 Extinction bias in photometric distance

Let $A\ge 0$ 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,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. ^[ref].

Ignoring positive interstellar extinction attributes too much dimming to distance, producing an overestimate: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 $d_{\mathrm{inferred}}=10\mathrm{pc}\times 10^{6/5}\simeq 158.489\mathrm{pc}$, 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.