A collision decided by a speed across the sky

Andromeda’s light is blueshifted. Almost every galaxy in the sky is moving away from the Milky Way, carried by the expansion of the universe, and Andromeda, the nearest large spiral, is one of the few that is approaching. After the Sun’s own orbit around the Galaxy is subtracted, its velocity along the line of sight is about 110 kilometres per second toward the Milky Way, at a distance of about 770 kiloparsecs. The two galaxies have been falling toward each other since the expansion around them was slowed and reversed by their combined mass — a history that can be run backwards to weigh the pair — and at that speed and distance they would meet in about five billion years.

Whether they merge when they meet is a different question, and the approach speed does not answer it. Two galaxies passing each other merge only if the passage transfers enough orbital energy into their internal motions, and how much it transfers depends above all on how close the passage comes. How close it comes depends on the pair’s angular momentum, and the angular momentum depends on the one component of Andromeda’s velocity that cannot be read off a spectrum: its motion across the sky.

The figure’s four curves share their start and their first few billion years almost exactly. Every one of them falls from 770 kiloparsecs toward closest approach in four to five billion years, because the time to fall is set by the energy, and the energy is dominated by the radial speed and the combined mass. They separate only near the bottom, where the sideways motion, negligible at large distances, becomes the whole of the velocity.

Why the arrival time is robust #

The time to first approach barely changes across the four curves, and the reason is the same one that makes the period of an orbit depend only on its energy. For a bound pair, the time spent falling from a given separation to the bottom of the potential depends on the orbital energy and the total mass, and the angular momentum only redistributes that time between the approach and the swing round the pericentre. The sideways speeds in question add at most a few per cent to the kinetic energy — 57 kilometres per second squared is a quarter of 110 squared — so the energy, and with it the arrival, is fixed by the radial speed and the mass.

That makes the arrival time the part of the forecast that the evidence pins down. For any sideways speed in the measured range, and for masses across their plausible range, the galaxies are closest between four and five billion years from now, about when the Sun, on its own evolution, will be starting to swell. It is the outcome at that moment that is open, not the date.

The asymmetry between the two components of velocity is worth holding onto because it is general. The radial component sets the timescale through the energy; the transverse component sets the geometry through the angular momentum; and the geometry of a close encounter is exquisitely sensitive to angular momentum while the timescale is not. A comet’s return date is known long before its closest approach to the Sun is, for the same reason, and so is the arrival of a spacecraft at a planet long before its aim point is.

A speed that has to be measured as an angle #

A velocity along the line of sight shifts spectral lines and is measured in an afternoon. A velocity across the line of sight moves the source on the sky, and at 770 kiloparsecs a sideways speed of 57 kilometres per second is an angular drift of about fifteen microarcseconds a year. Fifteen microarcseconds is roughly the width of a coin on the Moon seen from the Earth. Measuring it requires positions accurate to a fraction of that, repeated over years, against a reference frame that is itself not moving.

It has been done twice by different routes. In 2012 the Hubble Space Telescope measured the positions of stars in three fields of Andromeda against background galaxies, seven years apart, and combined them with the motions of Andromeda’s satellite galaxies. The result was a sideways speed of 17 kilometres per second, with an uncertainty as large as the value: consistent with zero. In 2019 and again in 2021, positions from the Gaia satellite for thousands of Andromeda’s brightest stars gave a larger value — about 57 kilometres per second in the first analysis and about 80 in the second, each with uncertainties of thirty to forty.

The two families of measurement are not formally in conflict, since their error bars overlap. But they place the value in different regions of a range where, as the first figure shows, the outcome changes qualitatively. Subtracting the Sun’s own motion adds a further uncertainty, since the Sun moves at about 250 kilometres per second around the Galaxy and its component across the line of sight to Andromeda has to be removed exactly — the same subtraction that turns a measured proper motion into a distance elsewhere in astronomy.

Angular momentum, and where it becomes everything #

The sideways speed matters through the angular momentum it carries. The pair’s angular momentum per unit reduced mass is the sideways speed times the separation — for 57 kilometres per second and 770 kiloparsecs, about forty-four thousand kiloparsec-kilometres per second. That quantity is conserved while nothing but the pair’s gravity acts, and at closest approach it sets a floor under the separation: the galaxies cannot come closer than the distance at which all their relative speed would have to be sideways to carry it.

