Executive summary
Ted Woodhouse wrote a report on one of the Mark IV images taken by Tom Droege. I asked him to send me the image, and his findings, so that I could double-check things. He very kindly and promptly replied. This Technical Note describes my analysis of the image; it focuses on the effects of the comatic PSF on photometry.
Table of Contents
The image is named "H3R1458.895". It is a V-band exposure taken by Tom Droege from his home in Batavia, through the first Mark IV camera. I believe that the image was taken on Sep 17, 1999. I do not know the exposure time, but I am guessing about 40 seconds (see section on limiting magnitude below). Here's a small picture of the image:
The image is centered around
RA = 02h 34m Dec = +05d 00'
in the tail of Cetus,
and is roughly 4.3 degrees on a side.
North is to the left, and East is down.
The plate scale is about 7.5 arcseconds per pixel.
The bright star near the center with vertical
bleed trails is nu Ceti = 78 Ceti,
magnitude V = 4.86.
This is not a raw image. Ted Woodhouse cleaned it up:
The image has a nearly uniform background, but there is a small variation with the following pattern: the upper-left (north-western) portion of the image, from rows 0 to 1600 and from cols 0 to 1600, is slightly fainter than the bottom and right (east and south) edges. The difference is small -- only about 1%. I don't know what causes this small variation, but I don't think that it has much of an effect on the rest of my analysis: the effects I describe appear at the twenty percent level, not the one percent level.
I created a histogram of the pixel values in the image, and found that
But there was one wierd feature of the pixel distribution: one particular value occurred about five times as frequently as the rest. Look at this section of the histogram:
Pixel value # of pixels
2520 41620
2521 42090
2522 42202
2523 42896
2524 205890 <--- this is the wierd one
2525 41490
2526 41385
2527 40919
2528 41611
2529 41444
I don't know why one particular value should be favored so strongly. There weren't any other peaks in the histogram in this vicinity. Ted, is it possible that your processing could have created this sharp peak? Or does the raw image have a similar excess of pixels with one particular value?
Using a set of Landolt standard stars in the image (see sections below for the details), I find that the image has a V-band sky brightness of about
sky brightness = 18.2 mag per square arcsec in V-band
This is not surprising for site in the suburbs.
At a dark site (such as Flagstaff), the sky might be
2 magnitudes or more fainter.
I wanted to find and measure properties of stars in the image. Using programs in the XVista suite, I checked the PSF and found that it was best at the center of the frame, and increasingly comatic (and large) as one moved away from the center.
near center: FWHM = 2.7, 2.7, 2.8 pix for 3 stars
1/2 to corner: 3.0, 2.9, 3.0
in a corner: 3.3, 3.2, 4.3
The FWHM values, above, don't emphasize enough how ugly the PSF becomes far from the optical axis. There are examples of the comatic PSF elsewhere.
Even at the center of the image, stars weren't round. Instead, they were stretched out East-West by about a two-to-one ratio relative to their North-South extent. I guess the camera wasn't tracking perfectly that night.
I ran the XVista program stars to detect objects in the image; it expects all objects to be round and star-like, so it probably doesn't do so well near the edges. I specified that it should find all objects with a central peak which was about 5 times higher than the noise in the sky, and accept any object with FWHM between 2.0 and 4.5 pixels. The result: 1,273 stars. I placed marks at the positions of all the detected objects, and looked briefly by eye at the image: it appeared to me that the program had found all the (non-saturated) stars down to a pretty faint limit. I could still make out by eye some stars which the program didn't find, so I wasn't digging into the noise. Good.
Next, I used the XVista program phot to measure
the flux of each star through circular apertures of radii 5 and 7
pixels.
I used a sky annulus with radii 15 and 25 pixels to estimate
a local sky value for each star.
Matching to the Tycho catalog
I selected stars from the Tycho catalog which fell within the rough boundaries of the image: there were about 400 of them.
