Editing Cosmic distance ladder (section)

== Extragalactic distance scale ==
{| class="wikitable" style="float:right; width:40%;"
|+Extragalactic distance indicators<ref>Adapted from {{cite journal |author1=George H. Jacoby |author2=David Branch |author3=Robin Ciardullo |author4=Roger L. Davies |author5=William E. Harris |author6=Michael J. Pierce |author7=Christopher J. Pritchet |author8=John L. Tonry |author9=Douglas L. Welch |title=A critical review of selected techniques for measuring extragalactic distances |journal=[[Publications of the Astronomical Society of the Pacific]] |date=1992 |volume=104 |pages=599–662 |number=678 |jstor=40679907 |bibcode=1992PASP..104..599J |doi=10.1086/133035 |doi-access=free}}</ref>
|- valign="top"
!Method
!Uncertainty for Single Galaxy (mag)
!Distance to [[Virgo Cluster]] ([[Parsec|Mpc]])
!Range (Mpc)
|- valign="top"
|Classical Cepheids
|0.16
|15–25
|29
|- valign="top"
|Novae
|0.4
|21.1 ± 3.9
|20
|- valign="top"
|Planetary Nebula Luminosity Function
|0.3
|15.4 ± 1.1
|50
|- valign="top"
|Globular Cluster Luminosity Function
|0.4
|18.8 ± 3.8
|50
|- valign="top"
|Surface Brightness Fluctuations
|0.3
|15.9 ± 0.9
|50
|- valign="top"
|Sigma-D relation
|0.5
|16.8 ± 2.4
|&gt; 100
|- valign="top"
|Type Ia Supernovae
|0.10
|19.4 ± 5.0
|&gt; 1000
|}

The extragalactic distance scale is a series of techniques used today by astronomers to determine the distance of cosmological bodies beyond our own galaxy, which are not easily obtained with traditional methods. Some procedures use properties of these objects, such as [[star]]s, [[globular cluster]]s, [[nebula]]e, and galaxies as a whole. Other methods are based more on the statistics and probabilities of things such as entire [[galaxy cluster]]s.

=== Wilson–Bappu effect ===
{{main|Wilson–Bappu effect}}
Discovered in 1956 by [[Olin Chaddock Wilson|Olin Wilson]] and [[Vainu Bappu|M.K. Vainu Bappu]], the [[Wilson–Bappu effect]] uses the effect known as [[spectroscopic parallax]]. Many stars have features in their [[Astronomical spectroscopy|spectra]], such as the [[Fraunhofer lines|calcium K-line]], that indicate their [[absolute magnitude]]. The distance to the star can then be calculated from its [[apparent magnitude]] using the [[distance modulus]].

There are major limitations to this method for finding stellar distances. The calibration of the spectral line strengths has limited accuracy and it requires a correction for [[interstellar extinction]]. Though in theory this method has the ability to provide reliable distance calculations to stars up to 7 megaparsecs (Mpc), it is generally only used for stars at hundreds of kiloparsecs (kpc).

=== Classical Cepheids ===

Beyond the reach of the [[Wilson–Bappu effect]], the next method relies on the [[period-luminosity relation]] of classical [[Cepheid variable]] stars. The following relation can be used to calculate the distance to Galactic and extragalactic classical Cepheids:

