= Flux-distance degeneracy
{title2=$(L,d)\mapsto(k^2L,kd)$}
One <observed radiative flux> cannot determine both <luminosity> and <Euclidean distance>: for every $k>0$, replacing $(L,d)$ by $(k^2L,kd)$ leaves $F$ unchanged. In particular, $4\times10^{26}\,\mathrm W$ at $40$ <parsecs> produces the same <observed radiative flux> as $10^{26}\,\mathrm W$ at $20$ <parsecs>. This is a concrete failure of <identifiability> in an <inverse problem>.
An independent <stellar parallax> fixes the distance and breaks this degeneracy; alternatively, a calibrated <standard candle> supplies the <luminosity>. <Interstellar extinction> can further reduce the <observed radiative flux>, and a <photometric passband> measures only a selected spectral range. At cosmological distances, use <luminosity distance> in the bolometric relation. Each inference must specify which <radiative flux> was measured and what propagation assumptions connect it to <luminosity>.
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