10 photons

Figure 2.2 The greater the number of photons, the better the signal-to-noise ratio. In this case study, the photon count across the face of the galaxy image varies from a mean value of 0.1 photons to a mean value of 10,000 photons. The signal-to-noise ratio equals the square root of the photon count.

Even after we have summed n samples, however, we still don't know the ex-act[ital] value of x because the mean photon count remains uncertain by Jk .

Suppose that instead of examining the same pixel in a sequence of images, we were to examine one image in an area of sky that has no stars. Although you might expect that all pixels making up that area of sky would have the same value, they do not. Just as a sequence of samples of one pixel varies, each pixel in a set of pixels that are side-by-side is an independent sample of sky brightness, and it obeys the same rule that a series of samples at the same location does.

This has a profound impact on the collection of astronomical data. If you take two "identical" images of the same object and compare them, you will find that they are not identical. They differ because they are independent samples of the mean photon rate, and each and every pixel in the two images has to follow the x±fx photon-counting rule.

It may not be immediately obvious from the mathematical symbols that the more photons in the signal, the better the signal quality. Why should this be so? Consider the following example: we have two signals, one consisting of 25 photons and the other consisting of 100 photons. In the 25-photon signal, the expected variation is

725 , or 5 photons; while in the 100-photon signal, the expected variation is yioo or 10 photons. Since the variation in the 100-photon signal is twice that of the 25-photon signal, does that mean the 100-photon signal is worse? In one sense it is—it has twice the standard deviation. However, the percentage variation in the 25-photon signal is 5/25 , or 20%; while in the 100-photon signal, the percentage variation is 10/100 , or 10%. Even though it has twice the standard deviation, as a ratio of the signal strength, the 100-photon signal has only half the standard deviation as the 25-photon signal.

To quantify signal quality, communications engineers invented the signal-to-noise ratio, or SNR. This ratio is nothing more than the signal divided by the noise. For signals in which photon counting is the primary source of noise, the signal-to-noise ratio is:

The SNR of the 25-photon signal above is 5, and the SNR of the 100-photon signal is 10. The greater the signal-to-noise ratio, the better the image quality. Note, however, that within a given image, the signal is not the same at all places. The sky background will have a lower signal level than the bright center of a galaxy, so it is meaningless to assign an SNR to an entire image because signal-to-noise ratio is meaningful for only one signal level. Nevertheless, astronomers sometimes quote an SNR for an image, and when they do, it refers to the SNR at the signal level of the sky background.