Astronomy 341 Fall 2012 Observational Astronomy Haverford College. CCD Terminology

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1 CCD Terminology Read noise An unavoidable pixel-to-pixel fluctuation in the number of electrons per pixel that occurs during chip readout. Typical values for read noise are ~ 10 or fewer electrons per pixel, and it is independent of exposure time. If there are many counts per pixel, then this noise will have a small impact on the total noise. But for faint sources, or when adding together a number of exposures, read noise can have a notable impact. Dark Current Over time, electrons are freed from the CCD material itself and collect inside individual pixels. These electrons are thermally generated; the cooler the chip, the lower the dark current. The signal and noise introduced by the dark current are dependent on exposure time. This dark current pools together with the electrons released by photons hitting the chip. The dark current of a chip (at a given temperature) can be measured by taking dark frames - exposures with the shutter closed. Dark current is typically negligible in modern instruments which at run at very cold temperatures. Gain The factor which relates the amount of charge collected in a pixel to the number output data units. Gain is quantified by electrons per Analog-to-Digital unit (e - / ADU). If the gain of a CCD is 1, then the number of counts that is assigned to a pixel (and that you read when you hover over that pixel with your mouse) is equal to the number of photons absorbed by that pixel. Values of gain are commonly ~ a few. Quantum Efficiency (QE) The efficiency with which photons hitting a pixel actually release an electron in that pixel. This is a function of wavelength. The quantum efficiency of all pixels on a CCD are not identical. This is a reason why it is necessary to obtain flat fields. Bias CCDs are designed so that they never read out a negative value of signal. To do this, they offset the actual value of signal by some amount - the bias. The bias of a CCD can be measured by taking bias frames - zero length exposures - the shutter remains closed. Flat field The gain and QE are slightly different from pixel to pixel in the CCD. A flat field is a way to normalize science exposures to divide out the effect of these variations. To

2 obtain a perfect measurement of the shape of a chip s response, a uniform illumination of the chip is needed. This is difficult to do; astronomers typically take many exposures of the sky during twilight or sunrise and median the exposures together to measure a flat. When processing flats, you need to first subtract off the bias and then normalize the flat field to have a median value of 1. Normalizing the flats is needed so that when you divide your science image by the flat, the number of counts is preserved. Reduced image Science images need to be corrected for systematic variations in gain and QE across the field-of-view and corrected for the bias. This is called reducing the data. A typical method of reducing data simply includes: Object = (Raw object - Bias)/(Flat field). This output object frame is the reduced image. CCD Statistics To understand CCD statistics, is to understand is the idea of Poisson statistics. The arrival of photons to a CCD pixel is a Poisson process. The conditions that must be satisfied for something to be considered a Poisson process are well articulated by WolframMathworld to be: 1. The numbers of changes in nonoverlapping intervals are independent for all intervals. 2. The probability of exactly one change in a sufficiently small interval /n is P=ν/n, where ν is the probability of one change and n is the number of trials. 3. The probability of two or more changes in a sufficiently small interval h is essentially 0. The numbers of events that occur from a Poisson process follow a Poisson distribution, with the probability of measuring x events given an expected average of µ: P(x; µ) = e µ µ x x! The variance of a Poisson distribution with an average number of events, µ, is also equal to µ.

3 Because they are zero-length exposures, the pixel-to-pixel fluctuations in bias frames result from readout noise. The bias frame is in units of ADU, and read noise is in units of electrons, so σ bias = RN/g Flat frames contain a substantial number of counts (i.e. have a high signal-to-noise but a number of counts below the non-linear regime of the chip) because they need to accurately characterize the shape of the flat field. The pixel-to-pixel fluctuations of the counts are thus dominated by the Poisson noise of the photons hitting each pixel. The number of electrons accumulated in the pixels thus varies like N 0.5. If gain is large, then a greater amount of charge (more photons) are required to yield a set number of ADU. There will then be a larger pixel-to-pixel fluctuation in signal, because the random fluctuations in signal are ~ N 0.5. Read noise does not substantially add to this. If the average value of the flat field in ADU (after subtracting the bias) is F, then σ flat = Fg g The quality of astronomical observations is a function of the signal-to-noise of the image. The signal-to-noise (S/N) is a function of the number of photons gathered for a specific measurement: S N = N γ Nγ + N pix (N sky + N dark + RN 2 ) In this expression, the capital N s are all total numbers of photons (electrons). If using this expression to compute the signal to noise of your observation, be sure to convert ADU to electrons using the gain. N γ is the number of photons (electrons) in a single pixel or from a set of pixels. For stellar photometry within an aperture, it would be the total number of photons (electrons) within the aperture. N pix is then number of pixels within the aperture. The other values of N are all per pixel. N sky is the expected number of photons per pixel from sky background, N dark is the expected number of photons per pixel from dark current and RN is the read noise per pixel. Because RN is the actual shot noise, instead of a Poisson variate, it is added in as RN 2

4 For bright sources, S/N N γ. For faint sources the other sources of noise in the denominator become as or more important. The sky brightness is typically the most important source of noise, after the Poisson fluctuations of photons arriving from the source itself. Lets get the expression above for S/N in terms of slightly more observable quantities. Typically, we know the flux (energy per area per second) of a star at the location of Earth from its apparent magnitude. The total number of photons from that source will be linearly proportional to time, as will the total number of sky photons. Using n to represent photon arrival rate (for the object it is over all pixels), and dropping the dark current (because its usually negligible): S N = n γ t nγ t + N pix (n sky t + RN 2 ) We can now easily see that S/N t exp in regimes where read noise isn t the dominant source of noise (most regimes). We can also solve for the integration time needed to obtain a measurement of some signal-to-noise, given the rates of sky photon arrival per pixel, object photon arrival per aperture, the number of pixels in the aperture, and the read noise: (n γ t + N pix (n sky t + RN 2 )) ( S N )2 = n 2 γt 2 n 2 γt 2 (n γ + N pix n sky )( S N )2 t N pix RN 2 ( S N )2 =0 This can be solved with the quadratic formula: t = B + B 2 4AC 2A Object and sky brightness We can use the apparent magnitude of the sky and of our objects to estimate the number of sky and object photons that should reach a CCD pixel given the collecting area of the telescope, the pixel area of the CCD, the photometric filter, and the throughput of the system. The website:

5 summarizes many quantities needed for S/N or exposure time calculations. This includes a worked out example of how to calculate the number of photons per second incident on a 1 m 2 aperture per second (at the top of the atmosphere) for a star with V = 23.9: This website also includes the approximate night sky brightnesses in magnitudes per square arcsecond in several filters, as a function of the phase of the Moon. This doesn t include light from man-made sources.