A bridge and a tail drawn by one force

Photographs of interacting galaxies show long, thin, curved streamers reaching out tens of kiloparsecs, sometimes joining the two systems and sometimes trailing away into nothing. The natural reading is that a collision has thrown material off.

It has not. Nothing collides — the stars of two galaxies pass through each other without a single encounter, because the spacing between stars is enormous compared with their size — and nothing is thrown off. Every star in a tidal tail is on its own orbit, following the same two masses as everything else.

The demonstration that this is enough was made in 1972 by Alar and Juri Toomre, with a computation small enough to run on the machines of the day and general enough that it still settles the question.

The restricted problem #

The computation is the restricted three-body problem, applied many times over.

Two masses — the primary and the companion — move under their mutual gravity on a Kepler orbit. A cloud of massless test particles is placed on circular orbits about the primary, and each particle is integrated in the field of both masses. Because the particles are massless they do not affect each other or the two galaxies, so the calculation is a set of independent two-centre problems and its cost is linear in the number of particles.

Written in a frame centred on the primary rather than on the centre of mass, the acceleration of a particle has three terms:

$$\ddot{\mathbf r} = -\frac{GM_1\mathbf r}{r^3} - \frac{GM_2(\mathbf r - \mathbf R)}{|\mathbf r - \mathbf R|^3} - \frac{GM_2\mathbf R}{R^3}.$$

The last is the indirect term — the acceleration of the frame itself, since the primary is being pulled by the companion too. Leaving it out is the classic error in this calculation: the disc then drifts bodily across the picture and the tails come out with the wrong shape.

Why the response is a stretch #

The reason a tail forms at all is the same reason a tide is a difference: what matters is not the companion’s pull but the variation of that pull across the disc.

The near side of the disc is pulled harder than the centre, the far side less hard. In the frame of the primary, that leaves a stretching field along the line to the companion and a compressing one perpendicular to it — the same quadrupole pattern that raises two tides on the Earth rather than one.

What makes a galaxy’s response so much more dramatic than the Earth’s is that the outer disc is loosely bound. The tidal force falls as the inverse cube of separation but the binding falls as the inverse square of radius, so the ratio grows outwards: material at fifteen kiloparsecs is far easier to lift than material at five. The near side of the outer disc is pulled towards the companion into a bridge; the far side is left behind into a tail; the inner disc barely notices.

The condition nobody expects #

The finding that made the 1972 paper decisive is not that the simulation produces tails. It is that it produces them only for a particular geometry.

The reason is a resonance rather than a force. In a prograde encounter, the outer particles are travelling in the same sense as the companion, so their angular speed and the companion’s are comparable and they stay near it for a substantial part of the passage. The perturbation acts in the same direction for a long time and accumulates.

In a retrograde encounter the particles sweep past the companion in the opposite sense, the relative angular speed is large, and each particle receives a rapid succession of impulses in changing directions which largely cancel.

This is the same mechanism, in a different guise, as the one that decides whether a resonance clears a gap or does nothing: a forcing that arrives in phase accumulates and one that does not, does not. Here the “in phase” condition is corotation with the perturber rather than a rational period ratio.

The arithmetic of an encounter #

Three numbers make the process concrete, and all three are within reach of an envelope.

How long it lasts. Two galaxies passing at a few hundred kilometres per second with a pericentre of twenty kiloparsecs are within a disc radius of each other for something like a hundred million years. The outer disc’s own orbital period at fifteen kiloparsecs is around four hundred million years. So the perturbation lasts a substantial fraction of an orbit — neither an instantaneous impulse nor a slow adiabatic squeeze, which is exactly the regime in which the response is largest and in which no simple approximation works.

How strong it is. The tidal acceleration across a disc of radius $r$ from a companion of mass $M_2$ at distance $D$ is about $2GM_2r/D^3$, against the binding acceleration $GM_1/r^2$. Their ratio is $2(M_2/M_1)(r/D)^3$, so for an equal-mass companion at three disc radii the tidal force is about seven per cent of the binding force at the disc’s edge — small, and applied coherently for a hundred million years, which is what makes it enough.

How far the material goes. A star lifted from fifteen kiloparsecs and given a modest fraction of the local circular speed follows a very eccentric orbit reaching a hundred kiloparsecs or more. Tails are long not because the material was thrown hard but because the outer potential is shallow, so a small change in energy buys a large change in apoapsis.

