How a telescope gathers light

Builds on: Sections S4 and O1.1
Reading time: 30 min
Exercises: 5
Olympiad problems: 0

At IOAA 2026 in Hanoi every student in the telescope round used the same instrument, a Newtonian reflector with a mirror 150 mm across and a focal length of 750 mm. The pupil of a dark-adapted eye is about 7 mm across, so that mirror collects about 459 times as much light as the eye does, and it shows stars about 6.7 magnitudes fainter than the faintest stars we can see without it. In this section we see what the parts of a telescope do, why the light it collects depends on the diameter of its mirror or lens, and why the focal ratio decides how bright the image of a nebula is in the focal plane, where a camera records it.

5.1.1Lenses and mirrors

Before you readMake a guess first

Two telescopes point at the same faint star. One has a lens 150 mm across and a focal length of 750 mm. The other has a lens 100 mm across and a focal length of 1000 mm. Which shows the star brighter?

Every telescope has an , the large lens or mirror at the front that collects the light. As Section S4 explained, the objective brings the parallel rays from a star together at a point a distance $F$ behind it, its focal length, and the plane through that point at right angles to the axis of the telescope is the focal plane. A camera can be put in the focal plane, or the image formed there can be looked at through an eyepiece, as Section 5.3 describes.

A telescope whose objective is a lens is a refractor. Glass bends blue light a little more than red light, so a single lens brings the colours to slightly different foci, and the image of a bright star has a coloured fringe. This fault is called chromatic aberration, and good refractors reduce it by combining two or three lenses of different kinds of glass. A telescope whose objective is a curved mirror is a reflector. A mirror reflects every colour in the same way, so it has no chromatic aberration, and since a mirror can be held up from behind it can be made far larger than a lens. Every large telescope of today is a reflector.

The focus of a mirror lies in front of it, in the path of the light coming in, so reflectors use a second, smaller mirror to send the light somewhere more convenient. In a Newtonian telescope, the kind used at IOAA 2026, a flat mirror at 45° near the top of the tube sends the light out through the side of the tube. In a Cassegrain telescope a convex mirror sends the light back down the tube, through a hole in the middle of the main mirror, to a focus behind it. The small mirror blocks a little of the incoming light, but it does not change the reasoning of this chapter, and we usually ignore it.

5.1.2Collecting light

The light a telescope collects from a star is all the light that falls on its objective. A star is so far away that its light arrives as a broad, even stream, so the amount collected in a given time is proportional to the area of the objective, which is $π D^{2} / 4$ for an objective of diameter $D$ . Comparing two telescopes, or a telescope and the eye, only the ratio of the diameters matters.

ResultLight gathered by a telescope(5.1)

\frac{L_{1}}{L_{2}} = {(\frac{D_{1}}{D_{2}})}^{2}

$L_{1}, L_{2}$	the light collected from the same star in the same time by two telescopes, or by a telescope and the eye
$D_{1}, D_{2}$	the diameters of their objectives, or of the pupil of the eye

The ratio is often called the light-gathering power of a telescope compared with the eye, using a pupil of about 7 mm. The light actually reaching the eye or the camera is a little less, since every mirror and lens loses some of it, as Section 5.5 shows.

Quick checkThe telescope of IOAA 2026

How many times as much light does a mirror 150 mm across collect from a star as an eye with a pupil 7 mm across?

times

On the magnitude scale of Section O1.1, each factor of 100 in brightness is a step of 5 magnitudes. A factor $x$ is therefore $5 \log_{100} x$ magnitudes, which by the rules of logarithms in Section S3 is $2.5 \log_{10} x$ , and a factor of 459 is $2.5 \log_{10} 459 = 6.7$ magnitudes. A star that is 6.7 magnitudes too faint for the naked eye is just bright enough to be seen through the telescope.

5.1.3The focal ratio

The of a telescope is its focal length divided by the diameter of its objective, $N = F / D$ , and it is written f/ $N$ . The telescope of IOAA 2026 has $750 / 150 = 5$ , so it is an f/5 telescope. The focal ratio sets the steepness of the cone of light that meets at the focus, as Figure 5.1 shows. A telescope with a small focal ratio is called fast, and one with a large focal ratio slow, from the exposure times that photographers need.

