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Imaging and • Now essentially always done with imaging arrays (e.g., Ay 122 - Fall 2006 CCDs); it used to be with single-channel instruments • Two basic purposes: 1. (photometry) Imaging and Photometry • photometry or S/N-like weighting • For unresolved sources: PSF fitting • Could be time-resolved (e.g., for variability) • Could involve polarimetry • Panoramic imaging especially useful if the surface density of sources is high 2. Morphology and structures (Many slides today c/o • Surface photometry or other parametrizations Mike Bolte, UCSC)

What Properties of Electromagnetic Measuring Flux = Energy/(unit time)/(unit area) Radiation Can We Measure? Real detectors are sensitive over a finite range of ! (or "). • Specific flux = (in ergs or photons) per unit are always measured over some finite bandpass. area (or ), time, (or ), Total energy flux: Integral of f" over -15 2 F = F" (")d" e.g., f! = 10 erg/cm /s/Å - a good spectroscopic unit # all • It is usually integrated over some finite bandpass (as in -1 -2 -1 photometry) or a spectral resolution element or a line Units: erg s cm Hz • It can be distributed on the sky (surface photometry, A standar!d unit for specific flux (initially in radio, but now e.g., ), or changing in time (variable sources) more common): 1 (Jy) =10"23 erg s-1 cm-2 Hz-1 • You can also measure the polarization parameters (photometry ! polarimetry, ! spectro- f" is often called the flux density - to get the power, one polarimetry); common in radio integrates it over the bandwith, and multiplies by the area ! (From P. Armitage) Fluxes and Magnitudes Johnson ! For historical reasons, fluxes in the optical and IR are measured in magnitudes: Gunn/SDSS m = "2.5log10 F + constant ! If F is the total flux, then m is the bolometric . Usually instead consider a finite bandpass, e.g., V band. ! e.g. in the V band (! ~ 550 nm): Some Common f ! mV = "2.5log10 F + constant Photometric Systems flux integrated over the range (in the visible) !! of for this band (From P. Armitage)

There are … and more … way, way … and more … too many … and more … photometric systems out there …

(Bandpass curves from Fukugita et al. 1995, PASP, 107, 945) Using Magnitudes Magnitude Zero Points

Consider two , one of which is a hundred times f! fainter than the other in some waveband (say V). = Alas, for the standard UBVRIJKL… # Lyrae system (and many others) the m1 = "2.5logF1 + constant magnitude zero-point in any band m = "2.5log(0.01F ) + constant is determined by the spectrum 2 1 of Vega ! const! = "2.5log(0.01) " 2.5logF1 + constant

= 5 " 2.5logF1 + constant U B V R I = 5 + m1 Source that is 100 times fainter in flux is five magnitudes ! -9 2 fainter (larger number). Vega calibration (m = 0): at ! = 5556: f! = 3.39 !10 erg/cm /s/Å f = 3.50 !10 -20 erg/cm2/s/Hz Faintest objects detectable with HST have magnitudes of " ! A more logical system is AB 2 " N! = 948 photons/cm /s/Å ~ 28 in R/I bands. The has mV = -26.75 mag magnitudes: AB" = -2.5 log f" [cgs] - 48.60 (From P. Armitage)

Photometric Zero-Points (Visible) Magnitudes, A Formal Definition

e.g., Because Vega (= # Lyrae) is declared to be the zero-point! (at least for the UBV… system)

(From Fukugita et al. 1995) Defining The Photometric Bands effective … where the atmospheric transmission windows are wavelengths (and the corresponding bandpass averaged fluxes)

Infrared Bandpasses Infrared Bandpasses IR Sky Backgrounds IR Sky Backgrounds

108 thermal IR OH can be resolved

I 105 !-4 Rayleigh scattered

273K blackbody (atmosphere, ) atmosphere 102 Zodiacal

1µ 2µ 10µ 100µ

Colors From Magnitudes Apparent vs. Absolute Magnitudes The color of an object is defined as the difference in the The is defined as the apparent mag. magnitude in each of two bandpasses: e.g. the (B-V) a source would have if it were at a distance of 10 pc: color is: B-V = mB-mV M = m + 5 - 5 log d/pc

Stars radiate roughly as It is a measure of the in some waveband. blackbodies, so the color For Sun: M"B = 5.47, M"V = 4.82, M"bol = 4.74 reflects surface . Difference between the # d & m and m " M = 5log10% ( Vega has T = 9500 K, by $ 10 pc' definition color is zero. the absolute magnitude M (any band) is a measure of the distance to the source (From P. Armitage) ! (From P. Armitage) Why Do We Need The Concept of Signal-to-Noise (S/N) This Mess? or: How good is that really?