{{in5}}<math> 5\log_{10}{d}=V+ (3.34) \log_{10}{P} - (2.45) (V-I) + 7.52 \,.</math><ref name=benedict2007>Benedict, G. Fritz et al. [http://adsabs.harvard.edu/abs/2007AJ....133.1810B "Hubble Space Telescope Fine Guidance Sensor Parallaxes of Galactic Cepheid Variable Stars: Period-Luminosity Relations"] {{Webarchive|url=https://web.archive.org/web/20160123123325/http://adsabs.harvard.edu/abs/2007AJ....133.1810B |date=2016-01-23 }}, ''The Astronomical Journal'', Volume 133, Issue 4, pp. 1810–1827 (2007)</ref> <br />
{{in5}}<math> 5\log_{10}{d}=V+ (3.37) \log_{10}{P} - (2.55) (V-I) + 7.48 \,. </math><ref name=majaess2011>Majaess, Daniel; Turner, David; Moni Bidin, Christian; Mauro, Francesco; Geisler, Douglas; Gieren, Wolfgang; Minniti, Dante; Chené, André-Nicolas; Lucas, Philip; Borissova, Jura; Kurtev, Radostn; Dékány, Istvan; Saito, Roberto K. [http://adsabs.harvard.edu/abs/2011ApJ...741L..27M "New Evidence Supporting Membership for TW Nor in Lyngå 6 and the Centaurus Spiral Arm"] {{Webarchive|url=https://web.archive.org/web/20170310051039/http://adsabs.harvard.edu/abs/2011ApJ...741L..27M |date=2017-03-10 }}, ''ApJ Letters'', Volume 741, Issue 2, article id. L2 (2011)</ref>

Several problems complicate the use of Cepheids as standard candles and are actively debated, chief among them are: the nature and linearity of the period-luminosity relation in various passbands and the impact of metallicity on both the zero-point and slope of those relations, and the effects of photometric contamination (blending) and a changing (typically unknown) extinction law on Cepheid distances.<ref name="stanekudalski1999">
{{cite arXiv |author1=Stanek, K. Z. |author2=Udalski, A. |date=1999 |eprint=astro-ph/9909346 |title=The Optical Gravitational Lensing Experiment. Investigating the Influence of Blending on the Cepheid Distance Scale with Cepheids in the Large Magellanic Cloud}}</ref><!--
--><ref name="udalski2001">{{cite journal |author1=Udalski, A. |author-link=Andrzej Udalski |author2=Wyrzykowski, L. |author3=Pietrzynski, G. |author4=Szewczyk, O. |author5=Szymanski, M. |author6=Kubiak, M. |author7=Soszynski, I. |author8=Zebrun, K. |bibcode=2001AcA....51..221U |title=The Optical Gravitational Lensing Experiment. Cepheids in the Galaxy IC1613: No Dependence of the Period-Luminosity Relation on Metallicity |date=2001 |page=221 |volume=51 |journal=Acta Astronomica |doi= |arxiv=astro-ph/0109446}}</ref><!--
--><ref name="ngeow2006">{{cite journal |author1=Ngeow, C. |author2=Kanbur, S. M. |title=The Hubble Constant from Type Ia Supernovae Calibrated with the Linear and Nonlinear Cepheid Period-Luminosity Relations |bibcode=2006ApJ...642L..29N |doi=10.1086/504478 |date=2006 |pages=L29 |volume=642 |issue=1 |journal=The Astrophysical Journal |arxiv=astro-ph/0603643 |s2cid=17860528}}</ref><!--
--><ref name="macri2006">{{cite journal |author1=Macri, L. M. |author2=Stanek, K. Z. |author3=Bersier, D. |author4=Greenhill, L. J. |author5=Reid, M. J. |bibcode=2006ApJ...652.1133M |title=A New Cepheid Distance to the Maser–Host Galaxy NGC 4258 and Its Implications for the Hubble Constant |doi=10.1086/508530 |date=2006 |pages=1133–1149 |issue=2 |volume=652 |journal=The Astrophysical Journal |arxiv=astro-ph/0608211 |s2cid=15728812}}</ref><!--
--><ref name="bono2008">{{cite journal |author1=Bono, G. |author2=Caputo, F. |author3=Fiorentino, G. |author4=Marconi, M. |author5=Musella, I. |title=Cepheids in External Galaxies. I. The Maser–Host Galaxy NGC 4258 and the Metallicity Dependence of Period–Luminosity and Period–Wesenheit Relations |bibcode=2008ApJ...684..102B |doi=10.1086/589965 |date=2008 |page=102 |volume=684 |issue=1 |journal=The Astrophysical Journal |arxiv=0805.1592 |s2cid=6275274}}</ref><!--
--><ref name=majaess2009b>{{cite journal |author1=Majaess, D. |author2=Turner, D. |author3=Lane, D. |bibcode=2009AcA....59..403M |title=Type II Cepheids as Extragalactic Distance Candles |date=2009 |page=403 |volume=59 |issue=4 |journal=Acta Astronomica |doi= |arxiv=0909.0181}}</ref><!--
--><ref name="madore2009">{{cite journal |author1=Madore, Barry F. |author2=Freedman, Wendy L. |title=Concerning the Slope of the Cepheid Period–Luminosity Relation |bibcode=2009ApJ...696.1498M |doi=10.1088/0004-637X/696/2/1498 |date=2009 |pages=1498–1501 |issue=2 |volume=696 |journal=The Astrophysical Journal |arxiv=0902.3747 |s2cid=16325249}}</ref><!--
--><ref name="scowcroft2009">{{cite journal |author1=Scowcroft, V. |author2=Bersier, D. |author3=Mould, J. R. |author4=Wood, P. R. |bibcode=2009MNRAS.396.1287S |title=The effect of metallicity on Cepheid magnitudes and the distance to M33 |doi=10.1111/j.1365-2966.2009.14822.x |date=2009 |pages=43–47 |issue=3 |volume=396 |journal=[[Monthly Notices of the Royal Astronomical Society]] |doi-access=free |arxiv=0903.4088}}</ref><!--
--><ref name="majaess2010">{{cite journal |author1=Majaess, D. |title=The Cepheids of Centaurus A (NGC 5128) and Implications for H0 |bibcode=2010AcA....60..121M |date=2010 |page=121 |volume=60 |issue=2 |journal=Acta Astronomica |doi= |arxiv=1006.2458}}</ref>

