Two spins that cancel, and an orbit that leaves

A disc’s size is set by the spin of the halo it formed in, and the spin, measured as a dimensionless parameter, is distributed the same way for haloes of every mass. That makes the specific angular momentum of a galaxy — its angular momentum per unit mass, $j$ — scale as the two-thirds power of its mass, and so it does: disc galaxies over three decades of stellar mass lie on a line of that slope in the $j$–$M$ plane, a relation first drawn by Fall in 1983. Elliptical galaxies lie on a parallel line, with the same slope, about five times lower.

The parallel lines say that morphology is not a matter of how much angular momentum a galaxy was given, since all galaxies of the same mass were given the same, but of how much it kept. That was the conclusion of the first essay on this subject, and it leaves the obvious question open. How is angular momentum lost, and why by a factor of five rather than two or fifty? Angular momentum is conserved; it can be moved but not destroyed. The answer turns out to be almost entirely vector addition.

Adding two arrows pointing anywhere #

Consider the simplest case: two disc galaxies of equal mass, each on the relation, merging. Each carries a spin vector — its total angular momentum — and the directions of the two vectors are unrelated, because each was set by the tidal field around its own halo long before the two met. The merged galaxy has twice the mass. What is its angular momentum?

If the orbit on which the two galaxies approached each other is set aside for a moment, the remnant’s angular momentum is the vector sum of the two spins. Two equal vectors at an angle $\theta$ add to a vector of length $2|\cos\tfrac{\theta}{2}|$ times either one. Aligned, they double; opposed, they cancel completely. Averaged over all orientations, equally likely in every direction, the sum is two thirds of the aligned value. The remnant’s specific angular momentum — the sum divided by the doubled mass — is therefore on average two thirds of what each progenitor had.

But the remnant has twice the mass, and the relation at twice the mass is higher, by two to the power two-thirds, a factor of 1.59. A galaxy that keeps two thirds of its specific angular momentum while needing 1.59 times as much to stay on the line falls to about 0.42 of the line. Computed over twenty thousand random orientations, the median remnant sits at 0.45 of the relation; the best case, exactly aligned spins, reaches only 0.63. No equal-mass merger of two galaxies on the relation can produce a remnant that is still on it, unless the orbit contributes.

The mass ratio matters as much as the orientation. A companion of a tenth of the mass carries a tenth of the mass and, being on the relation, a little under a twentieth of the angular momentum; it barely changes either total, and the remnant sits at 0.85 of the line. A companion of a third of the mass moves it to 0.63. The cancellation that turns a disc into a spheroid needs a partner of comparable mass, which is why the morphology–density relation — ellipticals concentrated in groups and clusters, where major mergers were once common — is at root a relation between environment and the mass ratios of past mergers.

Two generations of cancellation #

The most massive spheroids were not made by a single merger of two discs. They were assembled from galaxies that had themselves merged, so the progenitors of a late merger are already remnants, already below the line. When two such galaxies combine, the cancellation compounds.

Each generation of equal-mass mergers multiplies the typical height above or below the relation by about 0.42, so the median after two generations is 0.18 and after three 0.08. The spheroids’ relation, a fifth of the discs’, falls between one and two generations — close to two. A population of massive ellipticals assembled by two rounds of dry, major mergers from disc galaxies is expected to sit about where massive ellipticals are measured to sit, from nothing but the geometry of adding vectors in random directions.

The distributions also spread as they fall. Each merger multiplies the height by a random factor, so its logarithm performs a random walk and the population broadens with every generation. That is observed too: disc galaxies of a given mass differ in specific angular momentum by about a factor of two, while spheroids of the same mass differ by much more, and some of the most massive barely rotate at all.

Many small mergers, and why they do not add up #

If a merger with a companion a tenth of the mass moves a galaxy to 0.85 of the relation, ten such mergers might be expected to do what one major merger does. They do not, and the reason is that each one also moves the relation. After ten mergers of a tenth of the mass each, the galaxy’s mass has grown by a factor of about 2.6 and the relation at that mass is higher by a factor of 1.9; the angular momentum the companions brought, with their spins pointing in random directions, has mostly cancelled against itself rather than against the original spin, which survives nearly intact. The galaxy drifts down relative to the line, but slowly — a random walk with small steps, where a major merger is one large one.

