Cepheid variable#Use as a "standard candle"

The term Cepheid originates from the star Delta Cephei in the constellation Cepheus, which was one of the early discoveries.

File:Period-Luminosity Relation for Cepheids.png and type II Cepheids]]

On September 10, 1784, Edward Pigott detected the variability of Eta Aquilae, the first known representative of the class of classical Cepheid variables.{{Cite journal|last=Pigott|first=Edward|title=Observations of a new variable star|journal=Philosophical Transactions of the Royal Society |volume=75 |pages=127–136 |bibcode=1785RSPT...75..127P |year=1785 |doi=10.1098/rstl.1785.0007|s2cid=186212958}} The eponymous star for classical Cepheids, Delta Cephei, was discovered to be variable by John Goodricke a few months later.{{cite journal | author = Goodricke, John | title = A series of observations on, and a discovery of, the period of the variation of the light of the star marked δ by Bayer, near the head of Cepheus. In a letter from John Goodricke, Esq. to Nevil Maskelyne, D.D.F.R.S. and Astronomer Royal | journal = Philosophical Transactions of the Royal Society of London |year = 1786 | volume = 76 | pages = 48–61 | doi = 10.1098/rstl.1786.0002 | url = https://books.google.com/books?id=0YMFAAAAQAAJ&pg=PA48 | bibcode = 1786RSPT...76...48G | doi-access = free }} The number of similar variables grew to several dozen by the end of the 19th century, and they were referred to as a class as Cepheids.{{cite book |last1=Clarke |first1=Agnes Mary |title=Problems in Astrophysics |date=1903 |publisher=Adam & Charles Black |location=London, England |page=319 |isbn=978-0-403-01478-1 |url=https://books.google.com/books?id=8pQAAAAAMAAJ&pg=PA319}} Most of the Cepheids were known from the distinctive light curve shapes with the rapid increase in brightness and a hump, but some with more symmetrical light curves were known as Geminids after the prototype ζ Geminorum.{{cite thesis |bibcode=2015PhDT........45E |title=The Secret Lives of Cepheids: A Multi-Wavelength Study of the Atmospheres and Real-Time Evolution of Classical Cepheids |last1=Engle |first1=Scott |year=2015 |arxiv=1504.02713 |doi=10.5281/zenodo.45252 }}

A relationship between the period and luminosity for classical Cepheids was discovered in 1908 by Henrietta Swan Leavitt in an investigation of thousands of variable stars in the Magellanic Clouds.{{cite journal |bibcode=1908AnHar..60...87L |title=1777 variables in the Magellanic Clouds |author1=Leavitt, Henrietta S. |volume=60 |issue=4 |date=1908 |pages=87–108 |journal=Annals of the Astronomical Observatory of Harvard College |url=https://babel.hathitrust.org/cgi/pt?id=coo.31924056092228&view=1up&seq=103 }} She published it in 1912 with further evidence.{{cite journal |bibcode=1912HarCi.173....1L |title=Periods of 25 variable stars in the Small Magellanic Cloud |author1=Leavitt, Henrietta S. |author2=Pickering, Edward C. |volume=173 |date=1912 |pages=1–3 |journal=Harvard College Observatory Circular }} Cepheid variables were found to show radial velocity variation with the same period as the luminosity variation, and initially this was interpreted as evidence that these stars were part of a binary system. However, in 1914, Harlow Shapley demonstrated that this idea should be abandoned.{{cite journal | title=On the Nature and Cause of Cepheid Variation | last=Shapley | first=Harlow | journal=Astrophysical Journal | volume=40 | page=448 | date=December 1914 | doi=10.1086/142137 | bibcode=1914ApJ....40..448S | doi-access=free }} Two years later, Shapley and others had discovered that Cepheid variables changed their spectral types over the course of a cycle.{{citation | postscript=. | title=The variations in spectral type of twenty Cepheid variables | last=Shapley | first=H. | journal=Astrophysical Journal | volume=44 | pages=273 | year=1916 | bibcode=1916ApJ....44..273S | doi=10.1086/142295 | doi-access=free }}