Relative measurements • S/N = signal/error (If the noise is Gaussian, we are generally easier and speak of 3-$, 5- $, … detections. This translates into a more robust than the probability that the detection is spurious.) absolute ones; and often that is enough. • For a counting process (e.g., photons), error = " n, and An example: the Color- thus S/N = n / " n = " n (“Poissonian noise”). This is Magnitude Diagram the minimum possible error; there may be other sources The quantitative of error (e.g., from the detector itself) operational framework • If a source is seen against some back(fore)ground, then for studies of stellar 2 2 2 2 physics, evolution, $ total = $ signal + $ background + $ other populations, distances …

Signal-to-Noise (S/N) • Noise Sources:

R* " t # shot noise from source

2 Rsky " t " $r # shot noise from sky in aperture • Signal=R • t r * 2 2 time RN " $r # readout noise in aperture [RN 2 + (0.5 % gain)2] " $r2 # more general RN

detected rate in e-/second 2 I Dark " t " $r # shot noise in dark current in aperture • Consider the case where sky R = e&/sec from the source we count all the detected r * & e- in a circular aperture Rsky = e /sec/pixel from the sky with radius r. RN = read noise (as if RN2 e& had been detected) Dark = e_ /second/pixel

! S/N for object measured in aperture with radius r: npix=# of pixels in the aperture= #r2 Noise from the dark Side Issue: S/N %mag current in aperture

Signal R*t m ± "(m) = co # 2.5log(S ± N) 1 N 2 = c # 2.5log S 1± ) # gain& , 2 o [ ( S )] Noise R t R t n RN n Dark t n N + * " + sky " " pix + % + ( " pix + " " pix . = c # 2.5log(S) # 2.5log(1± ) * $ 2 ' - o S m %m Readnoise in aperture 2 (R* " t) ! 1 ! Noise from sky e- in aperture "(m) # 2.5log(1+ S / N ) 2.5 Note:in log +/- not N 1 N 2 1 N 3 = [ S $ 2 ( S ) + 3 ( S ) $ ...] symetric 2.3 ! All the noise terms added in quadrature N Fractional error #1.087( S ) Note: always calculate in e- This is the basis of people referring to +/- 0.02mag error as “2%”

!

Sky Background Scale ""/pix (LRIS - R : 0.218"/pix) Area of 1 pixel = (Scale)2 (LRIS # R : 0.0475"2 Signal from the sky background is this is the ratio of flux/pix to flux/" present in every pixel of the In magnitudes : NaD aperture. Because each instrument OH generally has a different pixel scale, I = I Scale2 I " Intensity (e-/sec) the sky is usually pix " Hg tabulated for a site in units of 2 #2.5log(Ipix ) = #2.5[log(I ) + log(Scale )] mag/arcsecond2. " (mag/ ) m = m # 2.5log(Scale2) (for LRIS - R : add 3.303mag) ˝ pix " Lunar and age U B V R I (days) R (m ) = R(m = 20) $10(0.4#m pix ) 0 22.0 22.7 21.8 20.9 19.9 sky pix 3 21.5 22.4 21.7 20.8 19.9 Example, LRIS in the R - band : 7 19.9 21.6 21.4 20.6 19.7 R =1890 $100.4(20#24.21) = 39.1 e- /pix/sec 10 18.5 20.7 20.7 20.3 19.5 sky 14 17.0 19.5 20.0 19.9 19.2 - R sky = 6.35e /pix/sec % RN in just 1 second

! S/N - some limiting cases. Let’s assume CCD with Dark=0, well sampled read noise. What is ignored in this S/N eqn? R*t 1 2 R " t + R " t " n + RN " n 2 [ * sky pix ( ) pix ] • Bias level/structure correction • Flat-fielding errors Bright Sources: (R t)1/2 dominates noise term * • Charge Transfer Efficiency (CTE) ! 1 R*t 2 0.99999/pixel transfer S/N " = R*t #t R t * • Non-linearity when approaching full well R t • Scale changes in focal plane Sky Limited ( R t > 3" RN) : S/N # * # t sky n R t ! pix sky • A zillion other potential problems

Note: seeing comes in with npix term !

