Whether it merges is a ratio of two times

A bridge and a tail are drawn out of a disc by one force, and the previous rung of this anchor showed that the whole spectacular morphology of an interacting pair follows from the tidal field of the companion acting on stars that are already in orbit. What it left open is the outcome. Two galaxies that have made those tails either fall back together and merge, or separate and never return.

The comparison that decides it #

The relevant number is not the impact parameter and not the relative velocity separately. It is the ratio of the time the encounter lasts to the time the stars take to go round.

If the encounter is fast compared with the internal orbital period, every star receives an impulsive kick and the galaxy is left with more kinetic energy than it had — heated, but not bound to anything. That is the impulse approximation, and it is the correct description when two galaxies pass at a thousand kilometres a second.

If the encounter is slow, the stars complete orbits during it and follow the perturbation adiabatically, giving back on the way out whatever they took on the way in. Almost no energy is transferred at all.

The interesting case is in between — when the encounter time is comparable to the internal period. Then the response is resonant rather than impulsive or adiabatic: stars whose orbital frequency matches the encounter’s are driven, they absorb orbital energy from the pair, and the orbit decays.

Why prograde matters and retrograde does not #

The resonance condition is what makes the sense of rotation matter, and the effect is dramatic.

Consider a star on the near side of a disc during a passage. If it orbits in the same sense as the companion moves, it stays near the companion for far longer than it would if it orbited the other way — its angular velocity partially cancels the companion’s apparent motion. The perturbation therefore acts in one direction for a long time and the star’s orbit is changed substantially.

If it orbits the other way, the companion sweeps past it quickly, the perturbation reverses sign, and the net effect over the passage is small.

That asymmetry is why the hero figure’s two panels look so different, and it explains an observational fact that would otherwise be puzzling: interacting pairs with long, thin tidal tails are common, and pairs with no tails at all are common, and the difference is not obviously in the geometry of the encounter. It is in the inclinations of the discs.

The energy has to go somewhere #

Whether the pair merges is an energy question, and the accounting is worth writing down.

Two galaxies approaching from a large separation arrive with some orbital energy. If they leave with the same amount, they are unbound and separate. To become bound, they have to leave with less — which means orbital energy has to be converted into internal energy of the stars.

The conversion happens in two ways, and both are covered by earlier rungs. The first is the resonant coupling above: stars driven at the encounter frequency absorb energy directly. The second is dynamical friction, in which each galaxy raises a wake in the other’s stars and is decelerated by it. The threshold is sharp enough to state simply. If the relative velocity at large separation is comparable to or smaller than the internal velocity dispersion of the galaxies, the encounter is slow enough to couple and the pair merges within a few passages. If it is much larger, nothing sticks.

Why clusters are collisionless and groups are not #

That criterion sorts environments rather than encounters, and the sorting is the reason galaxy evolution differs so sharply between the two.

In a group, the velocity dispersion is a few hundred kilometres a second — comparable to the internal dispersion of a large galaxy. Encounters are slow, coupling is efficient, and mergers are common. Most of the ellipticals in the universe are thought to have been made this way, and their light profiles differ from a disc’s in exactly the way a scrambling predicts.

In a cluster, the dispersion is a thousand or more. Encounters are fast, the impulse approximation applies, and instead of merging the galaxies are heated: a process called harassment, which strips their outer stars and thickens their discs without ever binding two of them together.

What a merger produces #

A merger between two comparable discs does not produce a bigger disc. The stars’ orbits are scrambled in a process that reaches a new equilibrium far faster than two-body relaxation could — because the potential itself is changing on the orbital timescale, so every star’s energy changes, and the system relaxes without any star ever having encountered another.

The result is a pressure-supported spheroid: a galaxy held up by disorder rather than by rotation. The gas behaves differently from the stars, and that difference is what makes mergers spectacular. Gas is dissipative: two streams that cross shock and lose energy, so gas falls to the centre rather than settling into a new spheroid. The result is a burst of star formation in the nucleus and, often, a period of accretion onto the central black hole.

A number for the criterion #

The verbal criterion — encounter speed against internal dispersion — can be turned into a length, and the length explains why the answer is so sharply divided by environment.