That last point is worth keeping. The tails in these figures reach several times the disc radius, and nothing in the calculation gave any particle a large velocity.

What the tails are for #

A tidal tail is not just a spectacle; it is a record, and three things are read from it.

The geometry of the encounter. The length, curvature and thickness of a tail depend on the pericentre distance, the mass ratio, the inclination of the disc to the orbital plane, and the sense of rotation. Matching a simulation to an observed pair constrains all of them, which is how the encounter that produced the Antennae or the Mice is reconstructed.

The mass of the halo. A tail’s stars are on nearly radial orbits reaching far outside the galaxy, so how far they get and how fast they are moving depends on the potential they are climbing out of. Tails are one of the few probes of a galaxy’s mass at a hundred kiloparsecs. The time since the encounter. A tail is a kinematic structure with a known expansion rate, so its length divided by its expansion speed gives an age — typically a few hundred million years for the systems that look most dramatic.

Why a tail is thin #

The length of a tail is easy to accept once the outer potential’s shallowness is granted. The thinness is the part that looks as though it needs a confining force, and there is none: the particles do not interact, and nothing holds a tail together.

The narrowness comes from where the material was taken from. The ratio of tidal force to binding force rises as the cube of radius, so the disc’s response is a very steep function of radius — the outermost annulus is lifted and the material a few kiloparsecs inside it is barely moved. What leaves is therefore drawn from a narrow range of initial radii, and every particle in that range was on nearly the same orbit, moving at nearly the same speed, when the perturbation arrived.

Nearly the same orbit, given nearly the same impulse, becomes nearly the same orbit afterwards. The escaping material occupies a one-parameter family rather than a volume, and a one-parameter family of orbits traces a curve through space. The tail is that curve, made visible by the stars sitting on it.

The small differences that do exist are what give the structure its shape over time. A particle lifted with slightly more energy reaches a slightly larger apoapsis and takes longer to get there, so the tail sorts itself along its own length by energy, with the most weakly bound material furthest out. That sorting is why a tail’s length divided by its expansion speed is an age, and why the surface brightness falls along it — the same number of stars is spread over a growing length.

It also explains why a tail thickens so slowly. The spread across the tail is set by the spread in the initial annulus’s orbital phase and by the small differences in the transverse kick, both of which are tiny; the spread along it is set by the energy differences, which the orbital period amplifies. A structure stretched in one direction and not in the others stays thin because the phase-space volume it occupies cannot grow, and stretching it along its length is therefore paid for by narrowing it across.

That last statement is general, and it is why the Sagittarius stream and the Magellanic Stream are thin for the same reason a tidal tail is. A stream is narrow because its progenitor was small in phase space, not because anything keeps it together.

What happens after the first pass #

The restricted calculation stops being adequate once the two galaxies begin to merge, and the reason is a process it explicitly excludes.

As the companion moves through the primary’s halo, it pulls the halo’s material into a wake behind it, and that wake pulls back. The result is dynamical friction: a drag proportional to the companion’s mass, which extracts orbital energy and shrinks the orbit. Chandrasekhar derived the effect in 1943 for a body moving through a uniform sea of stars, and it is the reason two galaxies that pass close once will usually pass again, closer, and eventually coalesce.

The restricted problem cannot show this, because it has no self-gravity: the halo particles are massless and their wake pulls on nothing. That is the honest limitation of the figures above, and it is why the modern versions of this calculation are full N-body simulations.

The end state is a spheroid. Two discs merging violently produce a system supported by random motions rather than by rotation, with a de Vaucouleurs profile — which is one of the strongest arguments that elliptical galaxies are merger remnants.

The observation behind the number #

Toomre and Toomre had four particular systems in mind and matched all four: the Antennae, the Mice, Arp 295 and M51 with its companion. Each match fixed the encounter’s parameters and the earlier state of the two galaxies.

What made the paper persuasive was not the individual matches but a statistical argument at the end of it. They counted the observed frequency of these systems, estimated how long a tail remains visible, and concluded that a substantial fraction of bright galaxies must have undergone such an encounter — and that ellipticals could plausibly be the remnants.

