Astronomy & Celestial Mechanics
the sky, computed
Galaxies

The brightness distance cannot touch

Flux falls as the inverse square of distance and so does solid angle, so their ratio does not fall at all. A galaxy's surface brightness is the same number wherever it is put, which means whole populations can be undetectable at any distance whatever.

Assumes Magnitudes and Distance ladder.

16 min read 8 figures Light is the only messengerMeasuring the unreachable

Almost everything about a distant object is dimmed by distance, and the dimming is the same everywhere: flux falls as 1/d21/d^2, which is the geometrical statement that a fixed number of photons spreads over a sphere whose area grows as the square of the radius.

There is one photometric quantity this does not touch, and its immunity has consequences that shaped what is in the catalogues and what is not.

What distance does, and does not, do to a galaxy. Three quantities against distance, each relative to its value at 5 Mpc, on logarithmic axes so that a power law is a straight line and its exponent is the slope. Flux falls with slope −2 and angular size with slope −1, both of which are ordinary. Their ratio has slope zero: a galaxy of surface brightness 23.5 magnitudes per square arcsecond has that surface brightness at every distance, and a sky of 22 is brighter than it at every distance too. The contrast against the sky — the quantity that decides whether the thing is detectable at all — is -1.5 magnitudes wherever it is put.
Fig. 1 Three quantities against distance, each relative to its value at 5 megaparsecs, on logarithmic axes so that a power law is a straight line and its exponent is the slope. Flux falls with slope −2 and angular size with slope −1. Their ratio — the flux per unit solid angle, which is what surface brightness means — has slope zero. A galaxy of 23.5 magnitudes per square arcsecond has that surface brightness at every distance, and a sky brighter than it is brighter than it everywhere.

The argument, which is two lines

An object of physical size DD at distance dd subtends an angle θ=D/d\theta = D/d, so the solid angle it covers is Ω∝θ2∝1/d2\Omega \propto \theta^2 \propto 1/d^2.

Its flux is F=L/4πd2∝1/d2F = L/4\pi d^2 \propto 1/d^2.

The surface brightness is the flux per unit solid angle:

S=FΩ∝1/d21/d2=constant.S = \frac{F}{\Omega} \propto \frac{1/d^2}{1/d^2} = \text{constant}.

Both quantities carry the same factor and it cancels exactly. Nothing about the object matters; nothing about the medium in between matters, provided it is transparent. Move a galaxy twice as far away and it becomes four times fainter and covers a quarter of the area, and every square arcsecond of it delivers precisely as many photons as before.

The same galaxy at 10, 30, 90 Mpc. A disc 30 kpc across, drawn at 10, 30, 90 megaparsecs. Its angular diameter falls as 1/d — 619″, 206″, 69″ — and its total flux as 1/d², so the far one delivers 1.2 per cent of the photons of the near one. Every disc is filled at exactly the same tone, and that is not an artistic choice: the flux per unit solid angle is the ratio of two quantities that both fall as 1/d², so it is the same number at every distance. A galaxy too faint to see against the sky is too faint at any distance.
Fig. 2 The same disc, thirty kiloparsecs across, drawn at ten, thirty and ninety megaparsecs. Its angular diameter falls as 1/d and its total flux as 1/d², so the far one delivers about one per cent of the photons of the near one. Every disc is filled at exactly the same tone, and that is not an artistic decision — it is the content of the previous paragraph. What changes with distance is how much sky the galaxy covers, not how bright that sky is.

Why that makes a galaxy undetectable

A telescope does not observe an object against blackness. It observes it against the night sky, which glows — from airglow in the upper atmosphere, from scattered starlight and sunlight, from zodiacal dust, and from unresolved faint galaxies. A dark site gives a sky of about 22 magnitudes per square arcsecond in the V band, which is a real and substantial brightness — in the units this collection measures brightness in, roughly what a twenty-second-magnitude star would deliver if it were smeared over one square second of arc.

