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

�

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

�

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

�

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

�

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

�

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