In 1913, Ejnar Hertzsprung attempted to find distances to 13 Cepheids using their motion through the sky.{{cite journal |last1=Hertzsprung |first1=E. |title=Über die räumliche Verteilung der Veränderlichen vom δ Cephei-Typus |journal=Astronomische Nachrichten |date=1913 |volume=196 |issue=4692 |pages=201–208 |trans-title=On the spatial distribution of variable [stars] of the δ Cephei type |language=de|bibcode=1913AN....196..201H }} (His results would later require revision.) In 1918, Harlow Shapley used Cepheids to place initial constraints on the size and shape of the Milky Way and of the placement of the Sun within it.{{cite journal |bibcode=1918PASP...30...42S |title=Globular Clusters and the Structure of the Galactic System |last1=Shapley |first1=H. |journal=Publications of the Astronomical Society of the Pacific |year=1918 |volume=30 |issue=173 |page=42 |doi=10.1086/122686 |doi-access=free }} In 1924, Edwin Hubble established the distance to classical Cepheid variables in the Andromeda Galaxy, until then known as the "Andromeda Nebula" and showed that those variables were not members of the Milky Way. Hubble's finding settled the question raised in the "Great Debate" of whether the Milky Way represented the entire Universe or was merely one of many galaxies in the Universe.{{cite journal |bibcode=1925Obs....48..139H |title=Cepheids in spiral nebulae |author1=Hubble, E. P. |volume=48 |date=1925 |pages=139 |journal=The Observatory }}

In 1929, Hubble and Milton L. Humason formulated what is now known as Hubble's law by combining Cepheid distances to several galaxies with Vesto Slipher's measurements of the speed at which those galaxies recede from us. They discovered that the Universe is expanding, confirming the theories of Georges Lemaître.{{cite journal|bibcode=1927ASSB...47...49L|title=Un Univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extra-galactiques|journal=Annales de la Société Scientifique de Bruxelles|volume=47|pages=49|last1=Lemaître|first1=G.|year=1927}}

File:VISTA finds hidden feature of Milky Way.jpg

In the mid 20th century, significant problems with the astronomical distance scale were resolved by dividing the Cepheids into different classes with very different properties. In the 1940s, Walter Baade recognized two separate populations of Cepheids (classical and type II). Classical Cepheids are younger and more massive population I stars, whereas type II Cepheids are older, fainter Population II stars. Classical Cepheids and type II Cepheids follow different period-luminosity relationships. The luminosity of type II Cepheids is, on average, less than classical Cepheids by about 1.5 magnitudes (but still brighter than RR Lyrae stars). Baade's seminal discovery led to a twofold increase in the distance to M31, and the extragalactic distance scale.{{cite journal|bibcode=1958AJ.....63..207B|title=Problems in the determination of the distance of galaxies|journal=Astronomical Journal|volume=63|pages=207|last1=Baade|first1=W.|year=1958|doi=10.1086/107726}}{{cite web|last=Allen|first=Nick|title=Section 2: The Great Debate and the Great Mistake: Shapley, Hubble, Baade|url=http://www.institute-of-brilliant-failures.com/section2.htm|archive-url=https://web.archive.org/web/20071210105344/http://www.institute-of-brilliant-failures.com/section2.htm|archive-date=Dec 10, 2007|work=The Cepheid Distance Scale: A History}} RR Lyrae stars, then known as Cluster Variables, were recognized fairly early as being a separate class of variable, due in part to their short periods.{{cite journal|bibcode=1918CMWCI.153....1S|title=No. 153. Studies based on the colors and magnitudes in stellar clusters. Eighth paper: The luminosities and distances of 139 Cepheid variables|journal=Contributions from the Mount Wilson Observatory|volume=153|pages=1|last1=Shapley|first1=Harlow.|year=1918}}{{cite journal |last1=Shapley |first1=Harlow |title=Studies based on the colors and magnitudes in stellar clusters. Eighth paper: The luminosities and distances of 139 Cepheid variables |journal=Astrophysical Journal |date=1918 |volume=48 |pages=279–294 |bibcode=1918ApJ....48..279S |doi=10.1086/142435 }}

