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Photometric Methods and Sources of Noise
1. CCD Detectors
2. Photometric measurements
3. Sources of Noise
Photometric detectors of the past: photomultipliers. Not
good for transit work because you could only observe
one star at a time.
In 1907 Joel Stebbins pioneered the use of photoelectric
devices in Astronomy
Charge Coupled Devices
The two dimensional
format makes these ideal
detectors for photometry
over a wide field.
CoRoT‘s 4 CCDs
Keplers‘s 42 CCDs
Reading out a CCD
A „3-phase CCD“
Figure from O‘Connell‘s lecture notes on detectors
Parallel registers shift the charge
along columns
There is one serial register at the
end which reads the charge
along the final row and records it
to a computer
Columns
For last row, shift is
done along the row
Important Parameters for Photometry
1.
Quantum Efficiency (QE) : fraction of incoming photons that are detected.
2.
Bandpass: wavelength region for which a CCD is sensitive. Not so important for
ground-based observations where you use filters, but important for spacebased observations (e.g. CoRoT) that use no filters
3.
Gain: Number of electrons needed to generate 1 data number (DN) of the
output of the device. Needed to convert your recorded counts to actual photons
4.
Charge Transfer Efficiency (CTE): Fraction of the total charge in a pixel that is
transfered during the readout process. This is something like 99.999%
5.
Readout noise: The noise introduced by reading out the device.
6.
Bias Level: Constant voltage value added to the data to ensure that there are
no negative values
7.
Dark Current (noise): Electrons caused by thermal motion in the device
8.
Full well: Maximum number of pixels that can be stored in a pixel before the
potential well overflows or too large for the Analog to Digital Converter (ADC)
9.
Linearity: Over what count range that the CCD output is proportional to the
exposure time.
Bandpass: The Quantum Efficiency as a function of Wavelength
The real power of CCDs is their high quantum efficiency
Values for a CCD used at McDonald
Observatory:
Gain = 0.56 ± 0.015 e–1/ADU
Readout Noise = 3.06 electrons
Bias level = 1024
Most of these parameters can be measured and this
should be done at the start of each observing run to
ensure that the device is performing as expected.
Bias Level
Overscan region
Pixel
Most CCDs have an overscan region, a portion of the chip that is not
exposed so as to record the bias level. You can use this so long as the
bias value is completely flat across the CCD.
The prefered way is to record a separate bias (a dark with 0 sec
exposure) frame and fit a polynomial 2-D surface to this. This is then
subtracted from every frame as the first step in the reduction. If the bias
changes with time then it is better to use the overscan region
Linearity
Mean Intensity
Take a series of frames of a low intensity lamp and plot the mean
counts as a function of exposure time
1.5 x 105
If the curve followed the red line at the high count rate end (and some CCDs do!) then you would know
to keep your exposure to under 150.000. Otherwise for brighter stars this can affect your photometry
CCD GAIN
For Photon statistics the variance, s = √Photons. Therefore s2
should be a measure of the number of detected photons
• Take a series of frames at with a constant light level
• Compute s for frames
• Change the exposure time and take another series of frames calculating
a new s
• Plot the observed mean intensity versus the variance squared (s2)
• The slope is a measure of the gain
Mean Intensity
CCD Gain
1.5 x 105
Signal-to-Noise Ratio
Readout Noise
0
1
3
10
Readout noise in electrons
Intensity
High readout noise CCDs (older ones) could seriously affect
your Signal-to-Noise ratios of observations. Readout noise is
not a concern with modern CCD systems.
Some Problems and Pitfalls of CCD Usage
Saturation
If too many electrons are produced (too high intensity level)
then the full well of the CCD is reached and the maximum
count level will be obtained. Additional detected photons
will not increase the measured intensity level:
65535
16-bit AD
converter
ADU
Exposure time
Blooming:
If the full well is exceeded then charge starts to spill over in
the readout direction, i.e. columns. This can destroy data
far away from the saturated pixels.
