Astronomy & Celestial Mechanics
the sky, computed
Galaxies

Where the sky is the measurement

The outskirts of a galaxy at 29 or 30 magnitudes per square arcsecond are a thousand times fainter than the night sky they are seen through. Photon noise averages down with exposure; an error in the sky subtracted from under them does not. A sky wrong by a tenth of a per cent draws a break into a disc that has none, or a halo around a galaxy that has none, and the telescope's own scattered light draws another.

Assumes Surface brightness, Photon noise and Seeing.

14 min read 6 figures Measuring the unreachableLight is the only messenger

A galaxy’s surface brightness does not change with its distance, and that single fact decides which galaxies can be seen: whatever is fainter than the sky per square arcsecond is hard to see at any distance at all. The inner parts of normal disc galaxies are about as bright as a dark night sky, a little under 22 magnitudes per square arcsecond. Their outer parts are not. By six or eight scale lengths from the centre a disc has fallen to 28 or 29, and the stellar halo around it, the debris of the small galaxies it has absorbed, sits at 30 and below. At 30 magnitudes per square arcsecond a galaxy is about two and a half thousand times fainter than the sky it is seen through.

Reaching that faint is not a problem of collecting enough light. It is a problem of subtracting the sky, and the two problems behave completely differently.

A sky wrong by a thousandth, and the disc that ends or does not. The surface-brightness profile of an exponential disc with a central surface brightness of 21.65 magnitudes per square arcsecond, against radius in units of its scale length, drawn straight — a pure exponential is a straight line in magnitudes — and then as it would be measured after subtracting a sky of 21.5 magnitudes per square arcsecond wrong by +0.1%, −0.1%, +0.3%, −0.3%. A sky left in bends the profile up into a shallower outer part, the shape called an anti-truncation; a sky taken out too hard bends it down into a break and then drops it out of existence where the residual goes negative. The +0.1% error departs by 0.2 magnitudes at 5.2 scale lengths, 27.3 magnitudes per square arcsecond; The −0.1% error departs by 0.2 magnitudes at 5.0 scale lengths, 27.1 magnitudes per square arcsecond; The +0.3% error departs by 0.2 magnitudes at 4.1 scale lengths, 26.1 magnitudes per square arcsecond; The −0.3% error departs by 0.2 magnitudes at 3.9 scale lengths, 25.9 magnitudes per square arcsecond. Real discs are observed to break both ways, at 24 to 27 magnitudes; every one of those breaks lies in the range where a sky error of a few tenths of a per cent would make one.
Fig. 1 A pure exponential disc, which is a straight line in magnitudes, and the same disc after subtracting a sky wrong by one or three parts in a thousand in either direction. Too little sky removed bends the outer profile up into a shallower tail; too much bends it down into a break and then drops it out of existence where the residual goes negative.

What the sky is made of

It is worth being clear about what is being subtracted, because its nature decides how well it can be known.

A dark night sky at a good site has a surface brightness of about 21 to 22 magnitudes per square arcsecond in visible light, and most of that is not astronomical. It is airglow: light emitted by atoms and molecules in the upper atmosphere, a hundred kilometres up, as they recombine after being ionised or excited by sunlight during the day. Airglow comes in bright emission lines — the green and red lines of oxygen, a forest of lines from the hydroxyl radical in the near-infrared — and its brightness changes by tens of per cent over an hour and varies across the sky in waves driven by the atmosphere’s own oscillations. The rest is zodiacal light, sunlight scattered by interplanetary dust, which is smooth and predictable but brightest near the ecliptic; the integrated glow of faint Milky Way stars and the light reflected by its dust; and, faintest of all, the summed light of every unresolved galaxy in the universe.

So the sky is a changing, structured foreground that does not end when astronomical twilight does, and its variations are in exactly the direction that matters: across the field, on the scale of a large galaxy, over the time of an exposure. A sky known to five parts in ten thousand over a whole galaxy is a sky whose spatial and temporal structure has been modelled at that level, not merely one that has been sampled.

An error that does not average down

Every measurement of faint light against a bright background has two kinds of error. The first is the random fluctuation in the number of photons, from the galaxy and from the sky. That error falls with exposure time: collect four times as many photons and the fluctuation relative to the signal halves. A long enough exposure can make it as small as desired.

The second is the error in the level of the sky itself. The sky is not measured at the position of the galaxy — the galaxy is in the way — but around it, and interpolated underneath. Any mismatch between the interpolated sky and the true sky under the galaxy is a constant offset added to every pixel, and it does not fall with exposure at all. A longer exposure measures the wrong sky more precisely.