I needed to strip out some bogus entries with magnitudes listed as "0.0" -- those must have been stars which had no listed magnitude in the Tycho catalog itself, and so had a default value of zero placed into the ASCII text file I was using. Removing these bogus entries is very important for success in matching, because the matching procedure I used selects only a small number of the brightest stars in the catalog ... and the bogus stars were (as far as the program could tell) the brightest!
After a little fiddling, I found 184 pairs of stars which appeared in both the Tycho catalog, and in the list of stars detected in the image. The matching program indicated
Once I had a matched set of stars, I was able to compare the Mark IV V-band instrumental magnitudes against the Tycho Vt-band magnitudes. I did not check for any color-dependent effects.
First, I calculated the mean difference between the two sets of magnitudes. The brightest stars in the image showed a significant difference from this mean -- I believe that they were saturated.
The mean difference was delta = 2.79 mag for the 5-pixel radius aperture, and delta = 2.69 mag for the 7-pixel radius aperture. The standard deviation from this mean was about 0.19 mag.
Now, this agrees with Ted's report. He also matched the stars in this image against Tycho stars. He used a 9x9-pixel square box to measure instrumental magnitudes in the image, rather than a circular aperture, but the size of his box was about the same as my small aperture. He found a scatter of about 0.19 magnitudes, just as I did.
What is the range of magnitudes visible in the image? It looks like stars brighter than V = 7.8 may be saturated. Looking at my list of stars detected in the image, I find that the faintest among them (after I add the offset to match the Tycho system) is about V = 14.3. This is roughly the same range as a Mark III image; perhaps it goes a bit fainter.
One question Tom asked us to check was: are there any systematic errors in photometry as a function of row or column? If so, it might indicate a problem with the electronics. Ted's report concluded No such effect. I thought so, too, at first ... but when I plotted the residuals as a function of row and column, I saw that there was a trend:
The TASS measurements are (relative to the mean offset)
The residuals can be fitted very neatly as a function of the total distance away from the center of the frame:
Note the size of the effect: the differences run from about -0.3 mag near the center of the frame to about +0.3 mag near the corners.
What could cause this effect? I thought of two things:
How to figure out which is correct? Or are both contributing to the problem?
Change in PSF across the frame
Here I check to see if the explanation for the trend in the photometric residuals might be a change in the PSF across the frame ... and conclude that it is the likely explanation.
The PSF certainly does grow in size, and change its shape, as one moves from the center of the image to the edges. How does that affect the aperture photometry? Well, I was simply adding up all the light within a circular aperture around the center of each star. One would expect that, as the PSF becomes more elongated and asymmetric, the fraction of a star's light which falls within a circular aperture should decrease.
So, for a small number of stars in different portions of the image, I measured manually the amount of light falling within several different apertures:
I picked two or three reasonably bright stars in each portion of the frame, and here's what I found:
star location f3/f10 f5/f10 f7/f10 f10/f10
-------------------------------------------------------
near center 0.89 0.99 1.00 1.00
0.89 0.99 0.99 1.00
0.92 1.00 1.00 1.00
1/2 to S edge 0.79 0.95 0.99 1.00
0.71 0.87 0.96 1.00
near N edge 0.68 0.83 0.89 1.00
0.68 0.83 0.91 1.00
0.70 0.85 0.93 1.00
in NE corner 0.61 0.78 0.89 1.00
0.58 0.78 0.89 1.00
0.61 0.78 0.89 1.00
The important point is that the fraction of a star's light within 5-pixel aperture shrinks by about 20 percent as one moves from the center of the frame to the corner. Actually, it's possible that the amount of light within the big, 10-pixel aperture might also be changing, which would lead to even larger residuals in the 5-pixel aperture.
This looks like a reasonable explanation for some of the systematic error -- it has roughly the right magnitude. Is there any way to check that this is the right explanation?
