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The cloud that cannot hold itself up

A cold cloud collapses when gravity beats pressure, and the mass at which that happens falls as the square root of the density — so the collapse makes the condition for collapse easier, over and over, until the gas can no longer get rid of the heat. That is why a cloud of ten thousand solar masses makes a cluster and not a star.

Assumes Interstellar medium, Hydrostatic equilibrium and Virial theorem.

17 min read 9 figures Scales that defeat intuitionMeasuring the unreachableSmall perturbations, large consequences

A cloud of molecular gas is held up by pressure and pulled in by gravity, and which of the two wins is a question about mass. Below some threshold the pressure holds; above it, nothing does. The threshold has a name and a formula, and the formula’s exponents are the whole of what follows.

MJ  =  (5kTGμmH)3/2(34πρ)1/2M_{\rm J} \;=\; \left(\frac{5kT}{G\mu m_{\rm H}}\right)^{3/2}\left(\frac{3}{4\pi\rho}\right)^{1/2}

Temperature enters to the three-halves and density to the minus one-half. The second of those is the one that matters, and it matters because of a sign: as a cloud contracts its density rises, so the mass required to be unstable falls. Collapse makes the condition for collapse easier to meet, and it keeps doing so.

The critical mass falls as the cloud contracts, with a slope of −0.50. The Jeans mass — the least mass of molecular gas that its own pressure cannot support — against number density, for gas at 10, 20, 50 K, both axes logarithmic. The curves are straight because the mass goes as T^(3/2) ρ^(−1/2), and the slope measured off the drawn 10 K curve is −0.500 against the −0.5 that exponent requires. The consequence is the one that matters and it is a matter of sign: a cloud collapsing at constant temperature moves to the right along one of these lines, so the mass it takes to be unstable keeps falling, and sub-regions that were individually stable when the collapse began become individually unstable during it. A giant molecular cloud at 100 particles per cubic centimetre has a Jeans mass of 99 solar masses; a dense core at 10⁵ has one of 1.70. That is why a cloud of ten thousand solar masses makes a cluster rather than a star, and why the question about star formation is not what makes gas collapse but what stops the fragmenting.
Fig. 1 The Jeans mass against density, for molecular gas at three temperatures, both axes logarithmic. The curves are straight lines of slope exactly minus one half, measured back off the drawing. A cloud collapsing at constant temperature moves to the right along one of them, so the bar keeps dropping underneath it: a giant molecular cloud at a hundred particles per cubic centimetre has a Jeans mass near a hundred solar masses, and a dense core at a hundred thousand has one under two.

Why the temperature stays put while the density does not

The whole argument depends on the collapse being isothermal, which is not obvious and is not true of most gases. Squeeze air and it heats up; squeeze a stellar interior and it heats up; squeeze molecular gas at ten kelvin and it does not, because it is an outstandingly good radiator at exactly the wavelengths its own compression heats it to.

Carbon monoxide is the workhorse. It has a rotational ladder whose lowest rung sits five kelvin above the ground state, so it is collisionally excited at temperatures where nothing atomic is, and it radiates at millimetre wavelengths to which the cloud is largely transparent. The heat of compression is therefore converted into photons that leave, rather than into pressure that resists. Over a wide range of density the gas contracts and its temperature barely moves.

That is why the curves above are straight lines and why the collapse slides along one of them rather than climbing off it. It is also why the interstellar medium’s cold phase is a prerequisite for any of this: only gas that has already found the cold branch of thermal balance, and then shielded itself enough to become molecular, is cold enough for the Jeans mass to be a stellar mass rather than a galactic one.

Two ways to ask the same question

The Jeans criterion, as usually derived, compares two energies in an infinite uniform medium, which is a slightly unsatisfactory object: an infinite uniform self-gravitating medium is not in equilibrium in the first place, a difficulty known politely as the Jeans swindle. A cleaner version asks about a real configuration.

Take an isothermal sphere in hydrostatic balance, held together by its own gravity and held in by an external pressure — the pressure of the warm medium around it, or of the larger cloud it sits inside. Solve for its structure, then ask how massive it can be before no solution exists.