These unresolved matters have resulted in cited values for the Hubble constant ranging between 60&nbsp;km/s/Mpc and 80&nbsp;km/s/Mpc. Resolving this discrepancy is one of the foremost problems in astronomy since some cosmological parameters of the Universe may be constrained significantly better by supplying a precise value of the Hubble constant.<ref name=tammannsandage2008>
{{cite journal |bibcode=2008A&ARv..15..289T |journal=Annual Review of Astronomy and Astrophysics |volume=15 |issue=4 |page=289 |doi=10.1007/s00159-008-0012-y |arxiv=0806.3018 |title=The expansion field: The value of H 0 |date=2008 |last1=Tammann |first1=G. A. |last2=Sandage |first2=A. |last3=Reindl |first3=B. |s2cid=18463474}}</ref><ref name=freedman2010>{{cite journal |bibcode=2010ARA&A..48..673F |journal=Annual Review of Astronomy and Astrophysics |volume=48 |pages=673–710 |doi=10.1146/annurev-astro-082708-101829 |arxiv=1004.1856 |title=The Hubble Constant |date=2010 |last1=Freedman |first1=Wendy L. |last2=Madore |first2=Barry F. |s2cid=119263173}}</ref>

Cepheid variable stars were the key instrument in Edwin Hubble's 1923 conclusion that [[Andromeda Galaxy|M31]] (Andromeda) was an external galaxy, as opposed to a smaller nebula within the Milky Way. He was able to calculate the distance of M31 to 285&nbsp;kpc, today's value being 770&nbsp;kpc.{{citation needed|date=January 2023}}

As detected thus far, NGC 3370, a spiral galaxy in the constellation Leo, contains the farthest Cepheids yet found at a distance of 29&nbsp;Mpc. Cepheid variable stars are in no way perfect distance markers: at nearby galaxies they have an error of about 7% and up to a 15% error for the most distant.<ref>{{Cite web |last=Welch |first=D. |title=Calibration and Uncertainties |url=https://ned.ipac.caltech.edu/level5/Jacoby/Jacoby3_5.html |website=[[NASA/IPAC Extragalactic Database]]}}</ref>

=== Supernovae ===
[[File:SN1994D.jpg|thumb|[[SN 1994D]] (bright spot on the lower left) in the [[NGC 4526]] galaxy. Image by [[NASA]], [[ESA]], The Hubble Key Project Team, and The High-Z Supernova Search Team]]

There are several different methods for which [[supernova]]e can be used to measure extragalactic distances.