Photometry With An Imaging Array Aperture Photometry

• Aperture Photometry: some modern ref’s I = I # n $ sky/pixel – DaCosta, 1992, ASP Conf Ser 23 " ij pix ij – Stetson, 1987, PASP, 99, 191 – Stetson, 1990, PASP, 102, 932 Total counts in Number of pixels in aperture aperture from source Determine sky ! Sum counts in all in annulus, pixels in aperture subtract off Counts in each pixel in aperture sky/pixel in central aperture m = c0 - 2.5log(I) Aperture Photometry Centers

• What do you need? • The usual approach is to use ``marginal sums’’.

– Source center " = I : Sum along columns x i # ij – Sky value j – Aperture radius

! j

i

• Find peaks: use $ & /$ x zeros Marginal Sums x • Isolate peaks: use ``symmetry cleaning’’ 1. Find peak • With noise and multiple sources you have to decide what is a source and to isolate sources. 2. compare pairs of points equidistant from center

3. if Ileft >> Iright, set Ileft = Iright • Finding centers: Intensity-weighted centroid " x " x2 # x i i # i i i 2 i 2 xcenter = ; $ = % xi " " # x i # i i i

! • Alternative for centers: Gaussian fit to &:

2 [#((x i #x c )/$ ) / 2] Sky "i = background + h•e • To determine the sky, typically use a local Solving for center Height of peak annulus, evaluate the distribution of counts ! in pixels in a way to reject the bias toward higher-than-background values. • DAOPHOT FIND algorithm uses marginal sums in subrasters, symmetry cleaning, • Remember the 3 Ms. reraster and Gaussian fit.

Mode (peak of this histogram) Sky From Minmax Rejection

Median (1/2 above, 1/2 below)

Arithmetic mean

#of pixels

Pixel value: 1000 200 180 180 Average with minmax rejection, reject 2 highest value averaging Counts lowest two will give the sky value. NOTE! Must normalize Because essentially all deviations from the sky are frames to common mean or mode before combining! Sometimes positive counts (stars and galaxies), the mode is the it is necessary to pre-clean the frames before combining. best approximation to the sky. • Radial intensity distribution for a faint : Aperture and growth curves Same frames as previous example

• First, it is VERY hard to measure the total light as The wings of a some light is scattered to very large radius. faint star are lost Perhaps you have most of to sky noise at a the light within this radius different radius than the wings of a bright star. Radial intensity distribution for a bright, isolated star.

Inner/outer sky radii Bright star aperture Radius from center in pixels

One Approach Is To Use “Growth Curves” Radial profile with neighbors • Idea is to use a small aperture (highest S against background and smaller chance of contamination) for Neighbors OK in everything and determine a correction to larger radii sky annulus (mode), trouble based on several relatively isolates, relatively bright in star stars in a frame. • Note! This assumes a linear response so that all point sources have the same fraction of light within a given radius. • Howell, 1989, PASP, 101, 616 • Stetson, 1990, PASP, 102, 932 $ mag for Growth Curves apertures n-1, n

Aperture 2 3 4 5 6 7 8

Star#1 0.43 0.31 0.17 0.09 0.05 0.02 0.00

Star#2 0.42 0.33 0.19 0.08 0.21 0.11 0.04

Star#3 0.43 0.32 0.18 0.10 0.06 0.02 -0.01

Star#4 0.44 0.33 0.18 0.22 0.14 0.12 0.14

Star#5 0.42 0.32 0.18 0.09 0.19 0.21 0.19

Star#6 0.41 0.33 0.19 0.10 0.05 0.30 0.12

cMean 0.430 0.324 0.184 0.094 0.057 0.02 0.00

Sum of these is the total aperture correction to be added to magnitude measured in aperture 1

0.05 Aperture Photometry Summary (n)] 1. Identify brightness peaks

aper 0

2. "Ixy # (sky•aperture area) n+1) -

( -0.05 xy

per Use small aperture a

[ !

ag -0.10 m 3. Add in ``aperture correction’’

$ ! determined from bright, isolated stars -0.15

5 10 15 20 25 30 Easy, fast, works well except for the case of overlapping images Aperture Radius Crowded-field Photometry • To construct a PSF: 1. Choose unsaturated, relatively isolated stars • As was assumed for aperture corrections, all point 2. If PSF varies over the frame, sample the full sources have the same PSF (linear detector) field • PSF fitting allows for an optimal S/N weighting 3. Make 1st iteration of the PSF • Various codes have been written that do: 4. Subtract psf-star neighbors 1. Automatic star finding 5. Make another PSF 2. Construction of PSF • PSF can be represented either as a table of 3. Fitting of PSF to (multiple) stars numbers, or as a fitting function (e.g., a • Many programs exist: DAOPHOT, ROMAPHOT, DOPHOT, STARMAN, … Gaussian + power-law or exponential • DAOPHOT is perhaps the most useful one: Stetson, wings, etc.), or as a combination 1987, PASP, 99, 191

