# Is the flux density independent of source-observer distance?

A particular observatory is providing the flux density of an astronomical source and only from this information, is it implicated that this particular parameter is unaffected by the distance between Earth and the source?

One way to see this is to consider that specific intensity is conserved. We then look at the following relation for flux density: $$S_{\nu}=\int_{\mathrm{source}} I_{\nu}(\theta,\phi)\cos\theta\;\mathrm{d}\Omega$$ where $$I_{\nu}(\theta,\phi)$$ is the specific intensity. Now, if a source is farther away, the solid angle it covers is smaller; therefore, we integrate over a smaller region of the sky. The flux density is then correspondingly smaller.
As an example, say we have a small source far away with constant specific intensity across a circle in the sky. In this case, we can write the flux density as $$S_{\nu}\simeq I_{\nu}\Omega$$ Using the small-angle approximation, the angular radius should scale approximately as $$\theta\approx R/d$$, with $$R$$ the physical radius and $$d$$ the distance to the source. The solid angle scales as $$\Omega\sim\theta^2\propto d^{-2}$$, so the flux density scales according to the inverse-square law, as you might expect.