imaging and photometry - space telescope science institutedlaw/ay122_lec4.pdf · 2017. 9. 4. ·...
TRANSCRIPT
Ay 122 - Fall 2006
Imaging and Photometry
(Many slides today c/o
Mike Bolte, UCSC)
Imaging and Photometry
• 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 unitarea (or solid angle), time, wavelength (or frequency),e.g., f! = 10-15 erg/cm2/s/Å - a good spectroscopic unit
• It is usually integrated over some finite bandpass (as inphotometry) 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 ! spectro-polarimetry); common in radio astronomy
(From P. Armitage)
Real detectors are sensitive over a finite range of ! (or ").
Fluxes are always measured over some finite bandpass.
!
F = F" (")d"#Total energy flux: Integral of f" over
all frequencies
Units: erg s-1 cm-2 Hz-1
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
Measuring Flux = Energy/(unit time)/(unit area)
Fluxes and Magnitudes
(From P. Armitage)
For historical reasons, fluxes in the optical and IR are
measured in magnitudes:
!
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.
f!
e.g. in the V band (! ~ 550 nm):
!
mV
= "2.5log10 F + constant
flux integrated over the range
of wavelengths for this band!
Some Common
Photometric
Systems
(in the visible)
Johnson !
Gunn/SDSS!
There areway, waytoo manyphotometricsystems outthere …
(Bandpass curves
from Fukugita et
al. 1995, PASP,
107, 945)
… and more …… and more …… and more …
(From P. Armitage)
Using Magnitudes
Consider two stars, one of which is a hundred times
fainter than the other in some waveband (say V).
!
m1 = "2.5logF1 + constant
m2 = "2.5log(0.01F1) + constant
= "2.5log(0.01) " 2.5logF1 + constant
= 5 " 2.5logF1 + constant
= 5 + m1
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 mV = -26.75 mag
Magnitude Zero Points
f!
!
Vega =# Lyrae
Alas, for the standard UBVRIJKL…system (and many others) the
magnitude zero-point in any bandis determined by the spectrum
of Vega ! const!
U B V R I
Vega calibration (m = 0): at ! = 5556: f! = 3.39 !10 -9 erg/cm2/s/Å
f" = 3.50 !10 -20 erg/cm2/s/Hz
N! = 948 photons/cm2/s/ÅA more logical system is AB"
magnitudes: AB" = -2.5 log f" [cgs] - 48.60
Photometric Zero-Points (Visible)
(From Fukugita et al. 1995)
Magnitudes, A Formal Definition
e.g., Because Vega(= # Lyrae) isdeclared to be
the zero-point!(at least for the
UBV… system)
Defining
effective
wavelengths
(and the
corresponding
bandpass
averaged
fluxes)
The Infrared Photometric Bands… where the atmospheric transmission windows are
Infrared Bandpasses Infrared Bandpasses
IR Sky Backgrounds
I
108
105
102
1µ 10µ 100µ2µ
!-4 Rayleigh scattered sunlight
273K blackbody
thermal IR
(atmosphere, telescopes)
OH can be resolved
Zodiacal lightatmosphere
IR Sky Backgrounds
Colors From Magnitudes
(From P. Armitage)
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 = mB-mV
Stars radiate roughly as
blackbodies, so the color
reflects surface temperature.
Vega has T = 9500 K, by
definition color is zero.
Apparent vs. Absolute Magnitudes
(From P. Armitage)
The absolute magnitude is defined as the apparent mag.
a source would have if it were at a distance of 10 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
!
m "M = 5log10
d
10 pc
#
$ %
&
' (
Distance modulus
M = m + 5 - 5 log d/pc
Why Do We Need
This Mess?
An example: the Color-
Magnitude DiagramThe quantitativeoperational frameworkfor studies of stellarphysics, evolution,populations, distances …
Relative measurements
are generally easier and
more robust than the
absolute ones; and often
that is enough.
