A galaxy that grows faster by eating small

A good test of whether a process is understood is whether its outcome can be worked out on the back of an envelope, before the simulation. The merger of two galaxies looks like the opposite of such a process. There is a violent phase of tidal stripping and dynamical friction, the orbital energy goes into the stars in complicated ways, and pictures of it are all tails and shells. Yet if neither galaxy holds much gas, the outcome can be worked out in two lines. It needs only that energy is conserved and that the remnant comes to rest in virial equilibrium, and it gives a result so different between big and small mergers that it may decide how the largest galaxies got their size.

The puzzle it may answer was opened by surveys in the mid-2000s, using infrared imaging from Hubble and the largest ground-based telescopes. The most massive galaxies at redshift two, already red and no longer forming stars, turned out to be astonishingly small. Pieter van Dokkum, Ignacio Trujillo and their colleagues measured half-light radii of about a kiloparsec at stellar masses of $10^{11}$ solar masses, a density in stars about a hundred times higher than in any comparable galaxy nearby. No such objects were then known in the local universe. So where did they go?

They have not gone anywhere. They are presumably inside today’s massive ellipticals. They would have to have grown about three to five times in size since then, while adding only a small amount to their mass, because the census of massive galaxies at that redshift says that they have not grown by more than a factor of about two. That is hard to do. Star formation cannot do it because these galaxies have stopped forming stars. The answer, on which most work has converged, involves mergers. But which kind?

Two lines of energy conservation #

For a stellar system in virial equilibrium the total energy is $E=-K$, the negative of the kinetic energy. With mass $M$ and a mean squared speed $\sigma^2$ (up to a structural constant),

$$E = -\tfrac12 M\sigma^2 ,$$

and, because the kinetic energy is half the potential energy, $\sigma^2\propto GM/r$.

A galaxy with $(M,\sigma^2)$ takes in a companion of mass $\eta M$ and mean squared speed $\varepsilon\sigma^2$. Suppose they fall in from a nearly parabolic orbit, so there is no energy on the orbit to add, and suppose nothing is lost. The final energy is the sum, and writing it in terms of final mass and speed gives

$$\frac{\sigma_f^2}{\sigma^2} = \frac{1+\eta\varepsilon}{1+\eta}.$$

Then $r\propto M/\sigma^2$ gives

$$\frac{r_f}{r} = \frac{(1+\eta)^2}{1+\eta\varepsilon}.$$

That is the entire calculation, as set out by Naab, Johansson and Ostriker in 2009 and in earlier work by Bezanson and others. It has no details of the encounter in it at all, and yet simulations bear it out to a surprising degree.

For an equal merger of two identical galaxies, $\eta=\varepsilon=1$, and everything doubles except the speeds: $r_f=2r$, $M_f=2M$, $\sigma_f=\sigma$. The remnant has twice the size and is half as dense in each of two senses, and it lies on a line of $r\propto M$.

The other limit is where the companions are small and cold, $\eta\varepsilon\ll\eta$. Then, for a small addition of mass, $\delta\ln r\approx 2\,\delta\ln M$, so that $r\propto M^2$.

It is easy to read the result as a claim about dynamical friction. In fact it comes from a statement about bookkeeping. A galaxy must end up with an energy not much different from its original one, as the companions were barely bound and brought little binding energy of their own; yet the mass is greater, so each star has less, and the system must be puffed up to be in balance. There is a fairly simple physical picture. A small companion is loosely bound compared with the main galaxy, so it is torn apart in the outskirts, and those stars sit a long way out, on wide orbits. They have added much to the mass-weighted extent and little to the energy.

The speeds fall as the size grows #

The same bookkeeping predicts the velocity dispersion, and that is a separate observable with a different systematic error.

Observationally, the velocity dispersions of compact high-redshift galaxies were hard to measure, because they are faint and the absorption lines must be separated from the continuum at a redshift that puts them in the near infrared. By the early 2010s a number of spectra had yielded dispersions of 300 to 500 kilometres a second, which is higher than almost anything nearby, as a small dense galaxy must have, and it confirmed that the small sizes were not an artefact of imaging. A compact galaxy of $10^{11}$ solar masses and a radius of a kiloparsec really is a dense object, with a dispersion consistent with its mass, and not a misidentified something else.

That matters, since the first response to the small sizes was disbelief. A galaxy might have had an extended, faint outer envelope, which would have been missed at the surface brightnesses available at that redshift because of the $(1+z)^4$ dimming, and that would make it look smaller than it is. Later and deeper images, stacked, found a small amount of envelope but not enough to change the conclusion. And dispersions are not dimmed in the same way, since they come from the central light. So the speeds are the independent check that the radius measurements were lacking.

