Sky value definition (GNU Astronomy Utilities)

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7.1.4.1 Sky value definition ¶

This analysis is taken from Akhlaghi and Ichikawa 2015. Let’s assume that all instrument defects – bias, dark and flat – have been corrected and the magnitude (see Brightness, Flux, Magnitude and Surface brightness) of a detected object, $O$, is desired. The sources of flux on pixel¹⁹⁶ $i$ of the image can be written as follows:

Contribution from the target object ($O_i$).
Contribution from other detected objects ($D_i$).
Undetected objects or the fainter undetected regions of bright objects ($U_i$).
A cosmic ray ($C_i$).
The background flux, which is defined to be the count if none of the others exists on that pixel ($B_i$).

The total flux in this pixel ($T_i$) can thus be written as:

$$T_i=B_i+D_i+U_i+C_i+O_i.$$

By definition, $D_i$ is detected and it can be assumed that it is correctly estimated (deblended) and subtracted, we can thus set $D_i=0$. There are also methods to detect and remove cosmic rays, for example, the method described in van Dokkum (2001)¹⁹⁷, or by comparing multiple exposures. This allows us to set $C_i=0$. Note that in practice, $D_i$ and $U_i$ are correlated, because they both directly depend on the detection algorithm and its input parameters. Also note that no detection or cosmic ray removal algorithm is perfect. With these limitations in mind, the observed Sky value for this pixel ($S_i$) can be defined as

$$S_i\equiv{}B_i+U_i.$$

Therefore, as the detection process (algorithm and input parameters) becomes more accurate, or $U_i\to0$, the Sky value will tend to the background value or $S_i\to B_i$. Hence, we see that while $B_i$ is an inherent property of the data (pixel in an image), $S_i$ depends on the detection process. Over a group of pixels, for example, in an image or part of an image, this equation translates to the average of undetected pixels (Sky$=\sum{S_i}$). With this definition of Sky, the object flux in the data can be calculated, per pixel, with

$$T_{i}=S_{i}+O_{i} \quad\rightarrow\quad O_{i}=T_{i}-S_{i}.$$

In the fainter outskirts of an object, a very small fraction of the photo-electrons in a pixel actually belongs to objects, the rest is caused by random factors (noise), see Figure 1b in Akhlaghi and Ichikawa 2015. Therefore even a small over estimation of the Sky value will result in the loss of a very large portion of most galaxies. Besides the lost area/brightness, this will also cause an over-estimation of the Sky value and thus even more under-estimation of the object’s magnitude. It is thus very important to detect the diffuse flux of a target, even if they are not your primary target.

In summary, the more accurately the Sky is measured, the more accurately the magnitude (calculated from the sum of pixel values) of the target object can be measured (photometry). Any under/over-estimation in the Sky will directly translate to an over/under-estimation of the measured object’s magnitude.

The Sky value is only correctly found when all the detected objects ($D_i$ and $C_i$) have been removed from the data.

Footnotes

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For this analysis the dimension of the data (image) is irrelevant. So if the data is an image (2D) with width of $w$ pixels, then a pixel located on column $x$ and row $y$ (where all counting starts from zero and (0, 0) is located on the bottom left corner of the image), would have an index: $i=x+y\times{}w$.

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van Dokkum, P. G. (2001). Publications of the Astronomical Society of the Pacific. 113, 1420.