A radius that is a measurement of the threshold

A planet has a radius: there is a surface, and past it there is nothing. A star nearly has one, since its photosphere is thin compared with the star. A galaxy has nothing of the kind. The light of a disc galaxy falls off exponentially with radius, by about a magnitude per scale length, and it goes on doing so past the edge of every image anyone has taken. The sky, not the galaxy, decides where the measurement stops, and a galaxy’s size is therefore a definition before it is a measurement.

The oldest definition is the simplest. Draw a contour of constant surface brightness — an isophote — at some chosen level, and quote the radius of that contour. The first large catalogues of galaxy sizes were built this way, at 26.5 magnitudes per square arcsecond on photographic plates in the 1950s and at 25 magnitudes per square arcsecond in the blue for the reference catalogues that followed, and “D25”, the diameter at the 25th-magnitude isophote, is still the size listed for most nearby galaxies. It is a well-defined quantity, easy to measure and reproducible from one image to another. It is also a mixture of two unrelated properties of the galaxy.

Two properties in one number #

For an exponential disc the surface brightness in magnitudes rises linearly with radius from its central value $\mu_0$, by 1.086 magnitudes per scale length h. The radius at which it reaches a limit $\mu_{\rm lim}$ is therefore

$$R_\mathrm{iso} = h\,\frac{\mu_\mathrm{lim} - \mu_0}{1.086}.$$

The scale length h is the galaxy’s size in any physically meaningful sense: it sets how its light and its mass are distributed. The factor after it is not about size. It measures how far above the threshold the galaxy’s centre sits, and a galaxy whose centre is a magnitude brighter has an isophotal radius nearly one scale length larger with no difference in its size at all.

The consequence for catalogues is immediate. A catalogue that includes every galaxy larger than some isophotal diameter includes every galaxy for which $h(\mu_\mathrm{lim}-\mu_0)$ exceeds a value, which is a joint cut on size and on central surface brightness. Galaxies of low central surface brightness are admitted only if they are very large, and galaxies whose centres are near the limit are not admitted at any size. That was the argument, made forcefully in the 1970s, that the narrow range of central surface brightness found in the early catalogues of disc galaxies — the famous near-constancy at 21.65 — might be a property of the catalogues rather than of the galaxies. The subsequent discovery of large populations of low-surface-brightness galaxies showed that it was largely so.

Why the thresholds were where they were #

The two classical limits were not arbitrary, and the reasons they were chosen explain what they were good for. Erik Holmberg’s 26.5 magnitudes per square arcsecond, in 1958, was close to the faintest level his photographic plates could reach reliably across a whole galaxy; his aim was a uniform comparison of the galaxies in his sample, and a limit near the plate’s own floor gave the largest radii the data could support. The later 25-magnitude level, adopted for the reference catalogues of bright galaxies, was set a magnitude and a half brighter so that it could be measured on shallower plates and by eye, across tens of thousands of galaxies, with an error of a few per cent.

Both were therefore limits of the instruments, chosen to be reproducible rather than to be physical. For comparing galaxies of similar type and similar central brightness they work well, since the dependence on $\mu_0$ drops out of a comparison between galaxies that share it. The trouble starts when the sample spans a range of surface brightness, or a range of redshift, or both — which is to say whenever the size is used as a physical quantity rather than as a label.

Holmberg’s radius was used for exactly such a purpose. His finding that satellite galaxies avoid the planes of their host discs — distributed preferentially near the poles — was measured in units of the host’s isophotal radius, and the effect has been argued over ever since, in part because satellites seen in projection close to a bright disc are hard to find against its light — a selection that depends on where the disc’s isophotes fall.

Which of two galaxies is bigger #

The dependence on central brightness means that a catalogue can rank two galaxies in the wrong order.

The larger disc carries only about half the light of the smaller — a low-surface-brightness disc spreads less light over more area — and its light is below the shallower limit over most of its extent. At 25 magnitudes it is the smaller galaxy; at 26.5 it is the larger; and the ratio of their true sizes, two, is recovered at neither.

This matters most for relations between size and other properties. The relation between a galaxy’s size and its luminosity, or its mass, or its rotation speed, is one of the basic scaling relations of galaxies — the size a disc has is the fossil of its halo’s angular momentum, and the scatter in size at fixed mass is a measurement of the spread of that angular momentum. An isophotal size compresses the range of sizes and correlates their errors with surface brightness, so a size–mass relation built from isophotal radii has a slope and a scatter that are partly properties of the threshold.

