Illustrated Physics
physics, drawn
Optics

The nebula no telescope can brighten

A telescope eight metres across gathers a million times more light than the eye, and a star seen through it is a million times brighter. The Orion nebula seen through it is not brighter at all. It is bigger, and spread over a bigger patch of the retina, and the two exactly cancel: no instrument made of lenses and mirrors can show an extended object brighter than the naked eye sees it. What a telescope does is decided by the small disc of light behind its eyepiece, the exit pupil, and by whether that disc is larger or smaller than the pupil of the eye looking through it.
15 min read 5 figures What stays the sameMade into an instrument

Assumes: The brightness no lens can increase · The invariant that is a count

The brightness no lens can increase proved, from the second law of thermodynamics, that no arrangement of lenses and mirrors can make an image brighter than its source: the radiance of light, its power per unit area per unit solid angle, is conserved through any passive optical system and can only be lost. The cone a fibre will accept, the cone light has to find to get out and the invariant that is a count applied the law to guides, to light trapped in dense materials and to modes, and the beams that add only if they differ found it forbidding the merger of identical laser beams.

All of those were about light sources and the devices that move their light around. This essay turns the law on the most familiar optical instrument of all, the one built specifically to make faint things easier to see: the telescope. The law says a telescope cannot make a faint extended object brighter, and that is exactly what anyone who has looked at a nebula through one discovers — and it is also what makes the telescope’s most important number not its aperture or its magnification but the diameter of the small disc of light behind its eyepiece.

Where the light goes

A simple astronomical telescope is two lenses. The objective, of focal length fof_o, gathers light from a distant object and forms an image at its focus; the eyepiece, of shorter focal length fef_e, takes that image and sends the light on in parallel bundles that the relaxed eye can focus. Light from a star on the axis arrives as a parallel bundle the width of the objective and leaves as a parallel bundle much narrower; light from a star slightly off the axis arrives at a small angle and leaves at an angle M=fo/feM = f_o/f_e times larger. That is the magnification.

Where a telescope puts the light it gathers. A telescope drawn as two thin lenses with a magnification of 4, low enough for the geometry to be seen, with an on-axis bundle of parallel light (blue) and one from a point 2.9° off the axis (red). The eyepiece forms an image of the objective, the exit pupil, a short distance behind it; every bundle that entered the objective passes through that small disc. Its diameter is the aperture divided by the magnification — here a quarter of it; for a 200 mm telescope at 20×, 10 mm. The eye, placed there, receives all the light the telescope collected if its own pupil is at least that wide — up to about 7 mm in the dark — and every bundle leaves at M times the angle it arrived at. The light from a whole patch of sky is squeezed into a pupil-sized beam and spread over a patch of the retina M² times larger; per unit of that patch, the brightness cannot rise.
Fig. 1 A telescope drawn as two thin lenses at a magnification of 4, with a bundle from a star on the axis (blue) and one from a star off it (red). Every bundle that entered the objective passes through the exit pupil, the image of the objective formed by the eyepiece, whose diameter is the aperture divided by the magnification.

Every bundle that entered the objective, from every direction in the field of view, passes through one small disc a short distance behind the eyepiece: the image of the objective formed by the eyepiece. That disc is the exit pupil, and its diameter is the aperture divided by the magnification, D/MD/M. A 200-millimetre telescope at twenty times has an exit pupil of ten millimetres; at a hundred times, two millimetres. The observer’s eye goes there, so that the eye’s own pupil coincides with the exit pupil and all the light passes into it.

The figure draws the geometry at a magnification of four, low enough to see. The on-axis bundle, as wide as the objective, converges to the focus, diverges again, and leaves the eyepiece a quarter as wide and still parallel. The off-axis bundle does the same around a different point of the focal plane and leaves at four times its original angle, crossing the axis at the exit pupil. The narrowing of the bundles and the widening of their angles are the same fact: the product of a bundle’s width and its angular spread, the étendue, cannot change.

