The ratio that is not √2

The speed that does not come back ends on a ratio with nothing in it: escape speed divided by circular speed is $\sqrt{2}$ at every distance from every body, because the mass and the radius cancel between the two square roots. It is the cleanest result about escape speed anywhere, and it is true of exactly one mass distribution.

The Sun orbits the centre of the Galaxy at about 233 kilometres a second. The escape speed from the Galaxy at the Sun’s position is about 530. That ratio is 2.27, not 1.41, and the discrepancy is sixty per cent.

Neither number is wrong. The $\sqrt{2}$ is.

Why the cancellation fails #

The $\sqrt{2}$ comes from dividing $\sqrt{2GM/r}$ by $\sqrt{GM/r}$, and both expressions assume that the mass relevant at radius $r$ is the same mass in both cases.

For circular speed it is the mass enclosed, and that is not an approximation: a spherical shell exerts no net force on anything inside it, so a star’s orbital speed is entirely deaf to whatever lies further out. For escape speed it is the depth of the potential well, which is an integral of the field along the whole path outward — and that path runs through everything.

So the two numbers are sensitive to different masses, and they coincide only when there is nothing outside the orbit to distinguish them. A point mass satisfies that trivially. The Sun and its planets satisfy it well enough that spacecraft navigation uses the $\sqrt{2}$ without comment. A galaxy does not satisfy it at all.

The ratio is a measurement, and of what #

Turn it round. If the ratio exceeds $\sqrt{2}$ by an amount set by the mass outside the orbit, then measuring the ratio measures that mass.

For a flat rotation curve the arithmetic is one line. A potential giving constant circular speed $v_c$ is $\Phi = v_c^2\ln r$, so the escape speed from radius $r$ to an edge at $r_t$ is $v_c\sqrt{2\ln(r_t/r)}$, and the ratio is $\sqrt{2\ln(r_t/r)}$ with the circular speed cancelled out of it entirely. The ratio does not measure a mass; it measures a radius, the distance at which the halo runs out.

A ratio of 2.27 puts that edge at $e^{2.27^2/2} \approx 13$ times the Sun’s distance, or about 110 kiloparsecs. That is a real result and it is the right order: the Milky Way’s satellite galaxies and the stellar streams they leave behind are found out to a few hundred kiloparsecs, and the virial radius estimated from the mass that is not the light is between 200 and 250.

It is also a result with a brutal sensitivity. The edge is an exponential of the square of the ratio, so a ten per cent error in the ratio is a factor of nearly two in the radius. Every kilometre a second in either speed is worth a great deal.

How an escape speed is measured at all #

Nothing escapes from the Galaxy in front of anybody, so the measurement is indirect and it is a measurement of an absence.

Take a large sample of stars near the Sun with measured space velocities and build the distribution of their speeds in the Galaxy’s frame. Stars faster than the local escape speed are not gravitationally bound and, on a timescale short compared with the Galaxy’s age, are not here. So the distribution has an upper limit, and the limit is the quantity wanted.

The fitting form is empirical. It is assumed that just below the limit the number of stars per unit speed goes as $(v_{\rm esc} - v)^k$, which is what a population in a smooth potential with a truncated energy distribution gives, and $k$ is taken from cosmological simulations of how haloes are assembled.

That importation is the method’s weakest joint, and its size is worth seeing rather than being told.

Why this measurement is made from the rarest stars #

The whole result rests on the fastest few stars in the sample, and that has consequences worth separating out.

The sample is tiny where it matters. A survey may contain millions of stars with velocities, and the ones above 300 kilometres a second — the ones carrying any information about the limit — number in the tens or low hundreds. The statistical error is therefore the error on a handful of objects, whatever the catalogue’s size.

A single wrong velocity moves the answer. A star whose distance is overestimated has its tangential velocity overestimated in proportion, and one spurious 600-kilometre-a-second star raises the fitted limit by more than the whole quoted error bar. Published analyses now cut hard on the quality of the astrometry for exactly that reason, and the cut itself is a choice that moves the answer.