Blooming
columns
Saturated
stars
Anti-blooming CCD can eliminate this effect:
Blooming
No blooming
Residual Images
If the intensity is too high this will leave a residual image. Left is a
normal CCD image. Right is a bias frame showing residual charge in
the CCD. This can effect photometry
Solution: several dark frames readout or shift image between
successive exposures
Fringing
CCDs especially back illuminated ones are bonded to a glass plate
SiO2
10 mm
Glue 1 mm
Glass
When the glass is illuminated by monochromatic light it creates a fringe
pattern. Fringing can also occur without a glass plate due to the
thickness of the CCD
l (Å)
6600
6760
6920
7080
7280
7460
7650
7850
8100
8400
Depending on the CCD fringing becomes important for wavelengths
greater than about 6500 Å. For example, for the Tautenburg TEST we
get better precision in a V filter rather than an R filter
Basic CCD reductions
• Subtract the Bias level. The bias level is an artificial constant
added in the electronics to ensure that there are no negative pixels
• Divide by a Flat lamp to ensure that there are no pixel to pixel
variations
• Optional: Removal of cosmic rays. These are high energy particles
from space that create „hot pixels“ on your detector. Also can be
caused by natural radiactive decay on the earth.
Flat Field Division
Raw Frame
Flat Field
Raw divided by Flat
Every CCD has different pixel-to-pixel sensitivity, defects, dust particles, etc that
not only make the image look bad, but if the sensitivity of pixels change with time
can influence your results. Every observation must be divided by a flat field after
bias subtraction. The flat field is an observation of a white lamp. For imaging one
must take either sky flats, or dome flats (an illuminated white screen or dome
observed with the telescope). For spectral observations „internal“ lamps (i.e. ones
that illuminate the spectrograph, but not observed through the telescope are
taken. Often even for spectroscopy „dome flats“ produce better results,
particularly if you want to minimize fringing.
Make sure you do not have multiple readout amplifiers!
Typical CCD readout times are 90 – 240 secs, depending on the size of the
CCD. This is for single amplifier CCDs. To reduce the readout time some
devices can have 4 channels (amplifiers) for readout:
Serial register with
one amplfier
Normal readout
4 Serial registers
with 4 amplfiers
4 Channel CCD
4 channel CCD cuts readout time by a factor of 4. Problem: each
quadrant usually behaves differently, with its own bias, flat field
response, etc. In the data reduction 4 channel CCDs have to be
reduced as if they were 4 independent frames.
CCD Photometry
CCD Imaging photometry is at the heart of any transit
search program
1. Color photometry
2. Aperture photometry
3. PSF photometry
4. Difference imaging
Filter Characteristics of Astronomical Photometry Systems
System
UBV (Johnson-Morgan)
Six-color (Stebbins-Whitford-Kron)
Infrared (Johnson)
uvbyb (Strömgren-Crawford)
Filter
U
l0
3650 Å
Dl1/2
700 Å
B
V
U
4400 Å
5500 Å
3550 Å
1000 Å
900 Å
500 Å
V
B
G
R
I
R
I
J
K
L
M
N
u
v
b
y
4200 Å
4900 Å
5700 Å
7200 Å
10,300 Å
7000 Å
8800 Å
1.25m
2.2m
3.4m
5.0m
10.4m
3500 Å
4100 Å
4700 Å
5500 Å
4860 Å
800 Å
800 Å
800 Å
1800 Å
1800 Å
2200 Å
2400 Å
0.38m
0.48m
0.70m
1.2m
5.7m
340 Å
200 Å
160 Å
240 Å
30 Å,150 Å
b
But first a few words about color photometry
From http://cas.sdss.org/dr5/en/proj/advanced/color/making.asp
Color indices are a measure of the shape of the black body curve and
thus the temperature. In transit searching you need to find the right kind
of stars (cool main sequence stars). Often you have to rely on color
photometry
For detecting transiting planets you
should avoid giant stars as well as
early-type main sequence stars
But for cool stars there is a degeneracy
between main sequence and giant stars.
