A galaxy held up by disorder

A spiral galaxy is held up by going round. Its stars are on nearly circular orbits in a common plane and in a common direction, and the speed of that circulation is what balances gravity — which is why a rotation curve is a mass measurement.

An elliptical galaxy is not held up by anything of the kind. Its stars orbit in every plane and in both directions, the net rotation is often close to zero, and the quantity that balances gravity is the spread of the velocities rather than their mean.

That change of support has a precedent in this collection, one field earlier and twenty orders of magnitude smaller. A star is held up by pressure, and pressure is the random motion of its particles. An elliptical galaxy is a star with a hundred billion particles, each of them a star.

The theorem, in the form that is actually used #

For any self-gravitating system that has settled down, the time-averaged kinetic and potential energies satisfy

$$2\langle T\rangle + \langle U\rangle = 0.$$

Write $\langle T\rangle = \tfrac12 M\langle v^2\rangle$ and $\langle U\rangle = -\alpha GM^2/R$ for some dimensionless $\alpha$ that depends on how the mass is distributed, and the mass follows:

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

That is the whole method. A velocity, a size, and a structure constant.

The structure constant is where the honesty is required. For a galaxy with a de Vaucouleurs profile, measuring $R$ as the half-light radius and $\langle v^2\rangle$ as three times the square of the line-of-sight dispersion — three, because only one of the three components is observed — the coefficient comes out close to five. Different profiles, different definitions of radius and different orbital anisotropies move it by tens of per cent, which sets the accuracy of the whole business at that level and no better.

Reading the velocity out of a smear #

An elliptical galaxy’s spectrum contains no individual star. What is recorded is the sum of a hundred billion stellar spectra, each Doppler-shifted by its own line-of-sight velocity.

Each star’s absorption lines are narrow. The sum of many such spectra, shifted by a distribution of velocities, has lines that are broadened by exactly that distribution — the observed spectrum is the intrinsic stellar spectrum convolved with the line-of-sight velocity distribution.

The dispersion is therefore obtained by fitting: take a template spectrum of the kind of star that dominates the light, broaden it by a trial distribution, compare with the observation, and vary the distribution. The output is not a single number but a profile, and its higher moments — the skewness and kurtosis of the velocity distribution — carry information about the shapes of the orbits.

The fourth root of radius #

An elliptical’s light is far more centrally concentrated than a disc’s, and the profile that fits it was found empirically by Gérard de Vaucouleurs in 1948: the surface brightness falls as $\exp[-(R/R_e)^{1/4}]$, up to constants.

The generalisation, due to José Sérsic, replaces the exponent $1/4$ by $1/n$, and the index $n$ then measures concentration: $n=1$ is an exponential disc, $n=4$ is de Vaucouleurs, and real galaxies run from about 0.5 to 10. The index correlates with luminosity — brighter ellipticals are more concentrated — which is one of the several correlations that make the elliptical population a one-parameter family to first approximation and something more complicated on closer inspection.

The reason the profile matters here is that it sets the structure constant in the virial estimate, and that it defines the radius. The half-light radius is not the edge of anything. It is the radius containing half the light in projection, and it is chosen because it is measurable and stable, not because anything happens there.

Rotation, where there is any #

The natural first guess about an elliptical galaxy is that it is flattened because it rotates, in the way a spinning ball of fluid is. That guess is testable and it turned out to be wrong, in one of the more decisive observational results of the 1970s.

If a system is flattened purely by rotation and its velocity distribution is otherwise isotropic, then its rotation speed and dispersion are related to its ellipticity by

$$\frac{v}{\sigma} = \sqrt{\frac{\varepsilon}{1-\varepsilon}}.$$

Measure $v$ and $\sigma$ for a sample of ellipticals, plot against ellipticity, and see whether they lie on that curve. The luminous ellipticals do not — they lie far below it, rotating far too slowly for their flattening. Their shapes are sustained not by rotation but by anisotropy: the velocity dispersion is larger along one axis than another, and a system can hold a flattened shape indefinitely that way with no rotation at all.

The fainter ellipticals do lie near the line, which is the first hint that “elliptical galaxy” names two populations rather than one — a division that modern integral-field surveys have made much sharper, into slow and fast rotators.

The observation behind the number #

Three quantities go into every point on the first figure, and each is worth naming.

The dispersion comes from the line broadening described above, measured within a defined aperture — usually one eighth of the half-light radius, so that different galaxies are compared over the same fraction of themselves. Aperture matters, because the dispersion varies with radius.

The half-light radius comes from fitting a Sérsic profile to the surface photometry, which requires deciding where the galaxy ends. That decision is not innocent: an elliptical’s outer profile falls off slowly, so the total light — and hence the half-light radius — depends on how far out the integration is taken.

And the distance, without which the radius is an angle. For nearby ellipticals this comes from surface-brightness fluctuations or from the fundamental plane itself, which makes the argument slightly circular and is a known difficulty rather than a hidden one.

