A dispersion inflated by orbits nobody resolved

A stellar system with no rotation is held up by disorder: its stars move on randomly oriented orbits, and the spread of their velocities measures the depth of the potential well they are moving in. Measure the spread, apply the virial theorem, and a mass falls out.

The measurement is a spread of radial velocities across a sample of stars, and it assumes each velocity is the velocity of a star’s centre of mass. For a star in a binary it is not.

A galaxy held up by disorder is the mechanism and an average that weighs what cannot be watched is the theorem. This essay is about a contaminant in the input that is negligible for large systems and dominant for small ones — and the small ones are where the interesting claims are.

Why it adds in quadrature and why that is the worst case #

A measured radial velocity is the system velocity of the star’s centre of mass plus its orbital velocity about a companion, if it has one. The two are independent, so the variances add:

$$\sigma_{\rm obs}^2 = \sigma_{\rm true}^2 + f\,\sigma_{\rm bin}^2,$$

with $f$ the binary fraction. Quadrature addition is the reason the effect is so sharply differentiated by system: where the true dispersion is much larger than the binary term the correction is negligible, and where it is much smaller the measurement is entirely binaries.

The two regimes are separated by the binary orbital velocity, which is a property of stars rather than of the system. It is of order one kilometre a second — the circular velocity at an astronomical unit around a solar mass — and it is essentially the same everywhere.

So a globular cluster with a dispersion of ten kilometres a second is unaffected, an ordinary dwarf spheroidal at nine is barely affected, and an ultra-faint dwarf at half a kilometre a second is measuring binaries.

It is worth noticing what quadrature addition implies for the direction of the error. Variances add, so the measured dispersion is always larger than the true one — never smaller. That gives the systematic a sign, and a sign in the direction that manufactures dark matter: an uncorrected dispersion overestimates the mass, overestimates the mass-to-light ratio, and makes a system look more dark-matter dominated than it is. Every claim in this area therefore has to be defended against the possibility that it is binaries, and the defence is a second epoch rather than an argument.

What it does to a mass #

A virial mass is proportional to the square of the dispersion, so a fractional error in the dispersion is twice that in the mass.

For an ultra-faint dwarf the numbers are stark. If the true dispersion is 0.3 kilometres a second and the binary contribution is 1.4, the measured value is 1.43 and the mass is inflated by a factor of over twenty. The mass-to-light ratio inflates with it, and the mass-to-light ratio is the whole basis for the claim that such systems are dark-matter dominated.

There is a second consequence that is easy to overlook and matters for how such results are read. The inflation depends on the ratio of the binary term to the true dispersion, so two systems with identical stellar populations and different dispersions are affected quite differently — and a comparison between them is not a comparison of like with like unless both were corrected. Several published trends of mass-to-light ratio with luminosity run in the direction that uncorrected binaries would produce, since fainter systems are colder, and disentangling the real trend from the systematic requires that the correction be applied uniformly across the sample.

The comparison with a globular cluster is worth making explicit because it is the natural control. A globular cluster has the same stars, the same binary fraction and roughly the same period distribution as a dwarf galaxy, and a dispersion ten to twenty times larger. Its measured dispersion is therefore essentially uncontaminated, and its mass-to-light ratio comes out at two or three in solar units — what stars alone give. That is the calibration: the technique demonstrably works where the binaries are negligible, and the systems where it fails are exactly the ones where they are not.

The fix, and why it works #

The correction is not a model. It is a second measurement.

A binary’s orbital velocity changes over its period — days to centuries — while the system velocity does not. So observing the same stars at several epochs separated by months separates the two: a star whose velocity changes is a binary and can be removed or fitted; a star whose velocity is steady contributes its system velocity.

Formally, the variance contributed by binaries falls as one over the number of epochs, because the orbital velocities at different epochs are uncorrelated while the system velocities are identical. Four epochs quarter the binary variance and leave the true variance untouched.

The practical procedure is better than that. With enough epochs the binaries are individually identified by their variability, and their orbits are fitted well enough to recover their centre-of-mass velocities. Then nothing is discarded and nothing is corrected statistically.

