The faint thing more magnification shows better

The nebula no telescope can brighten followed one of the stranger results in optics to its practical end. An extended object — a nebula, a galaxy, the sky itself — cannot be shown brighter through any telescope than it appears to the naked eye. The telescope collects more light, but it spreads that light over a proportionally larger image, and the conservation law behind it, the conservation of étendue in the brightness no lens can increase, allows no exception. At low magnification the surface brightness matches the naked eye’s; past the magnification at which the beam leaving the eyepiece is as wide as the eye’s pupil, it falls as the square of the magnification.

The natural conclusion is that faint extended objects should be observed at the lowest useful magnification, the one that just fills the pupil, since every step beyond it costs brightness. Experienced observers of faint galaxies do almost the opposite. To glimpse a small, faint galaxy at the limit of a telescope, they routinely increase the magnification well past that point, to exit pupils of two or three millimetres, and report that objects invisible at low power appear when the power is raised. Both statements are true. The resolution lies in what the eye actually responds to.

Three things magnification does #

Take a 200 mm telescope and an eye with a 7 mm pupil, dark-adapted. Consider what magnification changes about a faint extended object.

The surface brightness curve is the one the previous argument was about. The contrast line is the observation that matters most and is easiest to miss. Faint objects are seen against the sky, which is never perfectly dark: even far from cities, airglow and scattered starlight give it a surface brightness of its own. The telescope magnifies the sky’s light along with the object’s and dims both together, so the contrast — the object’s excess brightness divided by the sky’s — is exactly what it is to the naked eye, at every magnification. No optical arrangement can improve it. Only the third line, apparent size, rises.

So the question is whether size by itself helps an eye to see something. If the eye were a light meter, comparing the brightness of one patch with another, size would not matter. It is not a light meter.

What the eye adds up #

A dark-adapted eye has a threshold contrast for detecting a uniform patch on a uniform background, and the threshold depends on the patch’s size. For small patches the eye adds up the light over the patch’s area: halve the diameter and it needs four times the contrast. This is spatial summation, called Riccò’s law in its simplest form, and it comes from the way the retina pools signals from many rods into each nerve channel when light is scarce. Above some diameter, the summation stops being complete and the threshold flattens to a contrast that size no longer reduces.

The curves here are a model, chosen to have the features of the measurements made by H. Richard Blackwell in 1946, when a team of observers made hundreds of thousands of detection judgements for discs of many sizes against backgrounds from daylight down to starlight. Its slope of −2 for small discs, its flattening, and the way both rise and spread as the background dims are the measured features. The numbers — a large-disc threshold of four per cent against a dark sky, and a summation diameter of about two degrees — are representative values, and the conclusions below depend on the shape rather than on them. Two features carry the argument. At night the eye sums over a large area, of the order of degrees. And a dimmer background raises the threshold, roughly as the square root of the brightness falls, because at low light the eye is counting photons and their randomness sets a floor, the regime that the flash the eye counts reached at its limit.

Where the two laws come from #

Both features of the threshold follow from counting photons, which makes the model less arbitrary than a fitted curve. Suppose the retina pools the signals from a patch of a fixed area $A_u$, the summation unit, over a fixed time. The sky delivers to that patch a number of photons proportional to its brightness, $N \propto B A_u$, and like any count of independent arrivals it fluctuates by about $\sqrt{N}$. A faint object adds an excess $\Delta N$. For the eye to notice the excess reliably, it has to stand clear of the fluctuation, $\Delta N \gtrsim k\sqrt{N}$ for some factor $k$ of a few.

If the object is smaller than the unit, all its extra light falls inside one unit, and the excess is its contrast times the sky’s brightness times its own area, $\Delta N \propto C B A$. The threshold is then

$$C_{\text{th}} \propto \frac{\sqrt{B A_u}}{B A} = \frac{\sqrt{A_u}}{\sqrt{B}\,A},$$

which is both laws at once: inversely as the area, Riccò’s law, and inversely as the square root of the brightness, the law Hendrik de Vries and Albert Rose found for vision limited by photon noise. If the object is larger than the unit, only the part of it inside the unit counts, the area in the formula becomes $A_u$, and the threshold stops depending on size. The bend in the threshold curves is the object outgrowing a single pooling unit.

