A disc held open by what cannot be photographed

The Milky Way’s neutral hydrogen forms a layer about three hundred parsecs thick, in a disc thirty kiloparsecs across. That is a ratio of a hundred to one, which sounds thin and is not thin enough.

A layer of gas held up by its own thermal pressure against the gravity of the stars around it has a scale height that follows from two numbers: the sound speed and the vertical gravity. Put in the values and the answer comes out several times too small.

The calculation that fails #

Hydrostatic equilibrium in a vertical direction is the simplest equation in the subject: the pressure gradient balances the weight. In a layer thin compared with the disc’s radius, the gravity is approximately constant with height and equal to two pi G times the surface density of everything below.

For the solar neighbourhood that surface density is about fifty solar masses per square parsec of stars plus gas, giving a vertical gravity of a few times ten to the minus ninth in centimetre-gram-second units.

The warm neutral gas is at about eight thousand kelvin, giving a sound speed near eight kilometres a second. The scale height is the square of that divided by the gravity, and it comes out at about fifty parsecs.

The measured half-thickness of the neutral hydrogen layer is a hundred and fifty. The prediction is short by a factor of three, which is not a discrepancy to be absorbed into the surface density or the temperature — both are measured, and neither is wrong by a factor of three.

It is worth being clear about how the thickness is measured, because a layer seen from inside it is not obviously measurable at all. The route is the distribution of emission with galactic latitude combined with a rotation curve: gas at a given velocity is at a known distance along the line of sight, so its height above the plane follows from its latitude, and stacking many lines of sight builds the vertical profile. The method is indirect and it has been checked against external galaxies seen edge-on, which give thicknesses of the same order. The gravity is measured by an equally indirect route and one worth naming, because it is a classical problem. The vertical motions of a population of tracer stars are governed by the same potential the gas sits in, so measuring their velocity dispersion and their density profile gives the surface density directly — the Oort limit, computed in 1932 and refined ever since, and the earliest quantitative argument for unseen mass anywhere — the local ancestor of the mass that is not the light.

The three terms that were left out #

What is missing is that thermal pressure is not the only pressure the gas has.

The first addition is turbulence. Interstellar gas is in motion, with non-thermal velocity dispersions of five to ten kilometres a second measured directly from the widths of spectral lines — widths far broader than the thermal width of the gas at its measured temperature. Bulk motion exerts a pressure equal to the density times the square of the velocity dispersion, and at seven kilometres a second that is comparable to the thermal term.

The second is the magnetic field. A field of five microgauss exerts a pressure of about ten to the minus twelfth ergs per cubic centimetre, and that is again comparable.

The third is the cosmic rays. Their energy density near the Sun, measured directly by detectors above the atmosphere and corrected for the modulation the heliosphere imposes, is about one electronvolt per cubic centimetre; a relativistic gas exerts a pressure of a third of its energy density, and the resulting number is again in the same range.

Four pressures of comparable size is a striking observation and it is worth being careful about what it is not. It is not a theorem: nothing in the physics requires four independent quantities to agree within a factor of two. It is a measurement, made four times with four different instruments, and the agreement is the finding.

The four instruments are worth listing, because their independence is the point. The thermal term comes from spectral line ratios in a radio telescope. The turbulent term comes from line widths in the same data, after subtracting the thermal contribution. The magnetic term comes from Zeeman splitting and Faraday rotation, neither of which involves a line width. The cosmic-ray term comes from a particle detector on a spacecraft. No systematic error is shared between them, so their agreement cannot be an artefact of a common assumption.

One further check exists and it is a good one. The total pressure inferred by summing the four should equal the weight of the material above, since that is what hydrostatic equilibrium requires, and the weight is computed from a surface density measured independently of all four. The two agree to within the uncertainties, which is a closure test rather than a fit.

Why the near-equality might be real #

There are arguments for why the four should be comparable and none of them is decisive.

The magnetic and turbulent terms have a mechanism connecting them: turbulence in a conducting medium amplifies a field until the field’s energy approaches the turbulence’s, at which point the field resists further stretching. That is a saturation condition and it predicts rough equality of those two.

The cosmic rays have a different mechanism. A cosmic-ray population streaming through a magnetised medium faster than the Alfvén speed generates waves that scatter it, and the scattering limits the streaming — so the cosmic rays and the field are coupled, and their pressures are not independent.

The thermal term is the odd one out. Its value is set by the heating and cooling of the gas, which have nothing to do with the other three, and its rough agreement with them is the part that looks like a coincidence. The two-phase structure of the neutral gas is fixed by that same balance, and it is fixed at a pressure that has no reason to know about the field.