Be explicit about the arithmetic of the multi-epoch approach, because the gain is larger than the simple variance argument suggests. Two epochs separated by a year detect a binary if its velocity changed by more than the measurement error, which for periods from days to a few decades is most of them. Detected binaries are removed outright rather than corrected statistically, so the residual contamination comes only from the undetected ones — the very long periods, whose orbital velocities are small anyway, and the face-on orbits, whose radial component is small by projection. The residual after a well-designed campaign is therefore much smaller than the naive one-over-N scaling implies, and it is dominated by a part of the period distribution that contributes little.

The design question that follows is which epochs to take. Two visits a year apart detect long periods; two a week apart detect short ones. A campaign with a logarithmic spacing of intervals — a day, a week, a month, a year — covers the period distribution far better than the same number of visits spaced evenly, and it is the standard design for exactly this reason.

The sample is small, and that changes the statistics #

There is a second difficulty with the coldest systems that compounds the first, and it is worth separating because it is not a contaminant at all.

A dispersion estimated from $N$ stars has a fractional uncertainty of about one over the square root of twice $N$. An ultra-faint dwarf may have ten to thirty stars bright enough for a velocity, so the dispersion is uncertain by fifteen to twenty per cent before any systematic is considered, and the mass by twice that. The binaries then arrive on top of a measurement that was never going to be precise.

The small sample also breaks the tidy quadrature arithmetic. Subtracting an estimated binary variance from an estimated total variance can give a negative number, and frequently does when the true dispersion is small — the estimator is not bounded below by zero and the quantity it estimates is. Reporting the square root of a negative variance is impossible, and reporting zero is a boundary artefact rather than a measurement. This is the same difficulty a fitted eccentricity has at zero: a positive-definite quantity estimated by subtracting one noisy positive quantity from another.

The modern treatment avoids the subtraction entirely. Rather than measuring a dispersion and correcting it, the analysis writes a likelihood for all the velocities at all the epochs simultaneously, containing the system dispersion, the binary fraction, an assumed period and mass-ratio distribution, and each star’s measurement errors — and reports the posterior on the system dispersion after marginalising over everything else. That returns an honest interval, often an upper limit, and it never produces a negative variance because the parameter is bounded in the prior.

What it costs is a dependence on the assumed period distribution, which for an old, metal-poor population in a dwarf galaxy is taken from the solar neighbourhood because there is nowhere else to take it from. That assumption is the residual systematic, it is stated rather than hidden, and it is the right shape of answer for a measurement this marginal.

There is a further small-sample effect worth naming because it produces spurious detections rather than spurious masses. With twenty stars, one binary caught near its velocity extreme contributes a single outlier several kilometres a second from the mean, and a single outlier in twenty moves the dispersion enormously. The distribution of estimated dispersions across many such systems therefore has a long high tail, and the systems that appear most dark-matter dominated are preferentially the ones that got unlucky. The tallest peak in nothing at all is the same selection acting on a periodogram.

What was actually measured #

Three of them, and the second is the one that changed the field’s practice.

The binary fraction, measured. Multi-epoch surveys of dwarf spheroidals find binary fractions of thirty to sixty per cent among their red giants, comparable to the solar neighbourhood’s, with a period distribution that is not obviously different. So the contribution is calculable once the fraction is known, and the fraction has to be measured in the same system.

Dispersions that fell. Several ultra-faint dwarfs had published dispersions of a few kilometres a second from single-epoch data, and repeat observations reduced them substantially — in the most extreme cases to upper limits. The corresponding dark-matter masses fell by an order of magnitude, and at least one candidate galaxy was reclassified as a star cluster because its corrected dispersion was consistent with no dark matter at all.

And the classification depends on it. The distinction between a faint dwarf galaxy and a massive star cluster is, operationally, whether the system needs dark matter — that is, whether its dispersion exceeds what its stars alone would produce. For objects near the boundary that decision is made on a dispersion of about a kilometre a second, which is exactly where the binaries are.

There is also a fourth measurement worth recording because it is a check rather than a correction. If the excess dispersion in a system is caused by binaries, then the excess must correlate with quantities that track the binary fraction — and it does: the stars showing velocity variation between epochs are the ones responsible for the tails of the velocity distribution, and removing them removes the tails rather than narrowing the whole distribution. A dispersion inflated by real dark matter would not have that structure. So the shape of the velocity distribution, not merely its width, distinguishes the two — which is another instance of a shape carrying information a summary statistic does not.