==== Measuring a supernova's photosphere ====
We can assume that a supernova expands in a spherically symmetric manner. If the supernova is close enough such that we can measure the angular extent, ''θ''(''t''), of its [[photosphere]], we can use the equation

<math display="block">\omega = \frac{\Delta\theta}{\Delta t} \,,</math>
where ''ω'' is angular velocity, ''θ'' is angular extent. In order to get an accurate measurement, it is necessary to make two observations separated by time Δ''t''. Subsequently, we can use

<math display="block">\ d = \frac{V_{ej}}{\omega} \,,</math>
where d is the distance to the supernova, ''V<sub>ej</sub>'' is the supernova's ejecta's radial velocity (it can be assumed that ''V<sub>ej</sub>'' equals ''V<sub>θ</sub>'' if spherically symmetric).

This method works only if the supernova is close enough to be able to measure accurately the photosphere. Similarly, the expanding shell of gas is in fact not perfectly spherical nor a perfect blackbody. Also interstellar extinction can hinder the accurate measurements of the photosphere. This problem is further exacerbated by core-collapse supernova. All of these factors contribute to the distance error of up to 25%.

==== Type Ia light curves ====

[[Type Ia supernovae]] are some of the best ways to determine extragalactic distances, as introduced by Stirling A. Colgate.<ref>{{Cite journal |last=Colgate |first=S. A. |date=September 1979 |title=Supernovae as a standard candle for cosmology |url=http://adsabs.harvard.edu/doi/10.1086/157300 |journal=The Astrophysical Journal |language=en |volume=232 |page=404 |doi=10.1086/157300 |bibcode=1979ApJ...232..404C |issn=0004-637X}}</ref> Ia's occur when a binary white dwarf star begins to accrete matter from its companion star. As the white dwarf gains matter, eventually it reaches its [[Chandrasekhar limit]] of <math> 1.4 M_{\odot} </math>.

Once reached, the star becomes unstable and undergoes a runaway nuclear fusion reaction. Because all Type Ia supernovae explode at about the same mass, their absolute magnitudes are all the same. This makes them very useful as standard candles. All Type Ia supernovae have a standard blue and visual magnitude of

<math display="block">\ M_B \approx M_V \approx -19.3 \pm 0.3 \,.</math>
Therefore, when observing a Type Ia supernova, if it is possible to determine what its peak magnitude was, then its distance can be calculated. It is not intrinsically necessary to capture the supernova directly at its peak magnitude; using the '''multicolor light curve shape''' method ('''MLCS'''), the shape of the light curve (taken at any reasonable time after the initial explosion) is compared to a family of parameterized curves that will determine the absolute magnitude at the maximum brightness. This method also takes into effect interstellar extinction/dimming from dust and gas.

Similarly, the '''stretch method''' fits the particular supernovae magnitude light curves to a template light curve. This template, as opposed to being several light curves at different wavelengths (MLCS) is just a single light curve that has been stretched (or compressed) in time. By using this ''Stretch Factor'', the peak magnitude can be determined.<ref>{{cite journal |last1=Coelho |first1=R. |last2=Calv˜ao |first2=M. |last3=Ribamar |first3=R. |last4=Siffert |first4=B. |title=Standardization of type Ia supernovae |year=2015 |journal=European Journal of Physics |volume=36 |issue=1 |page=015007 |doi=10.1088/0143-0807/36/1/015007 |display-authors=1 |arxiv=1411.3596 |bibcode=2015EJPh...36a5007C |s2cid=119096479}}</ref>

Using Type Ia supernovae is one of the most accurate methods, particularly since supernova explosions can be visible at great distances (their luminosities rival that of the galaxy in which they are situated), much farther than Cepheid Variables (500 times farther). Much time has been devoted to the refining of this method. The current uncertainty approaches a mere 5%, corresponding to an uncertainty of just 0.1 magnitudes.