The mechanics of stellar pulsation as a heat-engine was proposed in 1917 by Arthur Stanley Eddington{{cite journal |bibcode=1917Obs....40..290E |title=The pulsation theory of Cepheid variables |author1=Eddington, A. S. |volume=40 |date=1917 |pages=290 |journal=The Observatory}} (who wrote at length on the dynamics of Cepheids), but it was not until 1953 that S. A. Zhevakin identified ionized helium as a likely valve for the engine.{{cite journal |author1=Zhevakin, S. A. |title=К Теории Цефеид. I |journal=Астрономический журнал |volume=30 |pages=161–179 |year=1953 }}

Cepheid variables are divided into two subclasses which exhibit markedly different masses, ages, and evolutionary histories: classical Cepheids and type II Cepheids. Delta Scuti variables are A-type stars on or near the main sequence at the lower end of the instability strip and were originally referred to as dwarf Cepheids. RR Lyrae variables have short periods and lie on the instability strip where it crosses the horizontal branch. Delta Scuti variables and RR Lyrae variables are not generally treated with Cepheid variables although their pulsations originate with the same helium ionisation kappa mechanism.

{{Main|Classical Cepheid variable}}

File:Delta Cephei lightcurve.jpg of Delta Cephei, the prototype of classical cepheids, showing the regular variations produced by intrinsic stellar pulsations]]

Classical Cepheids (also known as Population I Cepheids, type I Cepheids, or Delta Cepheid variables) undergo pulsations with very regular periods on the order of days to months. Classical Cepheids are Population I variable stars which are 4–20 times more massive than the Sun,{{cite journal |bibcode=1996JRASC..90...82T |title=The Progenitors of Classical Cepheid Variables |author1=Turner, David G. |volume=90 |date=1996 |pages=82 |journal=Journal of the Royal Astronomical Society of Canada}} and up to 100,000 times more luminous.{{cite journal |arxiv=0912.4864 |bibcode=2010Ap&SS.326..219T |doi=10.1007/s10509-009-0258-5 |title=The PL calibration for Milky Way Cepheids and its implications for the distance scale |date=2010 |last1=Turner |first1=David G. |journal=Astrophysics and Space Science |volume=326 |issue=2 |pages=219–231 |s2cid=119264970 }} These Cepheids are yellow bright giants and supergiants of spectral class F6 – K2 and their radii change by (~25% for the longer-period I Carinae) millions of kilometers during a pulsation cycle.{{cite journal |bibcode=1957MNRAS.117...85R |title=Radius variation and population type of cepheid variables |author1=Rodgers, A. W. |volume=117 |date=1957 |pages=85–94 |journal=Monthly Notices of the Royal Astronomical Society |doi=10.1093/mnras/117.1.85|doi-access=free }}

