# Flux density to flux

The definition of flux and flux density is fairly straightforward. However, I'm musing over the practical usage of it.

For one thing, if you observe a source through a filter, how would one go from the total observed energy/number of photons to the flux per unit wavelength? It's not a matter of simply dividing by the width of the filter -- a filter can have any exotic form of course.

So, how is this derived? Related, given a $f_{\nu}$ in some filter, what would be the way to go to total flux within that filter?

If the normalised filter response function is $R_{\nu}$ then the measured flux is $$F = \int f_{\nu} R_{\nu}\ d\nu$$ The integration is done over the frequency range of the filter.
If you measure a flux through a filter then the process cannot be inverted exactly. However the average flux density can be found by dividing the total flux by the effective frequency range $$\langle f_{\nu} \rangle = F/ \int R_{\nu} d\nu$$
• @John I don't understand your supplementary question. You cannot measure the luminosity of an object by observing through a single filter. All you can obtain is the average flux from the object in a particular wavelength range and that is done with eqn 2 of my answer. $F$ is what you measure and it depends on the true flux distribution of the object and the instrument response function $R_{\nu}$. Commented Oct 13, 2016 at 11:01
• I guess my question boils down to this: From this average $f_{\nu}$ that we have for a filter, how would one go about calculating the corresponding luminosity in this band, $L_{\nu}$? These are often used in literature. Commented Oct 13, 2016 at 14:01
• @John ?? $L_\nu = 4\pi d^2 f_{\nu}$, where $d$ is the distance and this assumes no extinction. Commented Oct 13, 2016 at 14:53
• I'm sorry for being so unclear - it's mainly because my issue was not quite clear to me either. It boils down to this: Suppose authors in a paper are talking about integrating a spectrum in a particular filter to obtain the luminosity of a galaxy, *in units of $L_{\odot}$. What does that mean? Is that an approximation of the bolometric luminosity based on the average flux in this filter? Not clear to me. Commented Oct 17, 2016 at 7:04