TN 0076: Analysis of Repeated Fields on Disk 16
Michael Richmond
Mar 14, 2001
Mar 15, 2001
Keywords: photometry
Introduction
Tom Droege used his Mark IV to acquire repeated observations of several
fields on Sep 28, 2000.
In this Technical Note, I describe my attempts to measure the
brightness of stars in these fields.
The bottom line is that I found a "floor" of noise at about
0.006 magnitudes, well above the photon noise for bright stars.
I don't know yet how much of this "floor" is due to my
chosen method (aperture photometry), my image reduction techniques,
or inherent properties of the images.
Observations
The data were taken from Tom's house in Batavia, Illinois,
on the night of September 28, 2000.
In short, Tom took "sets" of 10 images of a series of fields.
Each set consisted of 1 dark frame, followed by 9 frames of
a particular field.
All exposures were 50 seconds long.
After each set, the camera would move to a different field.
All fields were at the same declination: Dec = +0.3 degrees.
Tom wrote a subset of the night's images onto Data Disk 16.
As described in
his notes,
he wrote all the images for 2 particular fields,
plus a number of individual frames from other fields to serve
as sources for flatfield and dark frames.
I will call these two fields "A" and "B" in this Note.
They are
V-band center I-band center
RA Dec (J2000) RA Dec
Field A 359.9 +0.3 359.7 +0.3
Field B 39.0 +0.3 38.9 +0.3
Each image is about 4.2 degrees on a side.
The raw images are oriented so that rows run from East to West,
and columns run from South to North.
Processing the images
I used my
Mark IV pipeline
to reduce the images on the CD-Rom Tom distributed.
I had to perform several steps individually,
and edit the list of images by hand several times,
in order to accomodate the mode of observation for this night.
The basic steps were
- 1. create a list of images
- I added the image type, "dark" or "target", by hand.
In addition, the original (RA, Dec) values were inaccurate,
so I replaced them by hand after performing the astrometry
- 2. create a reference catalog of stars
- I used the Tycho-1 catalog as the astrometric reference;
later on, I used both Tycho-1 and the
TASS tenxcat
as photometric references.
The tenxcat catalog has positions which are
not good enough to serve as astrometric references
(as I discovered the hard way -- observed star lists
would fail to match against it).
- 3. create a "master dark"
- I combined a number of dark frames taken over the course
of the night. Their mean values varied by just 1 or 2 ADU,
which was much smaller than the standard deviation (about 20 ADU).
Evidently, Tom's temperature regulation worked well!
- 4. create "master flats"
- There were 19 frames of different fields taken in each
filter. Each frame had the master dark subtracted,
and was then scaled to match the others. I then chose
the median of all 19 values of each pixel.
The resulting flatfield frames are shown below.
Each image has North at right, East at top.
V-band: greyscale has white at 1200 ADU, black at 1400 ADU.
V-band: False-color version has arbitrary colors.
I-band: greyscale has white at 2500 ADU, black at 700 ADU.
I-band: False-color version has arbitrary colors.
- 5. Find stars in each image
- I chose thresholds as follows (the FWHM was about 3 pixels):
minsig 5.0
minfwhm 1.5
maxfwhm 5.0
minround -1.0
maxround 1.0
minsharp 0.3
maxsharp 0.9
- 6. Perform aperture photometry
- I used 2 apertures, of radius 4 and 8 pixels.
The sky was determined locally for each star, using an
annulus with radii 10 and 20 pixels.
- 7. Match detected stars against astrometric catalog
- The raw FITS headers had RA values which were initially
correct, but which continued to increase (incorrectly)
at the sidereal rate
throughout each set of images of the same field.
After the pipeline had identified the field in one image,
I edited the pseudo-log file used by the pipeline
to reflect the actual RA and Dec of each image.
- 8. Calibrate the photometry
- The nightly calibration for each passband included two terms:
- a zero-point magnitude
- a first-order color term, based on (raw V - raw I)
I ran solutions against two photometric catalogs:
one based on Tycho-1, using Arne Henden's equations
to convert Tycho Bt and Vt to V and I;
and one based on the Mark III tenxcat catalog.
There were small differences between the two solutions;
see more below.
