PHOTOMETRY USING SIMPLIFIED MAGNETUDE EQATIONS:
QUICK START EXAMPLES

Bruce L. Gary; Hereford, AZ; 2005.04.14

Abstract

This web page is for the user who has already read the "concepts and derivation" version and wants to try using my suggested simplified magnitude equations (SME) procedure for deriving standard magnitudes of stars from CCD images. Examples of both all-sky and differential photometry are presented using observations with my telescope. Procedures are given for calibrating a telescope/filter/CCD system.

Introduction

You may be reading this without having read the "concepts and derivation" web page that presents the underlying concepts for the procedures described in this web page. That's OK, but remember that eventually you will want to understand why the "photometry using simplified equations" procedures work, so I recommend the other web page as a future homework assignment: Concepts and Derivations.

The first few sections assume that the telescope/filter/CCD system has been calibrated (weeks or months before). The last few sections show how this system calibration can be performed.

Simplest Situation of All-Sky Photometry

Assume that observations of Landolt star fields weeks or months ago have led to the following "simplified photometry equations" for your telescope system:

Figure 1. Simplified magnitude equations for a 14-inch telescope (Celestron CGE-1400), Custom Scientific filters, Celestron f/7 focal reducer and SBIG ST-8XE CCD, located at a 4660-foot altitude site in Southern Arizona. The Cv and Cr equations are meant for estimating  V-magnitude and R-magnitude from unfiltered (clear filter) ofluxes. The symbol "F" is a star's flux (also referred to as intensity). The symbol "g" is exposure time [seconds]. The symbol "m" is air mass (usually represented as X). The star color symbol "C" is defined to be 0.57 * (B-V) - 0.3), which is a close approximation to V-R-0.31 (therefore, C ~ 0 for a typical star).

Consider an image of an asteroid in a star field that does not have calibrated reference stars. Here's an example of how the asteroid's flux is measured.

Figure 2. Flux measurement of an asteroid (15146) using MaxIm DL. The photometry aperture circles are centered on the asteroid and the real-time calculation of flux (referred to as "Intensity"by MaxIm) gives a value of 1698 counts. The SNR of 29 allows for the calculation of a SE on flux of 59 counts (1698 / 29). This image is a median combine of 3 images, each an unfiltered 180-second exposure.

The time of this observation is used to determine that the air mass for the asteroid was 1.32 (using TheSky). From Fig.1 we have:

(1)    Cv = 21.30 - 2.5 * LOG ( Fc / g ) - 0.14 * m + 0.52 * C

We have values for the following observables: Fc = 1698 +/- 59, g = 180 [seconds], and m = 1.32.  We don't know C, the asteroid's color. But knowing that its a main belt asteroid allows us to estimate B-V = 0.86 +/- 0.40, which means that C = +0.20 +/- 0.23. Therefore,

(2)    Cv = 21.30 - 2.5 * LOG ( 1698 / 180 ) - 0.14 * 1.32 + 0.52 * 0.20

Cv = 18.76 +/- 0.10

Notice that we've used an unfiltered image to estimate a V-band magnitude. The uncertainty of 0.10 has two sources: SNR = 29 and C = 0.20 +/- 0.23. The SNR source is a "stochastic uncertainty" whereas the asteroid color uncertainty is a systematic error. For the purposes of creating a "rotation light curve" during one observing session only the stochastic component needs to be used, so for that purpose Cv = 18.76 +/- 0.10.

Error Propogation for Above Example

I am a believer in the saying "A measurement is useless without an uncertainty estimate." Notice that I used the word "estimate." This is because all but one of the uncertainty sources are estimated calibration uncertainty; only the SNR is measured, and it's a stochastic component. The example, above, treated only one component of calibration uncertainty, the asteroid's unknown color. In a later section I will show how to measure color using the simplified magniutde equations, but for the balance of this section I want to illustrate a more complete error propogation analysis for the above asteroid example.

In the previous section I assumed that the constants and coefficients in equation (1) are without error. These values are never perfectly known. Their uncertainty leads to an uncertainty of the derived magnitude. These uncertainties belong to a category called "systematic calibration uncertainty." Here is a list of estimated uncertainties for terms in equation (1) :

Zcv = 21.30 +/- 0.07
Kcv =  0.14 +/- 0.02
Scv =  0.52 +/- 0.10

Let's assume that the above uncertainties are uncorrelated with each other, and propogate their errors. The following lists the SE uncertainty for the asteroid brightness for all sources:

Zcv SE            +/-0.070 mag
Kcv SE            +/-0.026 mag
Scv SE            +/-0.016 mag
Cvr SE            +/-0.120 mag
SE from SNR       +/-0.034 mag

Total SE (accuracy)= 0.146 mag accuracy)

Thus, we may conclude that the asteroid's V-band magnitude (based on unfiltered observations) is Cv = 18.76 +/- 0.15 magnitude.