A pressure-bounded cloud has a heaviest stable version, and it is 1.16 M☉. The mass of an isothermal sphere held together by gravity and held in by an external pressure, in the dimensionless form M P^(1/2) G^(3/2)/c_s⁴, plotted against how much denser its centre is than its edge. Every point is a genuine hydrostatic solution — the isothermal Lane–Emden equation integrated from the centre outwards and cut off at a radius — so the curve is a family of clouds that could exist, not a stability argument imposed on them. It has a maximum, at a density contrast of 14.0 and a dimensionless mass of 1.182, and past the maximum the same mass is served by two solutions of which the more centrally concentrated one is unstable. A cloud on the rising side that is squeezed harder settles at a smaller radius and stays; one past the peak that is squeezed has no solution to settle into and collapses. For molecular gas at 10 K squeezed by a pressure of 10⁵ K cm⁻³ the peak is at 1.16 solar masses, which is why the cores seen in nearby clouds are the mass of stars rather than the mass of clouds.
Fig. 2 The mass of such a sphere, in the dimensionless combination that scales out the sound speed and the gravitational constant, plotted against how much denser its centre is than its edge. Every point is a genuine hydrostatic solution — the isothermal Lane–Emden equation integrated outwards from the centre and truncated at a radius — so the curve is a family of clouds that could exist rather than a stability argument imposed on them. It has a maximum at a density contrast of fourteen, and past the maximum the same mass is served by two solutions of which the more concentrated one is unstable.

The maximum is the Bonnor–Ebert mass, and the physical reading of it is worth stating carefully. On the rising side, squeezing a cloud harder makes it settle at a smaller radius and a higher central density, and it stays. Past the peak there is no solution to settle into: the extra squeeze demands a configuration the family does not contain, and the cloud collapses. For gas at ten kelvin under the pressure inside a giant molecular cloud, that limit is a little over one solar mass.

That number is the reason star formation produces stars. Nothing in the argument mentions nuclear physics, or hydrogen burning, or the mass at which an object can shine. It is a statement about the sound speed of cold molecular gas and the pressure of the material around it, and it lands within a factor of a few of the mass of the Sun.

A core with a flat middle, and the singular solution it never reaches. The density profile of the critical isothermal sphere, in units of its own central density and of the radius at which it is truncated, on logarithmic axes. Two features are the whole of it. The centre is flat: inside about a fifth of the outer radius the density hardly changes, because gravity there has almost no enclosed mass to pull with, and this is what distinguishes a real core from the singular isothermal sphere. The outer part falls a little steeper than the −2 of that singular solution — measured off the drawing at −2.11 — because a truncated sphere crosses the singular profile rather than settling onto it, and the dashed line drawn for comparison has infinite central density and no stability question to ask about it at all. Observed cores in nearby dark clouds are fitted with exactly this shape, and the fitted density contrast is how a core is said to be near its critical state or far from it — a contrast well under fourteen means the thing is stable and will still be there in a million years.
Fig. 3 What the critical sphere looks like: density against radius, in units of its own centre and its own edge, on logarithmic axes. Two features are the whole of it. The middle is flat, because gravity there has almost no enclosed mass to pull with — which is exactly what distinguishes a real core from the singular isothermal sphere drawn dashed beside it, whose central density is infinite and which has no stability question to ask about. Cores observed in nearby dark clouds are fitted with this shape, and the fitted contrast is how a core is said to be near its critical state or far from it.

Fragmentation, and the floor underneath it

Return to the sign. A cloud a thousand times over the Jeans mass begins to collapse as a whole; as it does, its density rises and its own Jeans mass falls; and sub-regions that were individually stable at the start become individually unstable partway in. Each of those collapses, and inside each the same thing happens again.

The cascade is not gentle and it is not orderly, but its existence is forced by the exponent. Nothing has to be arranged for it to happen: any cloud that begins to collapse isothermally will, before it has contracted by a factor of a hundred in density, contain sub-regions that individually exceed a Jeans mass that has fallen by a factor of ten underneath them. The hierarchy of clumps and cores and condensations that a millimetre map of a nearby cloud shows is not a set of separate objects that happened to form near one another. It is one object partway through this process, photographed.