Larger, not brighter

Now consider an extended object, a nebula a degree across with a uniform surface brightness. To the naked eye it subtends a certain solid angle and sends a certain power into the eye’s pupil, spread over a certain patch of the retina. Through the telescope, the light arriving within the objective’s area — (D/p)2(D/p)^2 times more than the eye’s pupil pp admits — is collected and sent through the exit pupil. The image is magnified MM times in each direction, so it covers M2M^2 times more retina. If the exit pupil is no larger than the eye’s pupil, all the collected light enters, and the brightness on the retina per unit area changes by (D/p)2/M2=(D/Mp)2(D/p)^2/M^2 = (D/Mp)^2, the square of the ratio of the exit pupil to the eye pupil. That ratio is at most one.

How bright a nebula looks through a 200 mm telescope. The surface brightness of the image of an extended object — a nebula, a galaxy, the Moon — through a lossless 200 mm telescope, as a fraction of its brightness to the naked eye, against magnification on a logarithmic axis, for eye pupils of 7 and 5 mm. It is never above one. For a 7 mm pupil it is one up to a magnification of 28.6, where the exit pupil equals the eye's; below that magnification the exit pupil is wider than the eye's and part of the collected light falls on the iris; above it the image dims as the square of the exit pupil: 0.327 at 50×, 0.082 at 100×, 0.020 at 200×. The telescope makes the nebula larger, not brighter; at the magnification where its exit pupil matches the eye's it makes it as large as it can while keeping it as bright as it is.
Fig. 2 The surface brightness of an extended object through a lossless 200 mm telescope, relative to the naked eye, against magnification, for eye pupils of 7 and 5 mm. It is one up to 28.6× and 40×, where the exit pupil equals the eye’s, and falls as the square of the exit pupil beyond: 0.33 at 50×, 0.082 at 100×, 0.020 at 200× for the 7 mm pupil.

So the image is never brighter than the object seen with the naked eye, and it is exactly as bright — ignoring the ten or twenty per cent a real telescope loses in its glass and mirrors — only while the exit pupil is at least as wide as the eye’s. For a 200-millimetre telescope and a young dark-adapted eye with a seven-millimetre pupil, that holds up to a magnification of 28.6. Beyond it the exit pupil is smaller than the eye’s, the image is spread over more retina than the collected light can cover at the original brightness, and the surface brightness falls as the square of the exit pupil: to a third at fifty times, a twelfth at a hundred, a fiftieth at two hundred. Below 28.6 times the exit pupil is wider than the eye’s and the image is no brighter; the extra light falls on the iris and is wasted.

That is the result the thermodynamic argument promised, now seen through the instrument. A telescope trades area of collection for area of image. It can make the nebula bigger without making it dimmer, up to the magnification at which its exit pupil matches the eye’s, and that is all it can do for a surface.

Why stars are different

A star is not a surface. It is so far away that even the largest telescope cannot resolve its disc, and its light, however magnified, lands on a single point of the retina — or rather on the smallest spot the eye can form, the same size whatever the magnification. All the light the objective collects goes into that spot, so the star’s apparent brightness rises with the collecting area, by (D/p)2(D/p)^2.

Stars get brighter in a telescope, nebulae do not. The gain in brightness over the naked eye with a 7 mm pupil, on a logarithmic scale, against telescope aperture: for a point source such as a star (blue), (D/7 mm)², and for the surface of an extended source at the best magnification (red), one at every aperture. Binoculars of 50 mm: stars 51 times brighter, 4.3 magnitudes fainter visible; 200 mm: stars 816 times brighter, 7.3 magnitudes fainter visible; 1 m: stars 2·10⁴ times brighter, 10.8 magnitudes fainter visible; 8.2 m: stars 1.4·10⁶ times brighter, 15.3 magnitudes fainter visible. A star is a point, so all the light the aperture collects arrives on one point of the retina and its brightness rises with the area; a nebula's light is spread over an image that grows with the magnification, and the two growths cancel. That is why a telescope reveals faint stars by the million and shows the Orion nebula as a grey glow, never the colours of a long-exposure photograph.
Fig. 3 The gain in brightness over the naked eye for a star, (D/7 mm)², and for a nebula’s surface at the best magnification, never more than one, against aperture. Binoculars of 50 mm show stars 51 times brighter, 4.3 magnitudes fainter; a 200 mm telescope, 816 times; an 8.2 m telescope, 1.4 million times, 15.3 magnitudes fainter.