Ellipticals have haloes too, and it was harder to show #

A spiral galaxy announces its dark matter: the rotation curve is flat and the argument is over in one figure. An elliptical announces nothing, because a single dispersion measured inside the half-light radius is consistent with the stars alone.

Establishing that ellipticals have haloes therefore needed tracers further out than the light, and three were found.

Planetary nebulae. An old stellar population produces them continually, they are bright in one emission line, and their velocities are measurable individually at distances where an integrated spectrum is hopeless. They act as test particles out to several half-light radii.

Hot gas. Massive ellipticals hold an X-ray-emitting atmosphere at a few million kelvin, and its temperature and density profile give the enclosed mass by hydrostatic equilibrium — the same reasoning as for a cluster, applied one scale down.

Lensing. The mass inside an Einstein radius follows from an angle and three distances, with no assumption about equilibrium at all, and a massive elliptical is the commonest strong lens there is. All three agree that the mass-to-light ratio rises outwards, as it does in spirals. What the three do not agree about, at the ten to twenty per cent level, is the amount — which is the ordinary condition of this subject and is worth stating rather than smoothing over.

The plane the population lies on #

The virial expression relates three quantities — mass, radius and dispersion — and if galaxies were a one-parameter family only one of them would be free. They are not, but they are close to a two-parameter family, and the way that shows up is one of the most useful facts about ellipticals.

Rewrite the virial relation in terms of observables. Mass is $\propto \sigma^2 R_e$ and luminosity is $\propto I_e R_e^2$, where $I_e$ is the mean surface brightness inside the half-light radius. If the mass-to-light ratio were constant, then

$$R_e \propto \sigma^2 I_e^{-1},$$

and ellipticals would lie on a plane in the space of $\log R_e$, $\log\sigma$ and $\log I_e$. They do lie on a plane — the fundamental plane — and it is thin, with a scatter under twenty per cent in $R_e$.

But the plane’s tilt is not quite what constant mass-to-light predicts: the exponents come out near $R_e \propto \sigma^{1.4}I_e^{-0.8}$ rather than $\sigma^2 I_e^{-1}$. That discrepancy is real, it is measured, and it says that the mass-to-light ratio rises systematically with galaxy mass. A relation that was noticed as a distance indicator turned out to be a measurement of how galaxy populations differ along the sequence, which is a better thing to have.

Where the analogy with a star breaks #

A galaxy is collisionless and a star is not. In a gas, particles collide often enough to establish a local temperature, and the pressure follows from it. In a galaxy the time for a star’s orbit to be measurably changed by encounters with individual stars is far longer than the age of the universe, so nothing establishes a temperature and the velocity distribution is whatever the formation of the galaxy left behind.

That is the reason anisotropy is possible at all. A gas cannot be anisotropic for long, because collisions isotropise it. A galaxy can be, permanently, and the observed shapes of ellipticals are evidence that it is.

And the relaxation that does occur is violent. Donald Lynden-Bell’s violent relaxation is the process by which a system settling in a rapidly changing potential redistributes energies among its stars — collisionlessly, and in a few dynamical times. It explains why ellipticals look relaxed despite never having had time to relax by encounters, and its outcome is a profile close to the observed one.

The degeneracy, stated properly #

The mass–anisotropy degeneracy was named above and deserves to be set out, because it is the reason a dispersion is a weaker measurement than a rotation speed and because the same difficulty recurs everywhere a system is supported by disorder.

Write the anisotropy as the ratio of the tangential velocity dispersion to the radial one, conventionally as a parameter that is zero for isotropic orbits, positive for radially biased ones and negative for tangentially biased ones. The equation relating the observed dispersion profile to the enclosed mass — the spherical Jeans equation — contains that parameter and the mass on the same footing, and only one combination of them is constrained by a line-of-sight dispersion profile.

The consequence is concrete. A system whose stars are on predominantly radial orbits looks, in projection near its centre, like a system with more mass on isotropic orbits: the radial motions point along the line of sight there and inflate the measured dispersion. Further out the sign reverses, because at large projected radii the line of sight is transverse to the radius. So the shape of the observed profile does carry some information about anisotropy — and not enough, because the profile also depends on the mass distribution, which is what was wanted.

Two things break it in practice and neither is available everywhere. The higher moments of the line-of-sight velocity distribution — its kurtosis in particular — respond differently to anisotropy than to mass, so a spectrum good enough to measure a fourth moment adds a constraint. And a two-dimensional velocity field from integral-field spectroscopy constrains the orbit distribution far better than a single slit does, because it measures the dispersion along many position angles at once.

Where neither is available, a dispersion-derived mass carries an uncertainty of tens of per cent that no amount of signal-to-noise removes, and that is a statement about what the observable contains rather than about how well it was measured.

What the picture cannot show #

The virial mass is a total, not a profile. A rotation curve gives $M(<r)$ at every radius; a dispersion gives one number, and turning it into a profile requires modelling the orbits. Two systems with identical total masses and different orbital anisotropies produce different observed dispersions, and untangling that is the mass–anisotropy degeneracy — the elliptical galaxy’s version of the disc–halo problem.