What stops the cascade is a change in the physics rather than in the arithmetic.

Fragmentation stops at 0.005 solar masses, and the reason is opacity. The Jeans mass along a collapse rather than across a family of clouds. To the left of the knee the gas is isothermal at 10 K, because molecular material at these densities radiates away the heat of compression as fast as compression supplies it; the critical mass therefore falls, and the cloud fragments into pieces, and each piece fragments again. At 10¹⁰ particles per cubic centimetre the gas becomes opaque to its own cooling lines, the collapse turns adiabatic with an index of 1.67, and the temperature starts to rise as density to the 0.67. The Jeans mass turns with it. The minimum, read off the drawn curve at 0.0054 solar masses, is the smallest object this cascade can produce, and it is close to the hundredth of a solar mass usually quoted as the opacity limit. Nothing in the argument mentions hydrogen burning: the lower end of the stellar mass function is set by when a gas stops being able to get rid of heat, not by whether the result can ignite.
Fig. 4 The Jeans mass along a collapse rather than across a family of clouds. To the left of the knee the gas is isothermal and the critical mass falls, so the cloud fragments and the fragments fragment. Around ten to the tenth particles per cubic centimetre the gas becomes opaque to its own cooling lines, the collapse turns adiabatic, and the temperature begins to rise — at which point the three-halves power in the numerator starts to win and the Jeans mass turns round and climbs. The minimum read off the drawn curve is a few thousandths of a solar mass.

That minimum is the opacity limit for fragmentation, and it is the most satisfying number in this essay because of what is absent from its derivation. It says nothing about ignition. It is not the mass at which an object can burn hydrogen, or deuterium, or anything else. It is the mass below which a collapsing gas cannot make a smaller piece, because the piece would have to get rid of its compressional heat and no longer can.

The bottom of the observed stellar mass function sits near a tenth of a solar mass, which is an order of magnitude above the floor drawn here — and the objects between the two, the brown dwarfs, exist and are common. So the opacity limit is a real floor and it is not the only thing setting the low end; accretion after the fragment forms, and competition between fragments for the same gas, do the rest.

A pressure-bounded cloud has a heaviest stable version, and it is 4.65 M☉. The mass of an isothermal sphere held together by gravity and held in by an external pressure, in the dimensionless form M P^(1/2) G^(3/2)/c_s⁴, plotted against how much denser its centre is than its edge. Every point is a genuine hydrostatic solution — the isothermal Lane–Emden equation integrated from the centre outwards and cut off at a radius — so the curve is a family of clouds that could exist, not a stability argument imposed on them. It has a maximum, at a density contrast of 14.0 and a dimensionless mass of 1.182, and past the maximum the same mass is served by two solutions of which the more centrally concentrated one is unstable. A cloud on the rising side that is squeezed harder settles at a smaller radius and stays; one past the peak that is squeezed has no solution to settle into and collapses. For molecular gas at 20 K squeezed by a pressure of 10⁵ K cm⁻³ the peak is at 4.65 solar masses, which is why the cores seen in nearby clouds are the mass of stars rather than the mass of clouds.
Fig. 5 The stability curve for gas at twenty kelvin rather than ten. The critical mass rises — it goes as cs4c_s^4, so doubling the temperature is a factor of four — and the shape of the curve and the position of its peak are unchanged. The critical density contrast of about fourteen is a pure number, a property of the isothermal Lane–Emden equation and of nothing else, which is why it can be quoted without qualification while the mass it corresponds to cannot.

The clouds are over-pressured, and that is the point

A molecular cloud is not one of the phases of the interstellar medium in the sense that the neutral phases are. The distinction has an observational consequence. The neutral phases can be treated as a medium with a pressure and a temperature, and their properties are set by atomic physics and a radiation field. Molecular clouds have to be treated as objects, with masses and boundaries and lifetimes, and their properties are set by gravity — which is a different subject with different questions in it.

Are the clouds actually falling in?

