Estimating Photon Counts

For many observations, the dominant source of noise is simply that of arrival statistics for the finite number of photons which are recorded. If a telescope observes a star throughout a filter that admits 100 photons per second on average to the detector, a one second exposure will not always yield 100 photons. In fact, a large number of 1s observations would then yield a mean that tends to 100, but the standard deviation of the sample will tend towards 10, the square root of the number of photons in the mean sample.

This is a very general truth for samples in which the random process is the uncorrelated arrival of discrete entities, named for the mathematician Simeon-Denis Poisson. It applies to such disparate phenomena as counts of subway passengers, raindrops, radioactive decays, photon arrivals, etc. For a mean signal $S$ there is an associated noise $N$ with the simple relation that $N=\sqrt{S}$.

The "gold standard" of the quality of a measurement, at least the random part (systematic effects are something different altogether) is the ratio of the signal $S$ to the noise $N$, that is, the "Signal to Noise" ratio, denoted $S/N$ or SNR. And for a Poisson noise process, since the noise is simply the square root of the signal, the SNR is the square root of the number of photons collected.

Therefore, it is important to know how many photons one expects to collect in order to estimate in advance the SNR. Alternatively, from the SNR one can estimate the number of photons collected before even calibrating an instrument. One can plan, given an object's brightness, integration times which will yield an expected SNR in advance. This is an important part of advance planning in scientific observations.

For the purposes of estimation, the standard starting point is the determination that in green light, the star Vega (zero magnitude) delivers to the top of Earth's atmosphere about 1000 photons per second per square centimeter per Angstrom.

As an example, consider a 10s exposure of a tenth magnitude star to be recorded at the Hartung-Boothroyd 0.6m telescope through the Johnson V green filter. The telescope's collecting area is set by its circular primary mirror, which is about 2800 ${\rm cm}^2$ in area (actually somewhat less due to the central obscuration, but for estimation purposes these factors are secondary). The atmosphere absorbs a few percent. Each mirror reflection loses about 18% of the light. Filters, even in their passband, also lose around 15%. Reflection from uncoated transmissive surfaces robs between 4% and 8% per surface (remember, one lens or window has two surfaces!). So there's a lot to lose! Let's trace the path of light through our system onto a CCD camera:

  • Light passes through Earth's atmosphere: Efficiency: 0.85
  • Light reflects telescope primary: Efficiency: 0.80 (the coating is old)
  • Light reflects telescope secondary: Efficiency: 0.84 (the coating is in somewhat better shape
    (note: perfect freshly coated aluminum on glass only reflects 88% in green light)
  • Light passes through Johnson V filter: Efficiency: 0.80 (two surfaces + filter absorption)
  • Light passes vacuum window into CCD chamber: Efficiency: 0.92 (surfaces are uncoated)
  • Light interacts with CCD: Efficiency: 0.85 (our Andor CCD has high efficiency in green).

The product of all the efficiencies gives the total throughput: for the above example it's about 36%.

So for our tenth magnitude star, we start with a base rate of not 1000 photons/s/cm/ but four orders of magnitude less (each 2.5 magnitudes is a factor of ten), or 0.1 photons/s/cm/. The collecting area of the telescope, though, multiplies this by 2800, so our collection rate is 280 photons/s/. A Johnson V filter has a passband about 900 wide, so we collect in toto about 250000 photons/s. But the total throughput is 36%, so the amount actually recorded by our detector is about 90000 photons/s. In a ten second exposure, we would collect about 900000 photons from this star.

The intrinsic noise in this exposure would be the square root of the number of photons, or about 950. The SNR for this measurement is also about 950, if Poisson noise is the only noise term.

As a very general rule in astronomy, going above 200 SNR puts you into a regime where other systematic effects dominate. So even this short exposure of a (to human eye terms) faint star has already put us into the "maximum easy quality" regime of measurement, where sources of noise other than Poisson terms limit the measurement.

More detailed estimates

Johnson Central Width Flux at m=0 Reference
Filter Wavelength (nm) (nm) (Jy)
U 360 54 1810 Bessel
B 440 97 4260 Bessel
V 550 88 3640 Bessel
R 640 147 3080 Bessel
I 790 150 2550 Bessel
J 1260 202 1600 Campins
H 1600 368 1080 Campins
K 2220 511 670 Campins
g 520 73 3730 Schneider
r 670 94 4490 Schneider
i 790 126 4760 Schneider
z 910 118 4810 Schneider
Bessel, M.S., 1979, PASP 91, 589.
Campins, Rieke, Lebofsky, 1985, PASP 90, 896
Schneider, Gunn, Hoessel, 1983, AJ, 268, 476

Note also that $1 \;{\rm Jy} = 10^{-23}{\rm erg}\; {\rm sec}^{-1}{\rm cm}^{-2} {\rm Hz}^{-1} $ and $ 1 \; {\rm Jy} = 1.51 \times 10^7\; {\rm photons} \; {\rm sec}^{-1}{\rm m}^{-2}(d\lambda / \lambda)^{-1} $ where $d\lambda / \lambda$ is the ratio of the filter width $d\lambda$ to the central wavelength $\lambda$.

Gerardi, Bertelli, Bressan et al A&A 391 1 (2002)

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