From Relative Magnitudes to Fluxes
Nicole Fields

Now that all the relative magnitudes and errors from IRAF have been extracted from the .fits.mag.1 files, the relative magnitude of the source must be calibrated using the known magnitudes of the reference stars.

Source Star B magnitude (err) V magnitude (err) R magnitude (err)
Markarian 421 115.02 (0.03)14.36 (0.02)14.04 (0.02)
216.20 (0.04)15.57 (0.05)15.20 (0.03)
316.69 (0.03)15.77 (0.03)15.24 (0.03)
Markarian 501 113.55 (0.03)12.61 (0.02)12.11 (0.02)
214.10 (0.03)13.23 (0.02)12.79 (0.02)
315.98 (0.04)15.24 (0.02)14.80 (0.02)
416.05 (0.05)15.30 (0.02)14.96 (0.02)
516.27 (0.04)15.51 (0.02)15.08 (0.02)
616.82 (0.05)15.67 (0.04)14.99 (0.04)
1ES 1218+304 A13.13 (0.01)12.47 (0.01)12.11 (0.01)
B14.65 (0.01)14.01 (0.01)13.61 (0.01)
H 1426+428 A15.61 (0.03)14.16 (0.01)13.23 (0.02)
B15.47 (0.02)14.61 (0.01)14.17 (0.02)
C14.22 (0.02)13.46 (0.02)13.00 (0.02)
D16.14 (0.06)15.53 (0.04)15.02 (0.04)
1ES 1959+650 1 12.67 (0.04)12.29 (0.02)
2 12.86 (0.02)12.53 (0.02)
3 13.18 (0.02)12.27 (0.02)
4 14.53 (0.03)14.08 (0.03)
5Note: This is a variable star and was not used as a reference star.
6 15.20 (0.03)14.78 (0.03)
7 15.24 (0.03)14.79 (0.03)
1ES 2344+514 1 12.61 (0.04)12.25 (0.04)
2 14.62 (0.06)14.20 (0.05)
3 15.89 (0.08)15.40 (0.08)

Using the known magnitudes of the reference stars above, and the relative magnitudes of the reference stars given by IRAF from the photometry, it is possible to calculate the offset for each frame. This offset is calculated by subtracting the known magnitude from the relative magnitudefor each reference star, and then calculating the average. This average offset is then subtracted from the measured magnitude of the source to get the true apparent magnitude of the source.

The error on the magnitude of the source was calculated by determining the sample standard deviation of the offset and adding that in quadrature to the error on the magnitude of the source given by IRAF. In the few cases where only one reference star was available, then the error on the magnitude was calculated by propagating the errors on the known magnitudes of each reference star along with the errors on the relative magnitudes given by IRAF to get the error on the offset. This was then added in quadrature to the error on the magnitude of the source given by IRAF.

Once the magnitudes had been properly calibrated, they were then converted to fluxes. This was done using the following equation from Sparke & Gallagher.

Here m is in Vega/A0 magnitudes and F&nu is in Janskys. For the filters of our observation BVR with B and V Johnson filters and R a Cousins filter the following table gives the peak wavelength in Angstroms and the corresponding Zero Point.

&lambdaeff4360 Å5450 Å6410 Å
Zero Point ZP&nu-

Since magnitudes are on a log scale, it was slightly tricky to convert the error in magnitude to an error in flux. The following error formula is derived from the fact that magnitude differences are flux ratios.

All of these calculations were done with a C program whose code can be found here.

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