The Jeans argument says a cloud above the critical mass collapses. Observed giant molecular clouds are enormously above it — thousands of Jeans masses — and they are not collapsing at anything like free fall. If they were, the galaxy would convert its molecular gas into stars in about a free-fall time, some few million years, and it manifestly does not. The internal motions are turbulent and supersonic, and supersonic turbulence dissipates in about a crossing time, which is comparable with the free-fall time. So the support is not free: something has to keep stirring the cloud, and what does the stirring — outflows from the stars already forming, the passage of a spiral arm, supernovae — is one of the genuinely unsettled questions in the subject.

The distinction between a cloud that is supported and a cloud that is dissipating its support is not academic, because the two have the same appearance. Both show supersonic linewidths; both look bound by the virial test; and the test itself measures a snapshot of the kinetic energy without saying how long that energy has left. A cloud whose turbulence decays in one crossing time and a cloud whose turbulence is being replenished are the same measurement, and separating them takes an argument about what is driving rather than an observation of what is present.

There is a second reading of the same numbers, and it inverts the question. Perhaps the clouds are not supported at all, and are simply young: assembled recently, collapsing now, and dispersed by the first massive stars they make before more than a few per cent of the gas has been used. On that account the supersonic motions are not support but the collapse itself, seen in projection. The two pictures make different predictions about how the age of a cloud correlates with the fraction of it that has turned into stars, and distinguishing them is an active programme rather than a settled result.

What is not in doubt is the arithmetic of the outcome. Only a few per cent of the molecular gas in a cloud becomes stars before the cloud is dispersed, which is why the gas supply of a galaxy lasts a gigayear or two rather than a few million years.

Fragmentation stops at 0.005 solar masses, and the reason is opacity. The Jeans mass along a collapse rather than across a family of clouds. To the left of the knee the gas is isothermal at 10 K, because molecular material at these densities radiates away the heat of compression as fast as compression supplies it; the critical mass therefore falls, and the cloud fragments into pieces, and each piece fragments again. At 10¹⁰ particles per cubic centimetre the gas becomes opaque to its own cooling lines, the collapse turns adiabatic with an index of 1.40, and the temperature starts to rise as density to the 0.40. The Jeans mass turns with it. The minimum, read off the drawn curve at 0.0054 solar masses, is the smallest object this cascade can produce, and it is close to the hundredth of a solar mass usually quoted as the opacity limit. Nothing in the argument mentions hydrogen burning: the lower end of the stellar mass function is set by when a gas stops being able to get rid of heat, not by whether the result can ignite.
Fig. 6 The fragmentation calculation with the gas made stiffer once it becomes opaque — an adiabatic index of 1.4 rather than 5/3, which is what a diatomic gas gives once its rotational modes are excited. The minimum fragment mass rises from a hundredth of a solar mass to five thousandths… and the shape of the knee is what changes most: a softer equation of state turns the Jeans mass around less sharply. The opacity limit is a competition between two exponents, and both of them are properties of molecular hydrogen rather than of the cloud.

The same balance, one step further along

Once a fragment stops fragmenting and starts contracting as a single object, it becomes a different kind of problem: not a cloud in pressure balance with a medium, but a self-gravitating body in balance with its own thermal pressure all the way to the centre. From there the object is a star being assembled, and the questions become how fast it accretes, when its centre reaches ten million kelvin, and whether it ever does.

What the collapse leaves behind

The result of one cloud’s collapse is not one star and not a smooth distribution. It is a cluster, with a mass spectrum. Whether the mass function is inherited from the fragmentation described above — a cascade of Jeans masses producing a spectrum of fragment masses — or from the later competition between fragments for a shared reservoir, is not settled. The cores observed in nearby clouds already have a mass distribution resembling the stellar one, shifted upward by a factor of about three, which is suggestive and is not proof: the shift is exactly the efficiency factor a competitive-accretion picture would also produce.

The mass that is inferred from a molecule that is not there

Every mass in this essay is a mass of molecular hydrogen, and molecular hydrogen is essentially invisible.

The reason is a symmetry. A hydrogen molecule is two identical atoms, so it has no permanent electric dipole moment and no dipole rotational transitions. What it has are quadrupole transitions, whose lowest excited level sits about five hundred kelvin above the ground state — so a gas at ten kelvin has no molecules in it and emits nothing at all.