What minor mergers change is where the mass sits. Companions a tenth of the primary’s mass are torn apart by its tides before they reach the centre and deposit their stars at large radii, in an extended envelope. That is how the most massive ellipticals are thought to have grown in size by a factor of three or four since the universe was a quarter of its present age while growing in mass by much less: dry minor mergers build their outer parts from the outside in. The envelope carries whatever orbital angular momentum the companions’ debris retained, which is part of the reason the outer halo of an elliptical often rotates when its centre barely does. The tidal tails and bridges seen around interacting galaxies are the same debris caught before it has phase-mixed into a smooth envelope.

The orbit that carries more than the spins #

The calculation above set aside the orbit, and the orbit is not small. Two galaxies falling together on anything other than a perfectly head-on path carry orbital angular momentum about their common centre of mass, and for typical encounters it exceeds the two galaxies’ own spins together. If the stars of the remnant kept all of it, a merger would not reduce the specific angular momentum at all — it would raise it.

Where the orbital angular momentum goes is therefore the whole question, and it is decided by dynamical friction. Two galaxies do not merge by falling straight into each other; each is embedded in a dark halo many times its size, and as they pass through each other’s haloes they raise wakes of dark matter that drag them back. The drag removes their orbital energy and angular momentum and hands both to the dark matter, which is spread over a volume far larger than the stars. Whether they merge at all is set by that drag, and the same drag decides how much of the orbit the stars keep: in a merger of two gas-poor galaxies, almost none. The dark halo spins up; the stars at its centre end up with the two spins, partially cancelled, and little else.

Gas changes that. Gas collides with gas, shocks, and radiates away its energy while keeping much of its angular momentum, so the gas in a merger can settle back into a rotating disc carrying a large share of the orbit. Simulations of gas-rich mergers routinely regrow discs from the wreckage, and a galaxy that merges while it still has most of its baryons in gas can emerge as a disc again. The figure’s horizontal axis is thus, in effect, the gas fraction of the merger. At the left, dry mergers make spheroids; at the right, wet mergers rebuild discs. The same collision produces opposite morphologies depending on one number that the vector sum does not supply.

That is the resolution of a puzzle that ran for two decades. Early simulations of galaxy formation made discs far too small and too few, because the gas cooled into small clumps early, merged, and handed its angular momentum to the haloes through exactly this friction. The discs came out a factor of several below the Fall relation — spheroid-like. The fix was feedback: stellar winds and supernovae that keep gas hot and diffuse until late, so that it reaches the galaxy after most of the mergers are over and keeps its angular momentum. The relation’s two lines are, in this reading, a record of when each galaxy’s stars formed relative to its mergers.

Fast rotators and slow ones #

Ellipticals turn out to be two populations, and the division follows the arithmetic above. Early-type galaxies were long thought of as held up by disorder — random stellar motions rather than rotation — and some are. But integral-field spectrographs, which measure a velocity and a dispersion at every point across a galaxy’s face, showed that most of them rotate substantially.

The spin parameter $\lambda_R$ is the luminosity-weighted ratio of ordered to total motion, measured out to about one effective radius, and against the galaxy’s apparent flattening it divides early-type galaxies cleanly. About eighty-five per cent are fast rotators: flattened, rotating systems close to the line of an oblate body flattened by its own rotation. They are discs that have been heated rather than destroyed — the products of minor mergers, which barely move a galaxy off the relation, and of gas-rich major mergers that rebuilt their discs. The remaining fifteen per cent are slow rotators: round, barely rotating, often with a core whose stars rotate in a different direction from the outer parts, and concentrated almost entirely above about $2\times10^{11}$ solar masses, the most massive galaxies of all, in the centres of groups and clusters.

The slow rotators are what the vector sum leaves when the orbit is lost and the mergers are between equals of similar history. A kinematically decoupled core is the fossil of two spins that did not cancel everywhere: an inner component carrying one progenitor’s residual rotation, an outer one carrying another’s. Their location — only at the highest masses, only in the densest environments — is where the merger histories are dominated by several generations of gas-poor major mergers, and nowhere else is that true.