That is the whole reason telescopes reveal stars invisible to the eye. A pair of 50-millimetre binoculars makes every star 51 times brighter, reaching 4.3 magnitudes fainter than the naked eye; a 200-millimetre telescope, 816 times and 7.3 magnitudes; an 8.2-metre telescope, used with the eye, would make a star 1.4 million times brighter. The nebula’s surface gains nothing at any aperture. A large telescope shows it larger at the same brightness, which helps the eye to see detail in it, since the eye’s sensitivity to faint patches improves with their size; it does not make the nebula’s light any stronger.

It is also why a long-exposure photograph of the Orion nebula, glowing red and blue, looks nothing like the grey-green smudge an observer sees through any telescope. The colours are real, but the surface is too faint for the eye’s colour receptors, which need more light than its night-vision rods; a camera adds up photons over minutes, which the eye cannot, and no telescope can raise the surface brightness to where the eye’s colour vision would wake up.

Darkening the sky to find a star

The difference between points and surfaces has a use that every observer exploits. The night sky is not black: it glows faintly, from airglow, from scattered starlight and, in most places, from scattered streetlight, and that glow is a surface. A star is a point. Raise the magnification and the exit pupil shrinks, the sky’s surface brightness falls as the square of the exit pupil, and the star’s brightness does not change at all, because the star’s collected light still lands in one spot. The contrast between star and sky rises as the square of the magnification.

That is why a faint star that is invisible at low power can appear when the magnification is raised, against a sky that has turned from grey to black. The gain continues until the star stops behaving as a point: the turbulent atmosphere smears every star into a disc a second or two of arc across, and once the magnification makes that disc larger than the eye can resolve, the star is a small surface and dims with the sky. For most telescopes on most nights that happens somewhere between a hundred and two hundred times, and the faintest stars a given telescope shows are found near there, not at the lowest power where nebulae look best.

The same arithmetic explains a puzzle about daylight. With a bright eye pupil of two or three millimetres, the exit pupil of almost any binocular or telescope is wider than the eye’s, and the instrument is at its surface-brightness limit: daytime scenes look exactly as bright through it, only larger. Bright stars can nevertheless be seen in daylight through a telescope, which the naked eye cannot do, because the telescope at high magnification darkens the blue sky — a surface — while leaving the star — a point — as bright as its aperture allows. Venus and Jupiter, small enough to count as nearly points, can be found at noon this way.

The Moon through a small telescope

The bright extended objects show the law from the other side. The full Moon has a surface brightness that is already uncomfortable for a dark-adapted eye. Through a telescope at its minimum useful magnification it has exactly the same surface brightness, spread over a disc that may fill the field of view, and observers commonly find it dazzling and fit a neutral filter; at high magnification, with an exit pupil of a millimetre, its surface brightness drops fifty-fold and detail on it becomes easier to see. A planet works the same way. Jupiter’s cloud belts are a surface of moderate brightness, and the magnification that makes them large enough to see also dims them, which is why there is a best magnification for every planet and every telescope, set by the balance between the eye’s need for size and its need for light.

None of this is a property of any particular design of telescope. It is the conservation of radiance that the work a diluted beam will not do found fixing how much of sunlight’s quality survives its spreading out: what a lens or a mirror can change is how large an image is and how much light it carries, never the ratio between them.

