Transformation coefficients, NGC7790 & Steve Howell's Growth Curve Method
Although used less frequently than M 67, NGC 7790 is another standard star field for calculating transformation coefficients. As part of the XZ Cet "stellar experiment" I imaged NGC 7790 a few times in the early winter to use in transformation coefficient calculations. I spent much too much time measuring through different apertures sizes with different annulus sizes and comparing them to determine which seemed to give the best results on average ( lowest zero point standard deviation). After all that, I still had a lot of scatter – significantly more than I thought I should be getting from good photometry data. See the attached Tv coefficient plot of v-v0 vs. B-V as an example.
Conditions were far from ideal and the uncorrected 3000 mm focal length Dahl Kirkham telescope I used in conjunction with a fairly large format camera ~ 2000 x 3000 pixels at 0.9 microns (yielding ~ 30 x 20 min FOV) suffers significant distortion across the FOV. Further, there are a number of stars in the field that could suffer from crowding. That suggested that I should apply the growth curve method from Steve Howell's June 1989 PASP paper "Two-dimensional Aperture Photometry: Signal-To-Noise Ratio of Point-Source Observations and Optimal Data Extraction Techniques" to my instrumental magnitude measurements.
Before I recalculate everything all over again for all of the images, I would like to confirm that I am applying Steve Howell's method correctly. Please see the attached spread sheet.
1. I calculated the growth curves (delta magnitude vs measurement aperture) for the first 10 stars in the standard star list (from the AAVSO NGC 7790 standard field) in the image NGC 9970_001B1H180.fit - except for #7 which had an obvious crowding problem and an exceptionally large deviation from the average zero point residual - and plotted them in tab "Growth Curves (1-10)."
2. From those growth curves I selected those that seemed reasonably close to the "normal" shape that a growth curve should have, although you might argue about the curve of star 6. The selected growth curves are plotted in tab "Growth Curves Select (1-10)."
3. I plotted growth curves for all of the remaining stars in tabs "Growth Curves (11-20)" and "Growth curves (21-31)" and selected ones close to the "normal" growth curve shape as shown on tabs "Growth Curves Select (11-20)" and "Growth Curves Select (21-31)."
4. I averaged the growth values for the stars selected to have "normal" growth curves at each aperture from 2 to 20 pixels, on tab "Select Avgs" and plotted the result on tabs "Growth Avg Select" and “Growth Avg Select Zoom.” The resulting "normal" growth curve looked reasonable to my eye. There was a bit of wobble starting at 18 pixel aperture which could result from slight crowding in a number of the stars selected for calculation of the “normal” growth curve. Because of this slight “wobble” I decided to use the magnitudes at 17 pixel aperture radius as my stellar magnitudes on the corrected magnitude plots described below.
5. I plotted the error values (from the CCD equation) at each aperture and selected 6 pixels as the aperture to use as the base for growth. The growth values used for each star from their 6 pixel value are shown in column "CU" on the "Select Avgs" tab. The values in the “CV” column accomplish the same thing but starting with radius 7 they use an iterative method from the corrected instrumental magnitude at the next smaller aperture radius to facilitate pasting into column “AB” on tab “Photometry_NGC7790_B001_all-30-” in the rows applicable to image NGC 7790-001B1H180.fit.
6. The “corrected” (perhaps it would be better to call them “adjusted”) instrumental magnitudes for all stars are calculated in column “AC” on tab “Photometry_NGC7790_B001_all-30-” in the rows applicable to image NGC 7790-001B1H180.fit and are plotted on tabs “corr magns 1-10”, “corr magns 11-20” and “corr magns 21-31.”
7. The corrected magnitudes at 17 pixel aperture radius are summarized in Column “AC of tab “Corrected photometry & Residuals.”The zero point for the magnitude of each star the average zero point and the average zero point and the standard deviation of the residuals are calculated for the corrected and uncorrected magnitudes in columns “AE” through “AH” of the same tab. The average zero point for the corrected and uncorrected magnitudes are 19.762 and 19.806 respectively, a separation of only about 0.04 magnitudes. However the standard deviation of residuals for the corrected magnitudes only 0.050 magnitudes compared to 0.309 magnitudes. The uncorrected residuals are about 6 times as large as for the corrected magnitudes.
These results seem to indicate that using Howell’s method should reduce the scatter in my transformation plots significantly. However, before I go through the considerable effort of applying this method to all of the images from several nights, I want to be sure that I am doing it correctly. If anyone is experienced in using Howell’s growth curve method and can confirm if I have done it correctly or correct any errors I have made, I would appreciate it.
|Tv2Plot_NGC 7790_Trans_Coeffs_xls.pdf||71.41 KB|
|Photometry_NGC7790_B001_AllAps-30-40-0_R1-corrected .xlsx||593.8 KB|