The saying "A measurement is useless without an uncertainty estimate" uses the word "estimate" advisedly. This is because all but one of the uncertainty sources are estimated calibration uncertainty; only the SNR is measured, and it's the principal stochastic component. Estimating calibration uncertainty requires skill that can only be acquired through experience. I've had about 4 decades of this experience, but it shouldn't take a conscientious person that long to do the above SE estimations.

There are other sources of systematic uncertainty, and I'll merely list them here. I'll eventually create web pages that treat each of them, and update this web page with links when they're available.

Flux correction when using small "signal aperture" size (for improving SNR)
Interfering stars in the sky background annulus
Imperfect flat frames
Incorrect choice for x,y centroid location (serious for faint objects)

For now, let's live life dangerously, and assume that these additional systematic error sources are neglible.

Determining Color Using Iteration

In this section we're going to use V and B images to determine B and V magnitudes that don't rely upon B-V assumptions. In other words, the simplified magnitude equations can be used to determine B and V for any unknown star. This can be done using a spreadsheet for iterating from a starting B-V to a final, stable value.

Recall from Fig. 1 the following simplified magnitude equations for B and V.

(3) B = Zb - 2.5 * LOG ( Fb / g ) - Kb * m + Sv * Cbv

(4) V = Zv - 2.5 * LOG ( Fv / g ) - Kv * m + Sv * Cbv

For the balance of this section I will adopt values for the constants and coefficients in these equations based on the calibration of my telesope:

(5)    B = 19.13 - 2.5 * LOG ( Fb / g ) - 0.25 * m + 0.39 * C

(6)    V = 19.56 - 2.5 * LOG ( Fv / g ) - 0.17 * m - 0.09 * C

where you will recall that F = star flux, g = exposure time [seconds], m = air mass, and C (star color) is defined to be 0.57 * (B-V)-0.30).

On the right side of the equation everything is either known or measured, except star color, Cvr. When B and V images of the same star exist we have information about Cvr, and the trick is to arrive at an internally-consistent solution for B, V and Cvr.An iterative process can be used to determine these values. This is done by first assuming that Cvr =0 (which is true for the typical star, for which V-R ~ 0.30), then calculating a first iteration estimate for B and V (and therefore Cvr). After the initial iteration, producing B and V for a typical star Cvr value, the new B and V values are used in a second iteration (using a new Cvr = 0.57 * (B-V) - 0.30). Usually, only two or three iterations are necessary to achieve a stable solution. Let's use a specific example to illustrate the process.

Calibrating Telescope System

I will use observations from one night (2005.04.06 UT) to illustrate the use of spreadsheets for determining the SME constants for each filter.

Let's begin with how the observations were taken. MaxIm DL was used to control an SBIG ST-8XE CCD, a Celestron CGE-1400 telescope, a SBIG CFW-8 color filter wheel, and a Starizona MicroTouch wireless focuser. TheSky 6.0 was used to display the location of Landolt areas, which guided my selection of them for observing so as to produce a sampling of air mass values for the night. Each Landolt area was observed with filters B, V, R, I and C. A "sequence" of exposures was selected from previously-created sequences that specified filter, exposure time and number of exposures. A typical sequence consisted of 14 "lights" and 2 "darks." All exposure times were 10 seconds (unguided). The CCD cooler was controlled at -24 C the entire night.

Image analysis involved the calibration of raw images using flats and darks. The flats were taken at twilight (same observation night). Several flats were made using each filter, and they were averaged to produce a single flat for each filter. All flats had maximum counts within the range 20,000 and 33,000 and exposure times within the range 1 to 20 seconds. All dark frames for the night (a few dozen) were median combined to produce a single dark frame for use with all Landolt images. After calibrating the Landolt raw images for a sequence, they were median combined using auto-star alignment. This procedure produced a single image corresponding to an observing sequence of a Landolt area using a filter at one air mass value.

For example, Sequence #04 was made with a V-band filter of Landolt area "LA1242" (i.e., RA = 12:42) at an air mass value of 2.43 (as determined using TheSky), and it consisted of 12 useable images (only sharp images were used; I rejected those that suffered from tracking errors or image wander due to mountain waves). The next figure shows how star flux measurements of this #04 image was used to determine SME constants using an Excel spreadsheet.