There is a possible resolution, which is that the thermal pressure is not independent after all: star formation is regulated by the total pressure, supernovae supply both the heating and the turbulence, and the whole system settles into a state where the terms track one another. That is the self-regulation picture, and it is the current framework rather than a demonstrated result.

The picture has a testable consequence and the test is being run. If the terms are coupled through star formation, then a galaxy forming stars faster should have a proportionally larger total pressure and a correspondingly thicker or more turbulent gas layer. Surveys of nearby galaxies measuring gas velocity dispersion against star formation rate surface density do find a correlation, of about the right slope, with considerable scatter. That is encouraging rather than conclusive, and the scatter is where the interesting physics probably is.

There is also a limiting case that tests it from the other side. Dwarf galaxies with very low star formation rates have gas layers that are, relative to their own gravity, very thick — dispersions of eight or ten kilometres a second in systems whose gravity would support two or three. Something maintains turbulence there that supernovae cannot supply at the observed rate, and what that something is remains open.

What happens when the field is included properly #

Treating the field as an isotropic pressure is a convenience and it is wrong in a way that matters.

A magnetic field is not isotropic. It exerts a pressure across the field lines and a tension along them, and a field lying largely parallel to the disc plane supports the gas above it while also resisting being bent.

That combination is unstable. A horizontal field supporting weightless cosmic rays and heavy gas is top-heavy in the same sense a dense fluid over a light one is, and the gas can slide down the field lines into valleys while the buoyant field and cosmic rays rise into arches. The resulting Parker instability grows on a timescale of tens of millions of years, with a wavelength of about a kiloparsec. It has consequences that are observed. The gas collects in the valleys, where the column density is highest and molecular clouds form; the arches lift field and cosmic rays into the halo. Whether the instability is the dominant cloud-forming mechanism is argued, but the geometry — a magnetic field arching out of the plane over kiloparsec scales — is seen in edge-on galaxies as X-shaped polarised radio structures.

The instability also settles a question the isotropic treatment cannot even pose: whether the field helps or hinders. As a pressure it helps, by supporting the gas. As a set of rails it hinders, by letting the gas slide down into the valleys and concentrate. The net effect on the layer’s thickness depends on which wins, and the answer from simulations is that the field thickens the layer overall while creating dense structures within it — both effects at once, in the same medium.

The gas that is not in the layer at all #

A further complication is that the layer is not a single component in equilibrium, and treating it as one is a fiction.

The neutral hydrogen has at least two components with different thicknesses: a cold dense phase confined to about a hundred parsecs, and a warm diffuse one extending to four hundred or more. Ionised gas extends much further, to over a kiloparsec, and its scale height is larger than any thermal or turbulent argument gives.

And above all of it is a fountain. Supernovae drive hot gas out of the plane; it cools, condenses and falls back over tens of millions of years, and the high-velocity clouds seen falling toward the plane are the returning material.

So the “scale height” is a summary statistic for a system with several components and a circulation, and matching it with a single hydrostatic calculation is the kind of agreement that should be treated as an order-of-magnitude check rather than as a fit.

What keeps the turbulence going #

One term in the ledger has a problem the others do not, and it is worth its own section because it is the reason the whole arrangement is not obviously stable.

Turbulence decays. Energy injected at large scales cascades to small ones and is dissipated, and the decay time is roughly the eddy turnover time — for a layer a hundred parsecs thick stirred at seven kilometres a second, about ten million years. That is short compared with the age of the galaxy by three orders of magnitude, so the turbulence has to be driven continuously or it would have stopped long ago. The energy source is supernovae, and the arithmetic works. A supernova releases ten to the fifty-first ergs, of which a few per cent ends as bulk kinetic energy in the interstellar medium after the remnant has cooled; at a few per century in the galaxy that is enough to sustain the observed dispersion against decay.

It works with little margin, which is the interesting part. Estimates of the efficiency vary by a factor of a few, and the requirement is met to within about that factor. So supernova driving is the answer and it is not comfortably the answer, which is why alternative or supplementary drivers — gravitational instability of the disc, accretion of gas from outside, magnetorotational turbulence in the outer disc where supernovae are rare — continue to be argued for.

The magnetorotational option is the one worth flagging here, because it is the same instability that operates in accretion discs. A galactic disc is differentially rotating and magnetised, so the criterion that makes an accretion disc unstable is satisfied in a galaxy too. The growth rate is the orbital frequency, which in the outer disc is one per few hundred million years — slow, but faster than the gas is being used up, and in the outer regions there is nothing else on offer.

Why this matters outside the Milky Way #

The measurement matters because it is the only place all four terms can be measured, and the result is exported everywhere.