That is the whole difficulty in one sentence. For bright parts of a galaxy the sky error is irrelevant, because the galaxy is so much brighter than any plausible offset. For the faint outskirts the galaxy and the offset become comparable, and then the profile reported is the profile plus the error. The opening figure shows how it looks: the true disc is a straight line in magnitudes, and a sky mis-subtracted by a tenth of a per cent departs from it by a fifth of a magnitude at about five scale lengths, where the disc is only a few times brighter than the sky error — and diverges ever faster from there.

A depth set by how well the sky is known

The limiting surface brightness of an image is the level at which the galaxy equals the error in the sky, and it is a simple function of the sky’s brightness and the fraction to which that brightness is known.

The depth a sky allows, set by how well it is known. The faintest surface brightness at which a galaxy equals the error in the sky subtracted from under it, against the fractional accuracy of that subtraction, for three skies: bright moonlit sky at 19.5 magnitudes per square arcsecond, dark ground-based sky at 21.8 magnitudes per square arcsecond, zodiacal light from space at 23.3 magnitudes per square arcsecond. Every factor of ten in the sky's accuracy is 2.5 magnitudes of depth, and no amount of exposure time changes the line: photon noise averages down, and a systematic error in the sky does not. To reach 30 magnitudes per square arcsecond under a dark ground-based sky the sky has to be known to about 0.05% of itself, over the whole area of the galaxy and its surroundings — which is the reason the deepest images of galaxy outskirts are made by instruments designed around the sky rather than around the galaxy.
Fig. 2 The faintest surface brightness at which a galaxy equals the sky error, against the fractional accuracy of the subtraction, for a bright moonlit sky, a dark ground-based sky and the zodiacal light seen from space. Each factor of ten in accuracy buys 2.5 magnitudes of depth, and exposure time does not appear anywhere on the plot.

To reach 30 magnitudes per square arcsecond under a dark sky the sky has to be known to about five parts in ten thousand of itself — not at one point, but everywhere under a galaxy that may cover a large fraction of the detector. The sky is not flat at that level. The airglow in the upper atmosphere varies across the field and over minutes; scattered moonlight and starlight put gradients across the image; the detector’s own response has to be flattened to the same precision, since a flat-field error of a few parts in ten thousand is indistinguishable from a sky error of the same size.

That is why the deepest images of galaxy outskirts have come from instruments designed around the sky rather than around the galaxy: small, fast optics with very clean light paths, observing strategies that move the telescope by large amounts between exposures so that the sky can be measured on the same pixels the galaxy occupies at other times, and flat fields built from the night sky itself. Space helps in one way and not in another. The zodiacal light seen from above the atmosphere is fainter and far more stable than the airglow, which moves the line in the figure by one and a half magnitudes; but a space telescope’s own detectors and optics still have to be flattened to the same fraction, and its field of view is usually too small to contain the sky around a large galaxy at all.

There is one way to beat the sky with exposure after all, and its limits are instructive. If the sky error is different for every galaxy — random in sign and size from one image to the next — then averaging the profiles of many similar galaxies averages the sky errors down, while the galaxies’ true outskirts, which are the same sign every time, add up. Stacking a thousand galaxies from a shallow survey gains a factor of about thirty in the sky error, nearly four magnitudes of depth, and it was by stacking that the first statistical detections of faint halos around typical galaxies were made. What stacking cannot remove is any error that is the same for every galaxy: a systematic bias in how the sky was estimated near bright objects, or the scattered light of the telescope, which accompanies every galaxy in exactly the same proportion. The stacked halos were later found to contain a substantial scattered-light component for exactly that reason.

A disc that ends, or does not

The shapes the sky error draws are not arbitrary, and the ones it draws are exactly the shapes that disc galaxies are reported to have.

Disc galaxies are observed to have three kinds of outer profile. Some continue as a single exponential to the limit of the data. Some break, at a few scale lengths, into a steeper exponential beyond — a truncation, or down-bending break. And some break into a shallower exponential beyond — an anti-truncation, or up-bending break. In large samples of nearby discs the steeper breaks are the most common and the single exponentials the least, and the breaks are found at surface brightnesses between about 24 and 27 magnitudes per square arcsecond.

Every one of those breaks lies in the range where a sky error of a few tenths of a per cent would produce one. A sky taken out too hard draws a down-bending break; a sky left in draws an up-bending one. This does not mean the breaks are artefacts. Many are measured in data whose sky is known well enough to exclude it, the down-bending breaks coincide with changes in the colour of the stars and with the edge of recent star formation, and several are seen in edge-on discs where the sky problem is different. What it means is that every reported break has to be accompanied by an argument about the sky, and a catalogue of breaks is only as reliable as the least careful sky subtraction in it.

