PHOTOMETRY FOR DUMMIES: METHOD #3
Bruce L. Gary, Hereford Arizona Observatory (G95)
Last updated 2005.01.04

0.15 mag SE, Counts Equation Magnitudes (5 Minutes)

For this procedure you'll have to perform a once yearly calibration of your telescope/CCD system. If you change configurations, you'll have to repeat the calibration. I'll first show how to calculate magnitude under the assumption that a valid system calibration exists, then I'll describe that calibration observing session.

Procedure When System is Calibrated:

For my telescope system whenever the sky is cloudless I can convert a star's "intensity" to magnitude in a couple minutes without consulting a star catalog for a reference star's magnitude or observing a Landolt star field. The accuracy isn't very good, being ~0.15 magnitude, but it sure is quick! The procedure is straightforward, as summarized here (using MaxIm DL):

1) Move the photometer aperture pattern over a star and make sure the signal circle is large enough to "capture" most of the light from the object of interest (the asteroid),
2) Make sure the reference annulus does not have interfereing stars inside (if it does, either change the gap annulus width, change the sky background width, or use the pixel edit tool to delete them),
3) Note the star's "intensity" (taking care with faint objects to manually center the signal circle on the asteroid (and not the location yielding maximum intensity),
4) Note the time of the exposure,
5) Determine air mass using a planetarium program (such as TheSky 6.0),
6) Calculate magnitude using an equation of the following form:

R-mag = (20.0 +/- 0.1) - 2.5 * LOG10 ( INTr / g ) - (0.13 +/- 0.01 ) * m

where INTr = "intensity" of the asteroid using the R-band filter, g = exposure time, m = air mass. The constant 0.13 +/- 0.01 is my site's extinction {magnitudes per air mass). The constant 20.0 +/- 0.1 is unique to my telescope system and is the number that must be calibrated once per year (or whenever the configuration changes). Each filter has a different pair of constants, and I'm just illustrating the equation for converting intensity to magnitude for one filter.

Note 1: Notice that it was not necessary to consult a star catalog for obtaining a reference stat's magnitude. Reference stars are not used with this procedure.

Note 2: This procedure only works when the sky is cloudless. To assure that your observations are unaffected by clouds it is useful to check the sky background level on images taken at about the same air mass. This background level (after subtracting 100 counts) will be proportional to air mass, and the slope of this dependence will be greatest for the B-filter. Whenever I'm doing precision photometry I monitor the sky background level (comparing it with charts) and I also monitor an internet satellite IR image of my area to see when cirrus clouds are moving toward my site.

Note 3: The first constant term in the equation, above, is somewhat dependent upon the signal circle size. If the circle is smaller than the size used for the annual calibration (and that's a temptation, since your SNR for a faint object is best for smaller signal apertures), then you'll have to determine the response ratio. This is done using a bright, nearby star, where you note the ratio of intensities for a large aperture that captures >99% of the star light, and the intensity corresponding to the smaller aperture you want to use for the faint asteroid. Use this response ratio to adjust the asteroid's intensity before using the above equation.

Note 4: Notice that your asteroid magnitude will be more accurate when observing at high elevations, where air mass is small and the extinction term is smallest. It is difficult to know the exact value of your extinction coefficient on a particular night, and the greater the air mass the greater the systematic error this uncertainty will produce. At my site the clear sky extinction at zenith probably varies from 0.12 to 0.15 during the year. An observer at a lower altitude than my 4700 feet will have slightly higher extinction. For my site extinction values for the filters BVRIC (where C = unfiltered) are typically 0.28, 0.16, 0.13 and 0.09 magnitude per air mass.

Procedure for Annual System Calibration

The calibration observing session is meant to evaluate the first constant in the above equation, and possibly to check your assumed extinction per air mass coefficient. Whenever I conduct a system calibration I cycle through my 5 filters, BVRIC. Probably most asteroid searchers observe unfiltered, so you may ignore references to changing filters in the following description.

For the system calibration choose only a completely cloudless night with no greater than calm winds. When these two conditions are met the night is referred to as "photometric."

Start the night with dusk flat frames near zenith. I use a double T-shirt "diffuser" placed in front of the telescope aperture to eliminate the possibility of stars showing up on the flat frames. Choose exposure times that produce a maximum counts value for the entire image of no more than ~30,000 counts (in order to avoid saturation). Don't use images with exposure times <1 second (because shutters can introduce their own component of vignetting for short exposures). I use only flat frames with exposure times between 1 and 20 seconds. Using dark frames is a good idea, but some people consider this optional. Don't use a cooler for the CCD chip (or if you do, be sure the chip temperature is very stable). After the flats are complete for all filters, turn on the CCD cooler to a vlue that the TEC can reach in a reasonable time (a few minutes) and start focus observations.

Select Landolt areas to observe that are at low and high air mass. Try to position your FOV so that several Landolt stars are present. (As a convenience, I've produced a TXT-file of the Landolt stars for import into TheSky. During the import process TheSky creates a SDB-file for fast loading on subsequent uses of the program. Whenever TheSky is run it automatically includes the Landolt stars and displays their magnitudes when a star is clicked.) Choose exposure times that avoid saturation for all Landolt stars. For me 10 seconds works well. Make at least 4 exposures of each Landolt area, and make at least that many dark frame exposures. Calibrate the light images using many darks (median combined) and your flats (averaged).

In theory you can stop after obtaining exposures of a Landolt area at high air mass and another at low air mas, but it's prudent to repeat the pairing to be sure temporal trends of extinction are not present. Two pairs of Landolt area observations should take no more than 4 hours.

Analysis of these observations might be considered difficult by observers not used to the joys of spreadsheets. I make manual readings of intensity for all Landolt stars using a carefully chosen set of photometry dimensions. The signal circle must be large enough to accommodate >99% of the photons registered by the largest FWHM Landolt star for the night. Each image must be inspected to assure that nearby stars are not present in the sky reference annulus. If interfering stars can't be avoided, then edit them away! Don't be afraid of pixel editing, but be careful in using this tool to select new pixel values that are representative of the sky background level for that part of the image.

Record all intensities, and mid-exposure UT times, and use a planetarium program to determine air mass values for each image. Enter these data into a spreadsheet, and also enter the Landolt magnitudes. To veiw a sample spreadsheet layout, click SampleSpreadsheet. The concept to be implemented in the spreadsheet is to perform a least squares solution for that first term, unique to the telescope system, such that the equation produces magnitudes that agree with the Landolt magnitudes. With this as a guiding concept you are invited to create your own spreadsheet to solve for the best value for this one constant. You are also invited to solve for your site's extinction coefficient, making use of the fact that you have high and low air mass measurements of known stars. If you're ambitious, you can also try to detect a  temporal trend for extinction for this calibration observing session.

When the results of this telescope system calibration are used for the analysis of images taken on other nights there may be systematic errors of 0.1 magnitude. I don't understand these changes yet, but beware of systematic changes that might be even larger for unknown reasons. It is always prudent to use another method to check the magnitude determinations from this method if you want to be assured of 0.15 magnitude accuracy.