You should see 2 branches if you can
measure the color or brightness
Avoid stars with B–V values lower than 0.5
Color photometry is a poor persons way of getting a crude spectral type.
Done for faint stars or over a wide field where you can get classifications
of many stars
If all works well the B–V should tell you the luminosity class
Giants (most likely)
If you really want to get the spectral type of a star
get a spectrum!
From http://www.ucolick.org/~kcooksey/CTIOreu.html
Giant stars
Main sequence stars
For field stars the apparent magnitude does not tell you the
true luminosity. Therefore, color-color magnitude diagrams
are often employed, and infrared colors being the best
Photometry gives the spectral type as a K0 Main Sequence star
But this does not fit the spectrum
It is a giant!
Spectral
determination
Photometric
determination
Interstellar redening can affect the colors of stars. It is
best to take a spectrum
Aperture Photometry
Get data (star) counts
Get sky counts
Magnitude = constant –2.5 x log [Σ(data – sky)/(exposure time)]
Instrumental magnitude can be converted to real magnitude by
looking at standard stars
Aperture photometry is useless for crowded fields
Term: Point Spread Function
PSF: Image produced by the instrument + atmosphere = point
spread function
Atmosphere
Most photometric reduction
programs require modeling of
the PSF
Camera
Crowded field Photometry: DAOPHOT
Computer program developed to obtain accurate photometry of blended
images (Stetson 1987, Publications of the Astronomical Society of the
Pacific, 99, 191)
DAOPHOT software is part of the IRAF (Image Reduction and Analysis
Facility)
IRAF can be dowloaded from http://iraf.net (Windows, Mac)
or
http://star-www.rl.ac.uk/iraf/web/iraf-homepage.html (mostly Linux)
In iraf: load packages: noao -> digiphot -> daophot
Users manuals: http://www.iac.es/galeria/ncaon/IRAFSoporte/Iraf-Manuals.html
In DAOPHOT modeling of the PSF is done
through an iterative process:
1. Choose several stars as „psf“ stars
2. Fit psf
3. Subtract neighbors
4. Refit PSF
5. Iterate
6. Stop after 2-3 iterations
Original Data
Data minus stars found in first
star list
Data minus stars found in
second determination of star
list
Image Subtraction
If you are only interested in changes in the brightness (differential
photometry) of an object one can use image subtraction (Alard,
Astronomy and Astrophysics Suppl. Ser. 144, 363, 2000)
• Get a reference image R. This is either a synthetic image (point sources)
or a real data frame taken under good seeing conditions (usually your best
frame).
• Find a convolution Kernal, K, that will transform R to fit your observed
image, I. Your fit image is R * I where * is the convolution (i.e. smoothing)
• Solve in a least squares manner the Kernal that will minimize the sum:
S ([R * K](xi,yi) – I(xi,yi))2
i
Kernal is usually taken to be a Gaussian whose
width can vary across the frame.
In pictures:
Observation
Reference profile: e.g.
Observation taken under excellent
conditions
Smooth your reference profile
with your Kernel function. This
should look like your observation
In a perfect world if you subtract
the two you get zero, except for
differences due to star variabiltiy
These techniques are fine, but what happens when some light
clouds pass by covering some stars, but not others, or the
atmospheric transparency changes across the CCD?
You need to find a reference star with which you divide the flux
from your target star. But what if this star is variable?
In practice each star is divided by the sum of all the other stars
in the field, i.e. each star is referenced to all other stars in the
field.
T: Target, Red:
Reference Stars
T
A
C
B
T/A = Constant
T/B = Constant
T/C = variations
C is a variable star
Sources of Noise in Light Curves :
The Good, The Bad, and The Ugly
•
White Noise (The Good). Noise due to photon statistics that
does not produce false transit signals. If you want to decrease
your noise and improve your chances of detecting a transit, just
collect more photons. This is uncorrelated noise.
•
Red Noise (The Bad): Noise that is correlated and not random
often associated with atmospheric extinction. Collecting more
photons will not decrease your noise. Not only does red noise
mask signals, it can create false transit signals.
