellar densities is a few hundred thousand years, which by the standards of the objects involved is instantaneous.

What the criterion leaves out #

Magnetic fields resist compression and are not in it. A field threaded through a conducting gas is compressed with it, and the pressure it exerts grows as the gas is squeezed — so a threaded region is supported by something the dispersion relation above has no term for.

What the criterion leaves out first is a magnetic field. A field carried by a conducting fluid cannot get out of it, and it adds a stiffness that depends on direction — so the instability becomes anisotropic. A region can be stable across the field and unstable along it, which turns a question about a critical mass into a question about a critical mass-to-flux ratio, and that is the quantity star formation is actually organised around.

Rotation resists it in one direction. Angular momentum is conserved as a region contracts, so the spin rises and the centrifugal term with it. In a rotating disc the same competition gives a criterion of the same shape with the rotation rate in place of the sound speed — the Toomre condition — and the structure is identical: a stabilising response time against a collapse time, yielding a critical wavelength.

And a real medium is not uniform, not isothermal and not still. Supersonic motions provide an effective pressure while also creating dense regions whose local criterion is easier to meet, and the two effects work in opposite directions.

It leaves out turbulence too, and the usual repair is an estimate rather than a derivation. Turbulent motions are the nearest thing a real cloud has to a thermal population, except that they are correlated across a range of scales rather than at one — so substituting a turbulent velocity for the sound speed puts a number in the right place without justifying it. The criterion survives as a scaling and loses its precision.

Reading the criterion as an equilibrium condition #

There is a second way to arrive at the same wavelength that uses no waves at all, and comparing the two is instructive about what the criterion is.

Take a sphere of gas of mass $M$ and radius $R$ and apply the virial theorem: twice the kinetic energy plus the gravitational energy is zero in equilibrium. The kinetic energy is $\tfrac{3}{2}NkT$ and the gravitational energy is $-\tfrac{3}{5}GM^2/R$, and setting the balance gives a mass — the same Jeans mass, up to a numerical factor of order one.

There is a second route to the same criterion, and the difference between the two routes is instructive. The virial argument asks when the gravitational and thermal energies of a ball balance, rather than when a disturbance grows — so it produces a mass directly and says nothing whatever about which wavelength goes first. The two answers agree to within a factor of order one, and only the instability calculation says what the collapse looks like.

The factor of order one is the honest difference between the two derivations, and it is why quoted Jeans masses vary by a factor of two or three between textbooks: some use a sphere of diameter $\lambda_J$, some a cube of side $\lambda_J$, some the virial balance, and the three differ by $\pi/6$ and by numbers of that kind. The exponents are the physics and the prefactor is a convention — which is the usual situation for a criterion derived from a comparison rather than from a solution.

What the wave derivation adds, and the virial one cannot, is which disturbance grows and how fast. An equilibrium condition says whether a configuration can sit still; a dispersion relation says what happens when it cannot.

Why the collapse fragments, and where it stops #

The criterion has a consequence that follows from its own scaling and is more interesting than the criterion: applied to a collapsing region, it applies again, and again, to progressively smaller pieces.

Take a cloud that has just become unstable and start it collapsing. The density rises. The Jeans mass goes as $\rho^{-1/2}$ at fixed temperature, so it falls. Regions inside the cloud that were individually stable a moment ago now exceed their own local Jeans mass and begin collapsing on their own account, faster than the parent, because their free-fall time is shorter. Those sub-regions do the same to their own interiors, and so on down.

That is hierarchical fragmentation, and it is why a collapsing cloud produces a cluster of stars rather than one enormous one. It depends entirely on the temperature staying put while the density climbs, which for interstellar gas it does over a remarkable range: the collapse compresses the gas, the compression heats it, and the heat is radiated away by dust and by molecular lines almost as fast as it is generated. The gas is effectively isothermal over some fifteen orders of magnitude in density.

What stops the cascade is the failure of that cooling. Above about $10^{-13}$ grams per cubic centimetre the collapsing core becomes opaque to its own radiation, and the heat can no longer get out. The gas turns adiabatic, so the temperature now rises as $\rho^{2/3}$, and the Jeans mass — which goes as $T^{3/2}\rho^{-1/2}$ — becomes proportional to $\rho^{1/2}$ and starts increasing. Fragmentation stops immediately, because further compression makes the fragment more stable rather than less.

The mass at which that happens is the opacity limit for fragmentation, and it comes out at a hundredth of a solar mass or so — which is, satisfyingly, close to the smallest objects observed to form by collapse. The lower end of the stellar mass function may well be this number.

The general form of the argument is worth extracting, because it is not about the Jeans mass at all. A self-gravitating gas is unstable to collapse if its effective adiabatic index is below $4/3$ and stable above it, and $4/3$ is the value at which the pressure’s response to compression exactly matches gravity’s. Isothermal gas has an index of one and always loses. Monatomic gas with nothing to radiate has $5/3$ and always wins. Everything in star formation happens in the gap between those two numbers, and what moves a gas across it is whether it can get rid of heat.

The clouds that ought to have collapsed already #

There is an observation that the criterion on this page fails badly, and it is worth stating because the failure is a factor of a hundred rather than a factor of two.

A giant molecular cloud holds $10^5$ or $10^6$ solar masses of gas at about ten kelvin and a hundred particles per cubic centimetre. Its Jeans mass under those conditions is a few tens of solar masses. So the cloud is not marginally unstable — it exceeds its own criterion by three or four orders of magnitude, and the analysis on this page says it should be collapsing at very nearly the free-fall rate.