That argument turned interaction from a curiosity into a mechanism of galaxy formation, and it was made with a computation whose entire physical content is two point masses and Newton’s law.

What the picture cannot show #

No self-gravity, so no merger. The particles do not pull on each other, so the disc cannot form a bar, cannot respond collectively, and cannot sink. What the figures show is the first pass and nothing after it.

No gas. Real interactions drive gas to the centre and trigger enormous bursts of star formation; the most luminous infrared galaxies in the local universe are merging pairs doing exactly that. A dissipationless calculation says nothing about it.

No halo. The two masses here are points. Real galaxies have extended haloes, which change both the tidal field and the dynamical friction, and which make the encounter far more dissipative than a point-mass calculation suggests.

The particle count is small. One hundred and eighty particles in five rings is enough to show the structure and not enough to show its density; a modern calculation uses millions and reports surface densities rather than dots.

And what is drawn is starlight, where most of a tail is not. The particles here stand for stars, because stars are what the restricted problem follows. In the real objects the tails are far longer and far better defined in neutral hydrogen than in starlight — the gas sits further out in the disc to begin with, so it is lifted more easily, and it does not fade. A tail measured optically and the same tail measured at twenty-one centimetres are different lengths, and the second is the one that reaches far enough to weigh the halo.

And the panels are two-dimensional. The disc and the orbit share a plane here. Real encounters are inclined, and the inclination is one of the parameters a match to a photographed pair has to fit.

Where the Milky Way’s own record is #

This is not a phenomenon that happens to other galaxies. The Milky Way is doing it now, in two forms.

The Sagittarius dwarf is a small galaxy currently being pulled apart on an orbit that passes through the Galactic disc, and its debris forms a stream wrapping more than once around the sky. It is the tidal-tail calculation applied to a very unequal mass ratio, and because its stars can be resolved individually, the stream is a set of measured positions and velocities rather than a photograph.

The Magellanic Stream is a band of neutral hydrogen trailing the two Magellanic Clouds across a hundred and fifty degrees of sky, and its shape is a record of the Clouds’ orbit. Whether it is primarily tidal, primarily ram pressure from the Galactic halo, or both, is still argued.

Both are useful for a reason the external systems are not: a stream’s stars are on orbits in the Milky Way’s own potential, and their positions and velocities therefore measure that potential far out into the halo — including its shape, which a rotation curve cannot see at all. A stream is the closest thing available to a set of test particles released on purpose.

There is one more thing the restricted calculation gets right that a reader might not expect it to.

The two streamers are not symmetric, and the asymmetry is a prediction rather than an accident of the drawing. The bridge — material pulled towards the companion — is short, dense and quickly returned; the tail is long, thin and never comes back, because the material on the far side is lifted with more energy than it can lose again. The observable pair is therefore systematically lopsided, and which of the two is longer says which side of the disc faced the companion at closest approach. Matching that against a photograph fixes the geometry of an encounter that finished a hundred million years before anybody looked at it.

The generalisation #

The transferable lesson is about what a simulation is for: the value of a stripped-down model is not that it is realistic but that it isolates a claim.

The Toomres’ calculation contains no gas, no dark matter, no self-gravity and no collisions. If it had contained them, and had produced tails, it would have proved much less — because the question was precisely whether tails require any of those things. A model that excludes a mechanism and reproduces the observation is evidence that the mechanism is not needed, and that is a stronger conclusion than a realistic simulation can usually deliver.

The same logic runs through the rest of this collection. The restricted three-body problem is used for the same reason: not because a spacecraft has no mass but because setting it to zero shows that the interesting behaviour does not depend on it. A model is a claim about what can be left out.

One more encounter covers the intermediate mass ratio the essay’s two cases bracket.

Where the ladder goes next #

The next rung is what happens when the encounter is not a single pass: dynamical friction, the sinking of the companion, and the merger remnant — which is where the elliptical galaxies of the dispersion-supported ladder come from.

Later rungs on this anchor: dynamical friction derived and measured; the merger rate and how it is estimated from the fraction of galaxies in pairs; gas-rich mergers and the starbursts they drive; minor mergers as the mechanism that thickens discs; tidal dwarf galaxies formed in the tails themselves; the Milky Way’s own past encounters, written in its stellar streams; and the eventual merger with Andromeda, which is the same calculation run forwards on a system whose parameters are known.