•
Intrinsic Stellar Noise (The Ugly): Noise that is associated with
intrinsic variability on the star (e.g. spots or pulsations). This is
difficult to quantify and can be difficult to remove from your data.
It is often periodic and associated with stellar rotation,
oscillations, etc.
White Noise versus Red Noise
White noise
In Fourier space (frequency)
white noise has an amplitude
spectrum is constant as a
function of frequency. This is
the Fourier amplitude spectrum
of random noise
Red noise
In Fourier space correlated
noise has an amplitude
spectrum that has structure in
it. Often this rises to low
frequency and is thus called
„red“ noise. This is the
Fourier spectrum of the same
random noise as the right
panel, but with a trend.
Fourier Noise versus Noise
FT of a time series of random
noise
FT of a time series twice as long, but
with the same level of random noise
With constant noise in the time domain with rms s, the more data
you take, the noise level is still the same, i.e. s does not change. In
the Fourier domain the Fourier noise floor becomes less the more
data. This is why the more data you take, the easier it will be for you
to detect a periodic signal above the noise level (which is dropping).
Rule of thumb: a peak that has an amplitude 4 times the surrounding
noise level has a false alarm probability of 0.01 (99% chance it is a
real signal).
Sources of White Noise
photometric noise:
1. Photon noise:
error = √Ns (Ns = photons from source)
Signal to noise ratio = Ns/ √ Ns = √Ns
rms scatter in brightness = 1/(S/N)
Photon noise is often referred to as Gaussian, White, or
Uncorrelated noise (i.e. independent of other parameters like air
mass).
Note: your counts detected by your CCD need to be multiplied by
the gain to get real photons detected.
Sources of White Noise
2. Sky:
Sky is bright, adds noise, best not to observe
under full moon or in downtown Jena.
Ndata = counts from star
Error = (Ndata + Nsky)1/2
Nsky = background
S/N = (Ndata)/(Ndata + Nsky)1/2
rms scatter = 1/(S/N)
Nsky = 1000
Nsky = 100
Nsky = 10
rms
Nsky = 0
Ndata
Sources of White Noise
3. Dark Counts and Readout Noise:
Electrons dislodged by thermal noise, typically a
few per hour.
This can be neglected unless you are looking at
very faint sources
Readout Noise: Noise introduced in reading out the CCD:
Typical CCDs have readout noise counts of 3–11 e–1
(photons). This can also be neglected
Sources of White Noise
4. Scintillation Noise:
Amplitude variations due to Earth‘s atmosphere
s ~ [1 + 1.07(kD2/4L)7/6]–1
D is the telescope diameter
L is the length scale of the atmospheric turbulence
Incoming wavefront
Density/Temperature in cell diffracts
part of the wavefront away from the
telescope aperture
Star appears fainter
Density/Temperature in cell diffracts
part of the wavefront into the
telescope aperture
Star appears brighter
For larger telescopes the diameter of the telescope is much
larger than the length scale of the turbulence. This reduces the
scintillation noise. However, for transit searches using a large
telescope means you have to look at fainter stars.
Light Curves from Tautenburg taken with BEST (20cm)
star
Total
scintillation
Note: the scintilation noise from is what limits groundbased detection of terrestrial size planets. This is not
strictly „white noise“ in that it depends on the seeing
conditions at a given time at your observing sight. But
generally the limiting factor is not the scintillation noise
but other noise sources
A not-so-nice
looking light
curves from an
open cluster
Saturated
bright stars
CCD Counts
Saturation
Exposure time
CCD Counts
Saturation + nonlinearity
Exposure time
This can effect the
photometry of bright stars so
that you get a higher rms
scatter in spite of detecting
more photons
Red Noise: The Bad
Sources: Trends caused by changing airmass,
atmospheric conditions, telescope tracking, flat
field errors, fringing, etc. Usually it is combination
of several factors. Time scale of these is 2-3 hours
which is the same as transit timescales..