The mass is therefore measured through a tracer, and the tracer is carbon monoxide. It is asymmetric, so it has a dipole moment; it is abundant, being made of the two commonest elements after hydrogen and helium; and its lowest rotational transition is five kelvin above the ground state, which is exactly the range the gas occupies.

Converting a carbon monoxide brightness into a hydrogen mass requires a factor, and the factor is the least satisfactory number in the subject. It is not a straightforward abundance ratio, because the carbon monoxide line is optically thick — the emission comes from the surface of each clump rather than from its whole volume, so the brightness measures the number of clumps and their velocity spread rather than the amount of gas.

That it works at all is a coincidence of two effects cancelling. The line’s brightness is set by the cloud’s velocity dispersion, and the velocity dispersion is set by the cloud’s mass through the virial balance — so a quantity that measures surfaces ends up tracking a mass, provided the clouds are bound and roughly virialised.

The factor is calibrated three ways that share nothing: by measuring virial masses from linewidths and sizes, by counting gamma rays produced when cosmic rays strike the gas, and by measuring the dust and assuming a gas-to-dust ratio. The three agree within a factor of about two in the local Galaxy, and they diverge at low metallicity, where carbon monoxide is photodissociated in the outer parts of a cloud and a substantial mass of molecular hydrogen emits nothing.

Every molecular mass quoted anywhere carries that factor, and it is the dominant systematic in nearly every statement about how much star-forming gas a galaxy has.

The critical mass falls as the cloud contracts, with a slope of −0.50. The Jeans mass — the least mass of molecular gas that its own pressure cannot support — against number density, for gas at 10, 20, 50 K, both axes logarithmic. The curves are straight because the mass goes as T^(3/2) ρ^(−1/2), and the slope measured off the drawn 10 K curve is −0.500 against the −0.5 that exponent requires. The consequence is the one that matters and it is a matter of sign: a cloud collapsing at constant temperature moves to the right along one of these lines, so the mass it takes to be unstable keeps falling, and sub-regions that were individually stable when the collapse began become individually unstable during it. A giant molecular cloud at 100 particles per cubic centimetre has a Jeans mass of 99 solar masses; a dense core at 10⁵ has one of 1.70. That is why a cloud of ten thousand solar masses makes a cluster rather than a star, and why the question about star formation is not what makes gas collapse but what stops the fragmenting.
Fig. 7 The same three temperature curves over a narrower density range — ten to a million per cubic centimetre, which is the range a molecular cloud and its cores actually occupy. The slope is still exactly −0.500-0.500 and the whole span of the plot covers a factor of a thousand in critical mass. Between a cloud and a core the Jeans mass falls by three orders of magnitude at fixed temperature, and that is the entire reason a cloud fragments rather than collapsing as one object.

The only timescale there is

The Jeans criterion says whether a cloud collapses and says nothing about how fast. The answer to that is a single expression with one variable in it.

A pressureless sphere collapsing under its own gravity reaches infinite density after a time

tff=3π32Gρ,t_{\rm ff} = \sqrt{\frac{3\pi}{32 G\rho}},

and the striking feature is what is absent. There is no mass, no radius and no temperature — only the density. A cloud of a hundred solar masses and a cloud of a hundred thousand, at the same density, collapse in the same time.

That is what makes the fragmentation cascade coherent rather than chaotic. Every sub-region of a uniform collapsing cloud has the same density and therefore the same free-fall time, so they all collapse together rather than one after another — and the sub-regions that become denser than average run ahead, which is the instability the cascade is made of.

The numbers are short. Gas at a hundred particles per cubic centimetre falls in about three and a half million years; at a hundred thousand, in about a hundred thousand years. So a giant cloud’s overall collapse and a dense core’s are separated by a factor of thirty in time, which is why the hierarchy can be observed at several stages at once.

The quantity that is compared against it is the star-formation efficiency per free-fall time: the fraction of a cloud’s mass converted into stars in one such time. Measured across a wide range of environments it comes out at about one per cent, remarkably consistently.

