Magnitudes

Now that we have an aperture radius and annulus to isolate our star and subtract out the background, we can extract some data from our star images in order to calculate a magnitude for our star. Looking at Equation 3, we see that we need to find $N$, the number of photoelectrons coming from our star, and $t$, the exposure time that we used for each star. If we know the exposure time that we used for each star, then we just need to find the total number of photoelectrons collected from the star in order to obtain the flux for that star ( $\propto\frac{N}{t}$). Using the apertures and annuli we found before, we can create a mask of the second order sky corrected image with the given aperture and find the total signal within that aperture to get our total counts in DN. If we do this for two stars using the same telescope, we can determine, through relative photometry, the magnitude of one of the stars if we know the magnitude of the other star.

We measured 3 stars in the J-band (HD196573, HD201941, HD203856). HD201941 and HD 203856 are standard stars in the J-band and HD196573 is not. We took HD201941 as the calibration star (Star 2 in Equation 3) and tried to find the magnitudes of the other two stars with this calibration star using the process described in Section 4. A summary of our results is detailed in Table 1. The measured errors in our star images is given in Table 2 and the error propagation for the our magnitude is given in Equation 4.⁶ For this error in magnitude we assumed that the error from the standard star that we were comparing our star with is zero since the standard stars are measured with high accuracy. A figure of the histogram of our measured flux from our nine images of star HD196573 can be found on Figure 7(b).

$$\sigma_{m}^{2} = (1.087)^{2}\left(\frac{\sigma_{F}}{F}\right)^{2}$$ (4)

Table 1: Here is a table of the data that we extracted from our star images and the subsequent magnitudes that we calculated using this data. Note that all of the stars are taken in the J-band. The magnitude in parentheses is the magnitude we used for our calibration.

Star Name	N [$e^{-}$]	Exposure Time [sec]	Measured m$_{J}$	Actual m$_{J}$
HD201941	2.991e6	5	(6.7)	6.7
HD196573	2.326e6	0.66	4.696$\pm$0.016	N/A
HD203856	4.950e6	10	6.905$\pm$0.006	6.925

Table 2: Here is a table of the mean and SDOM of the nine images of each of the stars that we used for our photometry analysis.

Star Name	Mean [DN]	SDOM [DN]
HD201941	130,023	12,775.8
HD196573	101,141	11,608.6
HD203856	215,228	14,676.2