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 intensityweighted 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 (oneﬁfth to onetenth) of a pixel Use this center to then ﬁt, e.g., a twodimensional 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, wellexposed, unsaturated, isolated star out to 4 FWHM. Then use the magnitude difference between this aperture and your smaller aperture to correct all the photometry Curveofgrowth analysis
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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 groundbased observations, the PSF is usually a Gaussian
r2/22 2 2 I = I e , where r = (x x ) +(y y ) xy 0 0 0 or a Moﬀat proﬁle p 2 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 = D/d If the total luminosity of the galaxy in that area is L, then the received ﬂux is F = L/(4d2) So the surface brightness is
I(x)=F/2 = 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 In the Bband, 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). In NGC 1700, surface brightness falls by 9 Very good magnitudes/arcsecfit over 2 decades2 —in radius 4000x! — over 100x in radius Light in ellipticals is highly concentrated NGC 1700
The surface brightness falls 9 magnitudes from centre to outskirts: 109 falloff in projected luminosity!
The light in elliptical galaxies is quite centrally concentrated
Photometric calibration
To determine our photometric zeropoints, m = 2.5 log I + ZP 10 we need to observe objects — usually stars — of known magnitudes and colors at many different hour angles Photometric calibration
We need to correct three major effects: 1. overall magnitude offset: what magnitude corresponds to, say, one e–/s at X=1? 2. color shifts between your ﬁlters and the standard stars’ ﬁlters 3. atmospheric extinction
note that there will also be different kλ for differentcolored stars, due to the width of the broadband ﬁlter compared to the slope of kλ and the shape of the stars’ spectra
• The color terms come about through mismatches between the effective bandpasses of your filter system and those of the standard system. Objects with different spectral shapes have different offsets. Color terms Photometric calibration
Combining 1) and 2), we have
m = m + b + b c + b c2 + true 0,inst 0 1 2 ··· where c is the color of your object And 3) means V=v1+a0 m = m kX + k0cX 0,inst X where c is the color, k is the extinction coefﬁcient, k′ is the differential color–extinction RMS=0.055 coefﬁcient, and mX is the magnitude observed at airmass X just magnitude zeropoint
Photometric calibration
Combining 1) and 2), we have
m = m + b + b c + b c2 + true 0,inst 0 1 2 ··· where c is the color of your object And 3) means V=vinst+ c0 + c1X m = m kX + k0cX 0,inst X where c is the color, k is the extinction coefﬁcient, k′ is the differential color–extinction RMS=0.032 coefﬁcient, and mX is the magnitude observed at airmass X zeropoint and airmass Photometric calibration
Combining 1) and 2), we have
m = m + b + b c + b c2 + true 0,inst 0 1 2 ··· where c is the color of your object And 3) means
V=vinst+c0+c1X+c2(BV) m = m kX + k0cX 0,inst X where c is the color, k is the extinction coefﬁcient, k′ is the differential color–extinction RMS=0.021 coefﬁcient, and mX is the magnitude observed at airmass X zeropoint, airmass, and color
Photometric calibration
Thus, for a star of known mtrue and c observed at X, m = m + a + a X + a c + a cX + a c2 + true X 0 1 2 3 4 ··· Since each star satisﬁes this equation, a system of linear polynomial equations exist, and we can invert this system to get our necessary coefﬁcients ai Lists of standard stars can be found in papers by Landolt, Graham, and Stetson, and should be available at any observatory! I have used the data from 09.09.2015 to determine the photometric calibra tion for the BV R ﬁlters of the Gratama Telescope at the Blaauw Observatory. I used images of the ﬁelds SA 110, SA 35, and SA 38. The beginning of this night appears to have been reasonably photometric, although the scatter in the photometry is occasionally larger than the photon noise would suggest. I had to remove one Rband image of SA 38 whose photometry had a signiﬁcant o↵set from the other two Rband images. PhotometricI have ﬁt functions of the form calibration of mag mag = a + a c + a X + a cX, obs true 0 1 2 3 where magobs is the “instrumental magnitude” (the magnitude produced by 1 phot in IRAF, with a zeropoint of 25.0 corresponding to 1 e s ), magtrue is thethetrue magnitude (fromGratama Landolt 2009, 2013), c is the true color of the star,Telescope and X is the airmass at which the image of the star was taken. I performed this step for all three ﬁlters BV R and all three colors (B V ), (B R), and (V R). The results of these ﬁts are show in the ﬁgure on the next page. I then inverted the equations to determine the true magnitudes from the observedUsingmagnitudes anddata colors. In from the following 2015 equations, a capital letter (like “B”) represents the true magnitude (here in the B ﬁlter) and a small letter (likeSeptember “b”) represents the observed 9, magnitude I found (here again in the B ﬁlter).
(0.060 0.233X)( 0.048 + 0.607X) B = b (4.501 + 0.618X) (b r) 1.032 0.261X (0.028 + 0.028X)( 0.048 + 0.607X) R = r (4.549 + 0.011X) (b r) 1.032 0.261X (0.017 0.132X)( 0.132 + 0.436X) B = b (4.505 + 0.617X) (b v) 0.942 0.114X (0.074 0.018X)( 0.132 + 0.436X) V = v (4.637 + 0.181X) (b v) 0.942 0.114X (0.150 0.040X)(0.086 + 0.171X) V = v (4.644 + 0.173X) (v r) 1.055 0.106X (0.095 + 0.066X)(0.086 + 0.171X) R = r (4.558 + 0.002X) (v r) 1.055 0.106X
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Figure 1: The photometric solutions.
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The photometry “Golden Rules”
Always observe standard stars with colors bracketing the colors of the objects you want to calibrate Always observe standard stars at airmasses spanning the airmasses of your target exposures Only use very clear (photometric) weather! No clouds. The photometry “Golden Rules” Use blue ﬁlters at low X and least moon Save red ﬁlters for higher airmasses and more moon Try to work at X<1.5 On big telescopes (>2m) with CCD cameras, standard stars are very easy to saturate Use short exposures to get bright standard stars but not too short to avoid scintillation (>5–10 s) Use longer exposures to get faint standard stars