I have values of total infrared fluxes (SED) in literature expressed in Jy (Jansky) for galaxies. I want to assure that my infrared radial profiles of the same galaxies in MJy/sr (and other in Jy/pixel) give me the same integrated value as in the literature. How to get rid of the angular counterpart in steradian ? The data used to create radial profiles are HERSCHEL SPIRE 250um & 500um.
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$\begingroup$ Don't you just need to integrate over the solid angle/pixels? $\endgroup$– PrallaxSep 3, 2021 at 8:40
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$\begingroup$ Yes I think so, but the problem is to know how to do it when it is over the solid angle in steradian ? $\endgroup$– PsycOxSep 3, 2021 at 9:01
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Say that you have an image with $N \times N$ pixels that covers a region of sky of $\theta \times \theta$ steradians. For each pixel $i$ you have a value of flux density $J_i$, expressed in Jy/sr. The image contains a galaxy and you want to calculate its the total flux.
If this were a continuum problem, you would need to do an integral
$$F = \int_{\Omega_{gal}} J(\theta,\phi) d\Omega$$
Since the problem is discrete, you have to do a sum instead
$$F = \sum_{i=0}^M J_i \Delta \Omega$$
Where the pixels $i=0,..,M$ are the ones that contain the galaxy. To carry out the sum you only need to notice that $\Delta \Omega$, which is the solid angle covered by one single pixel, can be expressed as $\Delta \Omega = {\theta^2 \over N^2}$. And you are done:
$$F={\theta^2 \over N^2} \sum_{i=0}^M J_i$$