Unchanging galaxies that shrink with redshift #

The threshold’s influence becomes decisive at high redshift, for a reason that has nothing to do with galaxies.

In a static universe surface brightness does not depend on distance. In an expanding one it falls as $(1+z)^{-4}$: one factor of $1+z$ because each photon arrives with less energy, one because the photons arrive less often, and two because the relation between a galaxy’s physical size and its angular size, through the angular-diameter distance, differs from the relation between its luminosity and its flux by $(1+z)^2$. The dimming is purely geometric, and at a redshift of one it amounts to three magnitudes, at a redshift of three to six. It is the prediction the Tolman test checks, and it is well confirmed.

For an isophotal radius it is a shift of the threshold. Moving a galaxy to redshift z is equivalent to raising its central surface brightness by $10\log_{10}(1+z)$ magnitudes, so its isophotal radius falls by $9.2\log_{10}(1+z)$ scale lengths even if nothing about it has changed. The opening figure draws that: three unchanging discs, each shrinking steadily in isophotal radius and disappearing entirely from an isophotal catalogue at redshifts between one and three.

The real situation is more complicated in one way and simpler in another. More complicated, because a galaxy at high redshift is observed in a different part of its spectrum — a filter centred in the visible samples the ultraviolet of a galaxy at redshift two, where it may be brighter or fainter depending on how many young stars it has — and a magnitude measured in a band the source never had needs a correction that depends on a model of its spectrum. Simpler, because high-redshift galaxies are genuinely younger and their stars brighter, which partly offsets the dimming. The two effects are of comparable size and opposite sign, and neither is known well enough in any individual galaxy to separate from the other.

The numbers are large. A disc at the canonical central brightness, measured at 26.5 magnitudes per square arcsecond, has an isophotal radius of 4.5 scale lengths nearby, 1.7 at a redshift of one, and none at all at a redshift of two. A catalogue of isophotal sizes across that range would report a threefold shrinkage by a redshift of one for galaxies that had not changed, and would lose them entirely before the epoch at which most of the stars in the universe were formed.

A radius the galaxy defines for itself #

The way out of the threshold is to define a radius using only the shape of the profile, with no reference to its absolute level.

The standard construction, introduced by Vahe Petrosian in 1976, uses the ratio of the surface brightness at a radius to the mean surface brightness inside it. That ratio is one at the centre and falls outward. It contains no amplitude — any factor that dims the whole galaxy uniformly, whether distance, dust or the expansion of the universe, divides the numerator and the denominator alike — and the radius at which it reaches a chosen value is a property of the profile’s shape alone.

The largest imaging survey of the last generation adopted exactly this: every galaxy’s size and total flux were measured through an aperture of twice its Petrosian radius at a ratio of 0.2. That aperture captures essentially all of an exponential disc’s light and about four-fifths of a spheroid’s, because a spheroid’s light extends further relative to its core. The radius is immune to distance and to dimming, and it is not immune to morphology. Two galaxies with the same half-light radius but different profile shapes are given different Petrosian radii, and a sample that mixes discs and spheroids has sizes whose meaning differs from one galaxy to the next.

The half-light radius — the radius enclosing half the total light — is in one sense the natural size, since it depends on neither threshold nor shape. But it requires the total light, and the total light requires the outskirts, which are exactly the part that sits below the sky. In practice the total is obtained by fitting a model profile and extrapolating it to infinity, and the half-light radius inherits the model: a profile that falls more slowly than the model assumes has more light outside and a larger true half-light radius than the one quoted. A third construction, widely used in automated catalogues, is the first moment of the light profile — the light-weighted mean radius, due to Kron — with an aperture of a fixed multiple of it. It is computed from the pixels above the detection threshold, so it inherits a weaker version of the isophotal dependence: a galaxy whose outskirts fall below the detection limit has a smaller first moment than the same galaxy observed deeper, and its aperture captures a smaller fraction of its light.

Each definition of size trades one dependence for another. The isophotal radius depends on the threshold; the Petrosian radius on the shape; the half-light radius on the extrapolation; the moment radius on the detection. There is no definition that depends on nothing, because a profile that extends to infinity has no natural end.