Classical Cepheids are used to determine distances to galaxies within the Local Group and beyond, and are a means by which the Hubble constant can be established.{{cite journal |arxiv=astro.ph/0012376 |bibcode=2001ApJ...553...47F |doi=10.1086/320638 |title=Final Results from the Hubble Space Telescope Key Project to Measure the Hubble Constant |date=2001 |last1=Freedman |first1=Wendy L. |last2=Madore |first2=Barry F. |last3=Gibson |first3=Brad K. |last4=Ferrarese |first4=Laura |last5=Kelson |first5=Daniel D. |last6=Sakai |first6=Shoko |last7=Mould |first7=Jeremy R. |last8=Kennicutt, Jr. |first8=Robert C. |last9=Ford |first9=Holland C. |last10=Graham |first10=John A. |last11=Huchra |first11=John P. |last12=Hughes |first12=Shaun M. G. |last13=Illingworth |first13=Garth D. |last14=Macri |first14=Lucas M. |last15=Stetson |first15=Peter B. |journal=The Astrophysical Journal |volume=553 |issue=1 |pages=47–72 |s2cid=119097691 }}{{cite journal |arxiv=1004.1856 |bibcode=2010ARA&A..48..673F |title=The Hubble Constant |author1=Freedman, Wendy L. |author2=Madore, Barry F. |volume=48 |date=2010 |pages=673–710 |journal=Annual Review of Astronomy and Astrophysics |doi=10.1146/annurev-astro-082708-101829 |s2cid=13909389 }} Classical Cepheids have also been used to clarify many characteristics of the Milky Way galaxy, such as the Sun's height above the galactic plane and the Galaxy's local spiral structure.{{cite journal |arxiv=0903.4206 |bibcode=2009MNRAS.398..263M |doi=10.1111/j.1365-2966.2009.15096.x |title=Characteristics of the Galaxy according to Cepheids |date=2009 |last1=Majaess |first1=D. J. |last2=Turner |first2=D. G. |last3=Lane |first3=D. J. |journal=Monthly Notices of the Royal Astronomical Society |volume=398 |issue=1 |pages=263–270 |doi-access=free |s2cid=14316644 }}

A group of classical Cepheids with small amplitudes and sinusoidal light curves are often separated out as Small Amplitude Cepheids or s-Cepheids, many of them pulsating in the first overtone.

{{Main|Type II Cepheid}}

File:Kappa Pavonis TESS lightcurve.png, a Type II cepheid, recorded by NASA's Transiting Exoplanet Survey Satellite (TESS)]]

Type II Cepheids (also termed Population II Cepheids) are population II variable stars which pulsate with periods typically between 1 and 50 days.{{cite journal |bibcode=2002PASP..114..689W |doi=10.1086/341698 |title=The Cepheids of Population II and Related Stars |date=2002 |last1=Wallerstein |first1=George |journal=Publications of the Astronomical Society of the Pacific |volume=114 |issue=797 |pages=689–699|s2cid=122225966 }}{{cite journal |arxiv=0811.3636 |bibcode=2008AcA....58..293S |title=The Optical Gravitational Lensing Experiment. The OGLE-III Catalog of Variable Stars. II.Type II Cepheids and Anomalous Cepheids in the Large Magellanic Cloud |author1=Soszyński, I. |author2=Udalski, A. |author3=Szymański, M. K. |author4=Kubiak, M. |author5=Pietrzyński, G. |author6=Wyrzykowski, Ł. |author7=Szewczyk, O. |author8=Ulaczyk, K. |author9=Poleski, R. |volume=58 |date=2008 |pages=293 |journal=Acta Astronomica }} Type II Cepheids are typically metal-poor, old (~10 Gyr), low mass objects (~half the mass of the Sun). Type II Cepheids are divided into several subgroups by period. Stars with periods between 1 and 4 days are of the BL Her subclass, 10–20 days belong to the W Virginis subclass, and stars with periods greater than 20 days belong to the RV Tauri subclass.