A sky wrong by a thousandth, and the disc that ends or does not. The surface-brightness profile of an exponential disc with a central surface brightness of 23.5 magnitudes per square arcsecond, against radius in units of its scale length, drawn straight — a pure exponential is a straight line in magnitudes — and then as it would be measured after subtracting a sky of 21.5 magnitudes per square arcsecond wrong by +0.1%, −0.1%. A sky left in bends the profile up into a shallower outer part, the shape called an anti-truncation; a sky taken out too hard bends it down into a break and then drops it out of existence where the residual goes negative. The +0.1% error departs by 0.2 magnitudes at 3.5 scale lengths, 27.3 magnitudes per square arcsecond; The −0.1% error departs by 0.2 magnitudes at 3.3 scale lengths, 27.1 magnitudes per square arcsecond. Real discs are observed to break both ways, at 24 to 27 magnitudes; every one of those breaks lies in the range where a sky error of a few tenths of a per cent would make one.
Fig. 3 The same errors applied to a low-surface-brightness disc whose centre is at 23.5 magnitudes per square arcsecond. The profile departs from a straight line at only three scale lengths rather than five, because the whole galaxy is closer to the sky; for such a galaxy the sky error decides not only whether its outskirts break but how large the galaxy appears to be at all.

For galaxies whose centres are already faint, the problem moves inward and becomes a problem about the whole galaxy. A low-surface-brightness disc at 23.5 magnitudes per square arcsecond is affected by a tenth-of-a-per-cent sky error within three scale lengths, over most of its light. Its total magnitude, its scale length and its size are all measured through the same offset, and the population of such galaxies — which a catalogue selected by surface brightness underrepresents for a separate reason — is measured with errors that are systematic, correlated and in a known direction.

A halo, or a sky left behind

The faintest structure around a galaxy is its stellar halo: a diffuse envelope of old stars, many of them the remains of satellite galaxies torn apart by the galaxy’s tides and spread along their orbits. The fraction of a galaxy’s stars in its halo, and the halo’s profile, are a record of how many satellites it has absorbed and when, which is one of the few direct tests of how galaxies grow by merging. Measuring it means measuring a component at 29 to 32 magnitudes per square arcsecond, beyond the edge of the disc.

A stellar halo, or a sky left behind. Two ways for an exponential disc (21.65 magnitudes per square arcsecond at the centre) to acquire a faint outer component. Orange: a real stellar halo, with surface density falling as radius to the power −2, at 29 magnitudes per square arcsecond at ten scale lengths. Blue: no halo at all, but a sky subtraction that left 0.08% of a 21.8-magnitude sky behind. The halo takes over from the disc at 6.0 scale lengths, and the two agree to within a quarter of a magnitude from there out to 15.0 — the whole range over which an outer component would be claimed. The shape of a halo's decline is what distinguishes it from a constant, and it is measured exactly where the uncertainty in the constant is largest, so the stellar halo fraction of a galaxy is only as good as its sky.
Fig. 4 A disc with a real stellar halo, falling as the inverse square of radius, and the same disc with no halo but with eight parts in ten thousand of the sky left in. The halo takes over from the disc at six scale lengths, and from there out to fifteen the two profiles agree to within a quarter of a magnitude.

The two profiles in the figure are not quite the same shape. A halo falls with radius and a sky residual does not, so far enough out they must separate. But the separation happens where both are faintest and the data are noisiest, and over the range where the outer component is first detected — the range that decides whether a halo is claimed at all — a constant offset of the right size mimics it closely. A halo profile with a steeper slope is harder to imitate; a shallow one, easier. The halo fraction of a galaxy measured from integrated light is therefore constrained, in practice, by the sky model rather than by the photons.

The same limit governs the other faint structures that record a galaxy’s history. Tidal tails and bridges drawn out of interacting galaxies, the shells and plumes left around ellipticals by past mergers, and the streams of stars stripped from satellites all sit at 26 to 30 magnitudes per square arcsecond, and a census of how common they are around galaxies of a given kind is a census taken down to whatever limit the sky allowed. Two surveys that reach different depths find different fractions of disturbed galaxies, and the difference is a property of their skies rather than of the galaxies.

