Imaging and Photometry

Transcription

1 Imaging and Photometry Ay Fall 2006 Imaging and Photometry (Many slides today c/o Mike Bolte, UCSC) Now essentially always done with imaging arrays (e.g., CCDs); it used to be with single-channel instruments Two basic purposes: 1. Flux measurements (photometry) Aperture 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 Surface photometry or other parametrizations What Properties of Electromagnetic Radiation Can We Measure? Specific flux = Intensity (in ergs or photons) per unit area (or solid angle), time, wavelength (or frequency), e.g., f! = erg/cm 2 /s/å - a good spectroscopic unit It is usually integrated over some finite bandpass (as in photometry) or a spectral resolution element or a line It can be distributed on the sky (surface photometry, e.g., galaxies), or changing in time (variable sources) You can also measure the polarization parameters (photometry! polarimetry, spectroscopy! spectropolarimetry); common in radio astronomy Measuring Flux = Energy/(unit time)/(unit area) Real detectors are sensitive over a finite range of! (or "). Fluxes are always measured over some finite bandpass. Total energy flux: F = # F " (")d" Units: erg s -1 cm -2 Hz -1 Integral of f " over all frequencies A standard unit for specific flux (initially in radio, but now more common): 1 Jansky (Jy) =10 "23 erg s -1 cm -2 Hz -1 f " is often called the flux density - to get the power, one integrates it over the bandwith, and multiplies by the area (From P. Armitage)

2 Fluxes and Magnitudes For historical reasons, fluxes in the optical and IR are measured in magnitudes: m = "2.5log 10 F + constant If F is the total flux, then m is the bolometric magnitude. Usually instead consider a finite bandpass, e.g., V band. Johnson! Gunn/SDSS! f!! e.g. in the V band (! ~ 550 nm): m V = "2.5log 10 F + constant flux integrated over the range of wavelengths for this band (From P. Armitage) Some Common Photometric Systems (in the visible) There are way, way too many photometric systems out there and more and more and more (Bandpass curves from Fukugita et al. 1995, PASP, 107, 945)

3 Using Magnitudes Consider two stars, one of which is a hundred times fainter than the other in some waveband (say V). m 1 = "2.5log F 1 + constant m 2 = "2.5log(0.01F 1 ) + constant = "2.5log(0.01) " 2.5logF 1 + constant = 5 " 2.5logF 1 + constant = 5 + m 1 Source that is 100 times fainter in flux is five magnitudes fainter (larger number). Faintest objects detectable with HST have magnitudes of ~ 28 in R/I bands. The sun has m V = mag (From P. Armitage) f! Vega = # Lyrae Magnitude Zero Points Alas, for the standard UBVRIJKL system (and many others) the magnitude zero-point in any band is determined by the spectrum of Vega! const! U B V R I! Vega calibration (m = 0): at! = 5556: f! = 3.39!10-9 erg/cm 2 /s/å f " = 3.50!10-20 erg/cm 2 /s/hz A more logical system is AB " N! = 948 photons/cm 2 /s/å magnitudes: AB" = -2.5 log f " [cgs] 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)

4 Defining effective wavelengths (and the corresponding bandpass averaged fluxes) The Infrared Photometric Bands where the atmospheric transmission windows are Infrared Bandpasses Infrared Bandpasses

5 IR Sky Backgrounds IR Sky Backgrounds 10 8 thermal IR OH can be resolved I 10 5! -4 Rayleigh scattered sunlight K blackbody (atmosphere, telescopes) Zodiacal light atmosphere 1µ 10µ 100µ 2µ Colors From Magnitudes The color of an object is defined as the difference in the magnitude in each of two bandpasses: e.g. the (B-V) color is: B-V = m B -m V Stars radiate roughly as blackbodies, so the color reflects surface temperature. Vega has T = 9500 K, by definition color is zero. (From P. Armitage) Apparent vs. Absolute Magnitudes The absolute magnitude is defined as the apparent mag. a source would have if it were at a distance of 10 pc: M = m log d/pc It is a measure of the luminosity in some waveband. For Sun: M "B = 5.47, M "V = 4.82, M "bol = 4.74 Difference between the apparent magnitude m and the absolute magnitude M (any band) is a measure of the distance to the source # d & m " M = 5log 10 % ( $ 10 pc' Distance modulus (From P. Armitage)

