Imaging and Photometry • 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. Flux measurements (photometry) Imaging and 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 (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 = Intensity (in ergs or photons) per unit Fluxes are always measured over some finite bandpass. area (or solid angle), time, wavelength (or frequency), 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 frequencies • 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., galaxies), or changing in time (variable sources) more common): 1 Jansky (Jy) =10"23 erg s-1 cm-2 Hz-1 • You can also measure the polarization parameters (photometry ! polarimetry, spectroscopy ! spectro- f" is often called the flux density - to get the power, one polarimetry); common in radio astronomy 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 magnitude. 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 wavelengths 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 stars, one of which is a hundred times f! fainter than the other in some waveband (say V). Vega = 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 sun 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 Infrared 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 sunlight 273K blackbody (atmosphere, telescopes) atmosphere 102 Zodiacal light 1µ 2µ 10µ 100µ Colors From Magnitudes Apparent vs. Absolute Magnitudes The color of an object is defined as the difference in the The absolute magnitude 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 luminosity in some waveband. blackbodies, so the color For Sun: M"B = 5.47, M"V = 4.82, M"bol = 4.74 reflects surface temperature. Difference between the # d & apparent magnitude 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 Distance modulus 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 measurement 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%” ! Scale ""/pix (LRIS - R : 0.218"/pix) Sky Background 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 brightness 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.