Two galaxies of mass $M$ and internal dispersion $\sigma$ approaching with relative velocity $v$ at large separation become bound if the energy transferred exceeds their orbital kinetic energy. The transferred energy scales as the square of the tidal impulse, which for an encounter at pericentre $p$ goes as the inverse fourth power of $p$; setting that equal to the orbital energy gives a maximum pericentre for capture,

$$p_{\rm max} \;\sim\; R \left(\frac{\sigma}{v}\right)^{1/2},$$

with $R$ the galaxy’s own size. The dependence is weak in the velocity ratio and strong in nothing else — so a pair encountering at their own dispersion can be captured from a pericentre comparable to their size, and a pair encountering at three times it needs to come within about half that.

The reason the environmental division is sharp despite the weak power is that the cross-section goes as the square of the maximum pericentre and the number of encounters goes with the density, so the merger rate carries several factors that all point the same way. A cluster has ten times the dispersion of a group and roughly a hundred times the density, and the two effects nearly cancel in the encounter rate while both suppress the capture probability.

What was actually measured #

The mechanism cannot be watched. A merger takes a billion years and the observable is a snapshot.

The evidence is therefore a comparison of populations with simulations, and the strongest pieces are morphological.

Tidal tails are the first. A pair with tails as long as the galaxies are wide is a pair whose encounter was slow and prograde; the tails are made of material on nearly radial orbits that will fall back over the next billion years. Counting pairs with tails against pairs without measures the fraction of encounters that were slow.

The second is the merger rate itself, measured as the fraction of galaxies in close pairs at a given redshift. That fraction rises with redshift, roughly as the cube of one plus it, which is what the increase in density and the decrease in relative velocity together predict — though every survey draws a different sky, and a close pair at high redshift is a pair of objects a few tenths of an arcsecond apart. The third is kinematic. A merger remnant should retain a memory of the orbital angular momentum of the encounter, and some ellipticals do rotate slowly while others do not. The division between fast and slow rotators is one of the more robust results of integral-field spectroscopy, and it maps onto the difference between mergers of gas-rich and gas-poor progenitors.

The one thing the criterion cannot decide #

There is a case the ratio of times says nothing about, and it is the commonest case in the universe.

When a smaller galaxy encounters a larger one, the relevant internal dispersion is not one number: the small galaxy’s is small and the large one’s is large, so the same encounter is slow with respect to one and fast with respect to the other. The satellite is disrupted — its stars respond, its orbit decays, it is absorbed — while the primary barely notices, because for the primary’s stars the encounter was impulsive and weak.

That asymmetry is why minor mergers build galaxies without destroying them. The mass is delivered, the angular momentum is delivered, the satellite’s stars end up in the halo, and the primary’s disc survives with a somewhat thicker vertical profile than it had. A galaxy like this one has absorbed dozens of such objects and is still a disc.

The criterion in this essay is therefore a criterion about comparable pairs, and the comparable pairs are a small minority of interactions and a large majority of the transformations.

One consequence of that is a warning about counting. A survey that identifies interactions by looking for disturbed morphology is counting the encounters that were slow, prograde and comparable in mass — the ones that make tails. A survey that identifies them by looking for close pairs is counting every encounter regardless of geometry, most of which will not merge. The two methods give merger rates differing by factors of several, and the difference is not a disagreement about the data but about which of two different quantities the word means.

Reconciling them requires knowing the fraction of close pairs that actually coalesce, which is exactly the ratio this essay is about — so the observational calibration of the merger rate depends on the theoretical criterion it was supposed to test. That circularity is broken, slowly, by simulations that follow individual pairs and report what fraction of them merge within a given time, and the current answer is that a close pair at a projected separation of twenty kiloparsecs merges within about half a billion years rather more than half the time.

The evidence a merger leaves behind #

The criterion decides what happens; establishing that it happened requires reading a remnant, and the readable features are the ones the encounter throws outward rather than the ones it buries.