The window of useful magnification

The magnifications an eye can use. The exit pupil D/M of telescopes of 70, 200 and 400 mm aperture against magnification, on a logarithmic axis, with the band of eye pupil sizes from about 2 mm in daylight to 7 mm dark-adapted (shaded). Where a curve lies above the band the exit pupil is wider than the eye's and collected light is wasted on the iris: the lowest useful magnification is the aperture over 7 mm — 10× for 70 mm, 29× for 200 mm, 57× for 400 mm. Below the band the exit pupil is smaller than about 0.5–1 mm, the image is dim and diffraction blurs it more than magnification enlarges the detail: about twice the aperture in millimetres is the usual ceiling. Every telescope has a window of useful magnifications, set by the eye at one end and by the wavelength at the other.
Fig. 4 The exit pupil against magnification for 70, 200 and 400 mm telescopes, with the band of eye pupils from 2 mm in daylight to 7 mm dark-adapted. The lowest useful magnifications, where the exit pupil equals 7 mm, are 10×, 29× and 57×.

The exit pupil turns the choice of magnification into arithmetic. At the low end, the exit pupil should not be wider than the eye’s pupil, or light is thrown away: the minimum useful magnification is the aperture divided by seven millimetres, ten times for a 70-millimetre telescope, twenty-nine for 200 millimetres, fifty-seven for 400. Observers of faint nebulae choose eyepieces close to that magnification, which shows an object as large as it can be shown at its full natural brightness.

At the high end, the limit is set by diffraction. A telescope cannot resolve detail finer than about λ/D\lambda/D, and the eye can resolve about a minute of arc. Magnifying the telescope’s finest detail until it spans a minute of arc takes a magnification of roughly the aperture in millimetres; magnifying beyond twice that only enlarges the blur, while the exit pupil shrinks below half a millimetre and the image grows dim. How far apart two things have to be found the diffraction limit; the exit pupil translates it into the same units as the eye. Every telescope has a window of useful magnifications, about from D/7D/7 to 2D2D with DD in millimetres, and the exit pupil runs from seven millimetres down to half of one across it.

The eye’s own pupil sets one end of that window, and it changes with age. A young eye opens to seven or eight millimetres in the dark; at sixty, five is typical. The older observer’s lowest useful magnification on a 200-millimetre telescope is forty times, not twenty-nine, and binoculars with 7-millimetre exit pupils, popular for night use, deliver to an older eye no more surface brightness than ones with 5-millimetre exit pupils, at a higher weight. The same telescope is, in this respect, a different instrument for different eyes.

A wall is as bright from any distance

The deepest version of the result needs no telescope at all.

A wall looks equally bright from any distance. The brightness on the retina of a uniformly lit square of 1 m side, relative to its brightness from 1 m, against distance from 1 m to 10 km on logarithmic axes (red), and the total light the eye receives from it (blue), which falls as one over the distance squared. The total falls a millionfold by 1 km, but the image shrinks in proportion, so the light per unit of image — what the eye perceives as brightness — stays the same until the square becomes smaller than the eye can resolve, about one minute of arc, at 3.4 km. Beyond that it is a point, and points do dim with distance. The same rule is why the Sun's disc would look exactly as bright from Mars, smaller but not dimmer, and why a telescope cannot change it: a lens, like distance, trades the size of an image against the light in it, and the trade fixes the ratio.
Fig. 5 A uniformly lit square of 1 m side, seen from 1 m to 10 km: the total light entering the eye falls as one over the distance squared, but the brightness of its image on the retina stays the same until the square becomes too small to resolve, about 3.4 km away, and only then falls.

Walk away from a lit wall and the light entering the eye from it falls as the square of the distance, a millionfold over a kilometre. The wall does not look a millionth as bright. It looks the same brightness, smaller: its image on the retina shrinks in area by the same factor as the light falls, so the light per unit area of the image — what the eye perceives as brightness — is unchanged. Only when the wall is too small to resolve, beyond a few kilometres for a square metre, does it become a point, and points do dim with distance.

