Resolution and seeing

Builds on: Sections S4 and 5.3
Reading time: 30 min
Exercises: 4
Olympiad problems: 0

In the handle of the Big Dipper, Mizar and its faint neighbour Alcor are 11.8′ apart, and people with good eyes see them as two stars without a telescope. The telescope of IOAA 2026, 150 mm across, could in principle separate two stars only 0.92″ apart, about 770 times closer. On most nights it does not quite manage that, because the air above the telescope is never still. In this section we see what sets the finest detail a telescope can show, how the eye compares, and how the air and the pixels of a camera limit what we actually get.

5.4.1The diffraction limit

Section S4 showed that light passing through an opening of diameter $D$ spreads out, so that even a perfect telescope turns a star into a small disc surrounded by faint rings. By the two equal stars can just be told apart when they are $1.22 λ / D$ radians apart. For yellow-green light of wavelength 550 nm, near the middle of what the eye sees, the rule takes a form that is easy to use.

ResultResolution of a telescope(5.7)

\begin{aligned} θ_{min} & = 1.22 \frac{λ}{D} \\ θ_{min} & \approx \frac{138^{″}}{D / mm} at 550 nm \end{aligned}

$θ_{min}$	the smallest angle between two equal stars that the telescope can separate, in radians in the first form and in arcseconds in the second
$λ$	the wavelength of the light
$D$	the diameter of the objective, in millimetres in the second form

Observers of double stars often use Dawes' limit of

116^{″} / (D / mm)

, found by experience with pairs of equal stars. It is a little finer than the Rayleigh limit because the eye can see a slight dip between two overlapping discs.

Quick checkThe telescope of IOAA 2026

What is the diffraction limit of a telescope 150 mm across at 550 nm, in arcseconds?

arcseconds

Figure 5.3When two stars can be told apartinteractive

Separation of the stars

Brightness between the peaks

The figure shows the brightness along a line through the images of two equal stars A and B, seen through a telescope of diameter

D

in light of wavelength

λ

, as in Section S4. Each star alone, drawn thin, gives a bright centre and faint rings, and the heavy curve is their sum. The slider sets the separation in units of

λ / D

. At , the Rayleigh limit, the dip between the peaks falls to about three quarters of the peak brightness. At , close to Dawes' limit for a 550 nm star, the dip is only a few per cent deep, and below about the two images merge into one.

5.4.2The eye

Before you readMake a guess first

The pupil of a dark-adapted eye is about 7 mm across. Its diffraction limit is about 20″. How fine a detail can the eye actually see?

Diffraction alone would let the eye separate stars about 20″ apart, but the eye's lens blurs the image when the pupil is wide open, and the cells of the retina are too widely spaced to record fine detail, so the eye sees about 1′ at best. This is why Mizar and Alcor, 11.8′ apart, are an easy pair for the naked eye, and why a telescope must magnify an image enough for the eye to see all the detail in it. A telescope that resolves $θ_{min}$ needs a magnification of at least $1^{'} / θ_{min}$ to pass that detail on to the eye, which for the telescope of IOAA 2026 is $60 / 0.92 = 65$ times.

5.4.3Seeing

The air above a telescope is made of moving cells of slightly warmer and cooler air, and each bends the light a little differently. The image of a star dances about and spreads into a blurred disc, usually 0.5″ to 2″ across at good observatory sites and more elsewhere, and this blurring is called seeing. The size of the disc, measured as its full width at half its central brightness, is used as the measure of the seeing on a given night. To the naked eye the same effect makes stars twinkle.

Seeing sets a limit that does not depend on the telescope. On a night of 1″ seeing, a telescope 138 mm across already reaches that limit, and a larger one gathers more light but shows no finer detail. This is why the largest telescopes are built on high mountains where the air is steadiest, why many now use adaptive optics, a deformable mirror that is reshaped hundreds of times a second to undo the bending of the air, and why the sharpest pictures come from telescopes in space.

Example 5.6A 10 m telescope on Mauna Kea

The Keck telescopes on Mauna Kea have mirrors 10 m across, and the seeing there is often 0.5″. How much finer is their diffraction limit at 550 nm than the seeing?

Solution

By (5.7),

θ_{min} = \frac{138^{″}}{10 000} = {0.0138}^{″},

and $0.5 / 0.0138 = 36$ .

AnswerWithout adaptive optics the seeing blurs the image 36 times more than diffraction does.

5.4.4Pixels and sampling

A camera records an image as a grid of pixels, and pixels that are too large hide the detail that the telescope and the air let through. A common rule is that the pixel scale of (5.3) should be no larger than half the width of a star's image, so that each star covers at least two pixels across. On a night of 2″ seeing that means pixels of 1″ or less. Pixels much smaller than that only spread the same light over more pixels, each of which then adds its own noise, as Section 5.6 shows.

Common errorsSection 5.4

The formula $1.22 λ / D$ gives the angle in radians, and $λ$ and $D$ must be in the same unit.
A larger telescope on the ground does not show finer detail than the seeing allows. It still collects more light and shows fainter stars.
Magnification does not improve resolution. It only makes the detail that the telescope already resolves large enough for the eye to see.

ExercisesSection 5.44 questions

5.4.1
What is the diffraction limit of a 70 mm telescope at 550 nm?
arcseconds
5.4.2
How wide must a telescope be to separate two equal stars 0.5″ apart at 550 nm by the Rayleigh criterion?
mm
5.4.3
A telescope 400 mm across is used on a night of 1.5″ seeing. What sets the finest detail it shows?
5.4.4
A camera on a telescope of focal length 750 mm should have pixels covering 1″ each, to sample a night of 2″ seeing. How wide should the pixels be?
µm