= Extinction bias in photometric distance
{title2=$d_{\mathrm{inferred}}/d=10^{A/5}$}
= Distance overestimate from ignored extinction
{synonym}
Let $A\geq0$ be the <interstellar extinction> in magnitudes in the specified <photometric passband>. Compared with the same unobscured source at the same distance, its <radiative flux> is multiplied by $10^{-0.4A}$, so its <apparent magnitude> increases by $A$. Consequently,
$$
m-M=5\log_{10}\left(\frac{d}{10\,\mathrm{pc}}\right)+A,\qquad \mu_0=m-M-A.
$$
Here $\mu_0$ is the extinction-corrected <distance modulus>. The correction subtracts $A$ before using the distance formula. Caltech's NED distance-method documentation explicitly distinguishes an apparent <distance modulus> from an extinction-corrected <distance modulus>. https://ned.ipac.caltech.edu/Library/Distances/distintro.html[NASA/IPAC NED: Cepheid apparent and reddening-corrected distances]{ref}.
Ignoring positive <interstellar extinction> attributes too much dimming to distance, producing an overestimate:
$$
\frac{d_{\mathrm{inferred}}}{d}=10^{A/5}.
$$
As an illustrative toy, put the source with $M=0$ at $100$ <parsecs> behind $A=1$ magnitude of <interstellar extinction>. The measured <apparent magnitude> is $m=6$. Setting $A=0$ incorrectly gives $d_{\mathrm{inferred}}=10\,\mathrm{pc}\times10^{6/5}\simeq158.489\,\mathrm{pc}$, a factor $1.584893$ too large. Subtracting the known $A=1$ recovers $100$ <parsecs>. The diagram's fixed $A=1$ offset is a controlled illustration, not a model of how actual <interstellar extinction> grows with distance, and none of its values are <star> catalogue measurements.
Back to article page