Detecting an extended object means detecting a contrast against that background. And since the object’s surface brightness does not change with distance, neither does the contrast.

The consequence is stark: a galaxy fainter than the sky is undetectable at any distance whatever. Moving it closer does not help. Building a bigger telescope does not help either, in the way one might expect — a larger aperture collects more photons from the object and proportionally more from the sky, so the signal-to-noise improves only as the square root of the exposure and the aperture, and never by making a sub-sky object supra-sky.

What does help is a longer exposure, a darker sky, a detector with lower noise, and above all a careful treatment of flat-fielding and scattered light, because at this level the enemy is systematic rather than statistical.

What distance does, and does not, do to a galaxy. Three quantities against distance, each relative to its value at 5 Mpc, on logarithmic axes so that a power law is a straight line and its exponent is the slope. Flux falls with slope −2 and angular size with slope −1, both of which are ordinary. Their ratio has slope zero: a galaxy of surface brightness 26 magnitudes per square arcsecond has that surface brightness at every distance, and a sky of 22 is brighter than it at every distance too. The contrast against the sky — the quantity that decides whether the thing is detectable at all — is -4.0 magnitudes wherever it is put.
Fig. 3 Malin 1’s central surface brightness against the same sky. Four magnitudes below it is a factor of forty in flux per square arcsecond, so the object contributes about two and a half per cent of what the sky delivers to the same pixel — and the flat line says it will contribute exactly that at any distance whatever. That is the sentence “undetectable at any distance” drawn: the horizontal line is the object, the horizontal line above it is the sky, and no amount of moving the object along the axis brings the two closer together.

The arithmetic of an exposure

The claim that a longer exposure helps and a closer galaxy does not is worth turning into numbers, because the numbers explain why this field advanced when it did.

Suppose the sky delivers NN photons per square arcsecond per exposure and the galaxy adds a fraction ff of that. The signal is fNfN and the noise, if the sky is Poisson-limited, is N\sqrt{N}, so the signal-to-noise ratio per square arcsecond is fNf\sqrt N. Doubling the exposure doubles NN and improves the ratio by 2\sqrt 2; the same is true of doubling the collecting area.

For an object 4.35 magnitudes below the sky, f=0.018f = 0.018. Reaching a signal-to-noise of five per square arcsecond therefore needs N≈8×104N \approx 8\times10^4 photons per square arcsecond — which is a long exposure, and entirely achievable. Averaging over an area of a hundred square arcseconds relaxes it by a further factor of ten in the required NN.

So the limit is not photon statistics at all. It is whether the sky level can be known to a part in a hundred over the region of interest, and that is a question about flat-fielding, scattered light and the wings of bright stars. This is why the field’s advances have come from careful optics and careful reduction rather than from larger telescopes, and why a modest instrument built specifically for the purpose can outperform a large general-purpose one.

The population that was missed

Freeman’s 1970 result — that spiral discs have a central surface brightness clustered tightly around 21.65 blue magnitudes per square arcsecond — was taken for a decade as a law about galaxies.

Mike Disney pointed out in 1976 that it might instead be a law about telescopes. A galaxy is easiest to detect when its surface brightness is comfortably above the sky; a galaxy much fainter than the sky is invisible; and a galaxy much brighter than the sky per unit area is rare for other reasons. A catalogue assembled by eye from photographic plates would therefore contain a narrow range of surface brightness whatever the underlying population did.

He was substantially right, and the general form of his objection — that a survey’s boundary is a fact about the survey — is one this collection has now met in two fields. Deliberate searches for low-surface-brightness galaxies — using long exposures, careful sky subtraction, and later automated detection — found a large population that the earlier catalogues had passed over. Some of them are enormous: Malin 1, found in 1986, has a disc scale length of some fifty kiloparsecs and a central surface brightness around 26 magnitudes per square arcsecond, which is more than fifty times fainter than the sky.

