{{Short description|Logarithmic distance scale}} The '''distance modulus''' is a way of expressing distances that is often used in astronomy. It describes distances on a logarithmic scale based on the astronomical magnitude system.<ref name=":0">{{Cite book |last=Carroll |first=Bradley W. |title=An introduction to modern astrophysics |last2=Ostlie |first2=Dale A. |date=2017 |publisher=Cambridge University Press |isbn=978-1-108-42216-1 |edition=2nd |location=Cambridge}}</ref>
==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 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 magnitudes.{{Clarify|reason=The second part of the sentence is a bit confusing. Seems like circular logic. Not sure if needed at all.|date=August 2025}}
Absolute magnitude <math>M</math> is defined as the apparent magnitude of an object when seen at a distance of 10 parsecs. 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. In the above, the two magnitudes correspond to bolometric ones, i.e. measured across all wavelengths.<ref name=":0" /> In reality, detectors are more sensitive in specific frequency ranges, where other factors, like calibration or absorption, could play an important role.<ref>{{Cite book |last=Gallaway |first=Mark |title=An introduction to observational astrophysics |date=2020 |publisher=Springer |isbn=978-3-030-43551-6 |edition=2nd |series=Undergraduate lecture notes in physics |location=Cham, Switzerland}}</ref> Absorption 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 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 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 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 Smith, ''Introductory Astronomy and Astrophysics'' (1992, Thomson Learning) {{refend}}
{{DEFAULTSORT:Distance Modulus}} Category:Physical quantities