Material pulled off the outer parts of a galaxy during a close passage ends up on very long, very slow orbits, and it stays there. Tidal tails — thin, curved streams containing a few per cent of a galaxy’s stars — survive for around a billion years before they disperse or fall back, which is long enough that a substantial fraction of a merger’s aftermath is still visible.

They are also faint. A tail’s surface brightness is a hundredth or a thousandth of the galaxy’s centre, so detecting one requires a deep image with a well-controlled flat field, and the census of tidal features has grown by an order of magnitude as that became routine.

Shells are the other signature and they record a different geometry. A small galaxy falling in on a nearly radial orbit is disrupted and its stars pile up at the turning points of their orbits, producing a set of sharp-edged arcs at increasing radii. The spacing of the arcs is a clock: each shell corresponds to a different number of radial oscillations completed since the encounter, so counting them and knowing the potential gives the time elapsed.

So the remnants are datable, and the rate of mergers over cosmic time — the quantity that all of this is for — is measured by counting features whose lifetimes are known, in the same way a crater count measures a flux.

The criterion is a comparison of two timescales, and both the encounter geometry and the mass ratio move it, so it is worth drawing at two more settings.

When there is no gas #

The account of what a merger produces usually involves a burst of star formation, because the gas from both galaxies loses energy in shocks and falls to the centre. That requires gas, and the most massive galaxies do not have any.

A merger of two gas-poor systems — a dry merger — is a purely stellar event. Nothing radiates away energy, no starburst occurs, and the black holes are not fed. What happens is that two collections of stars combine, and the remnant is more massive and more extended than either progenitor, with a lower central density than a dissipative merger would produce.

That distinction matters for the growth of the largest galaxies. The most massive ellipticals contain stars that are uniformly old, so they cannot have formed their stars recently — and yet their masses are larger than any single progenitor could have reached. The resolution is that they assembled by dry mergers: their stars were made long ago in smaller systems, and the systems were combined afterwards.

The observational signature is a population growing in mass without growing in star formation, and it is seen: the number density of the most massive red galaxies has roughly doubled since redshift one while their stellar populations have simply aged.

A merger’s outcome therefore depends on what the galaxies were carrying as well as on the ratio of two times, and the same encounter that makes a starburst in a gas-rich pair does nothing visible at all in a gas-poor one.

Counting them, and why the rate is argued about #

The merger rate is the quantity all of this feeds into, and measuring it turns out to be harder than identifying any individual merger.

Two methods are used and they disagree.

Close pairs. Count galaxies with a companion within some projected separation and some velocity difference, divide by an assumed timescale for such a pair to merge, and the result is a rate. The difficulty is entirely in that assumed timescale, which comes from simulations, depends on the mass ratio and the orbit, and is uncertain by a factor of two — and it enters the answer linearly.

Morphological disturbance. Count galaxies showing tails, shells or asymmetry, and divide by the time such features remain visible. The difficulty is the same in a different place: the visibility time depends on the depth of the images and on what counts as disturbed, and different surveys applying different criteria to the same galaxies agree only roughly on which ones are disturbed at all.

Both methods also share a bias that is hard to remove. A merger in progress is bright — the starburst raises the luminosity — so a luminosity-limited sample contains mergers preferentially, and correcting for that requires knowing how much brighter they are, which depends on the gas fraction.

The published rates therefore span a factor of several at every redshift, and the shape of the rate’s evolution with time is better determined than its normalisation — which is the usual situation when a measurement is a count divided by a modelled timescale.

It is also why the two methods are now applied to the same samples deliberately, so that the timescale each assumes can be checked against the other rather than taken from a simulation alone.

And an equal-mass encounter followed through the sequence, which is the case the criterion says must merge.

Where the ladder goes #

The next rung is what happens to the two central black holes, which sink by dynamical friction until they form a bound pair and then stall — because the stars they need to eject in order to harden further run out. That stalling is the single largest uncertainty in predicting how many black hole mergers a gravitational-wave detector should see.

The other direction is the minor merger, where the mass ratio is ten to one or worse. Those are far commoner than major mergers and they do not scramble the primary at all; they thicken its disc, build its bulge, and deliver a stream of stars that can still be identified as a stream billions of years later.