Distance, in other words, does to an extended source exactly what a telescope does in reverse: it trades the size of the image against the light in it and leaves their ratio alone. The Sun’s disc seen from Mars would be smaller and no dimmer per unit area; a galaxy a hundred million light years away, which is still resolved, has the same surface brightness as one a tenth as far, until the stretching of space at cosmological distances dims both — by a factor that Tolman showed grows as the fourth power of one plus the redshift and that has been used to test whether the universe is really expanding. Olbers’ paradox, the question why the night sky is dark if every line of sight ends on a star, is this argument too: if it did, the whole sky would have a star’s surface brightness, and a telescope pointed at it would show nothing brighter.

Why the law cannot be beaten with a cleverer design

It is natural to suspect that some arrangement of optics could do better — a shorter eyepiece, a fibre bundle that tapers the image down, a fluorescent screen. None of the passive ones can, and the reason is the reason in the brightness no lens can increase: if an optical system could make an image of a warm surface brighter than the surface, it could heat something above the surface’s temperature using that surface alone, and the second law forbids it. A fibre taper that shrinks an image widens its angles in exactly the proportion needed to keep the radiance fixed, and the eye, with its fixed pupil, cannot accept the wider angles. A fluorescent screen absorbs and re-emits, which is no longer passive, and an image intensifier, which does brighten nebulae, adds energy from a battery: it counts the photons and makes new ones, the way a camera adds them up over time.

That is also why the hole that makes the sharpest picture gives a dim image and a lens a bright one: a lens of the same focal length gathers a far larger cone of light into each point of the image, and the image is as bright as the scene — which is the most any passive imager can manage — rather than dimmer by the ratio of the pinhole’s area to the lens’s.

What the pictures cannot show

The figures treat the telescope as lossless and the eye’s pupil as a sharp disc. Real telescopes lose ten to twenty per cent of their light in coatings and mirrors, and those with a central secondary mirror block part of the aperture, so their extended images are somewhat dimmer than the naked-eye limit drawn here. The eye is not a uniform detector: its sensitivity to faint extended objects depends on their angular size and on where on the retina they fall, which is why observers look slightly to one side of a faint galaxy, and why the larger, equally bright image a telescope gives is easier to see even though it is no brighter. The distance figure assumes a perfectly uniform square and an eye resolving one minute of arc; the transition from resolved to point-like is gradual in a real eye.

Still open: how faint a surface the eye can see

The eye’s threshold for a faint extended source depends on its size, its contrast against the sky and how long the observer looks, and the best measurements, made in the 1940s with careful observers in darkened rooms, are still the basis of the predictions observers use to decide whether a given galaxy will be visible in a given telescope. How those thresholds vary with age, between individuals and with the colour of the source, and how to combine them with the brightness of a real night sky, which light pollution has raised over most of the populated world by factors of several to hundreds, is still being worked out — with consequences for which objects remain visible at all from the places where most people live.

The habit worth carrying away is to ask whether a quantity is conserved before asking how to increase it. Radiance cannot be raised by lenses, so a telescope makes an extended object larger but never brighter per unit area: the image keeps naked-eye brightness while the exit pupil, D/M, is at least the eye’s pupil, and dims as (D/Mp)² beyond — while a star, a point, gains (D/p)². A wall looks as bright from a kilometre as from a metre for the same reason.

Part 8 of 8

This essay is one argument about Etendue. The others:

The brightness N lasers can be made into. The greatest radiance — power per unit area per unit solid angle — that N identical single-mode lasers can be combined into by lossless optics, as a multiple of one laser's, against N, for four arrangements. Laid side by side and passed through any lenses and mirrors, they fill N modes and are never brighter than one: the étendue of the combination is at least N times one beam's. Two can share a mode if their polarisations are crossed, so polarisation doubles it once. Beams of N different wavelengths can be overlapped into one spatial mode by a grating, and beams with their phases locked together can be made into one mode outright; either way the radiance grows as N, because the beams are no longer N copies of the same mode. The beams that add only if they differ Part 7 — Take two identical lasers and try to merge their beams into one beam twice as bright. No arrangement of lenses, mirrors and prisms will do it: the brightness of light cannot be increased by passive optics, and two copies of the same beam can only ever be put side by side. The escape is to make the beams not copies. Give them crossed polarisations and two will merge. Give each a different colour and a grating will stack dozens into a single beam. Lock their phases so that they stop being separate beams at all, and they add — but only as well as the gaps between them allow and only while their phases are held to a twentieth of a wavelength.