What distance does, and does not, do to a galaxy. Three quantities against distance, each relative to its value at 5 Mpc, on logarithmic axes so that a power law is a straight line and its exponent is the slope. Flux falls with slope −2 and angular size with slope −1, both of which are ordinary. Their ratio has slope zero: a galaxy of surface brightness 21.65 magnitudes per square arcsecond has that surface brightness at every distance, and a sky of 22 is brighter than it at every distance too. The contrast against the sky — the quantity that decides whether the thing is detectable at all — is 0.4 magnitudes wherever it is put.
Fig. 4 Freeman’s value, drawn against the same sky. At 21.65 the disc sits a third of a magnitude above the background — comfortably detectable on a photographic plate, and only just. That margin is Disney’s whole argument: a population whose central surface brightness clustered at exactly the value where detection becomes easy is a population selected for detectability, and the tightness of the clustering is evidence for the selection rather than against it. A law about galaxies and a law about plates predict the same histogram, and only one of them predicts that the histogram moves when the plates improve.

The observation behind the number

Measuring a surface brightness well is a different problem from measuring a flux well, and the difference is instructive.

A flux is a sum: count the photons from the object, subtract the sky, done. An error in the sky level adds a constant to a small number of pixels.

A surface brightness profile is a sum over annuli of increasing area, so an error in the assumed sky level adds a constant to a quantity that is itself falling exponentially — and beyond a few scale lengths the constant dominates. A sky subtraction wrong by one part in a thousand produces an entirely spurious outer profile, and outer profiles are exactly what the interesting questions are about.

Hence the peculiar discipline of low-surface-brightness work: dithering the telescope so that the sky can be constructed from the data themselves, masking every star aggressively, modelling the scattered-light halo of the optics, and treating any result within a magnitude of the sky as provisional. The claims that have been retracted in this field have almost all been sky-subtraction failures.

What it means for a survey

Every catalogue is defined by its selection function, and for extended objects that function has two variables rather than one.

An object enters a catalogue if it is bright enough in total to be measured and bright enough per unit area to be seen at all. The first limit is a flux limit and moves with distance; the second is a surface-brightness limit and does not. The two-dimensional nature of the selection is why the number density of galaxies is a harder quantity than it looks, and why the faint end of the luminosity function is the part that keeps moving. A survey that reaches faint total magnitudes but only bright surface brightnesses will find a different faint-end slope from one with the opposite balance, and the two can be reconciled only by modelling the selection in both variables.

What distance does, and does not, do to a galaxy. Three quantities against distance, each relative to its value at 5 Mpc, on logarithmic axes so that a power law is a straight line and its exponent is the slope. Flux falls with slope −2 and angular size with slope −1, both of which are ordinary. Their ratio has slope zero: a galaxy of surface brightness 23.5 magnitudes per square arcsecond has that surface brightness at every distance, and a sky of 24 is brighter than it at every distance too. The contrast against the sky — the quantity that decides whether the thing is detectable at all — is 0.5 magnitudes wherever it is put.
Fig. 5 The same galaxy against a sky two magnitudes darker — space, or an exceptional site with the airglow at a minimum. Now the object is half a magnitude above the background rather than a magnitude and a half below it, and everything changes: the contrast is positive, the detection is straightforward, and the galaxy enters the catalogue at every distance rather than at none. Darkening the sky is the only intervention that moves a horizontal line relative to another horizontal line, which is why it is the axis every advance in this field has come along.

What the missed population turned out to be

The galaxies that the surface-brightness limit was hiding are not a footnote to the census, and the two things they turned out to be are worth separating.

Large, diffuse discs. Malin 1 and its kind are enormous, gas-rich, slowly rotating discs whose stars are spread so thinly that the whole object sits below the sky. They are rare in number and they are not rare in mass — a galaxy with a fifty-kiloparsec scale length contains a great deal of hydrogen — and they extended the known range of disc properties by more than an order of magnitude in surface brightness.

