A ten-metre mirror that resolves like a ten-centimetre one

The resolution of a telescope in vacuum is $1.03\lambda/D$ and it improves without limit as the aperture grows. A 39-metre mirror at 500 nm resolves three milliarcseconds — a coin at four hundred kilometres.

Put that mirror under an atmosphere and it resolves one arcsecond, which is what a ten-centimetre mirror resolves. Not somewhat worse: the same, to within a per cent, and the same as every other aperture larger than about ten centimetres.

The atmosphere delivers the wavefront in pieces #

Light from a star arrives at the top of the atmosphere as a plane wave, to an accuracy no instrument can question: the source is effectively at infinity and the intervening space is empty.

It does not arrive at the ground that way. Air’s refractive index depends on its temperature, turbulent mixing produces cells of slightly different temperature at every scale from millimetres to hundreds of metres, and a wavefront crossing them accumulates path-length differences. By the time it reaches a mirror it is corrugated: flat over small patches, and mismatched between them.

The length that quantifies this is the Fried parameter $r_0$, defined as the diameter over which the wavefront’s root-mean-square phase error reaches about one radian. At a good site at 500 nm it is 10 to 20 centimetres; at a poor one, 5. It depends on wavelength as $\lambda^{6/5}$, which is why the infrared is easier, and on the airmass as $(\cos z)^{3/5}$, which is why nobody observes near the horizon.

An aperture smaller than $r_0$ sees one flat piece of wavefront and behaves exactly as it would in vacuum. An aperture larger than $r_0$ sees $(D/r_0)^2$ pieces, tilted with respect to one another, and forms $(D/r_0)^2$ separate images of the star scattered over an angle $\lambda/r_0$.

What the extra aperture is still buying #

A large telescope under an atmosphere is not a waste, and saying why is the useful part.

Photons, as $D^2$. Collecting area is the reason an 8-metre telescope reaches magnitude 27 and a 10-centimetre one reaches 13, and no atmospheric effect touches it — the magnitude scale runs backwards over sixty magnitudes and aperture is what moves an object along it. Sensitivity and resolution are separate goods and only the second is capped.

Speckles, as $(D/r_0)^2$. In a short exposure — under about ten milliseconds, before the pattern reshuffles — the image is not a blur at all. It is a scatter of sharp dots, each one a diffraction-limited image of the star formed by one coherent patch, each as narrow as the dashed curves above, spread over the seeing disc.

That second one is the whole reason the situation is recoverable. The information about fine structure has not been destroyed; it has been scrambled and then, in a long exposure, averaged away. Anything that avoids the averaging can get it back.

What was actually measured #

Seeing is not measured by looking at a picture and estimating how blurred it is. That estimate exists — the “seeing” quoted at an observatory is a full width at half maximum in arcseconds — but it is a derived quantity, and what the instrument records is something else entirely.

The standard instrument is a differential image motion monitor: a small telescope with two subapertures a fixed distance apart, each forming its own image of the same star through a prism that separates them. Both images dance, because each subaperture sees a differently tilted patch of wavefront. What is recorded is the variance of the separation between the two images, over a few thousand frames.

That variance is the observable. It is differential, so the telescope’s own tracking errors, wind shake and mount flexure cancel — both images move together under those and the separation does not change. And the theory of atmospheric turbulence gives a direct relation between the variance of the differential tilt over a known baseline and $r_0$, so a distribution of a measured angle becomes a length. The chain has the shape every measurement on this site has: an observable, a model, and a quantity that was never observed.

The number that gets published is therefore a statistic and not a measurement of any single thing. A seeing of $0.6''$ at a site means the differential image motion monitor’s variance, converted through a turbulence model, corresponds to an $r_0$ of about 17 cm, which through $0.98\lambda/r_0$ corresponds to a long-exposure width of $0.6''$ at 500 nm. Three conversions, one of which contains a model of the turbulence spectrum.

Two ways out, and they are opposites #

The remedies divide by whether they fight the atmosphere or accept it.

Adaptive optics fights it. A wavefront sensor measures the corrugation, a deformable mirror with hundreds to thousands of actuators applies the negative of it, and the corrected wavefront is flat again. It has to run faster than the atmosphere changes — a kilohertz — and it needs a bright enough reference source — a star of known brightness is not required, only a bright one — within the isoplanatic angle, a few tens of arcseconds, over which the corrugation is the same. Where no natural star is close enough, a laser is fired to excite sodium atoms at 90 km and make one.

