Distance modulus
= Distance modulus
{title2=$\mu$}
{wiki}
The distance modulus compares received brightness with intrinsic brightness through $\mu=m-M$, the difference between <apparent magnitude> and <absolute magnitude> in the same <photometric passband> and <magnitude zero point>. With no <interstellar extinction>, fixed emitted spectrum and <luminosity>, <isotropy> of emission, and the local <inverse-square law>, it determines <Euclidean distance>:
$$
\mu=m-M=5\log_{10}\left(\frac{d}{10\,\mathrm{pc}}\right),\qquad d=10\,\mathrm{pc}\;10^{\mu/5}.
$$
The ratio inside the <logarithm> is dimensionless. To see the factor $5$, use the <inverse-square law> in the <magnitude difference> formula:
$$
\frac{F(d)}{F_{10}}=\left(\frac{10\,\mathrm{pc}}{d}\right)^2,\qquad m-M=-2.5\log_{10}\left[\left(\frac{10\,\mathrm{pc}}{d}\right)^2\right].
$$
For example, $m=5$ and $M=0$ give $\mu=5$, hence $d=10\,\mathrm{pc}\times10=100\,\mathrm{pc}$. A tenfold increase in distance adds five magnitudes; an increase of one magnitude multiplies distance by $10^{1/5}\simeq1.584893$. The <distance modulus> is a <logarithmic scale> for distance, not a distance in length units. https://lco.global/spacebook/distance/what-is-distance-modulus/[Las Cumbres Observatory: What is distance modulus?]{ref}.
At $10$ <parsecs>, $\mu=0$ and $m=M$. Within $10$ <parsecs>, the <distance modulus> is negative; for example, $\mu=-5$ gives the positive distance $d=1\,\mathrm{pc}$. A negative <distance modulus> does not imply negative distance. At cosmological distances, the inference needs <luminosity distance> and a <K correction>; the simple local formula must not silently absorb these effects.
\Image[/distance-modulus-ideal-source.png]
{title=Toy source with absolute magnitude zero: inverse-square distance steps and the distance overestimate from ignoring one magnitude of extinction}
<Interstellar extinction> increases <apparent magnitude>. The observed <distance modulus> then includes both geometric dimming and attenuation; <extinction bias in photometric distance> explains how to separate them.