Where the picture stops #

Three of them, and the third is a caution about over-correcting. The mass that is not the light is what all of this is trying to weigh.

The binary contribution depends on the period distribution. Short-period binaries contribute large velocities and are easy to identify; long-period ones contribute small velocities and look like single stars for years. The residual contribution after a multi-epoch campaign comes from periods comparable to the campaign’s length, and estimating it requires a period distribution that has to be assumed.

The velocity precision has to beat the signal. For a dispersion of half a kilometre a second, individual velocities must be measured to a fraction of that, which for faint red giants in a distant dwarf is a substantial spectroscopic effort. Where the per-star error is comparable to the dispersion, it too adds in quadrature and has to be subtracted — and errors are often underestimated, which biases in the same direction as the binaries.

And removing binaries can bias the sample. Discarding every star that shows velocity variation removes binaries and also removes stars whose velocities were measured badly, which are preferentially the faintest. If the faintest stars are distributed differently — and in a system with mass segregation they are — the surviving sample is not a fair tracer.

One more belongs on that list, and it is about what a dispersion measures even when it is measured perfectly. The virial estimator assumes the system is in equilibrium, and an ultra-faint dwarf close enough to the Galaxy to be observed in detail is also close enough to be tidally disturbed. A system being stripped has an inflated line-of-sight dispersion from unbound stars along the line of sight, and that inflation has nothing to do with binaries and is not removed by repeat epochs. So the corrected dispersion is an upper limit for a second, independent reason, and separating the two requires the spatial distribution of the velocities rather than their spread.

Why a small system is harder than a large one #

The wider point is worth stating because it inverts a common intuition.

A measurement whose signal shrinks while its contaminant does not gets harder as the system gets smaller, and the contaminant here is a property of stars rather than of the system. Binary orbital velocities are the same in an ultra-faint dwarf as in a giant elliptical; the dispersion is a thousand times smaller in one than in the other.

That structure is common. The sky background is the same for a faint galaxy as for a bright one; the atmosphere’s dispersion is the same for a large telescope as for a small one; an instrument’s own polarisation is the same whatever it is pointed at. In each case the contaminant is fixed and the signal is not, so the difficulty is entirely in the ratio.

The corollary is that the interesting extremes of any population are the ones most exposed to systematics — because the extremes are where the signal is smallest. The ultra-faint dwarfs are the most dark-matter dominated systems known and the ones where the measurement is hardest, and those two facts are not independent: both follow from their being small.

One more observation about why the correction was slow to arrive. Single-epoch spectroscopy of a hundred stars in a dwarf galaxy is a night’s work on a large telescope; multi-epoch spectroscopy of the same hundred over three years is a programme. The scientific return of the second is invisible until it is done — it produces the same catalogue with smaller numbers — and the first produces a publishable dispersion immediately. That asymmetry in the incentives is not unique to this measurement, and it is the reason a known systematic can persist in a literature for a decade after everybody involved has agreed it matters. What changed the practice was not an argument but a case: a system whose dispersion fell by a factor of three, and whose interpretation changed from a dark-matter-dominated galaxy to an ordinary star cluster with no dark matter at all.

It is worth naming what the corrected measurements actually establish, since the essay has been about a systematic. After multi-epoch correction the classical dwarf spheroidals still have dispersions far above what their stars alone would produce — mass-to-light ratios of tens to hundreds — and those results are secure. What the binaries threatened was the extreme end: the ultra-faint systems with dispersions under a kilometre a second, whose mass-to-light ratios of a thousand or more are the strongest individual evidence for dark-matter-dominated objects. Those particular claims needed the correction and most of them survived it. The systematic was capable of manufacturing the result and it turned out not to have done so, which is a better outcome than either the alarm or the complacency would have predicted.

Where the ladder goes next #

Later rungs on this ladder start with the estimator itself: what a dispersion and a radius really constrain, why the virial coefficient depends on the orbital anisotropy, and how a mass within a particular radius is much better determined than a total. Above it again sits the profile — whether the inner density rises or flattens, which is an argument conducted entirely inside the innermost kiloparsec and which depends on exactly these measurements.