One per cent is the number the whole subject is trying to explain. A cloud in free fall with no support would convert most of its gas in one free-fall time, so the observed rate is a hundred times slower than the simplest possible expectation — and whether that factor comes from magnetic support, from turbulence, from feedback destroying the cloud early, or from the cloud never having been bound in the first place is the same open question this essay’s earlier section reached from the other side.

The critical mass falls as the cloud contracts, with a slope of −0.50. The Jeans mass — the least mass of molecular gas that its own pressure cannot support — against number density, for gas at 10, 20, 50 K, both axes logarithmic. The curves are straight because the mass goes as T^(3/2) ρ^(−1/2), and the slope measured off the drawn 10 K curve is −0.500 against the −0.5 that exponent requires. The consequence is the one that matters and it is a matter of sign: a cloud collapsing at constant temperature moves to the right along one of these lines, so the mass it takes to be unstable keeps falling, and sub-regions that were individually stable when the collapse began become individually unstable during it. A giant molecular cloud at 100 particles per cubic centimetre has a Jeans mass of 99 solar masses; a dense core at 10⁵ has one of 1.70. That is why a cloud of ten thousand solar masses makes a cluster rather than a star, and why the question about star formation is not what makes gas collapse but what stops the fragmenting.
Fig. 8 And the widest range the isothermal assumption can be stretched over — a hundred to a hundred billion. The curve is a straight line the whole way, which it must be, because MJ∝ρ−1/2M_J \propto \rho^{-1/2} has no scale in it. The straightness is the assumption: the moment the gas stops being isothermal the line bends, and where it bends is the previous figure and the smallest star that can be made.

Where the picture stops

The clouds are magnetised, and this essay has ignored it. The field threading a cloud contributes a pressure comparable with the thermal and turbulent ones, and it does something the others do not: it is anchored to the ionised fraction of the gas rather than to the neutrals, so it can only be shed slowly, by ions and neutrals drifting past one another. There is a magnetic critical mass in the same sense as a Jeans mass, and whether real cores are supported mainly by fields or mainly by turbulence was argued over for thirty years.

Isothermal is an approximation with a shelf life. The gas is isothermal because carbon monoxide radiates, and carbon monoxide freezes onto grains at the densities and temperatures of a dense core, which removes the coolant at exactly the point the argument needs it most. What takes over is dust continuum emission, and the transition is one of the places where the temperature is genuinely not constant.

And the collapse is not spherical. Angular momentum is conserved, and a cloud with any rotation at all cannot collapse to a point — it makes a disc, which then has to throw angular momentum outwards before anything can reach the centre. That problem is harder than everything above and it is where most of the interesting physics of star formation actually lives.

There is one more consequence of the isothermal assumption worth stating, because it is the reason molecular clouds are studied in carbon monoxide rather than in hydrogen. What keeps the gas at ten kelvin is line emission, and hydrogen has no line to emit at that temperature — it is symmetric, so it has no dipole moment and no rotational transitions at all. The cooling is done by the trace species, at an abundance of one in ten thousand, and the same trace species is what makes the cloud visible. The molecule that keeps the cloud cold is the molecule that lets it be seen, and the coincidence is not one: both facts are the same statement about which levels are accessible at ten kelvin.

One more reading of the criterion covers the temperature range from a prestellar core to a warm cloud.

The critical mass falls as the cloud contracts, with a slope of −0.50. The Jeans mass — the least mass of molecular gas that its own pressure cannot support — against number density, for gas at 5, 15, 100 K, both axes logarithmic. The curves are straight because the mass goes as T^(3/2) ρ^(−1/2), and the slope measured off the drawn 5 K curve is −0.500 against the −0.5 that exponent requires. The consequence is the one that matters and it is a matter of sign: a cloud collapsing at constant temperature moves to the right along one of these lines, so the mass it takes to be unstable keeps falling, and sub-regions that were individually stable when the collapse began become individually unstable during it. A giant molecular cloud at 100 particles per cubic centimetre has a Jeans mass of 99 solar masses; a dense core at 10⁵ has one of 1.70. That is why a cloud of ten thousand solar masses makes a cluster rather than a star, and why the question about star formation is not what makes gas collapse but what stops the fragmenting.
Fig. 9 The critical mass at five, fifteen and a hundred kelvin. It scales as the three-halves power of the temperature, so a factor of twenty in temperature is a factor of ninety in the mass that can collapse — which is why the coldest parts of a cloud are the parts that form stars.