Speckle interferometry accepts it. Take thousands of exposures short enough to freeze the pattern; each contains the full diffraction-limited information, scrambled. Averaging the images destroys it, but averaging the power spectra does not, because the scrambling is a phase and the power spectrum discards phase. Labeyrie showed in 1970 that the autocorrelation of the averaged power spectrum recovers structure at the diffraction limit.

Both work. Neither is free, and their costs are different in kind: adaptive optics needs hardware that is a substantial fraction of the telescope’s cost and a reference star; speckle methods need only a camera and lose an enormous fraction of the photons to the shutter, which restricts them to bright objects — and brightness is the one axis aperture buys outright.

Where the model stops #

The quadrature sum is a convenience. Combining $1.03\lambda/D$ and $0.98\lambda/r_0$ in quadrature is a standard approximation and it is not a theorem. The true long-exposure profile is the convolution of the Airy pattern with the seeing profile, and neither is a Gaussian; the resulting width is close to the quadrature sum near the two limits and departs by several per cent in between, which is exactly where the bend in the first figure is.

$r_0$ is one number for a phenomenon that has many. Turbulence has an outer scale — the largest eddy, tens of metres — beyond which the Kolmogorov spectrum does not apply, and it has an inner scale of millimetres. It is distributed in layers, at the ground, at the top of the boundary layer and at the tropopause, each moving at its own wind speed. A single $r_0$ compresses all of that into one length, and the quantities the compression throws away — the coherence time, the isoplanatic angle, the outer scale — are precisely the ones an adaptive optics system’s design depends on.

The wings are not Gaussian. A real long-exposure profile has a core close to a Gaussian and wings falling as roughly $\theta^{-11/3}$, which carry a substantial fraction of the light far from the centre. For photometry of a faint star next to a bright one, those wings are the whole problem, and a figure drawn with Gaussians has quietly removed it.

Above the atmosphere, none of this applies, and something else does. A space telescope is diffraction-limited and is limited instead by its own optics: manufacturing errors, thermal deformation, and the diffraction from its own support structure. Hubble’s first three years are the standing demonstration that removing the atmosphere removes one problem and not the concept of a problem.

The time nobody quotes #

The Fried parameter is the number sites are ranked by, and it is not the number that decides whether an adaptive optics system can work. That is the coherence time, and it is quoted far less often because it is harder to measure and less flattering.

The turbulent pattern above a telescope is not static; it is blown across the aperture by the wind. The time over which the wavefront stays essentially unchanged is therefore the coherence length divided by the wind speed at the height where the turbulence lives, and for a good site with a ten-centimetre coherence length and a twenty-metre-a-second jet stream that is about five milliseconds.

Five milliseconds is the whole budget. In that interval the system has to measure the wavefront, compute a correction and apply it — so the sensor’s integration, the computation and the mirror’s response all have to fit inside a fraction of it, and the loop has to run at a kilohertz or better.

That is what makes the reference star problem severe. A wavefront sensor integrating for a millisecond on a star collects a thousandth of the photons a one-second exposure would, and it has to divide them among hundreds of subapertures. The faintest usable natural reference is therefore around magnitude 12 to 14, and the sky density of such stars is low enough that only a per cent or so of the sky has one within the isoplanatic angle.

The coherence time also scales badly with wavelength — as $\lambda^{6/5}$, like the coherence length — so the infrared is easier in time as well as in space, which compounds the advantage already noted. Two of the three quantities that make adaptive optics hard improve together towards the red, and that is the whole reason the technique matured in the infrared and is still being fought for in the visible.

Why the site matters more than the mirror #

The plateau in the first figure is set by $r_0$, and $r_0$ is a property of a place. That single fact organises the whole geography of observational astronomy.

The best sites are high, dry, and — most importantly — above or beside a stable inversion layer that suppresses the boundary-layer turbulence which otherwise dominates. Mauna Kea, Paranal, La Palma, and the Chilean Atacama sites deliver median seeing between $0.6''$ and $0.8''$; Dome C in Antarctica, where the boundary layer is only 30 metres thick, delivers $0.3''$ above that layer and is otherwise nearly uninhabitable.

