An average that weighs what cannot be watched

Weighing a two-body system is easy in principle: watch it for a period, measure the separation, and Kepler’s third law returns the total mass. Weighing a cluster of a million stars is not, because nothing in it completes an orbit on any timescale anybody will observe, and no two of its members are recognisably a pair.

And yet clusters have masses, quoted to a factor much better than two, from measurements taken over a single night. The instrument that makes that possible is not a telescope. It is a theorem about averages, and its content is that a bound system cannot avoid arranging itself so that its kinetic energy is exactly half the magnitude of its potential energy — not at any instant, but on average.

The statement, and where the two comes from #

For a system of point masses interacting gravitationally, define the moment of inertia about the centre of mass, $I = \sum_i m_i r_i^2$. Differentiate it twice and the result is short:

$$\tfrac{1}{2}\ddot{I} = 2T + U,$$

with $T$ the total kinetic energy and $U$ the total gravitational potential energy, which is negative. The derivation is two lines of the equations of motion and Newton’s third law, and it holds instant by instant with no approximation whatever.

Now average both sides over a long time. If the system is bound — if it neither collapses nor disperses — then $I$ stays within limits, so its second derivative averages to zero, and

$$2\langle T\rangle = -\langle U\rangle.$$

The factor of two is not a convention. It comes from the potential going as $r^{-1}$: for a force law $r^{-n}$ the relation is $2\langle T\rangle = (n-1)\langle U\rangle$ in magnitude, and the inverse square is the case that gives two. It is the same exponent that makes orbits close, showing up in a different disguise.

What it is used for #

Rewrite it in the form a measurement can reach. For a system of total mass $M$ and some characteristic radius $R$, the potential energy is $U = -\alpha GM^2/R$ with $\alpha$ a number of order one that depends on how the mass is distributed, and the kinetic energy is $T = \tfrac12 M\langle v^2\rangle$. Then

$$M = \frac{\langle v^2\rangle R}{\alpha G}.$$

Everything on the right is measurable from a single night’s data. The mean square speed comes from Doppler shifts in the spectra of individual members — a line shift is a speedometer at any distance, and it does not care how far away the cluster is. The radius comes from the angular size and a distance. And $\alpha$ comes from a model of the density profile, which is the one place taste enters.

The condition, which is the whole difficulty #

The theorem requires the system to be bound and the average to be over long enough. Neither is checkable from a photograph, and both can fail.

The relevant clock is the crossing time: how long a member takes to traverse the system, roughly $R/\sqrt{\langle v^2\rangle}$. For a globular cluster that is a million years or so; for a galaxy cluster, about a billion. The average in the theorem is over many crossing times, and what supplies it is not the observer but the system’s own history: a cluster ten billion years old has crossed itself thousands of times, and the average has been taken whether or not anybody was watching.

That is the actual justification, and it is worth stating plainly because the theorem is often presented as though the observer were doing the averaging. The observer is doing nothing of the kind. The observer is betting that the system has already done it, and the bet fails exactly where it matters most — for a cluster that has recently merged, or that is still forming, or that is in the middle of being torn apart.

What Zwicky did with it #

The first use of the theorem on something outside the solar system is also the most consequential. In 1933 Fritz Zwicky measured line-of-sight velocities for eight galaxies in the Coma cluster, found a dispersion of about 1000 km/s, put it into the expression above, and got a mass some four hundred times the mass of the visible galaxies.

The number has since come down — modern values are nearer a factor of ten, because Zwicky’s distance scale was wrong by a factor of about eight and the cluster’s luminosity was underestimated — but the sign of the result and the argument that produced it have not changed at all. A cluster held together by the gravity of what can be seen would have a dispersion of a couple of hundred kilometres per second. Coma’s is five times that, and a system moving five times too fast for its own gravity would have dispersed in a crossing time, which is a hundredth of its age.

The strength of the argument is that it needs no model of a galaxy, no assumption about how light traces mass, and no orbit. It needs the system to be bound and old, and if it is not bound and old then it should not be there.

Half the energy, and where the other half went #

The theorem has a consequence that is more surprising than the theorem, and it is worth a section because it explains why gravitating systems behave unlike everything else.