Dwarfs, in enormous numbers. The faint end of the galaxy population is overwhelmingly low in surface brightness, so every improvement in this direction has added dwarfs. The Milky Way’s own satellite count has roughly quadrupled since 2005 for exactly this reason, and each new one is fainter and more diffuse than the last: the faintest known have absolute magnitudes around −1, meaning they emit less light than a single bright star while containing thousands of them spread over hundreds of parsecs.

Neither population changes the total stellar mass of the universe much — the light really is dominated by the bright galaxies, as the luminosity function requires. What they change is the count, and the count is what a theory of structure formation predicts most directly.

The same galaxy at 20, 40, 80 Mpc. A disc 60 kpc across, drawn at 20, 40, 80 megaparsecs. Its angular diameter falls as 1/d — 619″, 309″, 155″ — and its total flux as 1/d², so the far one delivers 6.3 per cent of the photons of the near one. Every disc is filled at exactly the same tone, and that is not an artistic choice: the flux per unit solid angle is the ratio of two quantities that both fall as 1/d², so it is the same number at every distance. A galaxy too faint to see against the sky is too faint at any distance.
Fig. 6 A disc twice the size of the previous one, drawn over a factor of four in distance. The angular sizes scale as they must, the tone does not change, and the sixty-kiloparsec object at eighty megaparsecs still subtends two and a half arcminutes — larger than most catalogued galaxies and fainter per square arcsecond than the sky. That combination is what a giant low-surface-brightness disc is, and it is the reason such objects were found by people looking for them rather than by anyone scanning plates: they are not small and faint, they are large and faint, which is a harder thing for an eye to miss and a harder thing for a threshold to catch.
The same galaxy at 5, 50, 500 Mpc. A disc 30 kpc across, drawn at 5, 50, 500 megaparsecs. Its angular diameter falls as 1/d — 1238″, 124″, 12″ — and its total flux as 1/d², so the far one delivers 0.0 per cent of the photons of the near one. Every disc is filled at exactly the same tone, and that is not an artistic choice: the flux per unit solid angle is the ratio of two quantities that both fall as 1/d², so it is the same number at every distance. A galaxy too faint to see against the sky is too faint at any distance.
Fig. 7 The scaling pushed to two decades. From five to five hundred megaparsecs the disc’s angular diameter falls by a factor of a hundred and its flux by ten thousand, and the tone is unchanged throughout. A survey that can see the first can see the third — provided it can resolve it at all, which is the separate constraint that eventually takes over: at 500 Mpc this disc is twelve arcseconds across, and below a few arcseconds a galaxy is compressed into the seeing disc and its surface brightness is no longer measured but averaged with the sky around it.

The one thing distance does change

There is an important exception, and stating it correctly is the difference between a rule and a superstition.

Surface brightness is distance-independent in a static, Euclidean, transparent universe. In an expanding one it is not: the photons arrive redshifted and time-dilated, and the solid angle relates to the physical size through the angular-diameter distance while the flux relates to it through the luminosity distance. The two differ, and the result is that surface brightness falls as (1+z)−4(1+z)^{-4}.

That is a steep dependence — at z=1z=1 it is a factor of sixteen — and it is the reason imaging high-redshift galaxies is so much harder than their total magnitudes suggest. It is also a testable prediction of expansion itself, and the test is worth naming: Richard Tolman proposed in 1930 that comparing the surface brightnesses of similar galaxies at different redshifts would distinguish an expanding universe from a static one with some other cause of redshift. The measurement is difficult, mostly because galaxies at different redshifts are not similar, and it comes out on the side of expansion.

What the picture cannot show

The figures here are Euclidean. The scaling drawn is the local one, valid where redshift is negligible, and that covers the range in which most of the galaxy census is done. The (1+z)−4(1+z)^{-4} dimming above is the correction that takes over beyond it, and it is not drawn.

Surface brightness is not one number for a galaxy. It varies with radius, which is the whole subject of a profile; the “central surface brightness” quoted in Freeman’s law is an extrapolation of the exponential fit inwards, not a measurement at the centre, where a bulge usually intervenes.

