= Observed radiative flux
{title2=$F=L/(4\pi d^2)$}
For a steady <point-like astronomical source> with <isotropy> of emission, in transparent space with negligible relativistic or cosmological effects, <conservation of energy> spreads its <luminosity> over a <sphere> of <surface area> $4\pi d^2$. Its bolometric observed radiative flux is
$$
F=\frac{L}{4\pi d^2},\qquad L=4\pi d^2F.
$$
This <inverse-square law> reflects geometric dilution: the same emitted <power> crosses increasingly large <spheres>. It does not require <light> to lose <energy> on the journey. For a fixed <luminosity>, doubling <Euclidean distance> quarters the <observed radiative flux>. https://science.nasa.gov/learn/basics-of-space-flight/chapter6-1/[NASA: Inverse-square propagation]{ref}.
With $L_0=10^{26}\,\mathrm W$ and $d_0=20$ <parsecs>, the result is $F_0\simeq2.09\times10^{-11}\,\mathrm{W\,m^{-2}}$. At $40$ <parsecs>, the same source gives $F_0/4\simeq5.22\times10^{-12}\,\mathrm{W\,m^{-2}}$. An ideal collecting area of $1\,\mathrm{m^2}$ at $20$ <parsecs> intercepts approximately $2.09\times10^{-11}\,\mathrm W$, before detector losses.
\Image[/stellar-distance-and-flux.png]
{title=Fixed luminosity gives inverse-square dimming; different luminosities can give equal observed flux; configuration diagram schematic, not to scale}
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