Where this goes next

The most any concentrator is allowed. The greatest concentration a receiver can be given, against the half-angle it accepts, both logarithmically, for a receiver in a medium of index 1. The upper curve is a three-dimensional concentrator, n²/sin²θ; the lower is a trough, which concentrates in one direction only, n/sin θ. The Sun's angular radius is 0.267°, so sunlight can be concentrated by at most 46165 times in a dish and 215 times in a trough — marked. Nothing about glass, mirrors, wavelength or aperture appears in either expression. The bound is thermodynamic: a receiver that accepts light out to θ also radiates out to θ, and the concentration at which what it emits balances what it absorbs is exactly where these curves are. The brightness no lens can increase A lens can make an image smaller and therefore hotter, and there is a temperature at which it stops — the temperature of the source. Every arrangement of glass and mirrors ever built obeys a bound that contains no wavelength, no aperture and no material — only the angle the receiver is allowed to accept — and the bound comes from thermodynamics rather than from optics. The étendue, tiled. The phase space of a one-dimensional optical system: position across a twenty-micrometre aperture, against the optical direction cosine n·sinθ, for a system accepting ±0.1. The shaded rectangle is what the beam occupies, and its area — 4 micrometre-radians — is the one-dimensional étendue, the quantity no arrangement of lenses can reduce. Divided by a wavelength of 500 nanometres it is 8, and the rectangle is tiled here with exactly 8 cells of area one wavelength each. That is the whole content of the count: a cell of phase-space area λ is the smallest patch a field can be confined to, because squeezing it in position spreads it in direction by the same relation that gives a slit its diffraction pattern, so an étendue is a number of modes and its conservation is the conservation of a count. The cells are drawn four wide and two high, which is arbitrary: only a cell's area is fixed, not its shape — a beam may be confined tightly in position and loosely in angle or the other way round, and the trade is what an optical system is for. The invariant that is a count Étendue is an area times a solid angle, and ray optics gives it no floor — nothing in a ray has a size. Divide it by the square of the wavelength and it becomes a number of modes, its conservation becomes the conservation of a count, and the count has a least value of one. That is where the ray bound hands over to diffraction, and the handover is the same number written two ways. Too small blurs by diffraction, too large by shadow. The size of the image of a distant point through a pinhole 100 mm from the screen, in 550 nm light, against the hole's diameter, both on logarithmic axes. The dashed lines are the two simple estimates: the hole's own shadow, as wide as the hole, and the diffraction spread, 2.44λf/d. The dots are the diameter holding half the light, computed from the Fresnel diffraction integral. They follow the shadow for large holes and the diffraction for small ones, and are least — 0.207 mm — for a hole 0.336 mm across, where the simple sum of the two estimates puts its minimum at 0.366 mm. At the best hole the Fresnel number d²/4λf is 0.51: the sharpest hole, judged by where half the light lands, lets through about half of the first Fresnel zone. The hole that makes the sharpest picture A pinhole camera makes a picture without bending any light: a small hole lets through, from each point of the scene, only the rays that reach one small patch of the screen. A smaller hole makes a sharper picture — until it does not. Below a certain size the hole spreads light by diffraction faster than it narrows the shadow, and the picture blurs again. The best hole balances the two, and at that balance it turns out to pass about half of the first Fresnel zone of the incoming wave: the sharpest pinhole is already beginning to behave like a lens.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

EtendueExit pupilHuman eyeMagnificationPoint sourceRadianceSurface brightnessTelescope