The steepness comes from how angular momentum enters the effective potential. The centrifugal barrier grows as the square of the angular momentum over the square of the distance, so for a nearly radial orbit, where the barrier only matters at small separations, the pericentre grows roughly as the square of the sideways speed. Doubling the sideways speed from 30 to 60 kilometres per second nearly quadruples the distance of closest approach. A measurement whose uncertainty is thirty kilometres per second therefore does not leave the pericentre uncertain by a factor of two; it leaves it uncertain between a collision of the discs and a passage outside both of them.

Friction turns a passage into a capture #

Two point masses on a bound orbit would return to the same pericentre forever. Galaxies are not point masses. Each sits inside a dark halo several hundred kiloparsecs across, and the haloes overlap long before the discs do. As Andromeda’s halo moves through the Milky Way’s, it gravitationally focuses the dark matter it passes into a wake behind it, and the wake pulls back. That is dynamical friction, and it drains the orbit’s energy fastest where the density is highest and the relative speed is comparable to the halo’s internal speeds — that is, near pericentre.

The second loop is the signature of friction. Without it, the orbit at 57 kilometres per second would return to 770 kiloparsecs and come back to a pericentre of a hundred again, and the galaxies would pass each other at intervals of about ten billion years indefinitely. With it, each passage costs a large fraction of the orbital energy, the apocentre shrinks, and the next pericentre is closer. The capture is not sudden; it is the accumulation of losses at a few close approaches, and the number of approaches it takes depends on how close the first one was.

The friction in this model uses the classical formula with a small Coulomb logarithm, appropriate for two haloes of comparable size, and the logarithm is the least well pinned part of the whole calculation. It moves the merger time of the wider orbits by billions of years. It hardly moves the head-on case, which merges on arrival whatever the friction, because there the orbit’s energy goes directly into the collision of the two galaxies.

The knife-edge between capture and escape runs through the middle of the Gaia range, and two more orbits either side of the measured value show where.

What divides the two is not a threshold in the sideways speed itself but where the first pericentre falls relative to the dense inner parts of the haloes. The density of a dark halo falls steeply with radius, as the cube of the distance at large radii, and the friction falls with it. A pericentre inside about a hundred kiloparsecs passes through dark matter dense enough to take a large fraction of the orbital energy in one passage; a pericentre at two hundred passes through matter eight times thinner and takes correspondingly little. The Gaia value sits near the transition, which is why its forecast is closer to a coin toss than either extreme.

The masses matter less than the angle #

The halo masses of the Milky Way and Andromeda are each uncertain by tens of per cent — the Milky Way’s between about 0.8 and 1.5 trillion solar masses, Andromeda’s similar or slightly larger. Mass enters twice. A heavier pair falls in faster and to a closer pericentre for the same velocities, and a heavier halo drags harder.

The asymmetry is the point. For a nearly head-on approach, the outcome is robust: every plausible mass gives a merger on the first passage, a little sooner or later. For a wide approach, the outcome is fragile: a thirty per cent change in the masses moves the merger from within the Sun’s lifetime to beyond the universe’s present age. The uncertainty in mass matters exactly as much as the sideways speed allows it to, and it is the speed that decides which regime the pair is in.

The fate as a probability #

With both uncertainties in hand, the question can be asked the only way it can be answered: draw many orbits from the measurements and count how many merge.

Neither curve is a prediction that the galaxies will or will not merge. They are statements about what is known, and the width of each is the width of the measurement. An analysis published in 2025 that included the pulls of the Large Magellanic Cloud and of the Triangulum galaxy, M33, along with the latest proper motions and masses, found the probability of a merger within ten billion years to be close to one half — between the two curves here, and near the Gaia one. The same analysis found that the Large Magellanic Cloud, by pulling the Milky Way sideways, makes the merger less likely, and that M33 makes it more likely. The Local Group is a problem of several bodies, and the pair is only its largest term.