For a star the focal ratio does not matter. All the light that the objective collects from a star lands in one tiny spot in the focal plane, so the star looks brighter only if the objective is larger. For an object that covers an area of the sky, such as the Moon or a nebula, the focal ratio matters a great deal. The image of the Moon in the focal plane is $F$ times its angle across, by the rule $s = F θ$ of Section S4, so its area grows as $F^{2}$ . The light that spreads over that area grows as $D^{2}$ . The light falling on each square millimetre of the image therefore goes as $D^{2} / F^{2}$ , which is $1 / N^{2}$ .

ResultFocal ratio and the brightness of an image(5.2)

\begin{aligned} N & = \frac{F}{D} \\ \frac{E_{1}}{E_{2}} & = {(\frac{N_{2}}{N_{1}})}^{2} \end{aligned}

$N$	the focal ratio of a telescope, written f/ $N$
$F, D$	the focal length and the diameter of the objective
$E_{1}, E_{2}$	the light falling on each unit area of the image of the same extended object, such as the Moon or a nebula, in two telescopes of focal ratios $N_{1}$ and $N_{2}$

The result holds for objects that cover many times the size of a star's image. The exposure time a camera needs for such an object is proportional to

N^{2}

, so an f/4 telescope needs a quarter of the time of an f/8 one.

Figure 5.1The cone of light behind an objectiveinteractive

Aperture DFocal length F

Focal ratio
Light gathered, compared with the eye
Image of the full Moon

The figure shows a telescope's objective of diameter

D

from the side, with widths drawn about four times larger than lengths so that the cone of light can be seen. Light from a star on the axis meets at the focus P, a focal length

F

behind the objective, and the heavy bar in the dotted focal plane is the image of the full Moon. The two sliders set

D

and

F

. The words below set both at once, to , to , where the Moon's image is twice as large and each part of it a quarter as bright, and to , which gives the same brightness per square millimetre as the first telescope on a Moon twice the size.

Example 5.1The Moon at the focus

The full Moon is 0.52° across. How large is its image in the focal plane of the telescope of IOAA 2026, and how much longer must a camera expose to record it if the same mirror had a focal length of 1500 mm?

Solution

In radians the Moon is $0.52 \times π / 180 = 0.00908$ rad across, so by $s = F θ$ its image is

s = 750 mm \times 0.00908 = 6.8 mm .

With a focal length of 1500 mm the focal ratio becomes $1500 / 150 = 10$ instead of 5, and by (5.2) each square millimetre of the image gets $(5 / 10)^{2} = 1 / 4$ as much light.

AnswerThe Moon's image is 6.8 mm across. At f/10 it would be 13.6 mm across, and a camera would need 4 times the exposure.

Common errorsSection 5.1

The focal length does not change how much light a telescope collects from a star. Only the diameter of the objective does, through its area.
The light collected grows with the square of the diameter. A telescope twice as wide collects $2^{2} = 4$ times as much light.
A small focal ratio makes extended objects such as nebulae brighter in the focal plane, where a camera records them. Seen by eye through an eyepiece, a nebula's brightness depends on the exit pupil of Section 5.3, and the faintest star visible depends on the diameter of the objective.

ExercisesSection 5.15 questions

5.1.1
One of the mirrors of the Very Large Telescope in Chile is 8.2 m across. How many times as much light does it collect from a star as an eye with a pupil 7 mm across?
times
5.1.2
A reflector has a mirror 200 mm across and a focal length of 1000 mm. What is its focal ratio $N$ ?
5.1.3
A nebula needs an exposure of 10 minutes with an f/4 camera lens. How many times as long must the exposure be with an f/8 telescope to record the nebula equally well?
times
5.1.4
A telescope's focal length is doubled while its mirror stays the same. What happens to a star and to a nebula in the focal plane?
5.1.5
How large is the image of the full Moon, 0.52° across, in the focal plane of a telescope of focal length 2000 mm?
mm