A few thousand clouds outrun a hundred billion stars

A galaxy is collisionless in the sense that matters: its stars are so numerous and so light compared with the whole that the deflections they give one another add up to nothing over a Hubble time. The relaxation time of the Milky Way’s disc, computed from its stars, is longer than the age of the universe by a factor of hundreds of thousands, and the stars move in the smooth potential of the whole as though the others were not there.

And yet the disc has visibly relaxed in one respect. Stars born recently move with the gas they formed from, a few kilometres a second from circular motion. Stars born ten billion years ago move with random velocities of forty or fifty kilometres a second. The disc has been heated, steadily, over its whole life — and the heating cannot be the stars’ doing.

The resolution is in the formula for the relaxation time itself, which contains the mass of the perturbers in a way that is easy to overlook. A star is not relaxed by the number of bodies it passes but by the mass of each one.

Why mass and not number #

The relaxation time comes from adding up many small deflections. A perturber of mass m passing at impact parameter b and relative speed v changes the star’s velocity by about $2Gm/(bv)$, in a random direction. The squares of those kicks add, because their directions are random, and the rate at which the squared velocity grows is the number of encounters per unit time times the square of each kick, integrated over impact parameters:

$$\frac{d\langle\Delta v^2\rangle}{dt} \propto n\,v \int \left(\frac{2Gm}{bv}\right)^2 2\pi b\,db = \frac{8\pi G^2 n m^2}{v}\ln\Lambda.$$

The number density n and the square of the mass appear together. Write the mass density as $\rho = nm$ and the rate becomes proportional to $\rho m$: at a given mass density, the heating rate is proportional to the mass of each perturber. Split the same mass into ten times as many pieces of a tenth the mass, and the kicks are ten times weaker, their squares a hundred times weaker, and there are ten times as many of them. The net heating falls by ten.

That is the whole of the opening figure. A disc in which the stars carry most of the mass and the molecular clouds a third as much is relaxed almost entirely by the clouds, because each cloud is a million times heavier than a star. The stars’ relaxation time is 350 thousand times the age of the universe. The clouds’ is about the age of the disc. A hundred-thousandth of the bodies, carrying less than half the mass, do essentially all the relaxing.

Lyman Spitzer and Martin Schwarzschild made this argument in 1951, before molecular clouds had been observed. They knew the disc had been heated and that the stars could not have done it, and they postulated massive gas complexes — clouds of a million solar masses — as the perturbers. The giant molecular clouds detected by their carbon monoxide emission two decades later turned out to have roughly the masses and the numbers the argument needed.

The clouds live in a thin layer #

The first estimate treats the clouds as point masses filling the disc uniformly. They do neither, and both departures weaken the heating.

They are confined to a layer near the Galactic plane, about fifty parsecs thick on each side, because they form from the gas and the gas is cold and thin. A star with a small random velocity oscillates through that layer and spends all its time inside it. A star that has been heated oscillates higher — to a height proportional to its vertical velocity — and spends a shrinking fraction of each oscillation where the clouds are.

The heating therefore weakens as it proceeds. A population heated from ten to thirty kilometres a second now meets the clouds a fifth as often as it did, and to heat it further takes correspondingly longer. The effect turns a heating law that would have grown as the cube root of time into one that grows as about the fourth root, and it makes the disc’s thickness self-limiting: the hotter a population is, the less it is heated.

The clouds are also not points. A molecular cloud is twenty parsecs or more across, and a star passing through or near one is not deflected by the full mass concentrated at its centre; the close encounters that dominate the Coulomb logarithm for point masses are softened away. The useful range of impact parameters runs from the cloud’s own size to the thickness of the layer, and its logarithm is close to one rather than five or ten. The heating rate drops by the same factor.

Heating that falls short #

With both corrections the clouds still heat the disc — but not enough.

The shortfall is a factor of two to three in velocity. It is not a matter of tuning: the cloud masses and the molecular gas density near the Sun are measured, and a population of more massive clouds helps only modestly.

Two independent facts confirm that the clouds are not the whole story. The measured relation between a population’s age and its velocity dispersion near the Sun rises more steeply than the clouds can manage — as the age to a power somewhere between 0.3 and 0.5 depending on the sample and the component, where confined clouds give 0.2 to 0.25. And the shape of the velocity distribution is wrong: clouds scatter stars roughly isotropically, which would leave the vertical and radial dispersions nearly equal, while in the disc the radial dispersion is about twice the vertical one.