==== Novae in distance determinations ====

[[Nova]]e can be used in much the same way as supernovae to derive extragalactic distances. There is a direct relation between a nova's max magnitude and the time for its visible light to decline by two magnitudes. This relation is shown to be:

<math display="block">\ M^\max_V = -9.96 - 2.31 \log_{10} \dot{x} \,.</math>
Where <math>\dot{x}</math> is the time derivative of the nova's mag, describing the average rate of decline over the first 2 magnitudes.

After novae fade, they are about as bright as the most luminous Cepheid variable stars, therefore both these techniques have about the same max distance: ~ 20 Mpc. The error in this method produces an uncertainty in magnitude of about&nbsp;±0.4

=== Globular cluster luminosity function ===

Based on the method of comparing the luminosities of globular clusters (located in galactic halos) from distant galaxies to that of the [[Virgo Cluster]], the [[globular cluster luminosity function]] carries an uncertainty of distance of about 20% (or&nbsp;0.4 magnitudes).

US astronomer William Alvin Baum first attempted to use globular clusters to measure distant elliptical galaxies.<ref>{{Cite journal |last=Baum |first=William A. |date=October 1955 |title=The Distribution of Luminosity in Elliptical Galaxies |url=http://iopscience.iop.org/article/10.1086/126829 |journal=Publications of the Astronomical Society of the Pacific |language=en |volume=67 |issue=398 |page=328 |doi=10.1086/126829 |bibcode=1955PASP...67..328B |issn=0004-6280|url-access=subscription }}</ref> He compared the brightest globular clusters in Virgo A galaxy with those in Andromeda, assuming the luminosities of the clusters were the same in both. Knowing the distance to Andromeda, Baum has assumed a direct correlation and estimated Virgo A's distance.

Baum used just a single globular cluster, but individual formations are often poor standard candles. Canadian astronomer [[René Racine]] assumed the use of the globular cluster luminosity function (GCLF) would lead to a better approximation.<ref>{{Cite journal |last1=Harris |first1=William E. |last2=Racine |first2=René |date=1979 |title=Globular Clusters in Galaxies |journal=Annual Review of Astronomy and Astrophysics |language=en |volume=17 |pages=241–274 |doi=10.1146/annurev.aa.17.090179.001325 |bibcode=1979ARA&A..17..241H |issn=0066-4146}}</ref> The number of globular clusters as a function of magnitude is given by:

<math display="block">\ \Phi (m) = A e^{(m-m_0)^2/2\sigma^2} \,</math>
where ''m''<sub>0</sub> is the turnover magnitude, ''M''<sub>0</sub> is the magnitude of the Virgo cluster, and sigma is the dispersion&nbsp;~&nbsp;1.4 mag.

It is assumed that globular clusters all have roughly the same luminosities within the [[universe]]. There is no universal globular cluster luminosity function that applies to all galaxies.

=== Planetary nebula luminosity function ===

Like the GCLF method, a similar numerical analysis can be used for [[planetary nebula]]e within far off galaxies. The [[planetary nebula luminosity function]] (PNLF) was first proposed in the late 1970s by Holland Cole Ford and David Jenner.<ref>{{Cite journal |last1=Jenner |first1=D. C. |last2=Ford |first2=H. C. |last3=Jacoby |first3=G. H. |date=January 1979 |title=Planetary nebulae in local group galaxies. VII - Spectrophotometry and filter photometry of M32-1 |url=http://adsabs.harvard.edu/doi/10.1086/156743 |journal=The Astrophysical Journal |language=en |volume=227 |page=391 |doi=10.1086/156743 |bibcode=1979ApJ...227..391J |issn=0004-637X}}</ref> They suggested that all planetary nebulae might all have similar maximum intrinsic brightness, now calculated to be M = −4.53. This would therefore make them potential standard candles for determining extragalactic distances.