Bugs discovered in the pipeline
In the course of my reductions, I discovered some bugs in
the pipeline software. I have fixed them, but the fixes
have not yet (3/14/2001) propagated to the packages
available for download.
- the do_stars procedure was using the raw frames as
input, not the dark-subtracted and flattened frames.
- the do_astrom procedure didn't handle properly
fields which "wrap-around" the RA = 0 = 360 boundary
- the phot command in
XVista
package had a small error in its calculation of noise
due to photons from the sky.
This didn't affect magnitude measurements; it only
caused the uncertainty estimates to be slightly too large.
- the do_stars procedure wasn't passing the gain
parameter to the phot program properly,
which caused uncertainty estimates to be incorrectly calculated
Accuracy as a function of position in the frame
One major concern with any wide-field imaging system is
its ability to measure stars near the edges of the field.
Early versions of the Mark IV suffered from coma
(due to an error in the construction of the lenses),
which increased sharply away from the field center.
Does this problem plague the current Mark IV as well?
For the two fields, "A" and "B", I calibrated each of the
9 images independently; that is, I didn't do any relative
photometry, but instead applied a the single photometric
solution for the entire night to each image.
I matched up stars in each field with reference photometric
catalogs: both Tycho and tenxcat.
I plotted the difference between the Mark IV and other magnitude
delta_mag = (Mark IV mag) - (reference mag)
as a function of the position of the star away from the center
of the field
delta_ra = (RA of star) - (RA of center of image)
delta_dec = (Dec of star) - (Dec of center of image)
All the magnitudes used to make the plots below were
measured through a small aperture:
a radius of 4 pixels, not very much larger than
the FWHM of the images (which ranged from 2.5 to 3.5).
In the plots below, the symbols
- + means "Mark IV magnitude was larger", hence fainter than reference
- o means "Mark IV magnitude was smaller", hence brighter than reference
The size of the symbol depends on the size of the difference in magnitude:
largest symbol: abs(delta_mag) > 0.10
next largest: 0.07 < abs(delta_mag) < 0.10
next largest: 0.03 < abs(delta_mag) < 0.07
smallest: 0.03 > abs(delta_mag)
First, plots showing the residuals for V-band measurements.
Field A, V-band, vs. Tycho:
Field A, V-band, vs. Tenxcat:
Field B, V-band, vs. Tycho:
Field B, V-band, vs. Tenxcat:
Now, residuals in I-band.
Field A, I-band, vs. Tycho:
Field A, I-band, vs. Tenxcat:
Field B, I-band, vs. Tycho:
Field B, I-band, vs. Tenxcat:
It is clear that there are systematic patterns in
the residuals.
The size of the residuals seems to be largest near
the edges of the frame (no surprise), though this
is not a hard rule.
In the V-band, Mark IV measurements are systematically
faint to the North, and bright to the South.
The pattern of the residuals is (suspiciously?) similar
to the pattern in the V-band master flatfield frame.
In the I-band, Mark IV measurements are systematically
bright to the North and West, faint to the South and East.
The pattern of the residuals does not appear related to
the pattern of the I-band master flatfield.
Differences between photometry in big and small apertures
I used simple aperture photometry to measure the brightness
of each star. Do the results depend on the size of the aperture?
Yes, of course!
The FWHM of the stellar images was
- V-band: FWHM = 2.5 - 3.3 pixels
- I-band: FWHM = 2.8 - 3.8 pixels
A good rule of thumb is to pick an aperture size which is close
to the FWHM. The more confident one is that the PSF doesn't change
across the frame, the smaller one can make the aperture.
I chose two apertures: radius of 4 pixels, radius of 8 pixels.
We expect that the precision of measurements through the
smaller aperture should be better, because they are contaminated
less by light from the sky.
On the other hand, one might expect that the accuracy of
measurements through the larger aperture might be better,
especially for bright stars, because the large aperture will
include (more nearly) all the star's light, even if the
PSF is slightly extended or asymmetric.
One way to compare the results is to look at the photometric
solution for the night.
The general photometric solution looks like this:
calibrated V = raw V + a + b*(raw V - raw I)
where a is a zero-point offset and b is a first-order
color term.