The heating the arms supply #

The missing heating is in the plane, and the spiral arms supply it. A spiral pattern is a wave of gravitational potential that sweeps through the disc, and a star meeting it away from corotation is given a kick that changes its random motion in the plane. A single steady pattern does little lasting damage, because the kicks are periodic and largely reversible. A succession of transient patterns, each growing and decaying over a few rotations, does a great deal: every new pattern meets the stars at a different phase, and the kicks add like a random walk. Simulations with transient spirals of the strength observed in disc galaxies heat the in-plane motions at about the rate the age–velocity relation requires.

Spirals are flat. They push stars around in the plane and scarcely at all vertically, because the pattern has no vertical structure on the scale of a star’s vertical oscillation. On their own they would produce a disc that is hot in the plane and cold out of it — thinner and more radially heated than the real one.

The two mechanisms together explain the shape. The spirals pump energy into the in-plane motions; the clouds, which scatter in all directions, take some of that energy and redirect it into the vertical. The ratio of vertical to radial dispersion settles where the redirection balances the pumping, at about a half, which is what is measured. The clouds are not the furnace of the disc’s heating. They are the converter that turns a flat heating into a three-dimensional one.

That division of labour also explains why old stars have migrated as well as heated. A transient spiral does something besides heating: at its corotation radius it can move a star’s whole orbit inward or outward without heating it at all, exchanging angular momentum with the pattern. The same transient arms that heat the disc stir it radially, and the old stars near the Sun include many born kiloparsecs away.

A relation that needs ages #

The heating is read from the age–velocity relation, and the relation is only as good as its ages. Measuring the velocity of a nearby star is now routine: astrometric satellites give positions and proper motions to microarcseconds, and spectrographs give radial velocities to a few hundred metres a second. Measuring its age is not routine at all, and the history of the subject is largely a history of age estimates.

The earliest relations used the strength of chromospheric emission as a clock, on the reasoning that a star’s magnetic activity declines as it spins down; that clock is the same one that starts by forgetting a star’s initial rotation, and it is reliable only for stars like the Sun and younger. Later work used the position of a star relative to the main-sequence turn-off — an age read off a bend in the colour–magnitude diagram — which works for subgiants and stars near the turn-off and requires a distance and a metallicity for each one. The most precise ages now come from the oscillation frequencies of red giants, which fix their masses and therefore their ages to ten or twenty per cent, and from those the relation can be followed to ten billion years with thousands of stars.

What the different clocks agree on is the shape: dispersion rising steadily with age, faster in the plane than out of it, with no sign of saturation until the oldest thin-disc stars, where the thick disc begins to contaminate the sample. What they do not agree on to better than a tenth is the exponent, because each clock has its own systematic error with age, and an error that grows with age tilts the relation. That is why the comparison with the cloud model above is made at the level of a factor of two in velocity rather than a precise slope.

Clouds that should have cooled #

There is a puzzle in the clouds’ own motion that the heating argument raises and has to answer. Relaxation drives a system towards equipartition of kinetic energy: heavy bodies should end up moving slowly and light ones quickly. The clouds and the stars exchange energy in every encounter, and the stars gain at the clouds’ expense — which is the same thing as the drag a heavy body feels from the lighter ones it passes. If the clouds were a permanent population, that drag would have cooled them to a random velocity a thousandth of the stars’, and with nothing left to give, the heating would have stopped.

The clouds move at about seven kilometres a second relative to circular motion — slower than the stars, but nothing like as slow as equipartition demands. They can keep moving because they are not permanent. A molecular cloud forms from the turbulent atomic gas, lives for a few tens of millions of years, and is dispersed by the stars it makes; its random velocity is the gas’s, re-set at every formation by the turbulence that supernovae and stellar winds keep driving. The energy that ends up in the stars’ random motions therefore comes, in the end, from the energy that massive stars put into the interstellar medium, routed through the clouds’ formation and the clouds’ gravity.

So the cloud heating is not quite the thermodynamics of a gravitating gas trying to reach equilibrium, in which, as for any self-gravitating system, losing energy makes it hotter. It is an engine. The stars feed the gas through supernovae, the gas makes clouds with fresh random motions, and the clouds pass some of that motion back to the stars, a little more with every generation.

The same arithmetic, applied to the dark halo #

The opening figure’s lesson — mass, not number — has a consequence well outside the disc.

The dark halo near the Sun has a mass density of about a hundredth of a solar mass per cubic parsec. If that mass were in elementary particles, its contribution to the disc’s relaxation would be nil. If it were in compact objects — black holes, say, of some mass m — every one of them passing through the disc would kick the stars, and the heating rate would be proportional to m. A halo made of heavy enough objects would heat the disc more than it is heated.