And it is defined per unit solid angle on the sky, not per unit area of the object. Those differ by the inclination: an inclined disc has more stars along each line of sight and is brighter per square arcsecond than the same disc seen face-on. Catalogues correct for it, and the correction assumes something about how transparent the disc is, which is not well known.

The same galaxy at 10, 30, 90 Mpc. A disc 5 kpc across, drawn at 10, 30, 90 megaparsecs. Its angular diameter falls as 1/d — 103″, 34″, 11″ — and its total flux as 1/d², so the far one delivers 1.2 per cent of the photons of the near one. Every disc is filled at exactly the same tone, and that is not an artistic choice: the flux per unit solid angle is the ratio of two quantities that both fall as 1/d², so it is the same number at every distance. A galaxy too faint to see against the sky is too faint at any distance.
Fig. 8 And the constraint the cancellation does not remove. A five-kiloparsec dwarf at ninety megaparsecs is eleven arcseconds across, which is a handful of resolution elements at ordinary seeing — so the surface brightness that is constant in principle is measured, in practice, after the object has been convolved with the atmosphere and blended with the sky around it. The flat line in every other figure here assumes the object is resolved. Once it is not, the galaxy’s light is spread into the background rather than measured against it, and the argument’s one exemption to distance quietly stops applying.

The rule has a contrapositive that is used daily and is worth stating.

If a measurement’s value changes when the object is moved further away, that measurement is not a property of the object. Apparent magnitude, angular size and total flux are all in that category, and none of them belongs in a physical statement without a distance beside it. Surface brightness, colour, and any ratio of two fluxes are in the other, and can be compared between a galaxy next door and one halfway across the observable universe with no distance ladder involved at all. Sorting a catalogue’s columns into those two lists before doing anything else with them saves a great deal of trouble.

The rule has a use as well as a consequence, and the use is a measurement nobody would have looked for.

The noise that is a distance

There is a measurement built on the exception to everything above, and it is worth describing because it turns the constancy of surface brightness into a distance indicator rather than an obstacle.

A galaxy’s image is not smooth. Each resolution element contains a finite number of stars, and that number fluctuates from element to element by Poisson statistics — so the image has a pixel-to-pixel variance over and above the photon noise.

Now scale the distance. The mean surface brightness is unchanged, by the whole argument of this essay. But each resolution element now covers a larger physical area and therefore contains more stars: the number goes as the square of the distance. Poisson fluctuations go as the square root of the number, so the fractional variation goes as one over the distance.

The image therefore gets smoother with distance, in a calculable way, and measuring how smooth it is measures how far away it is.

What is actually computed is the ratio of the variance to the mean, which has the units of a flux and is the flux of a typical star in the population — a quantity that can be compared with the same quantity in a galaxy of known distance. The method is called surface-brightness fluctuations, and it is one of the better rungs of the distance ladder for nearby ellipticals, reaching a few per cent.

Two corrections dominate its error budget and both are about the stellar population rather than the geometry. The fluctuation signal is dominated by the brightest stars in the population, which are the red giants, so the calibration depends on the population’s age and metallicity — and those are estimated from a colour, which is why the method is quoted with a colour-dependent zero point.

The second is contamination. Globular clusters, background galaxies and foreground stars all add variance that is not the signal, and each has to be identified and masked. That is why the method works best on smooth, dust-free ellipticals and not at all on spirals, whose real structure swamps the statistical fluctuation entirely.

The quantity that makes an extended object hard to detect is the same quantity whose graininess measures its distance, and both follow from one cancellation.

The generalisation

The pattern is one worth carrying: when two observables share a distance dependence, their ratio is a distance-independent property of the object, and any selection based on it is a selection on the object itself.

That is why surface brightness is a physical property of a galaxy in a way that apparent magnitude never is, and it is why a survey limited by it is biased against a kind of galaxy rather than against distant ones. The same logic explains why colours are useful — they are ratios of fluxes, so the distance cancels — and why a colour index measures a temperature at any distance at all.

