Photometry Observational Astronomy 2019 Part 9 Prof. S.C. Trager Photometry is the measurement of magnitudes from images technically, it’s the measurement of light, but astronomers use the above definition these days This sounds simple, but there are several steps required! We’ll focus here on photometry of objects from two- dimensional images (detectors) similar (if not all) steps are needed when using single- pixel detectors like photomultiplier tubes The process of photometry 1. Find the location(s) of the object(s) of interest (perhaps all) 2. Determine the background level(s) B of the object(s) (per pixel) 3. Calculate the integrated source intensity I for each object. To do this, sum S counts from N pixels. Then I = S N B − ⇥ The process of photometry 4. Determine the magnitude: m = 2.5 log I + C − 10 where C is the photometric zeropoint 5. Determine the photometric zeropoint(s), airmass correction(s), and color term(s) (if needed) using photometry of standard objects (almost always these are standard stars) Image centroiding In a digital (or digitized) image, the image of an object is spread over some number of pixels (hopefully!) To measure its flux, we want to know where its center is There are a few ways to do this... Centers • The usual approach is to use ``marginal sums’’. ρ = I : Sum along columns x i ∑ ij j € Image centroiding j i The easiest way to centroid an image is to use marginal distributions Imagine an image of a source with intensity Iij in ADUs in pixel (i,j) Then extract a 2L+1×2L+1 subarray containing – hopefully! – only the source Centers • The usual approach is to use ``marginal sums’’. ρ = I : Sum along columns x i ∑ ij j € Image centroiding j i Then j=L i=L Ii = Iij and Jj = Iij j= L i= L X− X− and the mean intensities in each direction are j=L 1 i=L 1 I¯ = I and J¯ = J 2L +1 i 2L +1 j i= L j= L X− X− Centers • The usual approach is to use ``marginal sums’’. ρ = I : Sum along columns x i ∑ ij j € Image centroiding j i Then the intensity-weighted centroids are i=L ¯ i= L(Ii I)xi x = − − for I I>¯ 0 c i=L ¯ i − P i= L(Ii I) − − and P j=L ¯ j= L(Jj J)yj y = − − for J J>¯ 0 c j=L ¯ j − P j= L(Jj J) − − P Image centroiding This method can locate the peak (center) of an object to a fraction (one-fifth to one-tenth) of a pixel Use this center to then fit, e.g., a two-dimensional Gaussian (for a star) to get a more precise center Background estimation It is always preferable to find the background as close to the object as possible When measuring magnitudes of compact objects (stars, small object galaxies), it is common to determine the background in an annulus sky annulus around the object Background estimation the annulus is often circular or elliptical, but can be a more complicated shape object not done when doing surface photometry sky annulus Background estimation Once the counts in the Mode (peak of this histogram) sky annulus have been measured, compute a Median (1/2 above, 1/2 below) histogram of these values Arithmetic mean then take the mode of #of pixels the distribution (the most probable value) to determine the background level per Counts Because essentially all deviations from the sky are pixel positive counts (stars and galaxies), the mode is the best approximation to the sky. Background estimation For surface photometry, you should fit a constant value or a plane or a surface to “blank” regions of the image and subtract this from the entire image Note: surface photometry is the measurement of magnitudes per unit area (on the sky) Determining source intensity There are two approaches to this step, depending on need and source type: aperture photometry PSF (fitting) photometry ...and there’s also surface photometry, which we’ll describe briefly below Aperture photometry In this case, we count the flux from the object and the sky within some aperture typically we use circular apertures for stellar photometry, but the apertures can be arbitrarily shaped for, say, complex galaxies Then the source intensity is I = I B N ij − ⇥ aperture aperture X Aperture photometry Problems... How big should the apertures be? Ideally, you’d like to get all of the light from your object... but even stars have very extended images Profile of a stellar image on a photographic plate (King 1971) The radius of this maximum S/N is often parameterized as the “Petrosian Aperture photometry radius” Note that a Gaussian profile contains 99% of its light