Photometric Calibration

• The photometric standard systems have tended to be zero- pointed arbitrarily. Vega is the most widely used and was original defined with V= 0 and all colors = 0. • Hayes & Latham (1975, ApJ, 197, 587) put the Vega scale on an absolute scale. • The AB scale (Oke, 1974, ApJS, 27, 21) is a physical-unit- based scale with: m(AB) = -2.5log(f) - 48.60 where f is monochromatic flux is in units of erg/sec/cm2/Hz. Objects with constant flux/unit frequency interval have zero color on this scale Photometric calibration Photometric Calibration

1. Instrumental magnitudes Counts/sec • To convert to a standard magnitude you need to observe some standard stars and m c 2.5log(I t) = 0 " • solve for the constants in an equation like: = c0 " 2.5log(I) " 2.5log(t) 2 minst = M + c0 + c1X + c2(color) + c3(UT) +c4 (color) + ...

minstrumental airmass Stnd mag zpt Color term ! ! Instr mag

Extinction coeff (mag/airmass)

coefficients: – Increase with decreasing wavelength – Can vary by 50% over time and by some amount during Photometric Standards a night – Are measured by observing standards at a range of • Landolt (1992, AJ, 104, 336) airmass during the night • Stetson (2000, PASP, 112, 995) • Fields containing several well measured stars of Slope of this similar brightness and a big range in color. The

line is c1 blue stars are the hard ones to find and several fields are center on PG sources. • Measure the fields over at least the the airmass range of your program objects and intersperse standard field observations throughout the night. Photometric Calibration: Standard Stars We often need to compare observations with models, Magnitudes of Vega (or other Low on the same photometric systems primary flux standards) are system transferred to many other, secondary standards. They are observed along with your main science targets, and (King, Anderson, processed in the same way. NGC 6397 Cool, Piotto, 1999)

Always, always transform

models to observational Intermediate system, e.g., by integrating metallicity model spectra through your (Bedin, Anderson, M4 King, Piotto, 2001) bandpasses

Alas, Even The “Same” Photometric This Generates Color Terms … Systems Are Seldom Really The Same … … From mismatches between the effective bandpasses of your filter system and those of the standard system. Objects with different spectral shapes have different offsets:

A is thus effectively (operationally) defined by a set of standard stars - since the actual bandpasses may not be well known. Surface Photometry Surface Photometry • The way to quantify the 2-dimensional distribution of light, e.g., in galaxies

Simple approach of aperture • Many references, e.g., photometry works OK for some – Davis et al., AJ, 90, 1985 purposes. – Jedrzejewski, MNRAS, 226, 747, 1987 mag=c - 2.5(cnts - #r2sky) 0 aper • Could fit (or find) isophotes, and the most common Typically working with much procedure is to fit elliptical isophotes. larger apertures • Isophotal parameters are: itself ( ), - prone to contamination µ - sky determination even more xcenter, ycenter, ellipticity ()), position angle (PA), the critical enclosed magnitude (m), and sometimes higher order - often want to know more than total brightness shape terms, all as functions of radius (r) or semi-major axis (a).

Start with guesses for xc, yc, fitted isophote R, ) and p.a., then compare true isophote the ellipse with real data all along the ellipse (all ' values) ' (I

'

Fit the (I - ' plot and iterate on xc, yc, p.a., and ) to minimize the coefficients in an expression like: Good isophote Iellipse- Iimage 0 I(')= I0 + A1sin(') + B1cos(') + A2sin(2') + B2cos(2')

' Changes to xc and yc mostly affect A1, B1, p.a. “ “ A2 ) “ “ B2 More specifically, #B1 Examples of Surface "(major axis center) = iterate the following: I $ Photometry of Ellipticals #A (1#%) "(minor axis center) = 1 ! Major axis surf. brightness profiles I $ Isophotal shape and orientation prof’s #2B2 (1#%) "(%) = " a0I $

2A2 (1#%) "(p.a) = 2 a0I $[( 1#%) #1] • After finding the best-fitting elliptical where : isophotes, the residuals are often &I interesting. Fit: I $ = &R a0 I = I0 + Ansin(n') + Bncos(n') already minimized n=1 and n=2, n=3 is usually not significant, but:

B4 is negative for “boxy” isophotes One can also measure the deviations B positive for “disky! ” isophotes 4 from the elliptical shape (boxy/disky)