A related point concerns how the local galaxies appear now. A bright elliptical has a two-part profile with a compact central body and an extended outer one. In these terms, the central part is the old compact galaxy and the extended part the accumulated minor debris. The cores of today’s massive galaxies have a density of stars similar to that of those early compact objects. The idea that the galaxy was built from the inside out has a direct image in that profile.

How far each path gets #

Now put the numbers together. The local relation for massive early types has about 3.6 kiloparsecs at $10^{11}$ solar masses and a slope of about 0.57 in mass. A compact galaxy at redshift two sits at about a kiloparsec with that mass, well below that line. It can move in the plane in only one direction, which depends on the mass ratio of its mergers.

A compact galaxy growing by equal mergers gets to the present relation only at a stellar mass nearly twenty times its starting value, an impossibly massive galaxy. The minor mergers do it in about three times. Considering the uncertainties, that is about what the observed mass growth allows. The census shows a growth in mass of about a factor of two since redshift two, in a population selected at fixed number density, and the comparison is about as close as such comparisons get.

The conclusion is often stated too strongly, so it should be stated carefully. The calculation does not show that minor mergers occurred. It shows that if the size growth was through mergers, then the mergers were probably mostly minor; and a companion’s dispersion is an assumed function of its mass. The $\varepsilon$ used in the figures assumes a companion scaling that is itself uncertain at low mass. Taking the extreme $\varepsilon=0$ raises the growth further, and a higher $\varepsilon$ reduces it. The results are fairly robust because the limiting exponent of 2 depends only on the companions being much less tightly bound than the main body. They are, for any reasonable relation.

Where the small companion’s stars end up #

The argument would be a dead end if the added mass ended up in the centre. There is a clean check here. A small, loosely bound companion is expected to come apart in the outer region of its host, because it is torn apart where its own mean density falls below that of the host.

That prediction is what the data show. Van Dokkum and colleagues followed galaxies at constant number density across redshift and found that the mass inside about 5 kiloparsecs barely changes between redshift two and today, while everything outside it grows. The mass density in the central region has stayed the same for ten billion years, while the galaxy has accumulated an envelope outside.

This gives the size-growth picture a kind of evidence that is unusual for galaxy evolution, because it is a statement of where and not merely of how much. In any process that had moved stars throughout the galaxy, the inner mass would have changed. Adiabatic expansion due to mass loss from the galaxy was once proposed as a rival, for instance. A strong outflow can puff up a compact galaxy, and it would expand all radii at once, the central density falling at the same time as the outer. The observation that it does not rules that out as the principal path.

Whether there are enough companions #

The remaining question is a matter of counting. Minor mergers take time, and the time available is about ten billion years. Are there enough companions at the right masses, around these galaxies, on merging orbits?

Pair counts tell how many companions there are within some small projected distance. To get a merger rate one needs an estimate of how long a pair takes to merge, which comes from simulations and is uncertain by factors of two or more. Several groups have done these counts around massive galaxies at redshifts from zero to about two and a half. Most of them conclude that the observed rate of mergers with companions down to a mass ratio of 1:10 can account for all of the size growth since a redshift of about one. For the period between redshifts of two and one, the size growth seems to be somewhat faster than the merger rate allows. So an additional process may be necessary at early times, or there may be more very small companions than the surveys can currently count.

Another contribution to size growth is purely apparent. New galaxies keep joining the red population as they stop forming stars, and later arrivals, having formed in a lower-density universe, are larger. So the average size of the sample grows even if no individual galaxy changes at all. This progenitor bias may account for perhaps half the growth in the average size at a given mass in some analyses. It makes the requirement on the individual galaxies smaller, without removing it, as a fixed-number-density selection demonstrates.

The other test comes from nearby. Compact galaxies at the early size have survived, in small numbers. Very massive, dense “relic” galaxies, such as NGC 1277 in the Perseus cluster, have been found nearby, old, with nearly the same structure as the early objects. Their rarity, a number density a thousand times lower than the population at redshift two, is what might be expected if most have been built on since and only a few escaped mergers, for instance because they fell into a cluster early, where encounters are too fast to merge.

What the virial argument assumes #

The remnant is fully virialised and has the same structure. A real remnant’s profile changes in the merger, and the structural factor connecting $r$ and $\sigma^2$ to the energy depends on it. Simulations indicate that the difference is about ten per cent in the case of a binary merger. Over many minor mergers the profile extends considerably and the radial profile index rises; the half-mass radius that the formula gives is not necessarily what an astronomer measures, as a half-light radius in a limited aperture.