within 10σ≈4 FWHM (because FWHM=2.355σ) But! As the aperture grows, S = aperture I ij increases, but so does Naperture×B (because N gets bigger) P therefore the noise increases, because N∝r2, where r is the radius (size) of the aperture therefore the maximum S/N occurs at some intermediate radius, depending on FWHM Aperture photometry If we restrict size to maximize S/N, we’re not measuring all of the flux Either measure and compare all objects through the same aperture ...or... Use the fact that the profile is the same for all stars (hopefully!) and measure a bright, well-exposed, unsaturated, isolated star out to 4 FWHM. Then use the magnitude difference between this aperture and your smaller aperture to correct all the photometry Curve-of-growth analysis 0.05 ] ) n ( r e p 0 a - ) 1 + n -0.05 ( r e p a [ g -0.10 a m ! -0.15 5 10 15 20 25 30 Aperture Radius Curve-of-growth analysis 0.05 ] ) n ( r e p 0 Sky over-subtracted a - ) 1 + n -0.05 ( r e p a [ g -0.10 a m ! -0.15 5 10 15 20 25 30 Aperture Radius Aperture photometry More problems... Cosmic rays or bad pixels contaminating your aperture just discard this object — why didn’t you take multiple images at slightly different pointings? Nearby stars (or other objects) contaminate your aperture not a problem for sky estimate because we used the mode but if your aperture is contaminated, need to discard object PSF (fitting) photometry To get around the problem of contamination, or more generically, for crowded fields, use the expectation that all stars have the same shape and vary only in brightness This means all the stars have the same point- spread function (PSF) beta is usually ≈2.5 PSF (fitting) photometry I0 is the peak intensity For ground-based observations, the PSF is usually a Gaussian r2/2σ2 2 2 I = I e− , where r = (x x ) +(y y ) xy 0 − 0 − 0 or a Moffat profile p β 2β 1 r 2 − I = I − 1+ xy 0 ↵2 ↵ ⇣ ⌘ PSF (fitting) photometry By assuming that the width of the stars (σ or α) is constant (or at least smoothly, slowly varying), we can fit Gaussians or Moffat profiles to every star, just varying I0 Then the magnitude difference between any two stars in the frame is I m m = 2.5 log 0,1 1 − 2 − 10 I ✓ 0,2 ◆ and so m = 2.5 log I − 10 0 PSF (fitting) photometry Advantages more accurate than aperture photometry robust against cosmic rays, bad pixels, neighboring objects can measure crowded fields because overlaps can be controlled for or, better, simultaneously fit Implemented in DAOPHOT (I+II), DoPHOT, ROMAPHOT, HST/DolPHOT (and others...) PSF (fitting) photometry Disadvantages Much more computationally expensive than aperture photometry Still need to perform aperture photometry on some stars to correct for light missed by PSF template how good is the template? how big is the template? Surface photometry Like aperture photometry on a large scale Determines intensity per unit (angular) area on the sky: SB = I B N /area ij − ⇥ aperture aperture ! X Often use elliptical apertures for this Surface photometry gives light profile and shape information (using parameters of aperture fits) Surface brightness profiles The surface brightness of a galaxy I(x) is the amount of light contained in some small area at a particular point x in an image Consider a square area with a side of length D of a galaxy at distance d. This length will subtend an angle α = D/d If the total luminosity of the galaxy in that area is L, then the received flux is F = L/(4πd2) So the surface brightness is I(x)=F/α2 = L/(4⇥d2)(d/D)2 = L/(4⇥D2) which is independent of distance note that this is not true at cosmological distances! 2 The units of I(x) are usually given in L pc− Often the magnitude per square arcsecond is quoted as the surface brightness: µ (x)= 2.5 log I (x) + constant λ − λ λ In the B-band, the constant is 27 mag/arcsec2, 2 which corresponds to 1 L pc− 0.4(µB 27) 2 Thus IB = 10− − L ,Bpc− Contours of constant surface brightness in an image are called isophotes NGC 7331 If we plot the surface brightness profile of NGC 7331 in magnitudes/arcsec2 as a function of radius, we find a straight line far away from the center This implies that its disk has an exponential profile: I(R)=I exp( R/h ) 0 − R Surface brightness profiles Surface brightness profiles Elliptical galaxies have very Surfacesmooth brightness profiles profile over 2 ordersfor giant ellipticalof magnitude galaxy: in asradius, f(R) and usually falling off as R1/4 f(R1/4).