The Concept of Signal-to-Noise (S/N)or: How good is that measurement really?
• S/N = signal/error (If the noise is Gaussian, wespeak of 3-$, 5- $, … detections. This translates into aprobability that the detection is spurious.)
• For a counting process (e.g., photons), error = " n, andthus S/N = n / " n = " n (“Poissonian noise”). This isthe minimum possible error; there may be other sourcesof 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)
• Signal=R*• t time
detected rate in e-/second
• Consider the case where we count all the detected e- in a circular aperture with radius r.
I
sky
r
r
• Noise Sources:
!
R* " t # shot noise from source
Rsky " t " $r2 # shot noise from sky in aperture
RN2" $r2 # readout noise in aperture
RN2 + (0.5 % gain)2[ ] " $r2 # more general RN
Dark " t " $r2 # shot noise in dark current in aperture
R* = e&/sec from the source
Rsky = e&/sec/pixel from the sky
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=# ofpixels in the aperture= #r2
!
R*t
R* " t + Rsky " t " npix + RN +gain
2
#
$ %
&
' ( 2
" npix +Dark " t " npix)
* +
,
- .
1
2
Signal
Noise
!
(R*" t)
2
All the noise terms added in quadratureNote: always calculate in e-
Noise from sky e- in aperture
Noise from the darkcurrent in aperture
Readnoise in aperture
Side Issue: S/N %mag
!
m ± "(m) = co# 2.5log(S ± N)
= co# 2.5log S 1± N
S( )[ ]= c
o# 2.5log(S) # 2.5log(1± N
S)
%mm
!
"(m) # 2.5log(1+ 1S /N)
=2.5
2.3N
S$ 12(NS)2 + 1
3(NS)3 $ ...[ ]
#1.087(NS)
Note:in log +/- notsymetric
This is the basis of people referring to +/- 0.02mag error as “2%”
Fractional error
Sky BackgroundSignal from the sky background ispresent in every pixel of theaperture. Because each instrumentgenerally has a different pixel scale,the sky brightness is usuallytabulated for a site in units ofmag/arcsecond2.
˝(mag/ )
19.219.920.019.517.014
19.520.320.720.718.510
19.720.621.421.619.97
19.920.821.722.421.53
19.920.921.822.722.00
IRVBULunarage
(days)
NaD OH
Hg
!
Scale""/pix (LRIS - R : 0.218"/pix)
Area of 1 pixel = (Scale)2 (LRIS#R : 0.0475"2
this is the ratio of flux/pix to flux/"
In magnitudes :
Ipix = I"Scale2 I" Intensity (e-/sec)
#2.5log(Ipix ) = #2.5[log(I") + log(Scale2)]
mpix = m"# 2.5log(Scale2) (for LRIS - R : add 3.303mag)
and
Rsky (mpix ) = R(m = 20) $10(0.4#m pix )
Example, LRIS in the R - band :
Rsky =1890 $100.4(20#24.21)= 39.1 e- /pix /sec
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.
!
R*t
R* " t + Rsky " t " npix + RN( )2" npix[ ]
1
2
Bright Sources: (R*t)1/2 dominates noise term
!
S/N "R*t
R*t
= R*t # t
1
2
Sky Limited
!
( Rskyt > 3"RN) : S/N#R*t
npixRskyt# t
Note: seeing comes in with npix term
What is ignored in this S/N eqn?
• Bias level/structure correction
• Flat-fielding errors
• Charge Transfer Efficiency (CTE)0.99999/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 allpixels in aperture
Determine skyin annulus,subtract offsky/pixel incentral aperture
Aperture Photometry
!
I = Iijij
" # npix $ sky/pixel
Total counts inaperture from source
Counts in each pixel in aperture
Number of pixels in aperture
m = c0 - 2.5log(I)
Aperture Photometry
• What do you need?– Source center
– Sky value
– Aperture radius
Centers
• The usual approach is to use ``marginal sums’’.