Figure 3. Part of an Excel spreadsheet showing observing Sequence #04 information. The blue cells at the top are SME constants (adjusted by the user), the blue cells on the left are exposure time, aperture flux fraction (described below), and air mass. The blue cells in the middle are measured star fluxes. Landolt BVRI magnitudes are shown (cells F11..I23). Columns "K" and "L" show B-V and V-R star colors (based on Landolt valeus). Column "M" is star color C = V-R-0.3 based on an equation for converting B-V to V-R. Column "O" is SME magnitude using the constants in cells G2..G5). Column "P" shows the difference between the SME V-magnitude and Landolt V-magnitude for stars 1 through 13. Cells with no flux values correspond to stars that were too faint (SNR<70).

In this spreadsheet note the light blue cells. These require user input or adjustment. The user must enter values for exposure time (B11), aperture star flux ratio (C11) and air mass (D11). The user must also enter star flux values in column J. The yellow cells provide feedback for the adjustments (described in the next paragraph). The Landolt magnitudes (F11..I23) can be copied from another sheet, so after they have been entered into a spreadsheet once it should not be necessary to enter them again. The other cells are calculated. The message I want to convey here is that once a spreadsheet has been created for using Landolt stars for calibrating the telescope system, and once the user is familiar with the spreadsheet, and, finally, once star fluxes have been measured and entered into the spreadsheet, the work of solving for SME constants is quite simple. The next paragraph explains this spreadsheet section in more detail.

The zero shift constant, Z, is at cell G2. It should be adjusted so that yellow cell P3 (average difference between SME V-magnitude and Landolt V-magnitude) is zero. Cell P2 makes this adjustment easy by suggesting the value that will accomplish this. For a single image it is not necessary to adjust zenith extinction (cell G3). When many stars are present the star color sensitivity coefficient, S, in cell G4 can be adjusted for minimum "RMS diff" in cell P4 (RMS difference bewteen SME magnitudes and Landolt magnitudes). For now we are not using W, the "air mass times star color" coefficient (cell G5).

Column "C" contains the number 0.978. This is the ratio of star flux using a small photometry aperture and a large one (9 pixel radius versus 14 pixel radius, for example). I like using an aperture that is as small as possible in order to permit the use of a small sky background reference annulus; this minimizes the chances of having interfering stars in the background reference annulus. For faint target stars the use of a small asperture increases SNR, which is another reason for having the option of choosing a small signal aperture. SME photometry requires the use of star fluxes that include the entire point-spread-function. The compromise I employ is to choose a small aperture for measuring star flux but correct these fluxes using a "flux response fraction" based on the measurement of a bright star using both the small aperture and a much larger one. This "flux response fraction," which I refer to by the symbol "f" (as in cell C11), must be determined for each image. The measured star fluxes (using the small aperture), shown in column "J", are adjusted by dividing by "f" in the magnitude equation.

I recently switched from using star colors based on B-V to a version based on V-R. All Landolt stars have B and V entries, but only ~1/4 have R and I entries. As explained in the companion web page star color V-R is correlated with B-V well enough for use by SME photometry, an empirical linear equation can be used to convert B-V to V-R with negligible error (usually). The conversion is accomplished using V-R = 0.01 + 0.57 * (B-V). After subtracting a typical V-R = 0.30 from the converted V-R, I refer to this new star color as Cvr or just plain C.  In the figure they are shown in column "M".  Column "N" shows the difference between my converted V-R-0.30 and Landolt's V-R-0.30, and the differences are small. (To see a graph showing Landolt V-R versus B-V, and the suitability of my conversion, click V-R vs B-V ).

As images are added to the analysis the spreadsheet will grow downward. As a convenience to the user graphs are also available for visualizing a good choice for K and S that are uninfluenced by outliers. This will be shown in the next figure.

At this point the reader should note that the above SME solution might be adequate for use with another image at the same air mass for converting a star's flux to a magnitude, provided we either knew or were prepared to assume its color. The accuracy with which this could be done appears to be 0.020 magntude, although this result is based on only 8 standard stars. As will be seen when we add more Landolt stars to the analysis (next figure), the apparent SE accuracy that can be achieved over a wide air mass and star color range is 0.017 magnitude. The next figure shows a larger area of the above spreadsheet after a total of 5 Landolt V-band images were included in the analysis.