Here, I use the tenxcat as the reference catalog.
aperture passband Nstar a +/- b +/- RMS
-----------------------------------------------------------------------
4 pix V 5985 -8.075 0.007 -0.064 0.006 0.051
8 pix V 4086 -8.050 0.010 -0.045 0.008 0.077
4 pix I 8050 -7.985 0.005 -0.094 0.004 0.096
8 pix I 7331 -7.791 0.006 -0.066 0.004 0.082
What does this mean?
- In the V-band, the results change little. Hence, we may conclude
that most of the light in the V-band PSF falls within
the smaller (and the larger) aperture.
- In the I-band, the results change quite a bit, a lot more than the
formal uncertainties in "b", for example.
The zero-point offset becomes smaller in size, as one would
expect if the larger aperture collects a significantly
larger fraction of the light.
We may conclude that a fair fraction of the light of the
I-band PSF falls outside the small, 4-pixel aperture.
Another way to examine the results is to compare the measurements
for the same star to each other.
We should have up to 9 independent magnitudes for each star.
We can calculate the standard deviation from the mean magnitude of
a star in all frames.
We hope that this quantity will be smaller for measurements
made through the smaller aperture.
Below, I show graphs of the standard deviation from the mean
magnitude as a function of calibrated magnitude.
I include only stars which were measured in 5 or more frames.
For each passband, there are two graphs: one for measurements
through apertures of radius 4 pixels, the other for
apertures of radius 8 pixels.
Ignore the points which have stdev = 0, down at the bottom
of the graph; they should not have been plotted.
V-band, 4 pixel aperture:
V-band, 8 pixel aperture:
I-band, 4 pixel aperture:
I-band, 8 pixel aperture:
Summary: just as one would expect, the internal scatter is smaller
for measurements through the smaller aperture.
The difference increases for faint stars, again as one would
expect.
Scatter in theory and in practice
Just how small should the scatter be in repeated
measurements of stars?
If there are no systematic errors in the detector or
reductions, then the answer should be given by simple
statistics. There is a single source of signal,
and three sources of noise:
- signal:
- photons from the star
- noise:
- statistical fluctuations in photons from the star
- statistical fluctuations in photons from the background sky
- statistical fluctuations in electrons in CCD readout
The signal-to-noise ratio for aperture photometry of stars with
a CCD in this simplest situation is:
signal = Star
noise = sqrt [ Star + N*(Sky) + N*(Readout^2) ]
where
Star is the number of electrons knocked free
by photons from the star (total)
Sky is the number of electrons knocked free
by photons from the sky (per pixel)
Readout is the readout noise from the CCD electronics
(per pixel)
N is the number of pixels within the aperture
If one knows the gain of the CCD electronics, one can convert the
ADU in an image to number of electrons on the CCD,
and hence calculate a priori both the signal and noise for
a particular star, observed in a particular sky, through a particular
aperture.
I've written programs to do this, including
a signal-to-noise calculator on the Web.
For these Mark IV exposures, we know some of the relevant parameters,
such as telescope size and exposure time;
but we don't know for sure several factors:
- overall quantum efficiency of lenses + filters + CCD
- sky brightness at Tom's site
- extinction due to the atmosphere at Tom's site
I made some reasonable guesses for these factors, then looked at
the numbers from the Disk 16 data and iterated until I found a
reasonable solution.
