CCDs and the signal-to-noise ratio

Builds on: Sections 5.2 and 5.4
Reading time: 40 min
Exercises: 4
Olympiad problems: 0

A camera on the telescope of IOAA 2026, exposing for a few minutes, records stars that no one could see through the same telescope by eye. The eye takes in light for only a fraction of a second at a time, and it turns only a small share of the photons that reach it into a signal, while the camera's sensor collects light for as long as its shutter is open and records most of the photons that fall on it. In this section we see how such a sensor works, why its counts are never exact, and how to tell whether a faint star stands out from the noise.

5.6.1How a CCD works

A CCD, short for charge-coupled device, is a thin slice of silicon divided into a grid of pixels. A photon that strikes a pixel can free an electron in the silicon, and during the exposure the freed electrons are held in that pixel. When the exposure ends, the charges are shifted across the chip row by row to an amplifier at its edge, which measures the number of electrons in each pixel in turn. The sensors in phones and most amateur cameras, called CMOS sensors, have an amplifier in each pixel instead, but they record light in the same way.

The fraction of the photons that free an electron is the quantum efficiency $η$ , often 80% or more for a good astronomical sensor at visible wavelengths. Up to the point where a pixel is full, the number of electrons is proportional to the light that fell on it, which lets astronomers measure brightnesses by counting. Three kinds of calibration frame are used to clean the counts. A bias frame, an exposure of zero length, records the offset the electronics add to every pixel. A dark frame, taken with the shutter closed, records the electrons that the warmth of the chip frees in each pixel, called the dark current. A flat frame of an evenly lit screen records how the sensitivity varies from pixel to pixel.

5.6.2Photon noise

Before you readMake a guess first

A pixel collects on average 10 000 electrons in an exposure. If the exposure is repeated many times, by about how much does the count vary from one exposure to the next?

Photons arrive at random times, like raindrops on a roof, so the number that lands in a given time is never quite the same. For random independent events of this kind, counts with an average of $N$ are spread about that average with a standard deviation of $\sqrt{N}$ . This spread is called photon noise, or shot noise, and it is the reason a faint star is hard to see in a short exposure. The signal-to-noise ratio of a measurement is the signal divided by its uncertainty, and for photon noise alone it is $N / \sqrt{N} = \sqrt{N}$ .

ResultPhoton noise(5.9)

\begin{aligned} σ & = \sqrt{N} \\ S / N & = \sqrt{N} \end{aligned}

$N$	the number of electrons counted from a source
$σ$	the standard deviation of that count from one exposure to the next
$S / N$	the signal-to-noise ratio when photon noise is the only noise

The count must be in electrons, the things actually counted. A camera reports its counts in its own units, and the gain of the camera turns them into electrons.

Quick checkDoubling the signal-to-noise ratio

A bright star gives a signal-to-noise ratio of 50 in a 10 s exposure. How many times as long must the exposure be to give 100?

times

5.6.3The CCD equation

Every pixel that holds a star's image also collects light from the glowing night sky, electrons from the dark current, and a little noise added each time the pixel is read, called the read noise. The star's light is measured by adding up the electrons in the $n$ pixels that hold its image and taking away what the sky and the dark current alone would have given. The noise of each of these parts adds in quadrature, as Section S3 explained for independent uncertainties.

ResultThe CCD equation(5.10)

\frac{S}{N} = \frac{N_{*}}{\sqrt{N_{*} + n (N_{S} + N_{D} + R^{2})}}

$N_{*}$	the electrons from the star, in all the pixels of its image
$n$	the number of pixels added up
$N_{S}, N_{D}$	the electrons from the sky and from the dark current in one pixel
$R$	the read noise of one pixel, in electrons

For a bright star the first term under the root wins, and the ratio is close to

\sqrt{N_{*}}

. For a faint star the sky usually wins, and the ratio is close to

N_{*} / \sqrt{n N_{S}}

. Either way, once the read noise is small compared with the rest, the signal-to-noise ratio grows as the square root of the exposure time.

Figure 5.4A star coming out of the noiseinteractive

Exposure

Electrons from the star
Electrons from the sky in the circle
Signal-to-noise ratio

The figure shows a patch of a CCD 15 pixels across holding the image of a star, with each pixel shaded by the number of electrons it holds in one random exposure. The star gives 400 electrons a second, the sky 20 electrons a second in every pixel, and the read noise is 5 electrons. The dashed circle holds the 13 pixels added up to measure the star, and the table gives its signal-to-noise ratio from (5.10). The slider sets the exposure time from 0.1 s to 100 s, and every change takes a fresh exposure. After the star is a faint spot in the speckle, after it stands out clearly, and after its signal-to-noise ratio has grown about

\sqrt{10}

times again.

Example 5.8Ten seconds on the star

For the star in Figure 5.4, the 13 pixels of the circle collect 285 electrons a second from the star, each pixel collects 20 electrons a second from the sky, the dark current is negligible, and the read noise is 5 electrons. What is the signal-to-noise ratio in a 10 s exposure?

Solution

In 10 s, $N_{*} = 2850$ , the sky gives $n N_{S} = 13 \times 200 = 2600$ , and the read noise gives $n R^{2} = 13 \times 25 = 325$ . By (5.10),

\frac{S}{N} = \frac{2850}{\sqrt{2850 + 2600 + 325}} = 37.5 .

AnswerThe signal-to-noise ratio is 37.5. The sky adds almost as much noise as the star itself.

Common errorsSection 5.6

The square-root rule applies to counts of electrons. Applying it to the numbers a camera reports, before turning them into electrons, gives the wrong noise.
The read noise enters squared and once for every pixel added up, so measuring a star over many more pixels than its image needs adds noise without adding signal.
A longer exposure raises the signal-to-noise ratio only as the square root of the time. An exposure four times as long only doubles it.

ExercisesSection 5.64 questions

5.6.1
A star gives 2500 electrons, and photon noise is the only noise. What is the signal-to-noise ratio?
5.6.2
5000 photons from a star reach a CCD with a quantum efficiency of 80%. What signal-to-noise ratio does photon noise allow?
5.6.3
A faint star gives 900 electrons spread over 9 pixels. Each pixel collects 100 electrons from the sky, the dark current is negligible, and the read noise is 10 electrons. What is the signal-to-noise ratio?
5.6.4
Which source of noise does not grow with the length of the exposure?