!
"x i = Iijj
# : Sum along columns
j
i
Marginal Sums
• With noise and multiple sources you have todecide 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 fromcenter
3. if Ileft >> Iright, set Ileft = Iright
• Finding centers: Intensity-weightedcentroid
!
xcenter
=
"x i
xi
i
#
"x i
i
#; $ 2
=
"ix
i
2
i
#
"i
i
#% x
i
2
• Alternative for centers: Gaussian fit to &:
• DAOPHOT FIND algorithm uses marginalsums in subrasters, symmetry cleaning,reraster and Gaussian fit.
!
"i = background+ h•e# (x i #x c )/$( )
2/ 2[ ]
Height of peakSolving for center
Sky
• To determine the sky, typically use a localannulus, evaluate the distribution of countsin pixels in a way to reject the bias towardhigher-than-background values.
• Remember the 3 Ms.
Counts
#of
pixe
ls
Mode (peak of this histogram)
Arithmetic mean
Median (1/2 above, 1/2 below)
Because essentially all deviations from the sky arepositive counts (stars and galaxies), the mode is thebest approximation to the sky.
Sky From Minmax Rejection
Pixel value: 1000 200 180 180
Average with minmax rejection, reject 2 highest value averaginglowest two will give the sky value. NOTE! Must normalizeframes to common mean or mode before combining! Sometimesit is necessary to pre-clean the frames before combining.
Aperture size and growth curves
• First, it is VERY hard to measure the total light assome light is scattered to very large radius.
Radial intensity distributionfor a bright, isolated star.
Radius from center in pixels
Inner/outer sky radii
Perhaps you have most ofthe light within this radius
• Radial intensity distribution for a faint star:
Same frames as previous example
Bright star aperture
The wings of afaint star are lostto sky noise at adifferent radiusthan the wings ofa bright star.
Radial profile with neighbors
Neighbors OK insky annulus(mode), troublein star apertures
One Approach Is To Use “Growth Curves”
• Idea is to use a small aperture (highest S againstbackground and smaller chance of contamination) foreverything and determine a correction to larger radiibased on several relatively isolates, relatively brightstars in a frame.
• Note! This assumes a linear response so that all pointsources have the same fraction of light within a givenradius.
• Howell, 1989, PASP, 101, 616
• Stetson, 1990, PASP, 102, 932
Growth Curves
0.000.020.0570.0940.1840.3240.430cMean
0.120.300.050.100.190.330.41Star#6
0.190.210.190.090.180.320.42Star#5
0.140.120.140.220.180.330.44Star#4
-0.010.020.060.100.180.320.43Star#3
0.040.110.210.080.190.330.42Star#2
0.000.020.050.090.170.310.43Star#1
8765432Aperture
$mag forapertures n-1, n
Sum of these is the total aperture correction to beadded to magnitude measured in aperture 1
Aperture Radius
$m
ag[a
per(
n+1)
- a
per(
n)]
5 10 15 20 25 30
-0.15
-0.10
-0.05
0
0.05Aperture Photometry Summary
1. Identify brightness peaks
!
Ixy
xy
" # (sky•aperture area)
!
2.
Use small aperture
3. Add in ``aperture correction’’determined from bright, isolated stars
Easy, fast, works well except for the case of overlapping images
Crowded-field Photometry
• As was assumed for aperture corrections, all pointsources 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 finding2. Construction of PSF3. 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 fullfield
3. Make 1st iteration of the PSF
4. Subtract psf-star neighbors
5. Make another PSF
• PSF can be represented either as a table ofnumbers, or as a fitting function (e.g., aGaussian + power-law or exponentialwings, etc.), or as a combination
Photometric Calibration
• The photometric standard systems have tended to be zero-pointed arbitrarily. Vega is the most widely used and wasoriginal defined with V= 0 and all colors = 0.
