Editing Distance modulus (section)

==Definition==

The distance modulus <math>\mu=m-M</math> is the difference between the [[apparent magnitude]] <math>m</math> (ideally, corrected from the effects of [[interstellar reddening|interstellar absorption]]) and the [[absolute magnitude]] <math>M</math> of an [[astronomical object]]. It is related to the luminous distance <math>d</math> in parsecs by:
<math display="block">\begin{align}
\log_{10}(d) &= 1 + \frac{\mu}{5} \\
\mu &= 5\log_{10}(d) - 5
\end{align}</math>

This definition is convenient because the observed brightness of a light source is related to its distance by the [[inverse square law]] (a source twice as far away appears one quarter as bright) and because brightnesses are usually expressed not directly, but in [[apparent magnitude|magnitudes]].

Absolute magnitude <math>M</math> is defined as the apparent magnitude of an object when seen at a distance of 10 [[parsec]]s. If a light source has flux {{math|''F''(''d'')}} when observed from a distance of <math>d</math> parsecs, and flux {{math|''F''(10)}} when observed from a distance of 10 parsecs, the inverse-square law is then written like:
<math display="block">F(d) = \frac{F(10)}{\left(\frac{d}{10}\right)^2} </math>

The magnitudes and flux are related by:
<math display="block">\begin{align}
m &= -2.5 \log_{10} F(d) \\[1ex]
M &= -2.5 \log_{10} F(d=10)
\end{align}</math>

Substituting and rearranging, we get:
<math display="block">\mu = m - M = 5 \log_{10}(d) - 5 =  5 \log_{10}\left(\frac{d}{10\,\mathrm{pc}}\right)</math>
which means that the apparent magnitude is the absolute magnitude plus the distance modulus.

Isolating <math>d</math> from the equation <math>5 \log_{10}(d) - 5 = \mu </math>, finds that the distance (or, the [[luminosity distance]]) in parsecs is given by
<math display="block">d = 10^{\frac{\mu}{5}+1} </math>

The uncertainty in the distance in parsecs ({{math|''δd''}}) can be computed from the uncertainty in the distance modulus ({{math|''δμ''}}) using
<math display="block"> \delta d = 0.2 \ln(10) 10^{0.2\mu+1} \delta\mu \approx 0.461 d \ \delta\mu</math>
which is derived using standard error analysis.<ref name="taylor1982">{{cite book
| first = John R. | last = Taylor
| year=1982
| title=An introduction to Error Analysis
| publisher=University Science Books
| location=Mill Valley, California
| isbn=0-935702-07-5
| url-access=registration
| url=https://archive.org/details/introductiontoer00tayl
}}</ref>