The alternative is not to measure integrated light at all. In the nearest galaxies the halo’s stars can be counted individually, star by star, with a space telescope or a very large ground-based one, and a count of resolved stars has no sky-subtraction problem: a star is either there or it is not. Resolved-star counts in the halos of a few nearby spirals have found halo fractions varying by more than a factor of ten from one galaxy to the next, which is itself a statement about how differently galaxies of the same mass have grown. Beyond a few megaparsecs the stars cannot be resolved and integrated light is the only option.

The galaxy’s own centre, scattered outward

There is a second systematic that behaves like the sky and is harder to remove, because it comes from the galaxy being measured.

A telescope does not image a point as a point. Most of the light from a star falls within an arcsecond or so, blurred by the atmosphere, but a small fraction is scattered much further — by dust on the optics, by micro-roughness of the mirror surfaces, by reflections inside the camera — into wings that extend to tens of arcminutes and fall roughly as the inverse square of the distance. A few per cent of all the light ends up in the wings. For a point source that is invisible; for a bright extended galaxy it means that a few per cent of the galaxy’s total light has been redistributed into a faint envelope around it.

A galaxy's own light, scattered into its outskirts. An exponential disc of central surface brightness 21 magnitudes per square arcsecond and scale length 15 arcseconds, and the light the telescope itself scatters out of it into the surroundings through the far wings of its point-spread function, modelled as a wing falling as r⁻² that holds 5% or 1% of all the light. The disc falls as a straight line; the scattered light falls only logarithmically in magnitudes, and it overtakes the disc at 133 arcseconds (8.9 scale lengths) and 164 arcseconds (10.9 scale lengths). Beyond that the measured outskirts are the galaxy's centre, redistributed. A few per cent of the light in the far wings is typical of real telescopes, and the wing is measured out to half a degree only by imaging very bright stars — so the faint halos and thick discs reported around edge-on galaxies at these levels have repeatedly been found, on re-analysis, to be largely scattered light.
Fig. 5 A disc and the light scattered out of it by a point-spread function whose far wing falls as the inverse square of radius and holds five or one per cent of the light. The disc falls as a straight line in magnitudes and the scattered light only logarithmically, so the scattered light overtakes the disc at nine and at eleven scale lengths.

Beyond the crossing, the “outskirts” measured are the galaxy’s bright centre, displaced. The scattered envelope has a profile that falls more slowly than any disc, and around an edge-on galaxy it produces exactly the thick, faint envelope above and below the disc that has been interpreted as a thick disc or a flattened halo. Several such detections have been reinterpreted after the far wings of the point-spread function were measured properly — which requires imaging very bright stars with the same telescope and camera, out to half a degree, to find the few per cent of light that is hiding there.

A galaxy's own light, scattered into its outskirts. An exponential disc of central surface brightness 21 magnitudes per square arcsecond and scale length 45 arcseconds, and the light the telescope itself scatters out of it into the surroundings through the far wings of its point-spread function, modelled as a wing falling as r⁻² that holds 5% or 1% of all the light. The disc falls as a straight line; the scattered light falls only logarithmically in magnitudes, and it overtakes the disc at 400 arcseconds (8.9 scale lengths) and 491 arcseconds (10.9 scale lengths). Beyond that the measured outskirts are the galaxy's centre, redistributed. A few per cent of the light in the far wings is typical of real telescopes, and the wing is measured out to half a degree only by imaging very bright stars — so the faint halos and thick discs reported around edge-on galaxies at these levels have repeatedly been found, on re-analysis, to be largely scattered light.
Fig. 6 The same wings around a galaxy three times larger on the sky. The crossings fall at the same 8.9 and 10.9 scale lengths: the scattered light is proportional to the total light, which grows as the square of the scale length, and it falls as the inverse square of radius, so in units of scale length the two cancel. A large nearby galaxy is not protected by its size; how far out its own light contaminates it is set only by its central surface brightness and by the telescope.

The correction is a deconvolution: model the galaxy, convolve the model with the measured point-spread function out to its full extent, and compare with the data. It works, but it needs the point-spread function to be known at levels of a millionth of the peak intensity, over a field wider than the galaxy, and for the same colours as the galaxy — since the wings depend on wavelength — and that measurement is harder than the one it is correcting.