• Hayes & Latham (1975, ApJ, 197, 587) put the Vega scaleon 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 frequencyinterval have zero color on this scale
Photometric calibration
1. Instrumental magnitudes
!
m = c0 " 2.5log(I • t)
= c0 " 2.5log(I) " 2.5log(t)
Counts/sec
minstrumental
Photometric Calibration
• To convert to a standard magnitude youneed to observe some standard stars andsolve for the constants in an equation like:
!
minst =M + c0 + c1X + c2(color)+ c3(UT) +c4 (color)2
+ ...
Stnd mag
Instr mag
zpt
Extinction coeff(mag/airmass)
airmass Color term
• Extinction coefficients:– Increase with decreasing wavelength
– Can vary by 50% over time and by some amount duringa night
– Are measured by observing standards at a range ofairmass during the night
Slope of thisline is c1
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. Theblue stars are the hard ones to find and severalfields are center on PG sources.
• Measure the fields over at least the the airmassrange of your program objects and interspersestandard field observations throughout the night.
Photometric Calibration: Standard Stars
Magnitudes of Vega (or othersystems primary flux standards) aretransferred to many other, secondarystandards. They are observed alongwith your main science targets, andprocessed in the same way.
(Bedin, Anderson,
King, Piotto, 2001)
NGC 6397
M4
(King, Anderson,
Cool, Piotto, 1999)
We often need to compare
observations with models,
on the same photometric
system
Low
metallicity
Intermediate
metallicity
Always, always transform
models to observational
system, e.g., by integrating
model spectra through your
bandpasses
Alas, Even The “Same” Photometric
Systems Are Seldom Really The Same … … From mismatches between the effective bandpasses ofyour filter system and those of the standard system.Objects with different spectral shapes have differentoffsets:
This Generates Color Terms …
A photometric system is thus effectively (operationally)defined by a set of standard stars - since the actualbandpasses may not be well known.
Surface Photometry
Simple approach of aperturephotometry works OK for somepurposes.
mag=c0 - 2.5(cntsaper- #r2sky)
Typically working with muchlarger apertures - prone to contamination - sky determination even morecritical - often want to know more thantotal brightness
• The way to quantify the 2-dimensional distribution oflight, e.g., in galaxies
• Many references, e.g.,– Davis et al., AJ, 90, 1985
– Jedrzejewski, MNRAS, 226, 747, 1987
• Could fit (or find) isophotes, and the most commonprocedure is to fit elliptical isophotes.
• Isophotal parameters are: surface brightness itself (µ),xcenter, ycenter, ellipticity ()), position angle (PA), theenclosed magnitude (m), and sometimes higher ordershape terms, all as functions of radius (r) or semi-majoraxis (a).
Surface Photometry
Start with guesses for xc, yc,R, ) and p.a., then comparethe ellipse with real data allalong the ellipse (all ' values)
Iellipse- Iimage
'
0 Good isophote
'
fitted isophotetrue isophote
(I
'
Fit the (I - ' plot and iterate on xc, yc, p.a., and ) to minimizethe coefficients in an expression like:
I(')= I0 + A1sin(') + B1cos(') + A2sin(2') + B2cos(2')
Changes to xc and yc mostly affect A1, B1, p.a. “ “ A2 ) “ “ B2
More specifically,
iterate the following:
!