Type II Cepheids are used to establish the distance to the Galactic Center, globular clusters, and galaxies.{{cite journal |arxiv=astro.ph/0306567 |bibcode=2003AcA....53..117K |title=The Optical Gravitational Lensing Experiment. Population II Cepheids in the Galactic Bulge |author1=Kubiak, M. |author2=Udalski, A. |volume=53 |date=2003 |pages=117 |journal=Acta Astronomica}}{{cite journal |arxiv=astro.ph/0606609 |bibcode=2006MNRAS.370.1979M |doi=10.1111/j.1365-2966.2006.10620.x |title=The period-luminosity relation for type II Cepheids in globular clusters |date=2006 |last1=Matsunaga |first1=Noriyuki |last2=Fukushi |first2=Hinako |last3=Nakada |first3=Yoshikazu |last4=Tanabé |first4=Toshihiko |last5=Feast |first5=Michael W. |last6=Menzies |first6=John W. |last7=Ita |first7=Yoshifusa |last8=Nishiyama |first8=Shogo |last9=Baba |first9=Daisuke |last10=Naoi |first10=Takahiro |last11=Nakaya |first11=Hidehiko |last12=Kawadu |first12=Takahiro |last13=Ishihara |first13=Akika |last14=Kato |first14=Daisuke |journal=Monthly Notices of the Royal Astronomical Society |volume=370 |issue=4 |pages=1979–1990 |doi-access=free |s2cid=25991504 |display-authors=8 }}{{cite journal |arxiv=0803.0466 |bibcode=2008MNRAS.386.2115F |doi=10.1111/j.1365-2966.2008.13181.x |title=The luminosities and distance scales of type II Cepheid and RR Lyrae variables |date=2008 |last1=Feast |first1=Michael W. |last2=Laney |first2=Clifton D. |last3=Kinman |first3=Thomas D. |last4=Van Leeuwen |first4=Floor |last5=Whitelock |first5=Patricia A. |journal=Monthly Notices of the Royal Astronomical Society |volume=386 |issue=4 |pages=2115–2134 |doi-access=free |s2cid=14459638 }}{{cite journal |arxiv=0912.2928 |bibcode=2010JAVSO..38..100M |title=RR Lyrae and Type II Cepheid Variables Adhere to a Common Distance Relation |author1=Majaess, D. J. |volume=38 |issue=1 |date=2010 |pages=100–112 |journal=The Journal of the American Association of Variable Star Observers}}{{cite journal |arxiv=0904.4701 |bibcode=2009MNRAS.397..933M |doi=10.1111/j.1365-2966.2009.14992.x |title=Period-luminosity relations for type II Cepheids and their application |date=2009 |last1=Matsunaga |first1=Noriyuki |last2=Feast |first2=Michael W. |last3=Menzies |first3=John W. |journal=Monthly Notices of the Royal Astronomical Society |volume=397 |issue=2 |pages=933–942 |doi-access=free |s2cid=13912466 }}

A group of pulsating stars on the instability strip have periods of less than 2 days, similar to RR Lyrae variables but with higher luminosities. Anomalous Cepheid variables have masses higher than type II Cepheids, RR Lyrae variables, and the Sun. It is unclear whether they are young stars on a "turned-back" horizontal branch, blue stragglers formed through mass transfer in binary systems, or a mix of both.{{cite journal|bibcode=2004A&A...424..927C|title=Bright metal-poor variables: Why Anomalous Cepheids?|journal=Astronomy and Astrophysics|volume=424|issue=3|pages=927–934 |last1=Caputo|first1=F.|last2=Castellani|first2=V.|last3=Degl'Innocenti|first3=S.|last4=Fiorentino|first4=G.|last5=Marconi|first5=M. |year=2004|doi=10.1051/0004-6361:20040307|arxiv=astro.ph/0405395 |s2cid=45306570}}{{cite journal|bibcode=2008AcA....58..293S|title=The Optical Gravitational Lensing Experiment. The OGLE-III Catalog of Variable Stars. II.Type II Cepheids and Anomalous Cepheids in the Large Magellanic Cloud|journal=Acta Astronomica|volume=58|pages=293|last1=Soszyński|first1=I.|last2=Udalski|first2=A.|last3=Szymański|first3=M. K.|last4=Kubiak|first4=M.|last5=Pietrzyński|first5=G.|last6=Wyrzykowski|first6=Ł.|last7=Szewczyk|first7=O.|last8=Ulaczyk|first8=K.|last9=Poleski |first9=R.|year=2008|arxiv=0811.3636 }}