What the figures idealise

Each figure applies one systematic to a perfect exponential disc and leaves everything else out. Real data suffer all of them at once: a sky with a gradient rather than a constant offset, a flat field that is wrong by a different amount in different parts of the detector, a scattered-light envelope added on top, and Galactic cirrus — faint, filamentary dust clouds in the Milky Way that scatter the light of the Galaxy’s stars towards the observer and appear, at 26 to 28 magnitudes per square arcsecond, as structure in the background of every deep image at high Galactic latitude. Cirrus is especially treacherous because it is not flat: it has filaments and arcs on the same angular scales as a galaxy’s outskirts, and a wisp of cirrus lying across a galaxy is indistinguishable from a tidal feature in a single colour. Its colours and its correlation with far-infrared maps of the dust are the only ways to tell. The effects do not add linearly in magnitudes, and a profile that looks clean may be the sum of two errors of opposite sign.

The figures also treat the sky as a single number known to some fraction. In practice it is estimated from the image itself, from pixels judged to be free of any source, and that judgement depends on how far out the galaxy is assumed to extend. A sky estimated too close to a galaxy includes some of the galaxy, which subtracts the outskirts from themselves and draws a truncation. The measurement and its background are not independent, and the error they share is exactly the one the figures describe.

Still open: how much of a galaxy lies below 30 magnitudes

The prediction from simulations of galaxy growth is that a few per cent to a few tens of per cent of a galaxy’s stars lie in its halo and outer disc, and that the fraction varies with the galaxy’s merger history. The measurements from integrated light, which have to contend with everything above, and the measurements from resolved stars, which do not, are now reaching the same levels for a handful of nearby galaxies and do not always agree. New surveys with very large fields and very careful sky modelling are designed to push the limit to 30 or 31 magnitudes per square arcsecond over the whole sky. Whether they reach it will be decided not by their collecting area but by how well they know the sky — and whether the galaxies’ outskirts then look like the truncations and halos reported so far is the test of every catalogue built at brighter levels.

Once the outskirts are reached, a quieter consequence of the same threshold becomes the issue: a galaxy’s size is usually quoted at a surface-brightness level, and a size defined that way depends on where the threshold was put.

Where this goes next

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. 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. What the error bar is made of, on a 1 m in 60 s. The four contributions to a photometric error, against the brightness of the star, for a 1-metre aperture, a 60-second exposure and a sky of 21 magnitudes per square arcsecond. The star's own photons give a line of slope exactly 0.2 — σ ∝ N^−1/2 and N ∝ 10^−0.4m, so a magnitude of extra faintness costs a fifth of a magnitude of precision, and no instrument changes that. The sky and the read noise are fixed counts, so their lines have slope 0.4, twice as steep, and they overtake the star at V = 18.25 — that crossing is the faint limit of the night, and it moves when the Moon rises rather than when the telescope changes. Scintillation is flat, because the atmosphere modulates a bright star and a faint one by the same fraction: at 4.09e-4 relative it is 0.44 millimagnitudes here and it is what caps the bright end, up to about V = 10.8. Below all of them is the systematic floor at 0.3 millimagnitudes, which is flat-fielding and colour terms and does not integrate down at all. The faint star is measured against a brighter sky For anything at the edge of detection the dominant source of noise is not the object. It is the sky in the same aperture, which is brighter than the star and is subtracted rather than measured — and once that is true, every rule of thumb about apertures, exposure times and image quality changes. A stream that is not the orbit it came from. 456 stars released in pairs from the two saddles of a 10⁵ solar-mass cluster over 4.0 billion years, integrated in a halo whose circular speed is 220 kilometres a second, drawn with the progenitor's own orbit. The cluster runs between 10 and 25 kiloparsecs and the orbit is the thin closed-looking curve; the stars are everything else. The point of the figure is the discrepancy. Stars leaving through the inner saddle are on slightly smaller orbits, turning round at a median of 24.77 kiloparsecs rather than the progenitor's 25.00, and therefore running ahead; stars leaving through the outer saddle reach 25.22 and fall behind. The whole spread is 1.8 per cent of the apocentre, which is the number worth carrying: an offset far too small to see in this drawing builds the entire stream, because it acts for four billion years. The two arms are therefore not merely displaced along the orbit, they are on different orbits, and the track a survey measures is a family of them rather than any single one. Fitting a Galactic potential by demanding that a stream lie along an orbit is wrong by exactly this much, and the size of the error grows with the mass of the progenitor, because the mass is what sets the distance between the two doors. A stream is not the orbit it came from A cluster torn apart by a galaxy leaves a thin trail of stars across the sky, and the obvious thing to do with it is fit an orbit. That is wrong by a knowable amount, because stars leave through two doors with a small energy offset and end up on a family of orbits rather than on one.

About the same objects

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Disc truncationExponential discLow-surface brightness galaxyThe point-spread functionScattered lightSky backgroundSky subtractionStellar haloSurface brightnessSystematic error