starlight, photoelectrons, centroids - ucb...
TRANSCRIPT
Starlight, Photoelectrons,&
CentroidsJames R. Graham
10/6/2009
Step 1: The Photon Path
Atmosphere
Filter
Detector
Telescope
Spectrum of Vega
Scattering & Absorption by theEarth’s Atmosphere
Mirror Reflectivity
Filter Transmission
Detector Efficiency
System Throughput
�
ηνFν
hνν 1
ν 2
∫ dν =
5.88 ×1010 γ s−1cm−2
Step 2: Systematic Errors
• Imaging detectors suffer from a number oferrors that must be corrected before the datacan be used for photometry
• Goal is to make the DNs from the FITSfiles proportional to the brightness of theastronomical source
Bias & Dark Current
• Even a zero second exposure gives non-zero DN
– Dark current masquerades as real signal– Dark current & bias (constants DC offset) can
be removed either by subtracting1. A dark frame of the same exposure time as the
science image—takes care of bias too, or2. An image of blank sky—takes care of bias & dark,
and also subtracts the sky brightness! (can be hardto find blank sky)
Relative Pixel Gain a.k.a. Flat Field
• Every pixel in the detector array has aslightly different response to light– Some pixels are more efficient than others
• Need to correct for pixel-to-pixel variationsby constructing a flat field– Make a flat field by observing a uniform source,
e.g., the twilight sky– Divide dark-subtracted images by the flat field
Moments
• For each star we can construct moments ofits light distribution– The first moment is
x = xi Iii∑ Ii
i∑
How Bright is that Star?
• The light from a star is spread over several pixels• How do we sum the light to get a measure of the
total signal from the star?1. Identify the location of the star (RDPIX)2. Select the associated pixels by making a mask3. Sum up the light (TOTAL)
– Subtract the sky background if necessary
Computing the Centroidskyval = median( px[wsky] )print, 'Median sky value = ',skyval
; compute the pixel centroids
xbar = total(mask*xx * (px-skyval) )/total(mask*(px-skyval))
ybar = total(mask*yy * (px-skyval))/total(mask*(px-skyval))
print,'<x> = ', xbarprint,'<y> = ', ybar
Step 3: Modeling the Noise
• What is the SNR of a given observation?• How do I choose and optimize the
photometric parameters– Exposure time required?– Aperture diameter?– Location and size of sky annuli?
How to Begin
• Write down an expression for the signal anduse error propagation to find the noise– Express results as signal-to-noise ratio vs.
photometric parameters
The Model
• The purpose is to estimate the noisecontributions– Often getting the answer to within a factor of
two is fine– Make simplifying assumptions—so long as you
can justify them
A Photometric Model
• What parametersdescribe themeasurement?
A Photometric Model• Star
– Brightness– Center (x0, y0)– Width (σ)
• Sky background in annulus– B
• Detector– QE, readnoise, dark current
• Aperture sizes– r1, r2, r3
r1
r2 r3
Photometric Model• Write down an expression for the signal, Si , in units of
photoelectrons– In an individual pixel
– Fi is the stellar signal = fi t at pixel i [e- ]• Different for every pixel
– Qi is the dark charge = ii t [e-] in a given pixel• The dark current iivaries from pixel to pixel• For SNR model assume constant
– Bi is the sky background = bit assumed uniform [e- ]• Varies from pixel to pixel, for SNR model assume constant
– Ei is the readout electronic offset or bias [e- ]• Varies from pixel to pixel, for SNR model assume constant
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Si = Fi + Bi +Qi + Ei
The Stellar Signal• The stellar signal is found by subtracting the background from Si and
summing over the N pixels that contain the star
• Error in FN is due to noise in in the signal itself, FN
• Noise due to dark charge, Qi
• Noise from the background, B• The read out noise σRO
�
Fi = Si − Bi +Qi + Ei( )FN = Fi
i=1
N1
∑ = Si − Bi +Qi + Ei( )i=1
N1
∑N1 = πr1
2
Noise Sources
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FN = Fii=1
N1
∑ = Si − Bi +Qi + Ei( )Background
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥ i=1
N1
∑
B = average sky/pixel & Q the average dark charge/pixel
σ F2 = FN
Poisson signalnoise
+ N1 B + Q + σ RO2( )
Poisson noisewithinr1
+ N1σ Sky2
σ Sky is the error in the skymeasured between r2 & r3
σ Sky2 = B + Q + σ RO
2( ) /N23 Every pixel between r2 & r3 contributes to the accuracy of thesky measurementN1 = πr1
2
Star
, N23 = πr32 −πr2
2
Sky
N1
N23
Noise Sources
• How do we choose r1, r2, r3?– Signal increases with N1
– Noise increases with N1 and decreases with N23
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FN = Fii=1
N1
∑ = Si − Ii + Bi + Ei( )Background
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥ i=1
N1
∑
B = average sky/pixel & Qd the average dark charge/pixel
σ F2 = FN
Poisson signalnoise
+ N1 B + Qd + σ RO2( )
Poisson noisewithinr1
+ N1 B + Qd + σ RO2( ) /N23
Poisson noisewithinr2 <r<r3
N1 = πr12
Star
, N23 = πr32 −πr2
2
Sky N1
N23
Signal-to-Noise
• How do we choose r1, r2, r3?– Signal increases with N1– Noise increases with N1 and decreases with N23
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SNR =FNSignal
FNSignal Noise
+ N1 B + Q + σ RO2( )
SkyDark&RONinstar aperture
+ N1 B + Q + σ RO
2( ) /N23
SkyDark&RONinskyaperture
An Example
• Suppose the stellar signal has a 2-dGaussian shape
– This tells us how FN changes with apertureradius
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Fi =F02πσ 2 exp −
12
riσ
⎛ ⎝
⎞ ⎠
2⎡
⎣ ⎢
⎤
⎦ ⎥ i
, ri2 = (x − x0 )
2 + (y − y0 )2
FN = 2π rFi0
r1∫ dr
StarProfile &Integral
�
Fi =F02πσ 2 exp −
12
riσ
⎛ ⎝
⎞ ⎠
2⎡
⎣ ⎢
⎤
⎦ ⎥ i
�
FN = 2π rFi0
r1∫ dr
SNR vs. r1
• F0 = 100 e-
• Bi = 100 e-
• Ii = 0 e-
• σRO = 10 e- rms• N23 >> N1