The slow rotators are also uniformly old and red. They sit at the top of the red sequence, with stellar populations that formed early and stopped forming, which is exactly the condition the argument requires: a galaxy whose gas was used up or expelled before its last major mergers had nothing left to settle back into a disc, and every merger after that was dry. A galaxy with fresh gas at the time of its last major merger can rebuild a disc and join the fast rotators, and the young stars in many fast rotators’ discs are that rebuilt component. Rotation and colour are therefore two records of one history — when the gas ran out relative to when the mergers happened — read from two different measurements.

What the vectors leave out #

The calculation is deliberately minimal. It gives every galaxy the same spin magnitude at a given mass, where real spins are distributed with a lognormal spread of about a factor of two, so real remnants are more widely spread than the figures show. It treats the orbit’s direction as unrelated to either spin, whereas galaxies falling along the same filament tend to have spins and orbits partly aligned with the filament, which reduces the cancellation somewhat. And it measures angular momentum as a total, when what is observed is the angular momentum of the stars within a few effective radii; mergers redistribute angular momentum outward as well as removing it, so a remnant can have a slowly rotating centre and a faster outer envelope.

Nor does the calculation know about stellar feedback, the gas fraction at the time of each merger, or the bars and spiral arms that move angular momentum outward within an intact disc. All of these change the height of a particular galaxy. None of them changes the central fact: two vectors in random directions add, on average, to two thirds of their aligned sum, and the relation that galaxies are measured against rises faster than that.

What is actually measured #

The two relations are not measured the same way, and the difference matters to what the factor of five means. For a disc galaxy the specific angular momentum is nearly a single number: the stars move on circular orbits, the rotation curve is close to flat beyond the inner few kiloparsecs, and for an exponential disc of scale length $R_d$ rotating at speed $v$, $j = 2vR_d$ to within ten per cent. Both quantities come from a rotation curve and a photometric profile, and the relation for discs is correspondingly tight.

For a spheroid the measurement is much harder. Most of its stars move on disordered orbits, so the net rotation is a small residual on top of a large dispersion, and it has to be mapped across the galaxy’s face in two dimensions. And most of a spheroid’s angular momentum lies at large radii, where the light is faint: for a de Vaucouleurs profile, half of the total angular momentum sits beyond about two effective radii, far outside the region an integral-field spectrograph covers. The spheroid relation was established by extrapolating the measured inner rotation outwards and checking the extrapolation against the velocities of planetary nebulae and globular clusters, which can be measured one at a time out to ten effective radii. The factor of five is secure; its value is uncertain by perhaps thirty per cent, most of it from that outer extrapolation.

That asymmetry cuts in the direction the merger argument predicts. If ellipticals carry angular momentum preferentially in their outer parts — in envelopes built by minor mergers, and in orbits partly retained from major ones — then the inner regions measured best will understate the total, and the true offset from the disc relation is somewhat smaller than the inner measurements alone would give.

The accounting of the orbit #

The same arithmetic runs in the other direction for the dark haloes. Every gram of angular momentum that a dry merger removes from the stars is added to the dark matter, which is why haloes that host ellipticals are expected to spin as fast as those hosting discs — and in simulations do. The factor of five between discs and spheroids is therefore not a loss from the universe but a transfer from the part of a galaxy that is seen to the part that is not. It can in principle be checked: the outer stellar halo and the globular clusters of an elliptical, which were stripped from the merging galaxies early and share the dark matter’s dynamics, should rotate faster than the stars at the centre. Measured with planetary nebulae and globular-cluster velocities out to five and ten effective radii, many do.

Still open: whether the orbit’s share can be measured #

The vector-sum argument explains the height of the spheroid relation, its greater scatter and the existence of slow rotators with a minimum of assumptions. What it cannot do is predict an individual galaxy, because the share of the orbit its stars kept is a property of its particular history — the gas fraction at each merger, the timing of its star formation, the friction exerted by its halo. That share is the one quantity that turns a merger into a spheroid or back into a disc, and it has never been measured directly for any galaxy. The closest approach is statistical: comparing the rotation of stars of different ages within the same galaxy, since stars formed before a merger should carry the cancelled spins and stars formed from the gas that settled afterwards should carry the orbit. Surveys that measure stellar ages and velocities together, spaxel by spaxel across thousands of galaxies, are beginning to separate those populations, and in them the orbit’s share of angular momentum becomes, for the first time, a number that can be read rather than inferred.