Models of galaxy formation compute a gas disc’s thickness in order to compute its density, and the density decides the star formation rate. Getting the thickness wrong by a factor of three gets the density wrong by the same factor and the rate wrong by more, since the rate depends on the density with an exponent above one. They also decide whether a disc is gravitationally stable, which depends on the velocity dispersion and therefore on the same non-thermal terms. A disc supported only by heat would be far more unstable than a real one, would fragment on a shorter timescale, and would form stars faster than galaxies are observed to — the same stability question that the spiral arms themselves depend on.

And the fraction of support that is non-thermal decides how a galaxy responds to feedback. A disc held up by cosmic rays behaves differently from one held up by thermal pressure, because cosmic rays diffuse rather than cooling — they cannot radiate their energy away, so the pressure they supply is not lost when the gas cools.

That last point is currently one of the more active areas in galaxy simulation, and its physical content is exactly the observation this essay started from: a pressure that emits nothing is a pressure that cannot be radiated away.

The one term that behaves differently #

Three of the four pressures share a property that the fourth does not, and the difference is beginning to look important.

Thermal pressure is lost when the gas cools, and gas cools efficiently. Turbulent pressure decays in an eddy turnover time. Magnetic pressure is dissipated by reconnection, slowly but not never. All three can be removed from a parcel of gas by processes internal to it.

Cosmic-ray pressure cannot. A relativistic proton loses energy to the gas only by rare nuclear collisions and by ionisation, on a timescale of hundreds of millions of years at interstellar densities, and it does not radiate at all. What it does instead is leave — diffusing along field lines, streaming outward at the Alfvén speed, and dragging the gas with it through the waves it generates. That combination — a pressure that cannot be radiated away and does escape slowly along field lines — is exactly what is needed to drive a wind out of a galaxy without first heating the gas to the point where it radiates the energy back out. Thermally driven winds are inefficient for that reason; cosmic-ray-driven winds are not, and simulations that include the term produce mass loss rates several times higher.

Whether real galaxies drive winds that way is not settled, and the calculation is sensitive to the diffusion coefficient, which is known for the Milky Way only through the ratio of spallation products described elsewhere. But the structural point is clean: of the four terms holding the disc open, the one that cannot be photographed at all is also the one that cannot be cooled away.

What is measured and what is inferred #

It is worth separating the tiers, because this subject mixes them freely.

Measured: the gas surface density, from twenty-one-centimetre and carbon monoxide emission. The stellar surface density, from star counts and from the vertical motions of tracer populations. The thermal temperature, from line ratios. The non-thermal line width, from line profiles. The layer’s thickness, from emission as a function of galactic latitude. The cosmic-ray energy density, locally, from particle detectors.

Inferred with a model: the field strength, from a rotation measure weighted by electron density or from the scatter of dust polarisation angles, neither of which is a clean volume average. The cosmic-ray pressure away from the Sun, extrapolated from synchrotron emission on an equipartition assumption that is itself questionable. The honest summary is that the failure of the thermal calculation is beyond doubt, the identity of the three missing terms is beyond doubt, and their relative sizes are known to a factor of two at best. That is enough to establish the essay’s claim — most of what holds a galaxy’s gas open does not shine — and not enough to compute a scale height from first principles. What has been established is a negative and a strong one: a galaxy’s gas layer cannot be understood as a warm atmosphere, and three quarters of what holds it open would be missing from any account written from an image.

The wider version of that negative is the reason the measurement is repeated in other galaxies whenever the data allow. If the four-way near-equality were a peculiarity of the solar neighbourhood it would be a curiosity; if it holds across galaxies of different masses, gas fractions and star formation rates, it is a regulated state and something is doing the regulating. The evidence so far is that gas velocity dispersions are remarkably uniform — eight to twelve kilometres a second in the outer discs of spirals of every kind, and rising only in the most vigorously star-forming systems — which is what a self-regulating arrangement would produce and is not what four independent quantities would.

What that uniformity does not settle is which term leads. A disc could be regulated by supernovae setting the turbulence, with the field and the cosmic rays following; or by the total pressure setting the star formation rate, with the supernovae following. The two run the same loop in opposite directions and predict the same equality.

Distinguishing them needs a system where one term is externally imposed rather than internally generated, and the candidates are galaxies being stripped as they fall into clusters, or discs stirred by an ongoing interaction. In both the turbulence has a source that is not supernovae, and the prediction is that the equality breaks — that a stripped disc should show a turbulent pressure out of proportion to its star formation rate. The measurements exist for a handful of systems and are not yet decisive. What they do establish is that a stripped disc keeps its dispersion after its star formation has fallen away, which is at least awkward for the simplest version of supernova driving. Whether it keeps it for long enough to matter is the measurement still to be made. A dispersion that decays in ten million years and a stripping event that lasted a hundred are separable in principle and not yet in practice.