Yes! The effect should be smaller if one measures the instrumental magnitudes in the Mark IV image through larger apertures. Here's a plot showing the residuals for 5-pixel apertures (red crosses) and 7-pixel apertures (black squares):
The lines drawn through the figure are unweighted linear fits.
aperture size slope of line scatter from line ---------------------------------------------------------------- 5-pixel 0.000463 mag/pixel 0.19 mag 7-pixel 0.000328 mag/pixel 0.14 mag
Using a larger aperture does decrease the systematic error, just as one would expect; the 7-pixel aperture decreases the size of the effect by about 50% relative to the 5-pixel aperture. The larger aperture also yields a somewhat smaller scatter.
What does this mean? It means that if one uses an aperture of fixed size to measure starlight in a Mark IV image, one will end up with a systematic error as a function of position on the image. So, what can one do about it? I see the following options (which aren't the only ones, of course):
Frankly, option number 2 -- a fixed, large aperture on the central portions
of each image -- may be the one I would pick in some circumstances.
Sure, it feels bad to ignore half the frame ... but for some
purposes, it might work out best.
One can still cover large areas on the sky -- one simply has
to decrease the space between the pointings, and take more pictures.
It turns out that the standard field PG 0231+051 falls within
the boundary of this image.
In fact, it falls near the center -- how nice!
Here's a closeup the of the field,
which is centered at (1063, 827), has 154 rows and 118 columns.
Note that I've rotated the picture by 90 degrees,
so that North is up and East to the left.
In this area, the stellar FWHM was approximately 2.7 pixels,
elongated (as the pictures shows) in the East-West direction.
Here are the catalog and measured properties of the marked stars.
Recall that the sky had mean 2524, stdev 40 ADU.
We see
One might estimate the limiting magnitude of this image
to be around V = 14.5, since it's hard to make a measurement of
star B, but easy for star D.
This agrees pretty well with the limit to which
the XVista stars program detected stars,
given a limit of 5-sigma detections.
To give the reader a better appreciation for the signal-to-noise
ratio of these stars, I show below radial profiles around
stars A (the brightest) and E (medium bright).
First, star A:
Now, star E:
I do not know the exposure time for this image.
For fun, I tried to figure it out, by calculating
the expected signal for a system with the
following properties:
I calculated the expected stellar signal and signal-to-noise
ratio for a star of V = 14.0, similar to star D in the
field above.
I found a reasonable match for an exposure time of 40 seconds,
with gain 1 electron/ADU.
Note added 12/21/1999: Tom Droege told me that the actual
exposure time was about 200 seconds, and the actual
gain somewhere around 2 or 3 electrons per ADU.
That leads to an overall efficiency not 40%,
but closer to 20%.
I have modified the tables below to use this new
information. MWR
Given these parameters, I can calculate the expected
magnitude at which the signal-to-noise (S/N) ratio
drops to any particular value, for any given exposure time.
Here's a table which may be of some use to Tom
(since the parameters are for his system and location),
and perhaps to others.
From a dark site, at which the sky has V = 20.5 mag/sq.arcsec,
the limits drop to
(60 sec exposure) V = 13.4 and V = 14.8,
(180 sec exposure) V = 14.4 and V = 15.7
Arne might find these numbers of interest ....
Photometry: Mark IV versus Landolt
Landolt
star V B-V peak flux 5-pix 5-pix TASS-Landolt
ADU ADU aper mag uncert
---------------------------------------------------------------------------
A 12.77 0.71 800 6340 15.50 0.09 2.73
B 14.74 1.45 140 570 18.1 ? ? 3.36
C 13.70 0.67 320 2890 16.38 0.19 2.68
D 14.03 1.09 250 1790 16.77 0.26 2.74
E 13.80 0.68 320 2340 16.54 0.22 2.74
Quantum Efficiency
Exposure time V mag at which V mag at which
(sec) S/N = 10 S/N = 3
------------------------------------------------------------
20 12.0 13.3
40 12.5 13.8
60 12.8 14.1
90 13.0 14.4
180 13.4 14.8