Since $2\langle T\rangle = -\langle U\rangle$, the total energy is

$$E = \langle T\rangle + \langle U\rangle = -\langle T\rangle.$$

The total energy of a bound gravitating system is minus its kinetic energy. Take energy away from such a system — let it radiate — and $E$ becomes more negative, so $\langle T\rangle$ increases. A self-gravitating system that loses energy gets hotter.

That is the reason a star can burn for ten billion years without a thermostat: contract it slightly and the released potential energy is split, half radiated and half retained as heat, so the interior warms and the reactions speed up until the loss is balanced. It is also why an orbit that is being braked speeds up — the same arithmetic, with two bodies instead of a million. It gives gravitating systems a negative heat capacity, which is a property no laboratory object has, and it is why a star cluster does not settle into a uniform temperature the way a gas in a box does. The core loses energy to the halo, contracts, and becomes hotter and denser rather than cooling — a runaway that ends in core collapse rather than in equilibrium. The virial theorem is the statement that this must happen, and the whole subject of cluster dynamics is the question of how fast.

Three routes, and why agreement matters #

Because the virial estimate rests on an assumption that cannot be tested from within itself, its value depends on being checkable from outside. It is, and by methods with nothing in common.

The X-ray gas in a cluster is a fluid in hydrostatic equilibrium in the same potential, so its temperature profile gives a mass by a completely different route — a pressure gradient rather than a time average. And the light of background galaxies bent past the cluster gives a third, which involves no equilibrium assumption at all, because a deflection angle is a statement about the potential right now.

Where the theorem is silently doing the work #

Once noticed, the virial relation turns up holding together arguments that do not appear to be about clusters at all.

A galaxy supported by disorder rather than by rotation is a virialised system, and its mass follows from its dispersion by exactly the expression above. So does the mass inside the radius at which a rotation curve is measured, by the ordered version of the same balance.

What “relaxed” actually requires #

The word doing the work in every application above is relaxed, and it is worth pinning down, because it means two different things that happen on wildly different timescales.

The first is what a cluster achieves in a few crossing times: the overall shape stops changing and the density profile settles. This is sometimes called violent relaxation, and it is driven by the changing potential during collapse rather than by anything two stars do to each other. It is fast — tens of millions of years for a globular cluster — and it is what makes the virial theorem applicable at all.

The second is genuine two-body relaxation, in which individual stars exchange energy in encounters until the velocity distribution approaches a Maxwellian and equipartition sets in. That takes far longer: the relaxation time exceeds the crossing time by roughly $N/(8\ln N)$, which for a globular cluster of $10^6$ stars is a factor of about ten thousand and for a galaxy of $10^{11}$ stars is a factor of $10^9$ — longer than the age of the universe by several orders.

The consequence is a division worth carrying: globular clusters are relaxed in the second sense and galaxies are not. A galaxy is collisionless. Its stars have never exchanged energy with one another in any meaningful way, and its velocity distribution is whatever violent relaxation left it, not a thermal one. So a galaxy can be virialised and anisotropic at the same time — held up by orbits that are radial in one place and tangential in another — while a globular cluster’s core has had time to become genuinely isothermal and to segregate its heavy stars towards the centre.

One galaxy in the wrong place #

The dispersion enters the mass as its square, and that turns a question about membership into the dominant systematic in the whole method.

A cluster is identified as a concentration of galaxies on the sky, and a galaxy on the sky may be anywhere along the line of sight. A foreground or background object included in the sample by mistake carries a velocity that has nothing to do with the cluster’s potential, and it can be thousands of kilometres a second away from the mean. One such interloper in a sample of twenty raises the measured dispersion substantially, and the mass by the square of that.

The obvious remedy — discard anything far from the mean — is circular, because how far is too far is exactly what the dispersion is supposed to measure. Iterating it converges, and it converges to a slightly different answer depending on where the clipping threshold is set, so a published cluster mass carries a dependence on a rejection criterion that is rarely quoted.