A small proportion of Cepheid variables have been observed to pulsate in two modes at the same time, usually the fundamental and first overtone, occasionally the second overtone.{{cite journal|bibcode=2008AcA....58..233S|arxiv=0809.1986|title=Double-Mode Classical Cepheid Models, Revisited|journal=Acta Astronomica|volume=58|pages=233|last1=Smolec|first1=R.|last2=Moskalik|first2=P.|year=2008}} A very small number pulsate in three modes, or an unusual combination of modes including higher overtones.{{cite journal|bibcode=2008AcA....58..153S|arxiv=0807.4182|title=The Optical Gravitational Lensing Experiment. Triple-Mode and 1O/3O Double-Mode Cepheids in the Large Magellanic Cloud|journal=Acta Astronomica|volume=58|pages=153 |last1=Soszynski|first1=I.|last2=Poleski|first2=R.|last3=Udalski|first3=A.|last4=Kubiak|first4=M.|last5=Szymanski|first5=M. K.|last6=Pietrzynski|first6=G.|last7=Wyrzykowski|first7=L.|last8=Szewczyk|first8=O.|last9=Ulaczyk|first9=K.|year=2008}}

Chief among the uncertainties tied to the classical and type II Cepheid distance scale are: the nature of the period-luminosity relation in various passbands, the impact of metallicity on both the zero-point and slope of those relations, and the effects of photometric contamination (blending with other stars) and a changing (typically unknown) extinction law on Cepheid distances. All these topics are actively debated in the literature.{{cite journal |arxiv=0806.3018 |bibcode=2008A&ARv..15..289T |doi=10.1007/s00159-008-0012-y |title=The expansion field: the value of H 0 |date=2008 |last1=Tammann |first1=G. A. |last2=Sandage |first2=A. |last3=Reindl |first3=B. |journal=The Astronomy and Astrophysics Review |volume=15 |issue=4 |pages=289–331 |s2cid=18463474 }}{{cite journal |arxiv=astro.ph/0603643 |bibcode=2006ApJ...642L..29N |doi=10.1086/504478 |title=The Hubble Constant from Type Ia Supernovae Calibrated with the Linear and Nonlinear Cepheid Period-Luminosity Relations |date=2006 |last1=Ngeow |first1=C. |last2=Kanbur |first2=S. M. |journal=The Astrophysical Journal |volume=642 |issue=1 |pages=L29–L32 |s2cid=17860528 }}{{cite journal |arxiv=0909.0181 |bibcode=2009AcA....59..403M |title=Type II Cepheids as Extragalactic Distance Candles |author1=Majaess, D. |author2=Turner, D. |author3=Lane, D. |volume=59 |issue=4 |date=2009 |pages=403 |journal=Acta Astronomica}}{{cite journal|bibcode=2007AJ....133.1810B|title=Hubble Space Telescope Fine Guidance Sensor Parallaxes of Galactic Cepheid Variable Stars: Period-Luminosity Relations|journal=The Astronomical Journal|volume=133|issue=4|pages=1810|last1=Benedict|first1=G. Fritz|last2=McArthur|first2=Barbara E.|last3=Feast|first3=Michael W.|last4=Barnes|first4=Thomas G.|last5=Harrison|first5=Thomas E.|last6=Patterson|first6=Richard J.|last7=Menzies|first7=John W.|last8=Bean|first8=Jacob L.|last9=Freedman|first9=Wendy L.|year=2007|doi=10.1086/511980|arxiv=astro.ph/0612465 |s2cid=16384267 }}{{cite arXiv |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 |author1=Stanek, K. Z. |author2=Udalski, A. |date=1999}}{{cite journal |arxiv=astro.ph/0109446 |bibcode=2001AcA....51..221U |title=The Optical Gravitational Lensing Experiment. Cepheids in the Galaxy IC1613: No Dependence of the Period-Luminosity Relation on Metallicity |author1=Udalski, A. |author2=Wyrzykowski, L. |author3=Pietrzynski, G. |author4=Szewczyk, O. |author5=Szymanski, M. |author6=Kubiak, M. |author7=Soszynski, I. |author8=Zebrun, K. |volume=51 |date=2001 |pages=221 |journal=Acta Astronomica}}{{cite journal |arxiv=astro.ph/0608211 |bibcode=2006ApJ...652.1133M |doi=10.1086/508530 |title=A New Cepheid Distance to the Maser-Host Galaxy NGC 4258 and Its Implications for the Hubble Constant |date=2006 |last1=Macri |first1=L. M. |last2=Stanek |first2=K. Z. |last3=Bersier |first3=D. |last4=Greenhill |first4=L. J. |last5=Reid |first5=M. J. |journal=The Astrophysical Journal |volume=652 |issue=2 |pages=1133–1149 |s2cid=15728812 }}{{cite journal |arxiv=0805.1592 |bibcode=2008ApJ...684..102B |doi=10.1086/589965 |title=Cepheids in External Galaxies. I. The Maser-Host Galaxy NGC 4258 and the Metallicity Dependence of Period-Luminosity and Period-Wesenheit Relations |date=2008 |last1=Bono |first1=G. |last2=Caputo |first2=F. |last3=Fiorentino |first3=G. |last4=Marconi |first4=M. |last5=Musella |first5=I. |journal=The Astrophysical Journal |volume=684 |issue=1 |pages=102–117 |s2cid=6275274 }}{{cite journal |arxiv=0902.3747 |bibcode=2009ApJ...696.1498M |doi=10.1088/0004-637X/696/2/1498 |title=Concerning the Slope of the Cepheid Period-Luminosity Relation |date=2009 |last1=Madore |first1=Barry F. |last2=Freedman |first2=Wendy L. |journal=The Astrophysical Journal |volume=696 |issue=2 |pages=1498–1501 |s2cid=16325249 }}{{cite journal |bibcode=2009MNRAS.396.1287S |doi=10.1111/j.1365-2966.2009.14822.x |title=The effect of metallicity on Cepheid magnitudes and the distance to M33 |date=2009 |last1=Scowcroft |first1=V. |last2=Bersier |first2=D. |last3=Mould |first3=J. R. |last4=Wood |first4=P. R. |journal=Monthly Notices of the Royal Astronomical Society |volume=396 |issue=3 |pages=1287–1296|doi-access=free |arxiv=0903.4088 }}{{cite journal |arxiv=1006.2458 |bibcode=2010AcA....60..121M |title=The Cepheids of Centaurus A (NGC 5128) and Implications for H0 |author1=Majaess, D. |volume=60 |issue=2 |date=2010 |pages=121 |journal=Acta Astronomica}}