6 Why Do We Need This Mess? Relative measurements are generally easier and more robust than the absolute ones; and often that is enough. An example: the Color- Magnitude Diagram The quantitative operational framework for studies of stellar physics, evolution, populations, distances The Concept of Signal-to-Noise (S/N) or: How good is that measurement really? S/N = signal/error (If the noise is Gaussian, we speak of 3-$, 5- $, detections. This translates into a probability that the detection is spurious.) For a counting process (e.g., photons), error = " n, and thus S/N = n / " n = " n ( Poissonian noise ). This is the minimum possible error; there may be other sources of error (e.g., from the detector itself) If a source is seen against some back(fore)ground, then $ 2 total = $ 2 signal + $ 2 background + $ 2 other Signal-to-Noise (S/N) Noise Sources: R * " t # shot noise from source Signal=R * t time detected rate in e-/second I r R sky " t " $r 2 RN 2 " $r 2 # shot noise from sky in aperture # readout noise in aperture [ RN 2 + (0.5 % gain) 2 ] " $r 2 # more general RN Dark " t " $r 2 # shot noise in dark current in aperture Consider the case where we count all the detected e- in a circular aperture with radius r. sky r R * = e & /sec from the source R sky = e & /sec/pixel from the sky RN = read noise (as if RN 2 e & had been detected) Dark = e _ /second/pixel

7 S/N for object measured in aperture with radius r: n pix =# of pixels in the aperture= #r 2 Noise from the dark current in aperture Signal R * t Noise (R * " t) 2 ) # R * " t + R sky " t " n pix + RN + gain 2 &, + % ( " n $ 2 ' pix + Dark " t " n pix. * - Readnoise in aperture Noise from sky e- in aperture All the noise terms added in quadrature Note: always calculate in e- 1 2 Note:in log +/- not symetric Side Issue: S/N m ± "(m) = c o # 2.5log(S ± N) [ ( )] = c o # 2.5log S 1± N S %mag = c o # 2.5log(S) # 2.5log(1± N S ) m %m "(m) # 2.5log(1+ 1 ) S / N = 2.5 N [ S 2.3 $ 1 ( N 2 S )2 + 1 ( N 3 S )3 $...] #1.087( N ) Fractional error S This is the basis of people referring to +/- 0.02mag error as 2% NaD Hg OH Sky Background Signal from the sky background is present in every pixel of the aperture. Because each instrument generally has a different pixel scale, the sky brightness is usually tabulated for a site in units of mag/arcsecond 2. (mag/ ) Lunar age (days) U B V R I Scale ""/pix (LRIS - R : 0.218"/pix) Area of 1 pixel = (Scale) 2 (LRIS # R : " 2 this is the ratio of flux/pix to flux/" In magnitudes : I pix = I Scale 2 " I " Intensity (e - /sec) #2.5log(I pix ) = #2.5[log(I ) + log(scale 2 )] " m pix = m # 2.5log(Scale 2 ) " (for LRIS - R : add 3.303mag) and R sky (m pix ) = R(m = 20) $10 (0.4#m pix ) Example, LRIS in the R - band : R sky =1890 $10 0.4(20#24.21) = 39.1 e - /pix/sec R sky = 6.35e - /pix/sec % RN in just 1 second

8 S/N - some limiting cases. Let s assume CCD with Dark=0, well sampled read noise. R * t What is ignored in this S/N eqn? Bright Sources: [ R * " t + R sky " t " n pix + ( RN) 2 " n pix ] 1 2 (R * t) 1/2 dominates noise term S/N " R *t R * t = R t * #t1 2 Sky Limited ( R sky t > 3" RN) : S/N # Note: seeing comes in with n pix term R * t n pix R sky t # t Bias level/structure correction Flat-fielding errors Charge Transfer Efficiency (CTE) /pixel transfer Non-linearity when approaching full well Scale changes in focal plane A zillion other potential problems Photometry With An Imaging Array Aperture Photometry: some modern ref s DaCosta, 1992, ASP Conf Ser 23 Stetson, 1987, PASP, 99, 191 Stetson, 1990, PASP, 102, 932 Sum counts in all pixels in aperture Determine sky in annulus, subtract off sky/pixel in central aperture Total counts in aperture from source Aperture Photometry I = " I ij # n pix $ sky/pixel ij Number of pixels in aperture Counts in each pixel in aperture m = c 0-2.5log(I)

9 Aperture Photometry What do you need? Source center Sky value Aperture radius Centers The usual approach is to use ``marginal sums. " x i = # I ij : Sum along columns j j i Marginal Sums With noise and multiple sources you have to decide what is a source and to isolate sources. Find peaks: use $ & x /$ x zeros Isolate peaks: use ``symmetry cleaning 1. Find peak 2. compare pairs of points equidistant from center 3. if I left >> I right, set I left = I right Finding centers: Intensity-weighted centroid 2 #" x i x i #" i x i i x center = # ; $ 2 i = #" % x i i i " x i i 2

10 Alternative for centers: Gaussian fit to &: [ " i = background + h e # ( (x i #x c )/$ ) 2 / 2] Height of peak Solving for center DAOPHOT FIND algorithm uses marginal sums in subrasters, symmetry cleaning, reraster and Gaussian fit. Sky To determine the sky, typically use a local annulus, evaluate the distribution of counts in pixels in a way to reject the bias toward higher-than-background values. Remember the 3 Ms. Mode (peak of this histogram) Sky From Minmax Rejection #of pixels Median (1/2 above, 1/2 below) Arithmetic mean Pixel value: Counts Because essentially all deviations from the sky are positive counts (stars and galaxies), the mode is the best approximation to the sky. Average with minmax rejection, reject 2 highest value averaging lowest two will give the sky value. NOTE! Must normalize frames to common mean or mode before combining! Sometimes it is necessary to pre-clean the frames before combining.

