Measurements of bright stars by Tom Droege's Mark IV cameras should have very small uncertainties. However, when we compare the magnitudes determined on different frames (either on the same night or different nights), we often find that even bright stars have a scatter of 0.02 or 0.03 magnitudes around their mean, especially in the I-band. See Tech Note 88 or Tech Note 85 for some examples. What could cause these errors?
One possibility is that for some reason -- improper flatfielding, or scattered light -- the apparent brightness of a star may depend on its position in the field. Yes, flatfielding is supposed to remove such effects, but there are many cases in which it does not.
A good way to check one's photometry is to take many images of the same field at slightly different positions. These "grid" observations put stars at many different places on the detector, making it simple to detect position-dependent errors.
Tom Droege used his Mark IV camera in Batavia, IL, to observe a region near Orion's belt on UT Nov 27, 2002. He took simultaneous, 60-second long V and I exposures in a 7x7 grid pattern covering about 4 square degrees (i.e. one full Mark IV field of view). He included these images together with a set of dark frames taken a week earlier in "Disk Set 24."
I processed these data with the Mark IV pipeline in the usual manner. The dark frames have a slightly different mean level in their overscan areas than the target frames; I used overscan data in columns 2051-2055 of the raw images to shift the masker darks before subtracting them from the target frames. I combined all the target frames in each passband with a median filter to create the master flatfields for the run. The pipeline extracted magnitudes using aperture photometry with a radius of 4 pixels, collated the V-band and I-band measurements of each star, then calibrated them (one image at a time) using Tycho stars within each frame.
The end result is a set of many V-band and I-band magnitudes for each star within the grid area. I calculated the mean value of each star's magnitudes, then determined the error from the mean for each individual measurement. A graph of the errors as a function of position on the images shows clear patterns:
These patterns are similar to, but not quite the same as the appearance of the master flatfield frames. Note also that the magnitude of the errors -- about +/- 0.06 mag -- is within a factor of two of the variation in the master flatfields. The master flats are brighter near the center and fainter in the corners, which would indeed cause the sort of residuals we actually find.
How can one correct for these errors? One approach is to fit a low-order polynomial function of position to the residuals, and then subtract the function from the raw instrumental magnitudes before further processing them. One can read about this approach in Manfroid, A&A Suppl., v. 113, p. 587 (1995).
I wrote a program which tries to minimize the residuals via a correction function as follows:
obs_mag m = true_mag M + z + a*(x-x0) + b*(y - y0)
+ c*(x-x0)^2 + d*(x-x0)*(y-y0) + e*(y-y0)^2
where
Using a sample of stars which appear in at least 8 frames (I) or 20 frames (V) of the 7x7 grid, and _not_ discarding saturated stars, I find the following values for the coefficients:
coeff V I
--------------------------------------------
a 2.1559E-5 -6.6695E-6
b -2.6171E-5 -3.4695E-5
c -6.6891E-8 -4.1831E-5
d 1.3353E-9 1.5551E-8
e -4.6399E-8 1.6295E-8
--------------------------------------------
Below are plots of these correction factors as a function of row or column across a frame.
V-band, row:
V-band, col:
I-band, row:
I-band, col:
Does this really help to reduce the scatter in measurements of the same star at different places? For one test, I ran ensemble photometry on a set of bright stars which appeared in many I-band frames of the grid. Look at the scatter as a function of instrumental magnitude for both raw and corrected datasets:
The "floor" in the scatter shrinks by roughly a factor of two, from 0.03 mag to 0.015 mag or so.
Another indication that we have improved the data: the "photom" program calculates the RMS between each star's calibrated magnitude (according to its photometric solution for the night) and the catalog magnitude. Compare the values for "raw" and "corrected" measurements:
V I
----------------------------------------------
raw RMS = 0.141 0.248
corrected 0.053 0.119
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Other steps one should take to verify that these corrections are valid:
Recent discussions indicate that a large part of the large-scale variations in signal across an image may be due to actual variations in the sky brightness. In that case, creating a flatfield out of night-sky images is a very bad thing to do, since will imprint a large-scale variation upon the photometry derived from "corrected" images. Several people have asked me to analyze the "raw" images, so I do so here.
The frames were subjected to the following processing:
A zeropoint which is more negative than average indicates that stars in its image were FAINTER than average -- which in turn indicates clouds. I decided that frames
0-4, 21, 28-30, 41-43in the grid sequence were cloudy, and discarded them from the following analysis. That left 37 of the original 49 frames.
I took photometry from the remaining "cloudfree" images and attempted to fit it to two models: a simple linear gradient in each direction
obs_mag m = true_mag M + z + a*(x-x0) + b*(y-y0)
and a linear gradient plus a radial gradient
obs_mag m = true_mag M + z + a*(x-x0) + b*(y-y0) + c*r
where r is defined as the radial distance from a fiducial point
r = sqrt( (x-x0)*(x-x0) + (y-y0)*(y-y0) )
I show below three graphs for each passband:
V-band: residuals from the mean
V-band: residuals from simple linear model
V-band: residuals from linear + radial gradient model
The raw residuals do not show a radial pattern, as one might expect if the lenses are vignetting strongly. Instead, it appears that stars near the Eastern edge of the frame are somewhat brighter than average, and those near the North and South edges are fainter than average. Neither of the simple models does much to reduce the residuals.
Let's now look at the I-band data. There are a few very large residuals in the graphs below, but they do not affect the overall solutions appreciably.
I-band: residuals from the mean
I-band: residuals from simple linear model
I-band: residuals from linear + radial gradient model
The raw residuals here take the form of a "yin-yang" pattern, with a patch of fainter-than-usual measurements near the center of the field. This might more reasonably be blamed on the optics, perhaps, than the V-band residuals. Note that the Eastern edge is again brighter than average, except for its Southern corner.
How large are these residuals? I find the following values for the variations in the brightness of a single star as it moves across the chip.
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Mean size of residuals over grid
V-band I-band
------------------------------------------------------------
raw 0.022 0.017
linear fit 0.025 0.016
=============================================================
The amplitude of corrections due to the linear model over the entire frame was about +/- 0.030 mag in V-band, and +/- 0.015 mag in I-band. Note that these are smaller than the corrections which had to applied to the flatfielded data (see the graphs labelled "Distorion corrections").
Another way to compare the results is to perform ensemble photometry on stars as they move across the chip during the grid sequence. I show below results which compare the unflattened data to that which has been flattened with night-sky flats, and (for I-band) data which has been flattened and then corrected for large-scale gradients.
V-band
I-band.
It seems clear to me that working with unflattened data is better than using data which has been flattened with night-sky flats.
The next step is to see how light-box flats work ....