And the fast stars are not a fair sample of anything. A star moving at 500 kilometres a second past the Sun has spent almost all of its life somewhere else, on a wildly eccentric orbit through the halo, and the halo’s velocity distribution is not the disc’s. Some of the fastest are not bound to the Galaxy at all: they were ejected by the black hole at the centre, or by a supernova in a binary, or arrived from a satellite galaxy. Every one of those is a star above the limit that the fit is trying to locate, and the fit has no way to tell them apart from the bound tail except by their speeds.

The stars that are above the line #

A distribution with an upper limit invites the question of what is found above it, and the answer is that something is.

Several hundred stars are known in the Galaxy whose speeds exceed any plausible local escape speed. They are not a tail; they are a separate population with a separate cause, and the causes are worth listing because each is a different physical process reading the same number.

Ejection by the central black hole. A binary star straying close to the hole at the Galactic centre can be torn apart, one component captured onto a tight orbit and the other flung outward at over a thousand kilometres a second. The ejected star’s speed is set by the binary’s own orbital speed amplified by the encounter, and the population traces back to the centre.

A supernova in a binary. When one member of a close pair explodes, the survivor is released at whatever orbital speed it had, which for a tight massive binary is hundreds of kilometres a second. These trace back to the disc rather than to the centre, and they are the more numerous kind.

An accreted dwarf galaxy. A satellite falling in on a radial orbit delivers stars at the speed its own infall gave them, which near the Sun can exceed the local escape speed while the stars remain bound to the Galaxy as a whole. These are the most awkward, because they are genuinely bound and genuinely above the local limit — the limit being local, and their orbits taking them far beyond it.

Each of those is a contaminant of the measurement in this essay and a measurement in its own right. The ejected ones time the black hole’s activity; the supernova ones measure the binary population that made them; the accreted ones map an infall. What is noise in one analysis is the whole signal in another, and the practical difficulty is that a catalogue of fast stars does not label which is which — the separation is made on the direction of travel and on chemistry, not on speed.

The same ratio in other systems #

The excess over $\sqrt{2}$ is a property of the mass distribution rather than of galaxies, so it is worth checking what it does elsewhere.

A globular cluster is centrally concentrated and has very little mass outside its own outskirts, so the ratio at its half-mass radius is close to $\sqrt{2}$ — around 1.5 to 1.7 — and a star’s escape from it is nearly a point-mass calculation. That is why the evaporation of clusters can be computed so simply, and why a cluster that boils itself away does so at a rate set by a two-body relaxation time rather than by any subtlety about the potential.

A galaxy cluster is the opposite. Its mass profile continues far beyond any galaxy in it, and the ratio at the position of a typical member galaxy is above 2.5. This is one reason cluster masses derived from the dispersion of galaxy velocities and from the maximum velocity observed have such different systematics: the first is an average over a well-sampled distribution and the second an extrapolation from its edge.

And the solar system, where the whole business began, is the case in which the ratio is $\sqrt{2}$ to within a part in a thousand — because 99.86 per cent of the mass is in one object at the centre and a planet’s orbit encloses essentially all of it. That earlier account was correct about the system it was written for, and the number of systems it is correct about is small.

The pattern across the three is simple to state and easy to forget. The $\sqrt{2}$ is a statement about how centrally concentrated a system is, disguised as a statement about gravity. Where a system’s mass is concentrated inside the orbit in question, it holds; where the mass keeps going, it is a lower bound; and how far above the bound the answer lands is the measurement.

What was actually measured #

For each star: a parallax, two proper motions and a radial velocity — five numbers from one wiggle plus a spectrum — converted into a space velocity in the Galaxy’s frame.

That conversion is where the assumptions live. It needs the Sun’s distance from the Galactic centre, its own velocity with respect to the Galaxy, and the local circular speed, and all three are themselves measurements with a few per cent on them. A star’s speed in the Galactic frame is therefore not an observation; it is an observation plus a model of where the observer is and how fast the observer is going.

And a parallax is not a distance. For the halo stars this matters more than anywhere: they are distant, their parallaxes are small fractions of their own errors, and inverting a noisy parallax systematically overestimates distance for the faintest — which systematically overestimates tangential velocity, which raises the fitted escape speed.

The modern answers cluster between 500 and 580 kilometres a second, and the spread between published analyses of nearly the same data is larger than any one of their error bars. That is the signature of a systematic rather than a statistical limitation, and the systematic is the one the figure above draws.

