OPTICAL PHOTOMETRY Observational Astronomy (2011) 1
The optical photons coming from an astronomical object (star, galaxy, quasar, etc) can be registered in the pixels of a frame (or image). Using a ground-based telescope, the light must pass through the Earth's atmosphere, suffer reflection at the mirrors, pass through the filter, and be absorbed by the optical camera (CCD = charge-coupled device: photoelectric effect photon-to-electron conversion). The signal is then digitised (converting electrons to counts) and registered (see Appendix I on pre-processing of optical frames) The astronomical task consists in removing the signature of the atmosphere and instrument, trying to recover the signal just before it reached the Earth's atmosphere (see the celestial sphere in Appendix II) INSTRUMENT: RED FILTER Filter systems Johnson (broad): UBVRI Stromgren (narrow): uvby Sloan (broad): ugriz CCD Observational Astronomy (2011) 2
ATMOSPHERE: EXTINCTION AIR MASS: X = ρ dh sec z / ρ dh ρ = air density, h = height Homogeneous and plane-parallel atmosphere: X = sec z E.g., X = 1 (z = 0º), X = 2 (z = 60º) h + dh h zenith dh sec z z (zenith angle) TO OBSERVER ATMOSPHERE: SEEING (blurring effect due to the atmospheric turbulence a point-like source is seen as an extended object) Many consecutive exposures of ~ 10 ms each Global exposure of ~ 10 2 seconds FWHM ~ 0.5-3 Observational Astronomy (2011) 3
MAGNITUDE AND COLOUR INDEX Neglecting atmospheric and instrumental perturbations, the photon flux at wavelength λ is F(λ) photons cm -2 s -1 Å -1. However, in real conditions, one measures the instrumental flux (e.g., using the r Sloan filter) f r (counts cm -2 s -1 ) = A(λ) T r (λ) F(λ) dλ, (1) where A and T r are the atmospheric and telescope (r Sloan passband) transmision, respectively. Here, T r incorporates the conversion from photons to counts The instrumental flux can be related to the photon flux at λ r (this basically corresponds to the effective or central wavelength of the filter transmission curve), f r (counts cm -2 s -1 ) = A(λ r ) F(λ r ) T r (λ) dλ. Thus, the true magnitude at λ r [Pogson law: r = 2.5 log F(λ r )] is derived from r inst = 2.5 log f r = r 2.5 log A(λ r ) + Z r, where Z r = 2.5 log T r (λ) dλ is the zero-point term Observational Astronomy (2011) 4
Extinction by the atmosphere can be described by a standard exponential law A(λ) = exp [ a(λ) τ], where τ = H X is the total thickness of the atmosphere (H = total height of the atmosphere, X = airmass) and a(λ) is the (linear) extinction coefficient per unit length, caracterizing the properties of the homogeneous medium and depending on the wavelength. Taking logarithms, at λ r we have an extinction term: 2.5 log A(λ r ) = k r X (Bouguer law). Here, k r = 2.5 log(e) a(λ r ) H is an extinction coefficient per unit airmass. This coefficient can vary with time (from season to season, from night to night and even within a single night) Putting all the pieces together, r inst = r + Z r + k r X, g inst = g + Z g + k g X (2) Multiplying Eq. (1) by the exposure time (time) and the effective area of the telescope (area), one can also work with the total counts C r, and redefine the instrumental magnitude, i.e., r inst = 2.5 log C r. The magnitudes r inst and r fulfil a relationship similar to those in Eq. (2): r inst = r + K r + k r X, where K r = 2.5 log [(time)(area) T r (λ) dλ] is the zero-point term Observational Astronomy (2011) 5
The difference between the magnitudes in two different passbands (e.g., g and r Sloan filters) is called a colour index (see Appendix III): g r = 2.5 log [F(λ g ) / F(λ r )] Observational Astronomy (2011) 6
IMAGE SHAPE ANALYSIS Point-like sources (stars o quasars) are seen as 2D distributions of light (atmospheric seeing) C ij i j This face-on spiral galaxy appears as an extended object (convolution of its intrinsic shape and the PSF) FWHM = 1.3 Point-Spread Function (PSF) = C ij /C C = i,j C ij Observational Astronomy (2011) 7
BACKGROUND AND APERTURE PHOTOMETRY Central pixel: C 00* = 2406 counts C ij* represents the raw counts (incorporating the background) in the pixel (i,j) Pixels around the star: mean (and median) background b = 650 counts. Hence, the true central counts (corresponding to the star) are C 00 = C 00* b = 1756 Photometric aperture (N pixels) Background (sky) annulus (M pixels). b is estimated from these M pixels The true counts inside the photometric aperture of radius R are C R = i,j C ij* Nb Observational Astronomy (2011) 8
APERTURE RADIUS AND DIFFERENTIAL PHOTOMETRY C ij i j R Basically 100% of the counts are included within a radius R = 3 FWHM. However, sometimes astronomers take other radii, e.g., R = FWHM or R = 2 FWHM (to avoid the noisy outer region). In these last cases, C R = p R C (p R is the the fraction of the total counts encircled by the aperture) A B The differential photometry has some important advantages over the absolute photometry, which requires to estimate the zero-point and extinction terms. One can use an aperture of radius R for two stars (e.g., the A and B stars in the r Sloan passband), and obtain r inst (A) r inst (B) = 2.5 log [C r,r (A)/C r,r (B)] = r(a) r(b). If one of both stars is well-identified, i.e., it does not vary over time and its r-band magnitude is known, the other star can be studied in a direct way Observational Astronomy (2011) 9