Where this ladder goes next

Later rungs on this anchor: the magnetic critical mass and ambipolar diffusion, which is the field-supported version of every argument here; the observation of a core, which means the millimetre continuum of cold dust rather than any line of gas, and the temperature that has to be assumed to turn one into the other; supersonic turbulence and its scaling laws, which are the reason a cloud has a linewidth at all; the freeze-out chemistry that removes the coolant; and the transition from a core to a protostar, where the collapse becomes an accretion problem and the object at the centre begins to argue back.

Where this goes next

Two densities in thermal balance at one pressure, 121 times apart. Thermal equilibrium for interstellar gas, drawn as pressure against density with both axes logarithmic. Every point on the curve is a temperature between 40 and 9000 K at which cooling exactly balances the 2·10⁻²⁶ erg per second per hydrogen nucleus that grain photoelectrons deliver. The curve is not monotonic: it rises to 5007 K cm⁻³, falls to 1597, and rises again, so a horizontal line anywhere between those two crosses it three times. At the 3000 K cm⁻³ of the local medium the three crossings are a warm phase at 0.47 cm⁻³ and 6354 K, a cold phase at 57 cm⁻³ and 52 K, and one in between drawn dashed because it cannot survive — a parcel there that is squeezed cools and keeps contracting, and one that expands heats and keeps expanding. The two survivors differ by a factor of 121 in density and by exactly the same factor in temperature — necessarily the same, because their product is the pressure both are held at — and yet they press on one another equally, which is why they can share the same volume of the disc indefinitely rather than mixing. Two temperatures, and nothing in between The gas between the stars is not one gas at one temperature. Thermal balance has three solutions at any ordinary interstellar pressure, the middle one cannot survive being nudged, and the two that can differ by a factor of over a hundred in both density and temperature — which is why a map of the interstellar medium looks like clouds rather than like fog. Inside a star that is holding itself up (polytrope n = 3). Temperature, density and pressure through a star, as fractions of their central values, against fractional radius. All three come from one numerical integration of the Lane–Emden equation at index 3, which is the hydrostatic balance written for a gas whose pressure is a power of its density. The inner half of the radius holds 90% of the mass, and the outer half is nearly weightless — which is why the load, and so the temperature, is concentrated where the burning is. A star is held up by its own weight A star has a central temperature because it has a central pressure, and it has a central pressure because everything above is pressing down. The nuclear reactions do not set that temperature — they obey it. One population, three integrals, two ends. The same mass function weighted three ways, each normalised so that the area under it is one, against mass on a logarithmic axis — so a share of the page is a share of the total. The light is a zero-age population's, every star still on the main sequence, which is the only age at which the top of the range is present at all. The steep curve on the left is the number of stars, which peaks at 0.08 M☉ and falls away because the function is steeper than m⁻¹; the middle one is the mass they carry; the one on the right is the light they emit, computed from this file's own main-sequence relation and peaking at 55.0 M☉ — 682 times further up the axis. The population that is counted and the population that is seen are two different populations. Above 8 M☉ there are 0.63% of the stars, 21% of the mass and 99% of the light. The slopes are α = 1.3 above 0.08 M☉, α = 2.3 above 0.5 M☉, and the low-mass end is where the honesty runs out: it has never been measured in another galaxy, and every mass inferred from a luminosity assumes it. Most stars are small, and most of the light is not The mass function is steep and the mass–luminosity relation is steeper, so counting stars and measuring their output are integrals of the same function dominated by opposite ends of it. Every mass inferred from a brightness passes through that mismatch.

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Bonnor ebert sphereDense coreExternal pressureFragmentationFree-fall timeInitial mass functionIsothermal collapseJeans massLane emden equationMolecular cloudOpacity limitVirial parameter