Editing Distance modulus

{{Short description|Logarithmic distance scale}}
The '''distance modulus''' is a way of expressing [[Distance|distances]] that is often used in [[astronomy]]. It describes distances on a [[logarithmic scale]] based on the [[Magnitude (astronomy)|astronomical magnitude system]].

==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>

== Different kinds of distance moduli ==
{{unreferenced | section|date=July 2023}}
Distance is not the only quantity relevant in determining the difference between absolute and apparent magnitude. Absorption is another important factor, and it may even be a dominant one in particular cases (''e.g.'', in the direction of the [[Galactic Center]]). Thus a distinction is made between distance moduli uncorrected for [[interstellar reddening|interstellar absorption]], the values of which would overestimate distances if used naively, and absorption-corrected moduli.

The first ones are termed ''visual distance moduli'' and are denoted by <math>{(m - M)}_{v}</math>, while the second ones are called ''true distance moduli'' and denoted by <math>{(m - M)}_{0}</math>.

Visual distance moduli are computed by calculating the difference between the observed apparent magnitude and some theoretical estimate of the absolute magnitude. True distance moduli require a further theoretical step; that is, the estimation of the [[interstellar absorption coefficient]].

==Usage==

Distance moduli are most commonly used when expressing the distance to other [[Galaxy|galaxies]] in the relatively nearby [[universe]].  For example, the [[Large Magellanic Cloud]] (LMC) is at a distance modulus of 18.5,<ref name="alvez2--4">{{cite journal | author=D. R. Alvez | title=A review of the distance and structure of the Large Magellanic Cloud | year=2004 | volume=48  | issue=9 | pages=659–665 | bibcode=2004NewAR..48..659A | doi=10.1016/j.newar.2004.03.001 | type=abstract | journal=New Astronomy Reviews | arxiv = astro-ph/0310673 }}</ref> the [[Andromeda Galaxy]]'s distance modulus is 24.4,<ref name="alvez2005">{{cite journal | author1=I. Ribas |author2=C. Jordi |author3=F. Vilardell |author4=E. L. Fitzpatrick | author5=R. W. Hilditch |author6=E. F. Guinan | title=First Determination of the Distance and Fundamental Properties of an Eclipsing Binary in the Andromeda Galaxy | year=2005 | volume=635  | issue=1 | pages=L37–L40 | bibcode=2005ApJ...635L..37R | doi=10.1086/499161 | type=abstract | journal=The Astrophysical Journal | arxiv = astro-ph/0511045 }}</ref> and the galaxy [[NGC 4548]] in the [[Virgo Cluster]] has a DM of 31.0.<ref name="graham1999">{{cite journal | author1=J. A. Graham |author2=L. Ferrarese |author3=W. L. Freedman |author4=R. C. Kennicutt Jr. |author5=J. R. Mould |author6=A. Saha |author7=P. B. Stetson |author8=B. F. Madore |author9=F. Bresolin |author10=H. C. Ford |author11=B. K. Gibson |author12=M. Han |author13=J. G. Hoessel |author14=J. Huchra |author15=S. M. Hughes |author16=G. D. Illingworth |author17=D. D. Kelson |author18=L. Macri |author19=R. Phelps |author20=S. Sakai |author21=N. A. Silbermann |author22=A. Turner | title=The Hubble Space Telescope Key Project on the Extragalactic Distance Scale. XX. The Discovery of Cepheids in the Virgo Cluster Galaxy NGC 4548 | year=1999 | volume=516  | issue=2 | pages=626–646 | bibcode=1999ApJ...516..626G | doi=10.1086/307151 | type=abstract | journal=The Astrophysical Journal | doi-access=free }}</ref>  In the case of the LMC, this means that [[SN 1987A|Supernova 1987A]], with a peak apparent magnitude of 2.8, had an absolute magnitude of −15.7, which is low by supernova standards.

Using distance moduli makes computing magnitudes easy. As for instance, a solar type star (M= 5) in the Andromeda Galaxy (DM= 24.4) would have an apparent magnitude (m) of 5 + 24.4 = 29.4, so it would be barely visible for the [[Hubble Space Telescope]] which has a limiting magnitude of about 30.<ref>{{cite journal |last1=Illingworth |first1=G. D. |last2=Magee |first2=D. |last3=Oesch |first3=P. A. |last4=Bouwens |first4=R. J. |last5=Labbé |first5=I. |last6=Stiavelli |first6=M. |last7=van Dokkum |first7=P. G. |last8=Franx |first8=M. |last9=Trenti |first9=M. |last10=Carollo |first10=C. M. |last11=Gonzalez |first11=V. |title=The HST eXtreme Deep Field XDF: Combining all ACS and WFC3/IR Data on the HUDF Region into the Deepest Field Ever|journal=The Astrophysical Journal Supplement Series |date=21 October 2013 |volume=209 |issue=1 |pages=6 |arxiv=1305.1931 |bibcode=2013ApJS..209....6I |doi=10.1088/0067-0049/209/1/6|s2cid=55052332 }}</ref> Since it is apparent magnitudes which are actually measured at a telescope, many discussions about distances in astronomy are really discussions about the putative or derived absolute magnitudes of the distant objects being observed.

==References==

{{reflist}}
{{refbegin}}
* Zeilik, Gregory and [[Elske Smith|Smith]], ''Introductory Astronomy and Astrophysics'' (1992, Thomson Learning)
{{refend}}

{{DEFAULTSORT:Distance Modulus}}
[[Category:Physical quantities]]

[[de:Absolute Helligkeit#Entfernungsmodul]]