11 Aperture size and growth curves Radial intensity distribution for a faint star: Same frames as previous example First, it is VERY hard to measure the total light as some light is scattered to very large radius. Perhaps you have most of the light within this radius The wings of a faint star are lost to sky noise at a different radius than the wings of a bright star. Radial intensity distribution for a bright, isolated star. Radius from center in pixels Inner/outer sky radii Bright star aperture Radial profile with neighbors Neighbors OK in sky annulus (mode), trouble in star apertures One Approach Is To Use Growth Curves Idea is to use a small aperture (highest S against background and smaller chance of contamination) for everything and determine a correction to larger radii based on several relatively isolates, relatively bright 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

12 $ mag for apertures n-1, n Growth Curves Aperture Star# Star# Star# Star# Star# Star# cmean Sum of these is the total aperture correction to be added to magnitude measured in aperture 1 $ mag[aper(n+1) - aper(n)] 0.05 Aperture Photometry Summary Identify brightness peaks 2. " I xy # (sky aperture area) xy Use small aperture 3. Add in ``aperture correction determined from bright, isolated stars Aperture Radius Easy, fast, works well except for the case of overlapping images

13 Crowded-field Photometry As was assumed for aperture corrections, all point sources have the same PSF (linear detector) PSF fitting allows for an optimal S/N weighting Various codes have been written that do: 1. Automatic star finding 2. Construction of PSF 3. Fitting of PSF to (multiple) stars Many programs exist: DAOPHOT, ROMAPHOT, DOPHOT, STARMAN, DAOPHOT is perhaps the most useful one: Stetson, 1987, PASP, 99, 191 To construct a PSF: 1. Choose unsaturated, relatively isolated stars 2. If PSF varies over the frame, sample the full field 3. Make 1st iteration of the PSF 4. Subtract psf-star neighbors 5. Make another PSF PSF can be represented either as a table of numbers, or as a fitting function (e.g., a Gaussian + power-law or exponential wings, etc.), or as a combination Photometric Calibration The photometric standard systems have tended to be zeropointed 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-unitbased scale with: m(ab) = -2.5log(f) where f is monochromatic flux is in units of erg/sec/cm 2 /Hz. Objects with constant flux/unit frequency interval have zero color on this scale

14 Photometric calibration Photometric Calibration 1. Instrumental magnitudes m = c 0 " 2.5log(I t) = c 0 " 2.5log(I) " 2.5log(t) Counts/sec To convert to a standard magnitude you need to observe some standard stars and solve for the constants in an equation like: m inst = M + c 0 + c 1 X + c 2 (color) + c 3 (UT) +c 4 (color) m instrumental Stnd mag zpt airmass Color term Instr mag Extinction coeff (mag/airmass) Extinction coefficients: Increase with decreasing wavelength Can vary by 50% over time and by some amount during a night Are measured by observing standards at a range of airmass during the night Slope of this line is c 1 Photometric Standards Landolt (1992, AJ, 104, 336) Stetson (2000, PASP, 112, 995) Fields containing several well measured stars of similar brightness and a big range in color. The 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.

15 Photometric Calibration: Standard Stars Magnitudes of Vega (or other systems primary flux standards) are transferred to many other, secondary standards. They are observed along with your main science targets, and processed in the same way. NGC 6397 Low metallicity (King, Anderson, Cool, Piotto, 1999) We often need to compare observations with models, on the same photometric system Always, always transform models to observational system, e.g., by integrating model spectra through your bandpasses M4 (Bedin, Anderson, King, Piotto, 2001) Intermediate metallicity Alas, Even The Same Photometric Systems Are Seldom Really The Same This Generates Color Terms 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 photometric system is thus effectively (operationally) defined by a set of standard stars - since the actual bandpasses may not be well known.