The system must be in equilibrium. A recent merger is not, and applying the theorem to one gives a mass that is simply wrong. The assumption is normally justified by a system looking regular and symmetric, which is weaker than it sounds.

And the coefficient is a model. Five is a good number for a de Vaucouleurs profile observed the standard way. It is not a constant of nature, and any mass quoted from a dispersion inherits its uncertainty.

One consequence of collisionless support deserves stating on its own, because it inverts an intuition carried over from gases.

A self-gravitating system that loses energy contracts, and the virial theorem then requires its velocity dispersion to rise. A galaxy that is somehow cooled gets hotter; there is no equilibrium it can settle into by radiating, because radiating is what makes it hotter. That negative heat capacity is why a star’s core heats as it shines, and it is why the long-term fate of any stellar system is to eject its members one at a time rather than to settle down. Disorder as a support mechanism has this peculiarity built into it, and nothing supported by pressure escapes it.

The extreme case, where the stars are almost incidental #

The method’s most striking application is to the smallest systems it reaches, and it is worth ending the observational half here because the arithmetic is the same and the answer is not.

A dwarf spheroidal satellite of the Milky Way contains a few hundred thousand to a few million stars, spread over a few hundred parsecs, with no gas and no rotation. Its stars are individually resolved and their velocities are measured one at a time rather than from an integrated spectrum, so the velocity dispersion is a genuine sample statistic rather than a deconvolved line width — typically a few kilometres a second, measured from a few hundred to a few thousand member stars.

Put that dispersion and that radius through the same virial expression and the mass comes out at ten to a hundred times the mass of the stars. The mass-to-light ratios run from tens to over a thousand, which is not a correction to a stellar population; it is a statement that the stars are a trace constituent of the object they sit in.

The measurement is delicate in ways an elliptical’s is not. A dispersion of a few kilometres a second is comparable with the orbital motions of binary stars among the members, so a single-epoch measurement inflates the dispersion and therefore the mass; repeat observations are needed to identify and remove the binaries. And membership matters enormously, since the systems are sparse and the Galactic foreground is not: one interloper at the wrong velocity among two hundred members moves the answer.

What makes them worth the trouble is the same thing that makes them hard. Their internal accelerations are the lowest measured anywhere, their dark-matter dominance is the highest, and the shape of their inner mass profiles is one of the few observational handles on what dark matter is. The virial argument was introduced here as a way of weighing a galaxy whose stars supply the support; in a dwarf spheroidal it weighs something the stars are merely embedded in, and the expression does not notice the difference.

The generalisation #

The step made here is one of the most general in physics: when the ordered motion of a system is not what supports it, the disordered motion is, and the two are handled by the same expression.

A star is supported by the pressure of a hot gas; a white dwarf by the pressure of a degenerate one, which is disorder that cannot be reduced by cooling; a planet’s atmosphere by the thermal motion of its molecules; an elliptical galaxy by the random motion of its stars; and a cluster of galaxies by the random motion of its galaxies. In every case the support is $\langle v^2\rangle$ and the balance is against $GM/R$.

What changes down that list is only what is doing the moving, and the range is remarkable: from a proton in a stellar core to an entire spiral galaxy crossing a cluster, the same three symbols hold the object up.

The profile that makes the fourth-root plot straight is the same at every size, and it is worth checking that the straightness is not an artefact of the one radius used above.

That scale-independence is what allows a single number to stand for a galaxy’s structure across four orders of magnitude in luminosity. A dwarf elliptical and a brightest cluster galaxy are drawn from the same family, differing in their half-light radius, their central surface brightness and their index — and the systematic trend in that third parameter, from index 1 at the faint end to index 8 or more at the bright end, is one of the cleanest statements the population makes about how it was assembled.

The trend runs the way a merger history predicts. Repeated dry mergers scatter stars to large radii without adding gas or forming stars, which builds an extended envelope and raises the index while leaving the total light almost unchanged. A galaxy that formed its stars in one dissipative collapse and was never disturbed keeps a low index. So the index is a record of how many times a galaxy has been hit, read off a photometric profile with no kinematics at all.

What it cannot do is date the events. A profile records the accumulated effect of a merger history and not its timing, and two galaxies with the same index may have reached it in one major merger or twenty minor ones. Separating those needs the kinematics the rest of this essay is about — and in particular the ratio of ordered to random motion, which a major merger destroys and a minor one does not.

Where the ladder goes next #

The next rung is the set of correlations that make ellipticals a two-parameter family — the fundamental plane, on which dispersion, radius and surface brightness lie in a tilted sheet whose tilt is itself a measurement of how mass-to-light ratio varies along the sequence.

Later rungs on this anchor: the mass–anisotropy degeneracy and how integral-field spectroscopy attacks it; slow and fast rotators as two formation histories; the fundamental plane as a distance indicator; violent relaxation and why the resulting profiles look the way they do; the dispersion profile of a dwarf spheroidal, where the dark matter dominates everywhere; and the connection between a galaxy’s central dispersion and the black hole at its middle, which has no business being as tight as it is.