So the whole argument of this essay is a statement about photon statistics. Magnifying past the full pupil spreads the object’s photons more thinly, but also spreads them over more of a pooling unit that collects the background’s photons regardless; while the image is smaller than the unit, enlarging it adds signal to the unit faster than dimming removes it. The noise pushed below the floor followed what can be done to light itself to beat the photon count. The eye’s answer is cruder and older: collect over a larger area and accept a coarser picture. That is also why the night eye sees no fine detail — the same pooling that makes faint patches visible blurs them, a trade that the eye a fly cannot afford met in a different form, where an eye too small for its own diffraction could not have both sharpness and sensitivity.

Dimmer and bigger #

Put the two together. Magnification beyond the full-pupil value does two things to a faint object’s visibility. It dims the image, which raises the threshold. It enlarges the image, which lowers the threshold while the image is still smaller than the summation area. Which wins depends on the exponents. Enlarging lowers the threshold as the square of the size; dimming by the square of the magnification raises it as roughly the first power of the magnification. Below the summation diameter, the enlargement wins.

At 28.6×, the full-pupil magnification, the galaxy’s image is about half a degree across and as bright as it will ever be. It is still too small: the eye’s threshold for a patch that size is above the galaxy’s contrast. Raise the power, and the image dims but grows, and the threshold falls faster than the dimming raises it. At 68× the galaxy crosses its threshold and is seen. The best view comes near 128×, where the image is a couple of degrees across and comparable to the summation area. Beyond that, enlargement no longer helps, and dimming takes over: by 234× the galaxy has faded below threshold again.

This is exactly the experience observers describe, including its least intuitive part, that a brighter image can show less. It also explains their rule of thumb. The best exit pupil for small faint objects is usually quoted as two or three millimetres, and the model’s 1.6 mm is in that range. It arises from the eye, not from the telescope. Roger Clark’s analysis for amateur observers in 1990 made the same argument with Blackwell’s data directly, and called the result the optimum detection magnification.

The visual system has another resource that the model leaves out. Observers use averted vision, looking slightly to one side so that the image falls on the rod-rich retina outside the fovea, and gain a factor of several in sensitivity. They also tap the telescope gently, because the eye is better at detecting a patch that moves. Both shift the threshold curve; neither changes the argument that size matters.

Which objects want which power #

The best magnification is the one that brings the image to roughly the summation size, so it depends mostly on how big the object is.

Large nebulae, which already cover more than the summation area at the lowest magnification, are best at the lowest power, exactly as the brightness argument said. Small ones — compact galaxies, small planetary nebulae, the faint companions of brighter galaxies — want much more. That is a single rule rather than two contradictory ones: magnify until the image is about as large as the patch of retina that pools its signal, and no further. The brightness argument was right about the objects it was usually applied to, and wrong as a general rule because it left out the eye.

The same reasoning applies on the ground. How far the air lets a mountain be seen worked out the distance at which a dark mountain fades into a bright sky, the meteorological range, with a fixed contrast threshold of a few per cent. That threshold is the large-object, daylight end of these curves. A small distant object — a boat, a tower — has to be judged against a higher threshold because it is small, and binoculars help to see it not by brightening it, which they cannot do, but by enlarging it towards a size the eye judges well.

What aperture is for #

If no telescope makes an extended object brighter, what is a bigger one for, when looking at faint galaxies?

The answer is that aperture buys magnification at a brightness. A galaxy needs a certain magnification to grow to a useful size. A small telescope can reach that magnification only with a small exit pupil and a dim image. A large telescope reaches the same magnification with a wider exit pupil and a brighter image, so at the size that suits the eye, its image is brighter. Visibility at the optimum grows a little faster than in proportion to aperture, and the galaxy in the figure crosses its threshold at about 186 mm. The surface brightness limit is not violated, because the comparison is at equal image size, not at equal magnification: the larger telescope does not exceed the naked eye’s surface brightness, it just stays closer to it while magnifying.