Atmospheric Extinction can affect colors of stars and photometric
precision of differential photometry since observations are done at
different air masses and these have a wavelength dependence
Major sources of extinction:
• Rayleigh scattering: cross section s per molecule ∝ l–4
• Aerosol Extinction
Major sources of extinction
All of these sources have a wavelength
dependence and one that depends on
the air mass
• Absorption by gases
Air Mass
For ground-based observations you have to worry about the airmass
of your observations. The airmass is the optical path length for light
coming from a celestial source
Air mass > 1
Air mass = 1
z
x
For a plane parallel atmosphere the air mass is X = sec Z, where z is
the zentith angle (0 is above).
However, the earth is not flat and there have been a
variety of formulae given to account for the curvature of
the earth.
These differ from the plane parallel approximation only for zenith
angles greater than 80 degrees (air mass > 5). One should never do
photometric observations at such a large air mass.
Wavelength
Atmospheric extinction can also affect differential photometry because
reference stars are not always the same spectral type.
A-star
K-star
Wavelength
Atmospheric extinction (e.g. Rayleigh scattering) will affect the A star more
than the K star because it has more flux at shorter wavelength where the
extinction is greater
Intensity
Atmospheric extinction could produce false detections:
Drop due to atmospheric
extinction
Intensity
Time
Transit detection algorithms
would detect these as a
transit
Time
Pont, Zucker, & Queloz, 2009, MNRAS, 373, 231
White (uncorrelated) noise
due to photon statistics
(simulated)
Red (correlated) noise
(simulated)
White + Red Noise
A real OGLE light curve
Standard deviation versus magnitude
for OGLE candidates (filled circles)
and for 10-point averages (triangles).
The stars represent the expected
position of the 10-point averages
assuming pure white noise . The solid
line is the expected dispersion of
individual points due to the white
photon noise, whereas the dashed
line shows the corresponding
dispersion for 10-point averages. For
most objects, the dispersion of 10point averages is much higher than
that which is expected for white noise,
especially for brighter magnitudes.
The dotted line shows the expected
dispersion of the 10-point means
according to the discussion in this
paper, with an amplitude of σr= 3.6
mmag for the red noise.
Red noise can influence your ability to detect transits.
White noise:
If s0 is your error and you have n points in your transit, the
„error“ of your transit detection is
s0
sd =
√n
The significance of your detection, Sd, is given by
Sd =
d
sd
=
d √n
s0
Where d is the transit depth
Red noise can influence your ability to detect transits.
Red noise:
You have to replace s0 with your covariance matrix
sd
2=
1
n2
S Cij =
i,j
s0
1
S
+
Cij
2
n
i≠j
n
Where Cij are the coefficients between the ith star and jth
light curve in your covariance matrix:
Photon
statistics
s11 s12 s13
s21 s22 ●
s31 ● s33
●
Red
noise
sn1
●
●
s1n
●
●
●
snn
sij is the covariance of xi
with xj
For white (uncorrelated
noise) you only have the
diagonal elements, all
others are zero
And your detection S/N
d
Sd =
√s
1
S Cij
+
2
n
n i≠j
0
As expected red noise decreases the signal in your
transit and decreases your ability to detect transits.
Removing Systematic or Red Noise
SYSREM: Corrects for systematic effects in light curves, like trends
due to color dependent atmospheric extinction. (Mazeh et al. 51 Peg
10th Anniversary Proceedings)
N : number of light curves (i.e. stars in your CCD frame)
M : Number of images taken to produce the light curve
rij : The residual of average-subtracted stellar image of the i-th
star of the j-th image taken at the aj-th airmass
ci : the best extinction coefficient for star i defined by the best
linear fit to the residuals as a function of air mass
ciaj : removed from rij
Best ci is the one that
minimizes the expression
sij = error of rij
SYSREM
Turning the problem around, since the atmospheric extinction
depends on air mass and weather conditions, we can ask what is
the most suitable airmass of each image given the known color of
each stars. We can look for aj that minimizes
given the set of {ci}. You can then use the new aj to find a revised
set ci. One iterates until one finds the best sets of {ci} and {aj}. Or
to find a global solution for both
SYSREM
Average periodograms
showing frequencies at
multiples of 1-d and at low
frequencies before
application of SYSREM
After application SYSREM
the frequencies at multiple
of 1-d are reduced as well
as the low frequency
components
Change in scatter as a
function of magnitude
(top) and original scatter
(lower)
SYSREM can reduce
the scatter by 5-30%
300 OGLE light curves
(each row is an CCD
image) before
SYSREM
And after
But Red Noise can also occur in Space-based data. This is
due to satellite „jitter“, cosmic ray events, etc.