There is no orbital energy. Bound and parabolic orbits are similar in this respect. A hyperbolic encounter that barely merges gives somewhat larger size growth. Dynamical friction shifts orbital energy into the host’s stars, and that is included in the energy balance, not a separate term.

There is no dark matter or gas in the sum. Some of the orbital energy goes into the dark halo, so the stellar system holds on to less of it than a gas-free stellar calculation assumes. Some simulations find that at very low mass ratios the dark matter’s share becomes significant. And gas-rich companions are not dry. Star formation in a deposited gas clump can shrink a galaxy’s central body rather than extend it. The observed gas fractions fall steeply below redshift one, so the approximation should get better with time.

Nothing here can show a merger. No curve in these figures is the path of an actual galaxy. They are sequences of equilibria. Faint shells and streams around ellipticals are an independent line of evidence that mergers do occur, and the surface brightness at which they appear is near the limit that a sky can allow.

A familiar arithmetic elsewhere #

A surprising link to something much smaller comes from one more view of the same inequality. A cluster of stars that has a star ejected in an encounter also conserves energy, with a mass that decreases and an energy that does not. There, a smaller, more tightly bound system results from losing a light body, the reverse of what happens here. In both cases what governs the outcome is what the light body carries per unit mass, not its share of the total. Adding what has little energy expands the system, removing it contracts, and only the virial theorem’s balance between the two changes.

Similarly, the argument explains why the same procedure applies at the opposite extreme of the system that gets hotter as it loses energy. A self-gravitating body has a negative heat capacity: extract energy and its particles speed up. A galaxy that adds zero-energy mass has a similarly surprising response. The mass is added with no energy and so per unit mass the energy has decreased in size; its stars slow down and its radius increases. Everything in this family of results comes from the virial theorem’s strict link of kinetic to potential energy, with its minus sign.

A plane that should be broken and is not #

There is one more thing that the arithmetic should be able to do, and it is a real constraint on the story. Massive early-type galaxies today lie on a tight relation between radius, velocity dispersion and surface brightness, the fundamental plane. Its small scatter has often been read as saying that their formation was orderly, since a chance sequence of mergers might be expected to shake them off it. Yet the virial theorem, used in the opposite direction, says that any relaxed remnant must lie on a similar plane regardless of history, for an equilibrium that the system reaches on its own account. What varies is where along it the galaxy ends up.

Mergers of equal mass move a galaxy along the plane in the direction that $r\propto M$ and fixed $\sigma$ prescribe. Minor mergers take it in the direction that $r\propto M^2$ and falling $\sigma$ prescribe. Both directions lie close to the plane, which is why such a tight relation is able to survive a violent history. The direction in which a real population has moved is an independent test, because the two tracks separate over a large range of mass, and the massive end of the plane has a measurable tilt and curvature. The curvature of the relation between stellar mass and dispersion above about $2\times10^{11}$ solar masses, where dispersions flatten while mass and size keep growing, looks like what such late mergers would do. Bernardi and collaborators have argued for this from survey data.

So the outcome is a sort of consistency. The plane does not record a galaxy’s merger history in any simple way, and that is why so much attention has gone to measuring size at redshift two directly. But it does show that the most massive galaxies have reached their sizes differently from the less massive ones, and at about the mass where the census of galaxies falls off.

Still open: the part of growth that came first #

The size growth from redshift one to now looks explained, within the error of pair counts. Going further back, the growth rates become very fast: in the first two or three billion years after the quenched galaxies appear, the observed average size nearly doubles in a billion years. Merger timescales of a few billion years are too slow. Near the highest redshifts at which compact quiescent galaxies are now known, measured by the James Webb Space Telescope out beyond redshift four, there may be a need for something other than minor mergers.

The new data also have a bearing on the compact galaxies’ origin. JWST has shown compact, dense, massive quiescent galaxies earlier than had been expected, and whether they form from gas-rich collapses that have a large share of star formation in a few hundred million years is not settled. Whether the rate of late accretion in today’s stellar halos, which can now be measured in nearby galaxies by counting resolved stars in their outskirts, matches the requirement is another check that is becoming possible. If a nearby massive elliptical’s outer stars turn out to be metal-rich, or to have come from a few large mergers, then the minor-merger view of the outer envelope will have to be adjusted. If they are metal-poor and old, from many small dwarfs, the argument here will have one more independent line in its favour.