The converse is the trap. A selection on a distance-dependent quantity produces a sample whose properties change with distance, which is Malmquist bias; a selection on a distance-independent one produces a sample that is missing the same objects everywhere. The first distorts a trend, the second removes a population, and the second is much harder to notice.

The two uses of the same cancellation are worth holding together, because they are the same fact read with opposite signs: constancy of the mean makes a faint galaxy invisible at every distance, and constancy of the mean is what makes the variance a clean function of distance alone.

Where the ladder goes next

The next rung is the population that the surface-brightness limit was hiding: what the low-surface-brightness galaxies turned out to be, how much of the total stellar mass they carry, and why the ultra-diffuse ones found in cluster surveys since 2015 are so awkward to account for.

Later rungs on this anchor: the sky background and where it comes from; the Tolman test and what it measures; surface-brightness fluctuations as a distance indicator, which turns the pixel-to-pixel noise of an unresolved galaxy into a measurement; isophotal radii and why they depend on the observer; the intracluster light, which is diffuse enough to be at the edge of detectability and carries a substantial fraction of a cluster’s stars; and the outer profiles of discs, where the sky subtraction is the entire experiment.

Where this goes next

The distance modulus. The difference between apparent and absolute magnitude against distance, on a logarithmic distance axis. It is a straight line of slope five per decade, passing through zero at ten parsecs — the definition of the absolute magnitude. Reading a distance off it requires the absolute magnitude, which is never measured and always inferred. A brightness is a distance only if something is known The inverse square law turns a brightness into a distance in one line. The line contains a quantity that has never been measured for any object outside the solar system. What each method can see. Planet mass against orbital distance, both logarithmic, with the detection threshold of each method drawn as the boundary it actually is. Radial velocity at 1 m/s needs mass rising as √a; astrometry at 20 µas needs it falling as 1/a, which is the only method that gets easier further out; a 100 ppm transit is a threshold on radius and so a horizontal line at about 1.4 Earth masses, cut off at 1.21 AU by the need for three transits in 4 years; direct imaging begins outside the diffraction limit, 0.6 AU at 10 parsecs for a 39 m aperture at 10 µm. The solar system is drawn on top: for two decades every one of its planets except Jupiter lay outside every region, which is the whole of what the early census was measuring. Every survey draws a different sky The first exoplanets found were enormous and impossibly close to their stars. That was not a discovery about planets. It was a measurement of what a 10 m/s spectrograph watching for three years is able to see. Tully–Fisher, from two assumptions and no fitting. 44 model galaxies spanning two and a half decades in luminosity, each built from a constant disc surface brightness of 380 L☉ per square parsec with 0.13 dex of scatter and a stellar mass-to-light ratio of 1.4 with 0.1 dex. Nothing about a luminosity–speed relation is put in: the scale length follows from the surface brightness, the mass from the light, and the speed from v² = GM/R. The construction makes L ∝ v⁴ exactly, so the true slope is −10.0 magnitudes per decade of speed. Least squares of magnitude on log speed returns -8.45, and the reverse regression -9.15: the scatter here is scatter in the speed at fixed luminosity, and error in the abscissa flattens a fitted slope, so neither fit returns the value the family was built with and the distance between them is the size of the effect. Residual scatter about the drawn line is 0.54 magnitudes. Observed slopes run from about −7.5 in blue light to −10 in the near infrared, where the mass-to-light ratio is steadiest — which is the same statement as the second assumption above. A line width that is a distance The width of a galaxy's hydrogen line depends on how fast it rotates, which depends on its mass, which is tied to its luminosity — so a quantity no distance enters gives an absolute brightness, and the distance follows from the brightness that is seen.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

The 8 of 24 essays linking to this one that name the most of the same objects.

The objects this essay names

Each one links to every other essay that touches it.

Detection limitFreeman's lawInverse square lawIsophoteLow-surface brightness galaxyMalmquist biasSelection effectSky backgroundSolid angleSurface brightness