Astronomer George Howard Jacoby and his colleagues later proposed that the PNLF function equaled:<ref>{{Cite journal |last=Jacoby |first=George H. |date=April 1989 |title=Planetary nebulae as standard candles. I - Evolutionary models |url=http://adsabs.harvard.edu/doi/10.1086/167274 |journal=The Astrophysical Journal |language=en |volume=339 |page=39 |doi=10.1086/167274 |bibcode=1989ApJ...339...39J |issn=0004-637X}}</ref>

<math display="block">\ N (M) \propto e^{0.307 M} (1 - e^{3(M^{*} - M)} ) \,.</math>
Where N(M) is number of planetary nebula, having absolute magnitude M. M* is equal to the nebula with the brightest magnitude.

=== Surface brightness fluctuation method ===

[[File:Galaxy cluster Abell 2218 gravitaitonal lens.jpg|thumb|Galaxy cluster]]
The following method deals with the overall inherent properties of galaxies. These methods, though with varying error percentages, have the ability to make distance estimates beyond 100 Mpc, though it is usually applied more locally.

The [[surface brightness fluctuation]] (SBF) method takes advantage of the use of [[Charge-coupled device|CCD]] cameras on telescopes. Because of spatial fluctuations in a galaxy's surface brightness, some pixels on these cameras will pick up more stars than others. As distance increases, the picture will become increasingly smoother. Analysis of this describes a magnitude of the pixel-to-pixel variation, which is directly related to a galaxy's distance.<ref name=tonry2001>{{Citation |last1=Tonry |first1=John L. |last2=Dressler |first2=Alan |last3=Blakeslee |first3=John P. |last4=Ajhar |first4=Edward A. |last5=Fletcher |first5=Andre B. |last6=Luppino |first6=Gerard A. |last7=Metzger |first7=Mark R. |last8=Moore |first8=Christopher B. |title=The SBF Survey of Galaxy Distances. IV. SBF Magnitudes, Colors, and Distances |journal=Astrophysical Journal |date=2001 |volume=546 |issue=2 |pages=681–693 |bibcode=2001ApJ...546..681T |doi=10.1086/318301 |arxiv=astro-ph/0011223 |s2cid=17628238}}</ref>

=== Sigma-D relation ===

The [[Sigma-D relation]] (or Σ-D relation), used in [[elliptical galaxy|elliptical galaxies]], relates the angular diameter (D) of the galaxy to its [[velocity dispersion]]. It is important to describe exactly what D represents, in order to understand this method. It is, more precisely, the galaxy's angular diameter out to the [[surface brightness]] level of 20.75 B-mag arcsec<sup>−2</sup>. This surface brightness is independent of the galaxy's actual distance from us. Instead, D is inversely proportional to the galaxy's distance, represented as d. Thus, this relation does not employ standard candles. Rather, D provides a standard ruler. This relation between D and Σ is

<math display="block"> \log (D) = 1.333 \log (\Sigma) + C</math>
where C is a constant which depends on the distance to the galaxy clusters.<ref>{{Cite journal |last=Dressler |first=Alan |date=1987 |title=The Dn-sigma relation for bulges of disk galaxies - A new, independent measure of the Hubble constant |journal=The Astrophysical Journal |language=en |volume=317 |page=1 |doi=10.1086/165251 |bibcode=1987ApJ...317....1D |issn=0004-637X |doi-access=free}}</ref>

This method has the potential to become one of the strongest methods of galactic distance calculators, perhaps exceeding the range of even the Tully–Fisher method. As of today, however, elliptical galaxies are not bright enough to provide a calibration for this method through the use of techniques such as Cepheids. Instead, calibration is done using more crude methods.