Mazeh et al. 2009, A&A, 506, 431
The residuals of CoRoT stars
from two exposures
Same as the right figure, but
taken 6 hours later
Simultaneous Additive and Relative SYSREM
(SARS) : Ofir et al. 2010, MNRAS,404, 90
A matrix of N (i=1,N) stars and M (j= 1, M) measurements and the
magnitude of the i-th star in the j-th frame is mij.
rij is the residuals of each star, i, in each frame, j, after subtracting the
mean or median
In SYSREM the residuals of intrinsically constant stars are modeled as :
rij = AjCi + noise
SARS assumes that magnitude dependent effects
stem from something that is additive to the flux.
rij = AjxijCA,i + RjCR,i + noise (where Aj from above has been
replaced by Rj, i.e. a relative contribution)
xij = 10(mj–mrel)
mrel is the average or median magnitude this form
of x can correct for magnitude dependent effects
that are additive in flux
We now have to minimize this
expression. In SYSREM the „model“ is
a variation with air mass
In SARS we have added an additional
term due to flux variations
Coefficients are found
iteratively the same way as
SYSREM
xij can be used to detrend
against any external
parameter: distance of star
from center of CCD, CCD
temperature, phase of
moon, etc.
Ratio of sSARS/sSYSREM (also known
Cleanset). Note that sSARS is lower
than for most stars
The above ratio as a function of
magnitude
The number of valid data points, M,
after rejecting outliers (red = SARS).
SARS rejects less data points.
The relative „discovery power“ of
SARS relative to SYSREM
Examples of SARS
processed CoRoT light
curves. These shallow
transits were not found
by the other detection
teams that used
SYSREM
Intrinsic Stellar Noise: The Ugly
Sources of Stellar Noise for a star like the Sun:
Oscillations
Spots and activity
Amplitudes: few to 10s of
percent
Timescales: days to
months (rotation period)
Convective granulation
Amplitudes: ~5x10–6
Amplitudes: ~10–7
Timescales: minutes
Timescales: hours to
years
For a star like the sun the intrinsic variability is several tenths of a percent which
could seriously compromise the detection of transiting terrestrial planets. For stars
more active than the Sun this could compromise the detection of hot Jupiters
Power spectrum of a giant star
Oscillation „noise“
Granulation noise
Example: CoRoT Light Curves (Aigrain et al. 2009, A&A, 506, 425)
Grey: Light curve before clipping
Black: Light Curve after clipping
Blue: Light curve after filtering for 1-d variations. This are intrinsic stellar
variations
Fourier filtering
For well sampled data you can use Fourier filtering to remove
stellar variability, or fitting functions (spline, polynomial) about the
transit depth. Whatever you do, be sure that your filtering method
does not introduce transit like events in your light curve.
Removing Periodic Signals: The simplest of all!
Related to orbital frequency
(harmonics)
13 c/d = 111 min
Orbital frequency
Transit signal
f3 f2
f1
data– f1
data– f1 – f2
data– f1 – f2 – f3
Sometimes noise can come from unexpected places:
The Radial Velocity error from the Tautenburg spectrograph
as a function of time. The circled point show a time when the
mean error was a factor of 4-5 higher
In the past the signal generator that drove the
telescope motor was a sine generator. A sine
function has a very clean Fourier spectrum:
Negative
frequencies
Positive
frequencies
This was replaced by a square wave generator. A
square wave has a messy Fourier spectrum with
lots of frequencies
One of these frequencies hit a resonance with the
CCD electronics and introduced noise