The consequence is that a site survey is worth more than a mirror upgrade, and observatories are built where they are for reasons that have nothing to do with the sky being darker or clearer. Cloud cover and darkness set how often one can observe; seeing sets how well. A telescope at a $1.5''$ site is not half as good as the same telescope at $0.75''$ — for a point source against a background, the signal-to-noise ratio goes as the reciprocal of the seeing disc’s area, so it is four times worse.

The other thing the air does to the light #

Everything above concerns the phase of the arriving wavefront. The air also modulates its amplitude, and the second effect limits an entirely different set of measurements.

A corrugated wavefront focuses and defocuses itself as it propagates: regions where the phase is curved converge, regions where it is curved the other way diverge, and after a few kilometres of travel the intensity across the beam is no longer uniform. That is scintillation, and it is why a star twinkles and a planet — which subtends enough angle to average over several patches — does not.

For imaging it is a minor nuisance. For photometry it is the dominant noise source on any bright star from the ground. The fractional intensity fluctuation scales as the aperture to the power minus two-thirds, so a larger telescope helps and not nearly as fast as counting statistics would, and it scales as the airmass to the power three-halves, so it is worse at low altitude.

The practical consequence is a floor. Ground-based photometry of a bright star reaches a precision of a few parts in ten thousand in an exposure of tens of seconds, and no amount of additional signal improves it, because the noise is multiplicative rather than Poissonian. That floor is why the transit of a small planet across a bright star is a space measurement, and why the ground-based technique that competes is differential photometry against a comparison star in the same field — which cancels the scintillation only if the two stars’ light has passed through the same turbulence, which requires them to be close together on the sky.

Adaptive optics does nothing for it. Correcting the phase across the aperture does not undo an amplitude modulation impressed kilometres above, and a system that delivers a diffraction-limited image still delivers it with a fluctuating brightness.

There is one arrangement that does help, and it is worth naming because it is the reason certain measurements are made from aircraft and balloons rather than from mountains. Scintillation is produced by propagation after the phase is corrugated, so it grows with the distance between the turbulent layer and the detector. A telescope above most of the atmosphere sees less of it not because the turbulence is weaker but because the beam has had less room to develop the intensity pattern.

That is a different reason for going up from the one that improves the seeing, and the two do not scale together. A site that is high because it is above the boundary layer improves the coherence length; an instrument that is high because it is above the jet stream improves the scintillation. Choosing between them depends on which measurement is being made, and the choice is usually made for the first reason by people who then discover they wanted the second.

What the picture cannot show #

The time. Every profile here is a long exposure, and the interesting structure lives at ten milliseconds. A figure of an averaged image has already thrown away the thing that makes recovery possible, which is that the average is of sharp things rather than of blurred ones.

The spatial structure. The seeing disc is drawn as a radial profile, and a real short-exposure image is a two-dimensional scatter of speckles with no radial symmetry whatever. The profile is what remains after averaging over both time and azimuth.

The colour. $r_0 \propto \lambda^{6/5}$ and the diffraction limit goes as $\lambda$, so the ratio of the two — the number of speckles — goes as $\lambda^{-12/5}$. Every curve here is at 500 nm, and at 2.2 microns the same 8-metre telescope spans thirty patches rather than six thousand, which is the entire reason adaptive optics was solved in the infrared first and is still hard in the visible.

Where the ladder goes next #

Later rungs on this anchor: the Kolmogorov turbulence spectrum, and how $r_0$ falls out of a structure function. The coherence time and the isoplanatic angle, which are what actually constrain an adaptive optics design. Laser guide stars, the cone effect, and the tip–tilt problem a laser cannot solve. Multi-conjugate adaptive optics, which corrects several layers at several conjugate heights and widens the corrected field. Lucky imaging, which is speckle interferometry’s cheap relative and simply discards the bad frames. And interferometry proper, where several apertures are combined at a baseline far larger than any mirror and the atmosphere is fought at each aperture separately.

Newton wrote in the Opticks that the only remedy was “a most serene and quiet Air, such as may perhaps be found on the tops of the highest Mountains above the grosser Clouds.” He was right about the remedy and wrong about the limit, in a way he could not have anticipated: the tops of the highest mountains give a factor of two, and the rest of the factor of a hundred had to wait for a computer that could reshape a mirror a thousand times a second.