Poissonian model: SIGNAL, ERROR and SIGNAL-TO-NOISE RATIO Consider a star and its signal (electrons) within an aperture of radius R. The signal S is given by S = i,j (S ij + B ij ) NB, where the sum is over N pixels in the aperture, B ij are the background e in the pixel (i,j), and B is the mean electronic background (e /pixel). Assuming Poissonian noise, the error in the measurement of S can be easily estimated from the equation σ S2 = i,j (S ij + B ij ) + N 2 σ B2 = S + NB + N 2 σ B2. To compute B, we take the mean level within the sky annulus containing M pixels, i.e., B = (1/M) k,l B kl. For this mean background, σ B2 = (1/M 2 ) k,l B kl = B/M. The uncertainty in S can be rewritten as σ S2 = S + NM σ B2 + N 2 σ B2 = S + N 2 σ B2 (1 + M/N). In general, the error in determining the background within the aperture is a small contribution to the total error (M >> N), so σ S (S + NB) 1/2. The signal-to-noise ratio is then S/N S/(S + NB) 1/2. Observational Astronomy (2011) 10
The number of e and the number of counts are related by the gain factor G (e /count). Thus, if we know the counts in the aperture (C R ) and background pixels (b), then S/N G C R / (G C R + G N b) 1/2. What about the (photometric) errors in C R = S/G and m R = 2.5 log C R? σ(c R ) = σ S/G = σ S /G = [C R /G + N(b/G)(1 + N/M)] 1/2 and σ(m R ) = 1.0857 σ(c R )/C R BIBLIOGRAPHY 1.- Optical Photometry by Sergio Ilovaisky, 5th NEON School (2006). Available at http://www.obs-hp.fr/www/ecole-ete/photometry.pdf. This includes a list of useful references (see also the collection of books in the Library of the Faculy of Sciences of the UC) 2.- A Beginner s Guide to Using IRAF by Jeannette Barnes (1993). Available at http://www.physics.ucf.edu/~yfernandez/iraf/beginnersguide.pdf. More information (help) is available at http://iraf.noao.edu/ or during the use of the different IRAF packages Observational Astronomy (2011) 11
Appendix I: Pre-processing of Optical Frames Basic instrumental reductions are applied to a frame before the data are analysed for astronomical/astrophysical purposes. This mainly includes: Bias Subtraction A small part of the frame is not related to the detection of light. The mean level (counts) in this region gives a measure of the average bias introduced by reading the CCD. Once the average bias has been computed, it should be removed from all pixels in the frame. However, there is no way to remove the random pixel-to-pixel fluctuations in the bias (read noise). Fortunately, the Poisson noise associated with photons often dominates this read noise Flat Fielding The response of individual pixels on the CCD is not constant. There are several reasons for this: the reduction in throughput near the edges of the field of view, the presence of dust on the CCD, the variation of the efficiency across the CCD, etc. To correct for this effect, one must acquire flat-field frames, e.g., twilight sky frames. Each flat-field frame corresponds to a constant illumination over the whole field of view, so a flat signal is expected for constant response of the pixels. Real flat-field frames do not contain constant signals, and this allows us to obtain the relative response of each pixel. In order to correct the science frames, one should divide each pixel by its relative response. This procedure effectively leads to normalise the array, or "flatten" it. Observational Astronomy (2011) 12
Appendix II: Celestial Sphere and Positions of Objects The position of an object (e.g., a star) in the sky can be described with just two numbers. Using the equatorial coordinate system, these two numbers are the so-called declination (δ) and right ascension (α). δ It measures the angle of an object above or below the celestial equator (objects in the northern celestial hemisphere have a positive declination) The celestial equator is the plane that contains the Earth s equatorial circle. The celestial poles are simple extensions of the terrestrial poles: north (N) y south (S). α It measures the angle of an object east of the location of the Sun at the moment of the March equinox. Right ascension is usually measured in hours (0-24 h) instead of degrees (0-360 deg) Observational Astronomy (2011) 13
Appendix III: g - r Colour and Surface Temperature Consider a model of a star consisting of a spherical blackbody with a surface temperature T and a radius R. If the star is located at a distance D from the Earth, its photon flux at wavelength λ is given by F(λ) = 2πc (R 2 /D 2 ) {λ 4 [exp(hc / λkt) 1]} -1. From this basic relationship, one can infer a colour index (or simply colour) g r = 2.5 log [F(λ g ) / F(λ r )] = - 2.5 log{λ r4 [exp(hc/λ r kt) 1]/ λ g 4 [exp(hc/λ g kt) 1]}, where λ g = 4686 Ǻ and λ r = 6165 Ǻ. Thus, an observational determination of the g - r colour allows one to estimate the surface temperature of the star. qualitative quantitative f(t) = [exp(hc/λ r kt) 1]/[exp(hc/λ g kt) 1]} = 10-0.4(g r) (λ g /λ r ) 4 Observational Astronomy (2011) 14