There is a familiar version of the same idea in photography, where the four cosines at the edge of a frame in the four cosines at the edge of a photograph are all about illumination falling off and nothing about what a viewer can detect. Optics decides how much light reaches each part of an image. What a viewer then sees in it is decided by a second, independent set of rules belonging to the detector, and the two have to be put together before any conclusion about visibility is drawn.

A detector with a size scale #

The deeper point is that the eye is a detector with a built-in spatial scale, and that a measurement is only optimised when the image is matched to the detector. Electronic cameras have the same issue in another form. A camera sensor has pixels of a fixed size, and astronomers choose a focal length so that a star’s image spans about two pixels, which is called matching the plate scale. Smaller images waste resolution; larger ones spread the light over more pixels than the signal needs, each one with its own noise. The eye’s summation area plays the part of a single, very large pixel at night, and the optimum detection magnification is plate-scale matching for that pixel.

For a camera, the matching is about noise per pixel, and the faint-object limit is set by photon statistics, as light arrives in lumps described. For the eye it is the same in the end: summation over a larger area collects more photons into each decision, and a larger patch can be detected at a lower contrast because more photons go into the comparison. The rods count, the retina pools, and magnification decides how many rods a faint object’s photons are shared among. That is the sense in which the threshold curve and the photon count are one fact.

Two eyes are a second pooling stage. Observing with both eyes through binoculars, each eye behind its own objective, lets the brain combine two independent photon counts of the same patch, and the threshold contrast falls by roughly the square root of two, the gain that averaging two noisy measurements always gives. A binocular viewer on a single telescope is a different case: it splits one beam between the eyes, each count holds half the photons, and the statistics give back exactly what the split took away, so whatever benefit observers report from such viewers has to come from comfort or from the brain rather than from counting. It is the same statistics one level up: more independent counts of the same thing, whether from more rods, more time or more eyes, lower the contrast that can be detected.

What the picture cannot show #

The threshold curves are a model, and their real counterparts vary from one observer to another by factors of two or more, with age (the pupil shrinks from about 7 mm in youth to 5 mm or less in old age, which moves the full-pupil magnification), with dark adaptation, fatigue and practice. The two-degree summation diameter and four per cent floor are illustrative values, and an observer with different ones would find a different best magnification for the same galaxy. The qualitative result — that the best magnification for a small faint object lies well past the full-pupil value, and that it falls as the object grows — holds across reasonable choices; the specific figures of 68×, 128× and 234× do not transfer.

Real galaxies are not uniform discs: they have bright cores and faint halos, and what is detected first is usually the core, whose size and contrast differ from the whole object’s. The sky’s brightness varies across a site and over a night. And the telescope’s own losses, its central obstruction, scattered light and the eyepiece’s quality, all lower the contrast slightly, which no magnification undoes. Most practically, at high magnification the atmosphere’s turbulence smears a small object, and on a night of poor seeing a compact galaxy may not gain from magnification at all, because its image is limited by the air rather than by size.

The model also treats detection as a fixed threshold, while the eye’s detection is probabilistic: an object at threshold is seen about half the time, glimpsed and lost. Observers’ reports of “averted vision only” objects are reports of that probability, and the figures here plot a ratio that ought to be read as rising odds rather than as a switch.

Still open: how the eye pools light at its faintest #

Blackwell’s measurements remain the standard data set, but they were made with a particular design — forced choices, fixed exposure times, particular observers — and modern work on rod vision has found that the pooling of rod signals in the retina is itself adaptive, with circuits that change how they combine signals as light falls. How large the summation area really is at the faintest light levels, how it differs between the fovea’s surroundings and the far periphery, and how much the brain’s integration over time adds to the retina’s over space, are still being measured. A model of visual detection good enough to predict, for a given observer, telescope and sky, exactly which faint galaxies will be seen and at which magnification, does not yet exist, though the elements of one do.

A telescope fixes contrast, cannot raise surface brightness and dims the image past the magnification that fills the pupil, but the eye’s threshold contrast falls as the square of apparent size up to a summation diameter of the order of degrees, so a small faint galaxy invisible at the brightest image — 28.6× in a 200 mm telescope — is seen from 68× to 234×, best near 128×, by a model threshold with the shape of the measured ones. The habit worth carrying is to ask what the detector responds to before optimising what reaches it, because the most light is not always the most signal.