Aperture Radius and Second Order Sky Correction

When examining the science frames, we see that our star covers a circular-like area with a width of a few pixels. In order to determine the flux received from the star of interest, we need to be able to define an aperture radius that encompasses the star. It is important to note here that the aperture radius cannot be too large or it will eventually be overwhelmed by the background sky radiation or encompass another star. Additionally, we cannot take a radius that is too small or it will not encompass most of the star's incoming flux. In order to choose an adequate radius for our star, we need to examine the Signal-to-Noise ratio (SNR) as we increase the aperture radius encompassing our star. An SNR plot for increasing aperture radius can be found on the Photometry Handout⁵. Based on the plot, we found that the best aperture to use would be an aperture with a radius of about 1.5$\sigma$. However, if we use a radius of 3$\sigma$, we will be able to enclose 99% of the flux coming from the star and the SNR using this radius is only slightly less than if we used 1.5 $\sigma$. Thus, for our stars, we picked an aperture radius of 6 pixels which corresponds to about 3$\sigma$. In addition to choosing an aperture to encompass our star, we can also did a second order correction to our sky image by taking an annulus around our star and subtracting the average value of the pixels in this annulus from our image. SNR analysis (which will be explained in Section 6.2) tells us that we can pick an arbitrarily large annulus to reduce the noise contribution by a very small amount. However, picking such a large annulus leads to other problems because it may end up encompassing another star. A good enough annulus for our purposes can be taken so that the area in our annulus is about twice that of the aperture. An image showing how this aperture and annulus looks for one of our stars is given in Figure 5(a). Figure 5(b) is a plot of the total flux enclosed as a function of aperture radius for the same star. We can see that after about 5-6 pixels, the total flux enclosed stops increasing and levels off, meaning that we've enclosed almost all of the flux of the star. Thus, our choice of a 6 pixel radius for our aperture is good.

Figure 5: The figure on the left (a) shows the aperture and annulus that we used for one of our star images. The plot on the right (b) shows how the total flux enclosed behaves as we increase the aperture radius. Notice that the plot approximately asymptotes at a radius of about 6 pixels.