There is an older use of galaxy sizes that the threshold defeated outright. If galaxies were standard rulers — objects of a known physical size — their angular sizes against redshift would trace the angular-diameter distance, and the famous turnover beyond a redshift of about 1.6, where objects of fixed size begin to look larger again, would be a direct measurement of the geometry of the universe. The test was proposed in the 1950s and attempted repeatedly with galaxies and with radio sources. It failed as a cosmological test, not because the geometry is wrong but because no size of a galaxy is fixed: an isophotal size shrinks with redshift through the dimming, an intrinsic size changes as galaxies grow, and the two effects are larger than the geometric one being sought. The standard ruler that finally worked was not an object at all but a length in the clustering of galaxies, the imprint of sound waves in the early universe, which has no surface brightness to fall below a threshold.

A radius in light is not a radius in mass #

Every definition above measures the distribution of light, and light is not mass. A galaxy’s older stars are redder and more centrally concentrated than its younger ones, and its dust is concentrated towards the centre too, so the same galaxy has a larger half-light radius in blue light than in red, and larger in red than in the near-infrared, where the light most nearly traces the stellar mass. A disc galaxy with an ordinary colour gradient can have a blue half-light radius a quarter larger than its infrared one.

The distinction becomes a physical one at high redshift, where it is known to matter. Galaxies grow inside-out: their centres form first and their outskirts later, so at any moment the outskirts hold younger, bluer, brighter stars per unit mass than the centre. The half-light radius is then larger than the half-mass radius by an amount that changes as the galaxy ages. Converting the light profile into a mass profile, pixel by pixel from its colours, has shown that part of the apparent growth of galaxies since a redshift of two is the ageing of their outskirts rather than the arrival of new stars there: the light has become more concentrated because the outer stars dimmed, not because the galaxy shrank at the centre.

That is a different kind of dependence from the threshold’s, but it has the same moral. A radius is a statement about a particular quantity measured in a particular way, and a comparison between radii is valid only when those are the same.

Even the categories galaxies are sorted into carry thresholds. The ultra-diffuse galaxies found in large numbers in nearby clusters since 2015 are defined by two cuts at once: a half-light radius larger than about 1.5 kiloparsecs and a central surface brightness fainter than about 24 magnitudes per square arcsecond. Both cuts sit close to where the population is densest, so the number of such galaxies counted in a cluster depends steeply on exactly where they are placed, and a galaxy just inside one survey’s definition may be just outside another’s. The class is physically interesting whatever its boundary, but its abundance is a statement about the boundary as well as about the galaxies.

What the figures leave out #

The figures use pure exponential discs, and real galaxies are not pure. They have bulges, whose steeper profile dominates the centre and raises the central surface brightness without changing the disc; they have breaks in their outer profiles, which change the isophotal radius if the break lies above the threshold; and they are seen at an inclination, which raises the surface brightness of a disc seen edge-on by up to a factor equal to its axis ratio and moves every isophote outward. Dust reddens and dims the centres of discs by different amounts in different bands, so the same galaxy has different isophotal radii in blue and in red.

They also treat the redshift dimming as the only change with distance. At high redshift the angular size of a galaxy of fixed physical size stops shrinking and begins to grow again beyond a redshift of about 1.6, because the angular-diameter distance reaches a maximum, and the resolution of the telescope becomes a larger fraction of the galaxy. A radius measured from a blurred image is biased in a direction that depends on the definition — upward for a half-light radius, downward for an isophotal one — and the blur has to be modelled before any definition can be applied.

Still open: how much galaxies have really grown #

Galaxies at high redshift are smaller than present-day galaxies of the same mass. That result has been established with half-light radii from model fits, which are not subject to the isophotal shrinkage drawn here, and it survives every correction that has been attempted: massive quiescent galaxies at a redshift of two have half-light radii three to five times smaller than their counterparts today. What is less settled is how much of the difference lies in faint outskirts that the model fits do not see. A galaxy that grew by accreting satellites into an extended envelope — the process that builds the stellar halos at 30 magnitudes and below — would have its growth concentrated exactly in the part of the profile that dimming and the sky make hardest to measure at high redshift. Deep images of a few compact galaxies at a redshift of two have found faint extended envelopes that the standard fits missed. Whether that is common decides whether the growth of galaxies since then is mostly real or partly a threshold moving through their light.