"(major axis center) =#B1
$ I
"(minor axis center) =#A1(1#%)
$ I
"(%) =#2B2(1#%)
a0$ I
"(p.a) =2A2(1#%)
a0$ I [(1#%)2
#1]
where :
$ I =&I
&R a0
• After finding the best-fitting ellipticalisophotes, the residuals are ofteninteresting. Fit:
I = I0 + Ansin(n') + Bncos(n')
already minimized n=1 and n=2, n=3is usually not significant, but:
B4 is negative for “boxy” isophotes
B4 positive for “disky” isophotes
! Major axis surf. brightness profiles
Isophotal shape and orientation prof’s"
Examples of Surface
Photometry of Ellipticals
One can also measure the deviationsfrom the elliptical shape (boxy/disky)
Disky and Boxy Elliptical Isophotes Calculate mean and RMS pixel intensity forannulus, toss any values above mean + nRMS
From your isophotal fits, you can then construct the best2-dimensional elliptical model for the light distribution
… And subtract it from the image to reveal anydeviations from the assumed elliptical symmetry
Saturated core
Hidden disk
Non elliptical structuresBad job of clipping
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 theirmeasured properties
• If done properly, essentially all information isextracted into a more useful form; but not always…
• Key steps:1. Object finding (there is always some spatial “filter”)2. Object measurements / parametrization3. Object classification (e.g., star/galaxy)
Thresholding is an alternative to peak finding. Look forcontiguous pixels above a threshold value.
• User sets area, threshold value.• Sometimes combine with a smoothing filter
Deblending based on multiple-pass thresholding
Image (De-)Blending and Segmentation
How are the individual sources -or source components - defined?
Easy Not so easy
• One very common program for panoramic photometry
is Sextractor– Bertin & Arnouts, 1996, A&AS, 117, 393
• Not for good a detailed surface photometry, but goodfor classification and rough photometric and structuralparameter derivation for large fields.
Steps:
1. Background map (skydetermination)
2. Identification of objects(thresholding)
3. Deblending
4. Photometry
5. Shape analysis
SextractorFlowchart
Star-Galaxy separation
• Galaxies are resolved, stars are not
• All methods use various approaches to comparingthe amount of light at large and small radii.
I
Pixel position
Star
galaxy
m1-
m2
m1
stars
galaxies too noisy
Star-Galaxy Separation
• Important, since for most studies you want either stars(or quasars), or galaxies; and then the depth to which areliable classiffication can be done is the effectivelimiting 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 concentrationvs. magnitude (perhaps more than one), and/or somemeasure of the PSF fit quality (e.g., *2)
• For more advanced approaches, use some machinelearning method, e.g., neural nets or decision trees
Sextractor Star/GalaxySeparation
• Lots of talk about neural-net algorithms, but in the end it isa moment analysis.
• “Stellarity”. Typically test it with artificial stars and find itis very good to some limiting magnitude.
s-g going bad at R~22
Typical Parameter Space for S/G Classif.
Stellar locus#
Galaxies
(From DPOSS)
More S/G Classification Parameter
Spaces: Normalized By The Stellar Locus
Then a set of such parameters can be fed into an automated classifier(ANN, DT, …) which can be trained with a “ground truth” sample
Automated Star-Galaxy Classification:
Artificial Neural Nets (ANN)
Input:
variousimageshapeparameters.
Output:
Star, p(s)
Galaxy, p(g)
Other, p(o)
(Odewahn et al. 1992)
Automated Star-Galaxy Classification:
Decision Trees (DTs)
(Weir et al. 1995)
Automated Star-Galaxy Classification:
Unsupervised Classifiers
No training data set - the program decides on the number of classespresent in the data, and partitions the data set accordingly.
Star
Star+fuzz
Gal1 (E?)
Gal2 (Sp?)
An example:AutoClass(Cheeseman et al.)
Uses Bayesianapproach inmachinelearning (ML).
This applicationfrom DPOSS(Weir et al. 1995)
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 signalover 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 frequencyinformation 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 differentseeing conditions
Exampleof seeingvariationsin theground-based data(from thePQ 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 (surfacephotometry)
• The arcana of the magnitudes and many differentphotometric systems … !
• The S/N computation - many sources of noise• Issues in the photometry with an imaging array: object
finding and centering, sky determination, aperturephotometry, PSF fitting, calibrations …
• Surface photometry and isophote fitting lore …• Star-galaxy classification in panoramic imaging• The effects of the seeing: flux measurements and
morphology