----------------------------------------------------------------------- Estimating Signal-To-Noise Ratio of a Point Source Measurement for IRAC ----------------------------------------------------------------------- Updated to apply to warm IRAC observations on April 13, 2010 Note that a tool to calculate the Signal-To-Noise Ratio and other quantities mentioned in this memo is available at https://irsa.ipac.caltech.edu/data/SPITZER/docs/files/spitzer/snirac_warm.pro https://irsa.ipac.caltech.edu/data/SPITZER/docs/files/spitzer/snirac.pro When planning observations that need to measure the flux density or variations in the flux density of a point source to a certain precision, it is important to include the Poisson noise of the source in the estimation of the signal-to-noise ratio. The online sensitivity estimator, SENS-PET, that determines the detection signal-to-noise ratio (how significant a source is compared to the background) only includes the read noise of the instrument and the Poisson noise of the background in the calculation of the uncertainty. This memo discusses in detail how to estimate the uncertainty in a measurement of the source flux density for the benefit of observers who are interested in variations in source brightness (as in observations of extrasolar planet transits, etc). The uncertainty in the measured flux density of a point source in a single exposure is sigma^2 = sigma_rn^2 + sigma_s^2 + ap_fac * sigma_bg^2 + sigma_sys^2 where sigma_rn is the read noise of the detector, sigma_s is the Poisson noise of the source + background, sigma_bg is the Poisson noise of the subtracted background alone, ap_fac is a scaling factor accounting for the noise in the annulus used to estimate the background to subtract, and sigma_sys is any systematic/confusion error. In this memo, sigma is calculated for a single exposure and in the limit where sigma_sys = 0; therefore, sigma decreases as the square-root of the number of exposures. The number of pixels (npix) contributing to the source is the size of the aperture used (~28.3 pixels for a 3 pixel radius circular aperture). Then the read noise for the source is sigma_rn^2 = npix * sigma_readnoise^2, where sigma_readnoise is the read noise in electrons for the desired frametime (see Table 6.4 of the SOM). The term sigma_s^2 is the Poisson noise of the source plus the background in the aperture used. Then, sigma_s^2 = e_s + npix * e_bg, where e_s is the signal of the source in electrons and e_bg is the signal of the background per pixel in electrons. Typically, the source flux density (fd) is given in micro-Jy. To convert to electrons detected, use e_s = fd(micro-Jy) * GAIN * EXPTIME / (scale * FLUXCONV * apcorr) where scale (34.98) converts from micro-Jy to equivalent MJy/sr, so that the the correct number of electrons are determined when using the FLUXCONV and GAIN values supplied; FLUXCONV converts from MJy/sr to DN/s; GAIN converts from DN to electrons; and, EXPTIME is the effective exposure time used. Note that GAIN, EXPTIME and FLUXCONV can all be found in the headers of a BCD image. GAIN and FLUXCONV are constants for a given IRAC channel. 1./apcorr is the fraction of source flux enclosed in the default aperture. Simplifying the expression, e_s = fd * EXPTIME * fd_to_e where fd_to_e = 0.700, 0.580 electrons / (micro-Jy * sec) for channels 1 and 2, respectively. Likewise, the number of electrons contributed by the background is given by e_bg = background_sb * EXPTIME * sb_to_e where the background_sb is an estimate of the surface brightness of the background in MJy/sr (such as the estimate given by Spot) and sb_to_e is the scaling between surface brightness and electrons. The term sb_to_e = 27.546, 22.869 electrons / (MJy/sr * sec) for channels 1 and 2, respectively. To be conservative, assume that the background subtraction used in either aperture photometry or PSF fitting contributes noise from the background summed over the number of noise pixels for the source. Then, ap_fac = npix and sigma_bg^2 = e_bg / nback = background_sb * EXPTIME * sb_to_e / nback where nback is the number of pixels used in a reasonably-sized background annulus. A reasonable estimate of the noise (in electron units) in a single exposure is then sigma^2 = sigma_rn^2 + EXPTIME * (fd * fd_to_e + npix * background_sb * sb_to_e + npix * background_sb * sb_to_e / nback). To be conservative, scale the noise upward by 30%-50% to account for systematics, etc. The signal-to-noise ratio for one exposure is simply fd * fd_to_e * EXPTIME / sigma.