Figure 4. Screen shot of an Excel spreadsheet showing a "solution" for V-band observations of Landolt star fields. Sheets like this are present for each filter (see tabs at bottom). The two charts on the right are used to determine zenith extinction and the star color coefficient, as explained in the text. The lower-left chart shows equation V-magnitude errors versus star color (V-R-0.3), from which it was determined that the RMS discrepancy with Landolt V-magnitudes = 0.017 magnitude based on 47 Landolt stars. Additional explanations are in the text.

Let's analyze the additional material in this figure, one step at a time. Some of this description will be a repeat of material presented in the companion web page's "Calibrating a Telescope System."

Notice the new data for images #15 and #22 (others are present below the screen shot boundary). A nice range of air mass values is present, extending from 1.2 to 2.43. Note slightly different aperture response ratios, "f".

The chart in the upper-right shows a parameter "K * air mass" versus "air mass." The slope of the fitted dashed line corresponds to the zenith extinction coefficient for V-band, Kv. The data is merely a plot of column Q versus column D, where column Q is an equation for K*m (solved for using the SME for V):

(7)    K*m = Z - V - 2.5 * LOG ( Fv / g ) +S * C

where C is the star color V-R-0.31 (derived from 0.57 * (B-V) - 0.30). The fitted dashed line is specified by the zenith extinction value specified by the user in cell G3.

If extinction had changed during the observing period it wouldn't produce such a well-behaved (highly correlated) plot as seen here.

The lower-right chart is an analgous version used to determine the color sensitivity parameter, S, where "S * C" is plotted versus C. The slope of the fitted dashed line corresponds to the value for S, which in this case is -0.07. The fitted line is determined by the value in cell G4, and it must hinge through the origin, at 0,0. The lower-left chart is a plot of SME V-magnitude minus Landolt V-magnitude, versus star color. The RMS discrepancy with Landolt is 0.017 magnitude (based on 47 Landolt stars), and there is no residual dependence upon star color.

Similar spreadsheet analyses were performed for B, R and I, as well as C (clear filter). The next several panels of graphs are for B, V, R and I filter data.

Figure 5. Screen shots of Excel spreadsheet showing a graphs of parameters related to zenith extinction, star color sensitivity, and residuals from truth (Landolt).

This set of graphs show a progression from high zenith extinction at B-band to low extinction at I-band. Notice the different star color dependencies.

The unfiltered fluxes were used to estimate V-magnitude and R-magnitude, represented by the symbols Cv and Cr. The solutions produced the following SME equations:

(8)     Cv = 21.261 - 2.5 * LOG ( Fc / g ) - 0.138 * m + 0.688 * C;  RMS = 0.028, N = 35

(9)     Cr = 20.940 - 2.5 * LOG ( Fc / g ) - 0.138 * m - 0.025 * C;  RMS = 0.025, N = 27

When the SME constant solutions for this observing date are combined with results from 3 other observing dates the following equations are been obtained:

Figure 6. Summary of SME constants derived from observations of several Landolt areas on March 11, April 2, April 6 and April 9, 2005. These equations use star color parameter Cvr = 0.57 * (B-V) - 0.30, which is a close approximation to V-R-0.30. The "air mass times star color term" was not used for this analysis (W was set to zero).

The reason the "air mass times star color" term was omitted from this analysis is that several dozen standard stars are needed to distinguish bewteen the effects of that term from the preceding star color term, and there weren't enough stars during this single night of observations to solve for both W and S. The W term can't be too important, because the RMS performance was good for a wide range of star colors and air mass values.

Based on the RMS performances for the SME solutions for this observing session it is fair to say that any stars in images taken during this observing session could be asssigned BVRI magnitudes with an accuracy of <0.03 magnitude, provided their star color was known. And if star color wasn't known, it could be determined with an accuracy of ~0.05 magnitude using an iteration procedure described above and on the companion web page. When C has an uncertainty of 0.05 magnitude, the largest uncertainty this produces for a magnitude using the SME is for B-band, and for that band the B-magnitude uncertainty attributable to an uncertain C is 0.015 magnitude. TheV-magnitude uncertainty attributable to a 0.05 magnitude uncertainty in C is 0.003 magnitude, and for R-band and I-band the uncertainties are 0.09 and 0.00 magnitude. Therefore, the iteration procedure for establishing star color is accurate enough to have negligible effect on the SME magnitude determinations (based on this night's observations).

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This site opened:  March 25, 2005 Last Update:  April 14, 2005