Disky and Boxy Elliptical Isophotes Calculate mean and RMS pixel intensity for annulus, toss any values above mean + nRMS From your isophotal fits, you can then construct the best 2-dimensional elliptical model for the light distribution

Saturated core

Hidden disk

Bad job of clipping Non elliptical structures … And subtract it from the image to reveal any deviations from the assumed elliptical symmetry Panoramic Imaging Image (De-)Blending and Segmentation How are the individual sources - • Generally used to study populations of sources (e.g., or source components - defined? faint counts; star clusters; etc.) • Commonly done in (wide-area) surveys • Image (pixels) ! catalogs of objects and their measured properties

• If done properly, essentially all information is Easy Not so easy extracted into a more useful form; but not always… • Key steps: Thresholding is an alternative to peak finding. Look for 1. Object finding (there is always some spatial “filter”) contiguous pixels above a threshold value. 2. Object measurements / parametrization • User sets area, threshold value. 3. Object classification (e.g., star/galaxy) • Sometimes combine with a smoothing filter Deblending based on multiple-pass thresholding

• One very common program for panoramic photometry is Sextractor – Bertin & Arnouts, 1996, A&AS, 117, 393 • Not for good a detailed surface photometry, but good Sextractor for classification and rough photometric and structural Flowchart parameter derivation for large fields. Steps: 1. Background map (sky determination) 2. Identification of objects (thresholding) 3. Deblending 4. Photometry 5. Shape analysis Star-Galaxy Separation

Star-Galaxy separation • Important, since for most studies you want either stars (or quasars), or galaxies; and then the depth to which a • Galaxies are resolved, stars are not reliable classiffication can be done is the effective limiting depth of your catalog - not the detection depth • All methods use various approaches to comparing – There is generally more to measure for a non-PSF object the amount of light at large and small radii. stars • You’d like to have an automated and objective process, Star with some estimate of the accuracy as a f (mag) 2 – Generally classification fails at the faint end m - 1 galaxy • Most methods use some measures of light concentration m I vs. magnitude (perhaps more than one), and/or some measure of the PSF fit quality (e.g., *2) • For more advanced approaches, use some machine m 1 learning method, e.g., neural nets or decision trees galaxies too noisy Pixel position

Sextractor Star/Galaxy Typical Parameter Space for S/G Classif. Separation • Lots of talk about neural-net algorithms, but in the end it is a moment analysis. Stellar locus • “Stellarity”. Typically test it with artificial stars and find it # is very good to some .

Galaxies

(From DPOSS) s-g going bad at R~22 More S/G Classification Parameter Automated Star-Galaxy Classification: Spaces: Normalized By The Stellar Locus Artificial Neural Nets (ANN)

Output: Input: Star, p(s) various image shape Galaxy, p(g) parameters.

Other, p(o)

Then a set of such parameters can be fed into an automated classifier (ANN, DT, …) which can be trained with a “ground truth” sample (Odewahn et al. 1992)

Automated Star-Galaxy Classification: Automated Star-Galaxy Classification: Decision Trees (DTs) Unsupervised Classifiers No training data set - the program decides on the number of classes present in the data, and partitions the data set accordingly.

Star An example: AutoClass (Cheeseman et al.) Uses Bayesian Star+fuzz approach in machine learning (ML). Gal1 (E?) This application from DPOSS Gal2 (Sp?) (Weir et al. 1995) (Weir et al. 1995) Seeing - A Key Issue • The “image size”, given by a convolution of the atmospheric turbulence, and instrument • Affects two important aspects of imaging photometry: Example 1. S/N: because a larger image spreads the available signal of seeing over more noise-contributing pixels variations • Defines the detection limit in the Good seeing • Should have pixel scale matched to the expected image quality, ground- to at least Nyquist sampling 2. Morphological resolution: as high spatial frequency based data information is lost (from the • Defines the star-galaxy classification limit PQ survey) • Can be recovered only partly by image deconvolutions, which require a good S/N, sampling, and a well measured PSF • A tricky issue is how to combine data obtained in different seeing conditions Mediocre seeing

Summary of the Key Points • Photometry = flux measurement over a finite bandpass, could be integral (the entire object) or resolved (surface photometry) • The arcana of the magnitudes and many different photometric systems … ! • The S/N computation - many sources of noise • Issues in the photometry with an imaging array: object finding and centering, sky determination, aperture photometry, PSF fitting, calibrations … • Surface photometry and isophote fitting lore … • Star-galaxy classification in panoramic imaging • The effects of the seeing: flux measurements and morphology