Problems for Chapter 5

Builds on: Sections 5.1 to 5.8
Reading time: 40 min
Exercises: 0
Olympiad problems: 1

Olympiad problems on telescopes rarely test one formula at a time. A typical problem gives a telescope, a camera and a target, and asks for one result after another, each built on the last. The past problems in Sections 5.3, 5.5, 5.7 and 5.8 each used the ideas of one or two sections. The problem in this section, from IOAA 2024, needs the plate scale, the pixels of a camera, the magnitude scale and the CCD equation in a single chain.

5.9.1From textbook to olympiad

In the problem below, an astronomer photographs the faint globular cluster Palomar 4, and from the signal-to-noise ratio of the picture we are asked to find how dark the sky was. The method of Section S2 gets us from the chapter to the solution in the steps this problem needs, with a check at the end.

We start by drawing it. The cluster is a small round patch 72″ across, which the telescope turns into a disc about half a millimetre wide on the sensor, covering many thousands of tiny pixels. Every one of those pixels also collects light from the sky, adds a little dark current and adds read noise when it is read.
We name the physics. Part (a) is the plate scale of (5.3). Part (b) is geometry, the area of the disc divided by the area of a pixel. Part (c) is the CCD equation, (5.10), with the sky as the one unknown.
We work backwards from the target. To find the sky's surface brightness we need the sky's count $N_{sky}$ , which the CCD equation gives once we know the cluster's count $N_{GC}$ and the noise of the camera. $N_{GC}$ in turn needs the cluster's total magnitude and the number of photons a star of that magnitude delivers.
We use every given. The width and centre of the V band, the photon flux of a zeroth-magnitude star, the quantum efficiency, the read noise and the dark current each appear once in the chain. The pixel size enters through part (b), whose answer is reused in part (c), as earlier parts usually are.
We check the answer. A dark site has a sky of about 21 to 22 magnitudes per square arcsecond in the V band, and a result far outside that would point to a slip.

Olympiad problemsPast IOAA problems, restated1 problem

2024Photographing Palomar 420 points

Adapted from IOAA 2024 (Rio de Janeiro), theory problem 6. We have restated it in our own words and kept the original data. The solution was last checked on 20 October 2026.

An astronomer photographs the globular cluster Palomar 4 in the V band from a site with no light pollution. The cluster appears as a disc 72.0″ across with an even surface brightness of 20.6 magnitudes per square arcsecond, which means that each square arcsecond of it is as bright as a star of magnitude 20.6. The telescope is a reflector 305 mm across at f/5, with a camera at its prime focus whose quantum efficiency is 80% and whose square pixels are 3.80 µm wide. The V band is centred on 550 nm and is 88.0 nm wide, and a star of magnitude 0 delivers 10 000 photons per nanometre of the band, per square centimetre, per second.

(a) What is the plate scale, in arcminutes per millimetre? (3 points)

(b) About how many pixels does the image of the cluster cover? (4 points)

(c) A 15 s exposure gives the cluster a signal-to-noise ratio of 225. The camera has a read noise of 5 electrons per pixel and a dark current of 6 electrons per pixel per minute. Find the surface brightness of the sky, in magnitudes per square arcsecond. (13 points)

I solved it without the solution

AnswerThe plate scale is 2.25′ per millimetre, the cluster covers about 15 400 pixels, and the sky's surface brightness is about 21.9 magnitudes per square arcsecond, as expected for a dark site. The official solution, which rounds the photon flux in step 4 more coarsely, finds a sky count of 55 250 electrons and the same surface brightness to one decimal place.