If it were, the arithmetic is easy and the answer is wrong. The molecular gas in the Galaxy amounts to roughly $10^9$ solar masses with a free-fall time of a few million years, so stars would form at something like a couple of hundred solar masses a year. The measured rate is between one and two.

So molecular clouds convert about one per cent of their mass into stars per free-fall time, and something is holding up the other ninety-nine. The candidates are the three terms the dispersion relation does not have. Magnetic fields thread the clouds and are measured to be roughly strong enough to matter, though whether they are quite strong enough is a long argument about a difficult measurement. Supersonic turbulence supplies a pressure that the sound speed underestimates by a factor of several, and simultaneously creates the dense regions where collapse does happen — which is why turbulence both suppresses the global rate and sets the local one. And the stars that do form blow the surrounding gas apart with radiation, winds and eventually supernovae, so the cloud is destroyed before it has finished collapsing.

The honest position is that all three contribute, that their relative weights are the central open question of the subject, and that the Jeans criterion is right about what happens and silent about how much. It identifies the instability. It does not identify the regulator, and the regulator is what the observed rate measures.

The same sign change elsewhere #

The structure — a restoring term that competes with one of the opposite sign, with the balance decided by wavelength — recurs, and recognising it is worth more than the criterion itself.

In a stratified fluid. A parcel displaced in a stably stratified layer oscillates at the buoyancy frequency; reverse the density gradient and the same expression gives an imaginary frequency and the Rayleigh–Taylor instability. One sign change, two entirely different phenomena, one algebra.

In a plasma, where screening decides what a charge can reach and several instabilities are exactly this shape: a dispersion relation whose right-hand side changes sign when a drift, a current or a temperature anisotropy exceeds a threshold, with a growth rate that is the imaginary part of the same $\omega$.

And in the early universe, where the same competition — with radiation supplying the pressure — decides which density fluctuations survive to become structure. The pressure disappears at recombination, when the photons stop interacting with the baryons, and the Jeans mass falls by ten orders of magnitude in a few thousand years. Nothing about the gravity changes at that moment; what changes is what is resisting it.

The same sign change happens in a fluid nobody would call astrophysical. A parcel displaced in a stratified fluid oscillates about where it started, at a frequency set by how the density varies with height; make the stratification unstable and the restoring coefficient goes negative. A negative restoring coefficient is not a slower oscillation. It is a growth, and it is the identical piece of algebra.

Why the sign of a squared frequency is worth watching #

The habit this essay is really about is smaller than the criterion and more portable.

A great deal of physics arrives as a dispersion relation — a formula for $\omega^2$ in terms of a wavenumber and the properties of a medium. When that formula is a sum of terms, the question worth asking first is not what the frequency is but whether any term can make the sum negative, because a negative $\omega^2$ is not a slower wave. It is a different phenomenon with a different vocabulary: growth rather than propagation, an e-folding time rather than a period, and a threshold rather than a speed.

The terms that can do it are recognisable. A restoring force proportional to displacement gives a positive contribution; a force that grows with displacement gives a negative one; and any term that does not carry a factor of $k^2$ competes with the ones that do at a wavelength that can be solved for. Self-gravity is the second and third at once, which is why it produces an instability with a definite scale rather than at every scale or none.

Once the shape is recognised, the criterion falls out without solving anything: set the sum to zero and read off the wavelength. Everything else in this essay — the numbers, the scalings, the exponents — is that one step, evaluated.

It is worth watching the sign of a squared frequency for exactly this reason. A positive one gives everything the word oscillator implies — a resonance, a peak, a phase lag behind a drive. Make it negative and none of those exist: there is no resonance to be driven, no peak, and no steady state for anything to lag behind. The equation looks almost the same and describes a different world.

What the pictures cannot show #

The analysis is linear. Everything here is a first-order perturbation, so it says which disturbances begin to grow and nothing about what they become. The collapse that follows is not homologous — the centre runs away first — and the linear growth rate stops describing it almost immediately.

The gas is isothermal and ideal. A real medium’s temperature depends on density through a balance of heating and cooling, and the effective exponent is near isothermal without being equal to it. The criterion is sensitive to that only through the sound speed; what is sensitive to it is everything that happens afterwards, which is somebody else’s subject and not this essay’s.

The free-fall time assumes no pressure at all. A region collapsing with pressure takes longer than the quoted time, and the quoted time is a lower bound rather than a prediction.

And nothing here is a cloud. The marked densities are where interstellar gas is found, and they are there to give the axes a scale rather than to describe an object. What a real cloud does with this instability — how it fragments, what stops it, and what emerges — is astrophysics with its own numbers, and is not derived from a dispersion relation.

The ladder from here #

Later rungs on this anchor: the Bonnor–Ebert sphere, which is the largest isothermal cloud a given external pressure can hold up and is the equilibrium the Jeans analysis is a perturbation of; the criterion in an expanding background, where the swindle is repaired and the growth becomes a power law; the magnetic case, with the mass-to-flux ratio; and the Toomre criterion, where rotation rather than pressure supplies the stabilising response.

The neighbouring ladders are the wave equation, which this is one subtracted term away from, the size at which a body becomes round, which is the same competition in a solid, and the ball of gas that heats up as it cools, which is what a self-gravitating gas does once it has started.