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I am currently working on a project on which I am calculating the luminosity of galaxies by using the SDSS data. They use a system of asinh magnitudes, and according to their website they use a system of softening parameters as it follows in https://www.sdss.org/dr12/algorithms/magnitudes/#asinh:
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I used their cmodel magnitude that they advise for galaxies' photometry. For example, I have the r magnitude of this galaxy that is 14.68, and I am trying to find its luminosity. They say that to convert to flux density, one must follow the following equation:

S = 3631 Jy * f/f0, where for the r band the AB conversion and shift is minimal.

However, when I plug the numbers into the equation:

formula

The distance of the galaxy is 63.3 Mpc But it seems that when I plug the numbers, the final value does not match, how is that? And yes, I converted to the units including the Jansky to 10^-26 W m^-2 Hz^-1 and the Mpc to m.

I wish to find the luminosity of the galaxy in terms of solar luminosity and I know that one solar luminosity unit is the same as 3.827e+26 W.

I was only able to find the correct answer when I did the following equation:

formula

Where:

formula

If anyone knows what I am doing wrong point out, please. Is it because flux intensity is different from flux?

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