What the merger would make #

If the pair does merge, the product is not a larger spiral. The discs of both galaxies are thin because their stars move on nearly circular orbits in one plane, and a merger of two discs of comparable mass scrambles those orbits into a roughly spheroidal system held up by random motions rather than rotation — a galaxy held up by disorder, the shape of an elliptical. Simulations of the pair show the remnant settling within a couple of billion years of the final coalescence into a smooth, reddish, gas-poor elliptical of about three trillion solar masses in total.

The gas in the two discs would be compressed at each close passage and form a burst of new stars, and some of it would fall to the centre and feed the two central black holes. Those black holes, of four million and about a hundred million solar masses, would sink toward the centre of the remnant by the same friction that merged their galaxies, form a bound pair, and then face the problem of the last parsec: whether enough stars can be thrown out of their way for them to spiral together and merge in a burst of gravitational waves. The galaxy merger is the first and best-determined step of a chain whose later steps are much less certain.

What the model leaves out #

The integration treats each galaxy as a smooth spherical halo, with no disc, no bulge and no internal response. A real close passage raises tides in both discs, draws out bridges and tails, and puts orbital energy into internal motions faster than the friction formula alone describes, so the merger times for close passages are, if anything, shorter than those drawn. Full N-body simulations of the pair, which follow these responses, find the same division into near-head-on and wide approaches.

It leaves out the rest of the Local Group. M33, a tenth of Andromeda’s mass, orbits Andromeda; the Large Magellanic Cloud, perhaps a tenth of the Milky Way’s, is falling into the Milky Way now and has already pulled it measurably. Each changes the pair’s effective velocities and masses by amounts comparable to the uncertainties, and the published probability of about one half comes from including them.

It leaves out the expansion of the universe and the mass outside the pair, both small at these distances over these times. And it leaves out the most-asked question about the event, which is what happens to the Sun: nothing violent. Stars in both galaxies are separated by distances millions of times their own sizes, and a merger of two galaxies is a merger of two potentials, in which almost no pair of stars ever comes close. The Sun would be flung into a new orbit in the combined galaxy, and the night sky would change completely, over hundreds of millions of years.

What was actually measured #

Three numbers go into the calculation and only one is a direct measurement in the ordinary sense. The radial velocity is a Doppler shift, measured to a few kilometres per second. The distance comes from Cepheids and the tip of the red giant branch, to about three per cent. The sideways velocity is a change in position of a few hundredths of a milliarcsecond over a decade, measured against quasars and background galaxies, and converted to a speed with the distance and the Sun’s own velocity.

The masses are not measured at all in that sense. They are inferred from the motions of satellites and globular clusters and halo stars around each galaxy, from the timing of the pair’s own approach, and from rotation curves extrapolated beyond the discs. Each method returns a mass within some radius, and the halo mass beyond that radius — which is what the friction drags against — is a model.

The pivotal quantity, then, is the smallest and least certain. It is also the one whose error is hardest to state honestly, because it depends on modelling the internal rotation of a galaxy seen nearly edge-on, where the stars that are measured are moving in their own orbits at speeds ten times the drift being sought. That is typical of orbital problems: the radial motion fixes when and the transverse motion fixes how, and the transverse motion is always the harder one to see. Kepler’s problem was solved from angles rather than distances for the same reason, and the fate of the Local Group is being decided by astrometry, not spectroscopy.

Still open: a velocity measured to ten kilometres a second #

The sideways speed is converging, slowly, as Gaia’s baseline lengthens and as Hubble and its successors return to the same fields. A measurement good to ten kilometres per second would place the pair unambiguously on one side of the divide the second figure draws: below about thirty, a near head-on encounter in under five billion years; above about fifty, a wide first passage and a merger delayed or missed. The measurements so far favour the second, but not decisively, and the answer depends on a systematic question about how well the motion of Andromeda’s bright stars represents the motion of Andromeda’s centre of mass — stars in a rotating disc move across the sky at hundreds of kilometres per second in their own right, and the disc’s rotation has to be modelled out before the galaxy’s drift is left. The same question, with a different galaxy, is what separates the Hubble and Gaia values for the Large Magellanic Cloud’s orbit, and resolving it for both would fix the geometry of the Local Group’s next few billion years.