Cedric Lacey and Jeremiah Ostriker turned this round in 1985. Black holes of about two million solar masses, they pointed out, would heat the disc at just the observed rate and with just the observed growth as the square root of age — an explanation of the disc’s heating by the halo itself. The idea was striking and it did not survive. The heating it predicts is isotropic, and the disc’s ratio of vertical to radial dispersion is not; the observed age–velocity relation is not cleanly a square root; and the spiral and cloud mechanisms, once modelled properly, account for most of the heating without help.

What survives is the bound. Whatever mechanism heats the disc, the disc is no hotter than it is, and that caps the heating any population of compact halo objects can have delivered. With all of the halo in such objects, they cannot weigh more than about a million suns each; with a fraction f of the halo in them, the cap rises as one over f.

It is a remarkable constraint to get from so little. The thickness of the Milky Way’s disc, measured from the vertical motions of nearby stars, limits the mass of individual objects in a dark halo that has never been seen, at a level that other methods — the survival of wide binary stars, the absence of microlensing events of long duration — have since tightened from other directions. The disc is a detector, and its readout is how hot it has become.

A test the stars cannot fail and the clouds can #

There is one more feature of cloud heating that distinguishes it from every alternative, and it concerns what the clouds do to each other’s neighbours rather than to the disc as a whole.

A cloud passing a star cluster heats the cluster as well as the disc around it. A cluster is a bound system with an internal velocity dispersion of a kilometre a second or less, and a passing cloud’s tidal force pumps energy into its internal motions — the impulsive heating that also disrupts globular clusters passing through the disc. Open clusters in the solar neighbourhood are destroyed in a few hundred million years, much faster than their own internal relaxation would do it, and encounters with molecular clouds are the leading reason. The same clouds that warm the disc over billions of years dismantle its clusters in hundreds of millions.

That is a consistency check with no free parameter left. The cloud masses and numbers that are needed to supply the vertical part of the disc’s heating are the masses and numbers that should disrupt open clusters at the observed rate. Both come out within a factor of a few of what is measured, and the agreement — two effects of one population of clouds, measured in completely different ways — is better evidence for the clouds’ role than either effect alone.

What the model leaves out #

The heating calculation is a single population of identical clouds in a uniform layer, with a constant mass density and a constant star-formation history. The real cloud population has a broad mass spectrum, in which the most massive clouds dominate the heating as they dominate the mass; it is concentrated in the spiral arms rather than spread evenly; and its total mass has changed over the disc’s life, probably decreasing as the gas was consumed. Each of these changes the numbers by factors of order unity and none of them removes the shortfall, which is the result of the geometry of the thin layer rather than of any detail.

The disc stars’ vertical oscillation is treated as harmonic, in a midplane density that is held fixed. As stars are heated they reach heights where the disc’s own density has fallen and the oscillation is no longer harmonic, which changes the fraction of time spent in the layer at large dispersions. And the age–velocity relation the calculation is compared with is itself uncertain at large ages, because the oldest disc stars blend into the thick disc, a population that was probably heated violently and early rather than gradually.

The general point: weigh the perturbers, do not count them #

The arithmetic that makes the stars ineffective and the clouds effective is a theorem about any random walk driven by kicks proportional to a mass. It is the reason a few massive substructures in a dark halo can heat a galaxy’s disc while a smooth halo of particles cannot; the reason a cluster’s few black holes, sunk to its centre, dominate its central dynamics; and the reason a single massive satellite sinking through a halo is felt by the halo while the same mass in a stream of small ones is not. In each case the question to ask of a population is not how many there are but how the mass is divided, and the answer is weighted entirely towards the heavy end.

The stars of the disc have been there all along, and they have done nothing to one another. Its few thousand heaviest clouds, which come and go on timescales of tens of millions of years, have been doing almost all the work.

Still open: how much of the disc’s heat came from outside #

The account here is of heating from within: spirals, a bar, molecular clouds. The disc has also been hit from outside. Satellite galaxies have fallen through it, a major merger early in its history probably made the thick disc, and the dark halo is not smooth but lumpy with subhaloes too dark to contain stars. Each passing lump heats the disc by the same arithmetic as a cloud, with a mass up to a billion times larger and an encounter rate far smaller. How much of the present dispersion of old stars is the slow, internal heating of the clouds and arms, and how much the scar of a few large external impacts, is not settled — and the answer is a measurement, through the disc’s temperature, of how much substructure the dark halo contains.