Here's my best guess at the theoretical signals and noises for
stars measured on Data Disk 16 in V-band with an aperture of radius 4 pixels:
photons from a mag zero star per sq.cm. per second: 855000.000000
stellar magnitude starting point: 7.000000
stellar magnitude ending point: 15.000000
magnitude step size: 0.500000
telescope diameter (cm): 10.000000
quantum efficiency of system (0-1): 0.250000
CCD pixel size (arcsec): 7.500000
CCD readnoise (electrons): 15.000000
sky brightness (mag/sq.arcsec): 18.100000
airmass: 1.400000
extinction coeff (mag/airmass): 0.200000
exposure time (sec): 50.000000
seeing FWHM (arcsec): 20.000000
aperture radius (arcsec): 30.000000
fraction of star's light inside aperture: 0.9992
number of pixels inside aperture: 50.27
mag 7.00: star 1027064.53 sky 136571 read 11309.72 -> S/N 947.52
mag 7.50: star 648033.91 sky 136571 read 11309.72 -> S/N 726.38
mag 8.00: star 408881.76 sky 136571 read 11309.72 -> S/N 547.98
mag 8.50: star 257986.95 sky 136571 read 11309.72 -> S/N 404.95
mag 9.00: star 162778.76 sky 136571 read 11309.72 -> S/N 292.05
mag 9.50: star 102706.45 sky 136571 read 11309.72 -> S/N 205.17
mag 10.00: star 64803.39 sky 136571 read 11309.72 -> S/N 140.52
mag 10.50: star 40888.18 sky 136571 read 11309.72 -> S/N 94.11
mag 11.00: star 25798.69 sky 136571 read 11309.72 -> S/N 61.90
mag 11.50: star 16277.88 sky 136571 read 11309.72 -> S/N 40.18
mag 12.00: star 10270.65 sky 136571 read 11309.72 -> S/N 25.83
mag 12.50: star 6480.34 sky 136571 read 11309.72 -> S/N 16.49
mag 13.00: star 4088.82 sky 136571 read 11309.72 -> S/N 10.49
mag 13.50: star 2579.87 sky 136571 read 11309.72 -> S/N 6.65
mag 14.00: star 1627.79 sky 136571 read 11309.72 -> S/N 4.21
mag 14.50: star 1027.06 sky 136571 read 11309.72 -> S/N 2.66
mag 15.00: star 648.03 sky 136571 read 11309.72 -> S/N 1.68
Since the uncertainty in a measured magnitude is roughly equal to
the reciprocal of the signal-to-noise ratio, we expect an
uncertainty of 1 percent = 0.01 mag at about V = 10.5, and
an uncertainty of 4 percent = 0.04 mag at about V = 12.0.
Looking at the plots of internal scatter versus magnitude
above, we see that these are close to the measured values.
Good.
But -- what about the very bright stars?
The theoretical calculations indicate that a star of mag V = 8.0
ought to have an uncertainty of 0.002 mag.
Yet the actual measured uncertainty is significantly
larger, as can be seen in these closeups.
The theoretical scatter is shown in small red dots,
and the measured scatter in large green dots.
V-band:
I-band:
Clearly, the actual measurements don't agree with theoretical predictions.
The scatter reaches a "floor" of about 0.005 mag, regardless of the
brightness of the star.
Does ensemble photometry reduce the scatter?
One source of error in the photometric measurements is a variation
in the zero-point of each image, due to changing transparency
(i.e. clouds) or changes in the response of the detector
(i.e. temperature-dependent effects).
Since I reduced each of the images of a field independently,
any small zero-point variation between the fields would cause
a "synchonized" scatter in magnitude measurements.
For example, all the stars in frame 2 might be 0.002 mag
fainter than those on frame 1, and all the stars on frame 3
another 0.001 mag fainter still.
In theory, if one can detect such zero-point shifts from frame
to frame, one can remove them,
and reduce the scatter.
One way to find and fix this sort of error is through
ensemble photometry.
You can find a very good paper on the topic by
R. K. Honeycutt,
"CCD Ensemble Photometry on an Inhomogeneous Set of Exposures",
Publications of the Astronomical Society of the Pacific,
vol 104, 435 (1992).
The Astrophysics Data System has a
link to the paper.
The basic idea is to look for small shifts in the brightness of
a bunch of stars together, from one frame to another, and then
add (or subtract) just the right amount to the magnitudes for each frame
to remove the shifts.
As Honeycutt's paper shows, the procedure can reduce the scatter
by factors of two or more.
So, I applied this method of analysis to the stars detected
at least 5 times in each field.
Did it help?
As the following plots show, the answer is "not signficantly".
In each plot, the red dots denote scatter for measurements
calibrated independently, and the blue crosses denote
scatter for measurements made via ensemble photometry.
I applied the ensemble photometry to the raw, instrumental
magnitudes, and then shifted the results to match the
calibrated magnitude of one star in the field
(yes, this is less than optimal).
V-band
I-band
I interpret these plots to indicate that frame-to-frame
zero-point shifts are not the dominant source of noise
in the measurements.
For reference, here are the frame-to-frame offsets determined
by the ensemble photometry procedure:
V-band in blue triangles, I-band in red dots.