The better remedy uses the geometry. A cluster’s members occupy a characteristic region in the plane of projected radius against velocity: near the centre the range of possible velocities is wide, and it narrows outward, because a galaxy far from the centre in projection is also far from it in space and therefore in a shallower part of the potential. The boundary of the occupied region is a curve whose amplitude is set by the escape velocity, and it is measurable directly from a large enough sample.

That is the caustic technique, and its virtue is exactly the one the virial theorem lacks: it makes no equilibrium assumption. An escape velocity is a statement about the potential right now, and a galaxy at the boundary is one that is barely bound, not one that has been averaged over anything. The two methods therefore fail in different circumstances, and they are applied to the same clusters for that reason.

It is worth adding what the two methods share, since it is the thing neither can escape. Both work from line-of-sight velocities, so both are measuring one component of a three-dimensional motion and both are integrating along a line that passes through the cluster’s front and back as well as its middle. A galaxy seen near the centre in projection may be anywhere along that line, and its velocity is weighted into the estimate as though it were at the centre. That projection is not a source of noise to be averaged down; it is a systematic that a larger sample does not touch, and correcting it requires exactly the density profile the mass estimate was supposed to avoid assuming.

The theorem’s whole content is that a time average exists, so the honest test of it is to run the same system for much longer and watch whether the average settles or wanders.

Telling a relaxed cluster from one that is not #

Since the theorem’s condition cannot be checked from within the measurement, it is checked from outside — and there are several ways, none of them conclusive alone.

Substructure. A cluster that has recently swallowed a group carries the group’s galaxies as a kinematically distinct clump: a region of the sky whose local mean velocity and local dispersion differ from the cluster’s overall values. Testing for that is a standard statistical exercise on the positions and velocities together, and it flags a large fraction of clusters — a third or more — as having substructure at some level.

Morphology of the hot gas. The X-ray emitting gas responds to the potential faster than the galaxies do, so a disturbed potential shows up first as an irregular, asymmetric X-ray image, sometimes with a sharp edge where infalling gas has been shocked or stripped. A round, centrally peaked X-ray halo is the signature of a cluster left alone for a few billion years.

Offsets. In a relaxed cluster the brightest galaxy sits at the centre of the X-ray emission and at the centre of the mass reconstructed from lensing. In a merging one they separate, because the galaxies and the dark matter pass through each other while the gas is slowed by its own pressure.

That last case is worth stating for its own sake. A cluster caught mid-collision has its gas displaced from its mass, and the displacement is measurable — the gas visible in X-rays sitting between two concentrations of lensing mass that have carried on. That configuration is the cleanest demonstration that the mass is not the gas, and it is available only in systems where the virial theorem does not apply at all.

The value of a cross-check is that it is unavailable to the method being checked, and every item above is a measurement the virial estimate cannot make about itself.

None of these checks is available for a single cluster observed once, which is the situation most published masses come from.

What the picture cannot show #

The theorem has a shape that a figure cannot carry, and it is worth naming.

Every figure here draws a ratio against time, and the quantity in the theorem is a limit as that time goes to infinity. What a real observation has instead is one moment, plus an argument about the system’s age. The uncertainty on a virial mass is therefore not a measurement error and is not on the graph — it is the residual risk that the system is not what the theorem requires, and it is estimated from simulations of clusters that are not relaxed rather than from any property of the data.

There is a second gap. The theorem constrains $\langle T\rangle$ and $\langle U\rangle$, and $\langle U\rangle$ depends on the whole mass distribution, including the part that is not producing the velocities being measured. So a virial mass is a mass of everything within the region the tracers occupy and says nothing about anything outside it. In a cluster with an extended halo — which is every cluster — that is a systematic underestimate whose size can only be had from a model. The theorem is nevertheless the first thing anyone reaches for, and deservedly so: it is the only method that turns a night of spectroscopy into a mass.

And the divergence that makes the average necessary in the first place, since a system whose trajectories could be predicted would not need one.

Where the ladder goes next #

The obvious next rung is the failure mode: what a system that is not relaxed does to a virial mass, and how the discrepancy is estimated. Past that is the tensor form of the theorem, which relates the kinetic and potential energies component by component rather than in total, and which is what turns a measured flattening into a statement about whether a galaxy is held up by rotation or by anisotropy — a question the scalar version cannot even pose.