The number at the Sun, arrived at three ways #

It is worth setting the three independent routes to the Galaxy’s mass beside one another, because they do not agree in a way that is informative rather than embarrassing.

From the escape speed. The tail fit gives 500 to 580 kilometres a second locally, and converting that into a mass requires a profile shape as well — the escape speed constrains the potential’s depth at one radius, and a mass needs the whole run. Assuming a standard profile, the answer is a virial mass around $1.0$ to $1.5\times10^{12}$ solar masses.

From the satellites. The Galaxy’s dwarf companions and globular clusters have measured positions and, increasingly, full space velocities. Applying the virial relation between their motions and the mass holding them gives a mass of the same order, with the dominant uncertainty being whether the sample is in equilibrium — which for a system whose largest satellites arrived recently is genuinely unclear.

From the approach of Andromeda. The two large galaxies of the Local Group are approaching at 110 kilometres a second after a Hubble time of separation, and requiring gravity to have turned an initial recession into that approach gives the total mass of the pair. This is the oldest of the three and it is beautiful, because it uses no model of either galaxy at all — only that they started together and are now approaching.

The three land within a factor of about two of one another, which sounds poor and is the honest state of the subject. What makes the comparison useful is that their systematics are unrelated: the first depends on an exponent from simulations, the second on an equilibrium assumption, the third on a tangential velocity that was unmeasurable until recently and that turned out to be small. Three methods agreeing to a factor of two, with no shared assumption, is a stronger statement than any one of them quoting ten per cent.

Where the model stops #

The potential is not spherical. Everything here treats the Galaxy as a sphere so that the escape speed is a function of radius alone. The disc is not spherical and the halo is probably not either, so the escape speed depends on direction as well as distance, and the fitted number is an average over whatever directions the sample’s stars happen to be leaving in.

There is no edge. A flat rotation curve’s potential diverges logarithmically, so an escape speed exists only because the mass does stop somewhere — and where it stops is not observed, it is defined. Different analyses define it differently: some at the virial radius, some where the density falls to a multiple of the cosmic mean, some by fitting a profile that has a finite total mass built in. The quantity being measured depends on a convention, and comparing two published escape speeds requires knowing which.

Escape from the Galaxy is not escape. The Milky Way is falling toward Andromeda and both sit in a group that is bound. A star that leaves the Galaxy at 530 kilometres a second is still bound to the Local Group, so the “escape speed” is the speed to leave one member of a system and not the system — which is the same nesting-of-wells caveat the earlier account makes about leaving the Earth and not the Sun.

And the tail is a model of a distribution, not of a star. No individual star in any of these samples is known to be near the limit. The fit is a statement about where a curve has to stop, made from objects that are all comfortably inside it.

The generalisation #

The useful shape here is what happens to a ratio when its two halves stop depending on the same thing.

The $\sqrt{2}$ is exact and it is exact because of a cancellation, and a cancellation is a coincidence of dependences rather than a law. Escape speed and circular speed both go as the square root of a mass over a radius, and their ratio is clean only while the mass in one is the mass in the other. The moment the two quantities measure different masses — enclosed against total — the cancellation fails, and the size of the failure is the difference between the two masses.

A ratio that was chosen because it cancels is a ratio that becomes a measurement when it stops. That is a better use of it than the original, since a quantity that is constant everywhere carries no information and a departure from it carries all of it. The same move is what makes the departure of a rotation curve from the Keplerian fall a mass measurement, and the departure of a cluster’s galaxies from the speed its light could hold another.

The second reading is less encouraging and it is about what the measurement rests on. The information about the escape speed lives entirely in stars that are nearly unbound, which are the rarest objects in the sample, the ones most likely to have been mismeasured and the ones least likely to belong to the population being modelled. The quantity is measured by the objects least representative of it, and there is no way round that: a star deep in the well says nothing about how deep the well is.

Still open: what is left at infinity #

What comes next leaves the threshold behind. Escape is a boundary and no mission is designed to sit on one, so what an interplanetary departure is actually specified by is the speed still remaining when the pull has run out — and that quantity behaves nothing like a speed, because energies add and speeds do not.

Beside it lies the potential itself, treated as an object rather than as a number: the well’s shape rather than its depth, which is what a stream of stars torn from a satellite galaxy traces out and what a stream that is not the orbit it came from is a warning about reading too directly.