16 Surface Photometry Simple approach of aperture photometry works OK for some purposes. mag=c 0-2.5(cnts aper - #r 2 sky) Typically working with much larger apertures - prone to contamination - sky determination even more critical - often want to know more than total brightness The way to quantify the 2-dimensional distribution of light, e.g., in galaxies Many references, e.g., Surface Photometry Davis et al., AJ, 90, 1985 Jedrzejewski, MNRAS, 226, 747, 1987 Could fit (or find) isophotes, and the most common procedure is to fit elliptical isophotes. Isophotal parameters are: surface brightness itself (µ), x center, y center, ellipticity ()), position angle (PA), the enclosed magnitude (m), and sometimes higher order shape terms, all as functions of radius (r) or semi-major axis (a). Start with guesses for x c, y c, R, ) and p.a., then compare the ellipse with real data all along the ellipse (all ' values) true isophote fitted isophote ' (I I ellipse - I image 0 Good isophote ' Fit the (I - ' plot and iterate on x c, y c, p.a., and ) to minimize the coefficients in an expression like: I(')= I 0 + A 1 sin(') + B 1 cos(') + A 2 sin(2') + B 2 cos(2') Changes to x c and y c mostly affect A 1, B 1, p.a. A 2 ) B 2 '

17 More specifically, iterate the following: After finding the best-fitting elliptical isophotes, the residuals are often interesting. Fit: I = I 0 + A n sin(n') + B n cos(n') already minimized n=1 and n=2, n=3 is usually not significant, but: B 4 is negative for boxy isophotes B 4 positive for disky isophotes "(major axis center) = #B 1 I $ "(minor axis center) = #A (1#%) 1 I $ "(%) = #2B (1#%) 2 a 0 I $ "(p.a) = where : 2A 2 (1#%) a 0 I $ [(1#%) 2 #1] I $ = &I &R a 0 Examples of Surface Photometry of Ellipticals! Major axis surf. brightness profiles Isophotal shape and orientation prof s " One can also measure the deviations 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

18 From your isophotal fits, you can then construct the best 2-dimensional elliptical model for the light distribution Saturated core Hidden disk And subtract it from the image to reveal any deviations from the assumed elliptical symmetry Bad job of clipping Non elliptical structures

19 Panoramic Imaging Generally used to study populations of sources (e.g., faint galaxy 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 extracted into a more useful form; but not always Key steps: 1. Object finding (there is always some spatial filter ) 2. Object measurements / parametrization 3. Object classification (e.g., star/galaxy) Image (De-)Blending and Segmentation How are the individual sources - or source components - defined? Easy Not so easy Thresholding is an alternative to peak finding. Look for contiguous pixels above a threshold value. User sets area, threshold value. 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 for classification and rough photometric and structural parameter derivation for large fields. Sextractor Flowchart Steps: 1. Background map (sky determination) 2. Identification of objects (thresholding) 3. Deblending 4. Photometry 5. Shape analysis

20 m 1 -m 2 Star-Galaxy separation Galaxies are resolved, stars are not All methods use various approaches to comparing the amount of light at large and small radii. stars galaxies m 1 Star galaxy I too noisy Pixel position Star-Galaxy Separation Important, since for most studies you want either stars (or quasars), or galaxies; and then the depth to which a reliable classiffication can be done is the effective limiting depth of your catalog - not the detection depth There is generally more to measure for a non-psf object You d like to have an automated and objective process, with some estimate of the accuracy as a f (mag) Generally classification fails at the faint end Most methods use some measures of light concentration 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 learning method, e.g., neural nets or decision trees Sextractor Star/Galaxy Separation Lots of talk about neural-net algorithms, but in the end it is a moment analysis. Stellarity. Typically test it with artificial stars and find it is very good to some limiting magnitude. Typical Parameter Space for S/G Classif. Stellar locus # Galaxies s-g going bad at R~22 (From DPOSS)

21 More S/G Classification Parameter Spaces: Normalized By The Stellar Locus Automated Star-Galaxy Classification: Artificial Neural Nets (ANN) Input: various image shape parameters. Output: Star, p(s) Galaxy, p(g) 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: Decision Trees (DTs) Automated Star-Galaxy Classification: Unsupervised Classifiers No training data set - the program decides on the number of classes present in the data, and partitions the data set accordingly. (Weir et al. 1995) Star Star+fuzz Gal1 (E?) Gal2 (Sp?) An example: AutoClass (Cheeseman et al.) Uses Bayesian approach in machine learning (ML). This application from DPOSS (Weir et al. 1995)

22 Seeing - A Key Issue The image size, given by a convolution of the atmospheric turbulence, and instrument optics Affects two important aspects of imaging photometry: 1. S/N: because a larger image spreads the available signal over more noise-contributing pixels Defines the detection limit Should have pixel scale matched to the expected image quality, to at least Nyquist sampling 2. Morphological resolution: as high spatial frequency information is lost Defines the star-galaxy classification limit 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 Example of seeing variations in the groundbased data (from the PQ survey) Good seeing 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