These unresolved matters have resulted in cited values for the Hubble constant (established from Classical Cepheids) ranging between 60 km/s/Mpc and 80 km/s/Mpc. Resolving this discrepancy is one of the foremost problems in astronomy since the cosmological parameters of the Universe may be constrained by supplying a precise value of the Hubble constant.{{cite conference |bibcode=2009AIPC.1170...23M |doi=10.1063/1.3246452 |chapter=The SH0ES Project: Observations of Cepheids in NGC 4258 and Type Ia SN Hosts |title=AIP Conference Proceedings |conference=Stellar Pulsation: Challenges for Theory and Observation: Proceedings of the International Conference. AIP Conference Proceedings |volume=1170 |date=2009 |last1=Macri |first1=Lucas M. |last2=Riess |first2=Adam G. |last3=Guzik |first3=Joyce Ann |last4=Bradley |first4=Paul A. |pages=23–25}} Uncertainties have diminished over the years, due in part to discoveries such as RS Puppis.

Delta Cephei is also of particular importance as a calibrator of the Cepheid period-luminosity relation since its distance is among the most precisely established for a Cepheid, partly because it is a member of a star cluster{{cite journal|bibcode=1999AJ....117..354D|title=A HIPPARCOS Census of the Nearby OB Associations|journal=The Astronomical Journal|volume=117|issue=1|pages=354–399|last1=De Zeeuw|first1=P. T.|last2=Hoogerwerf|first2=R.|last3=De Bruijne|first3=J. H. J.|last4=Brown|first4=A. G. A.|last5=Blaauw|first5=A.|year=1999|doi=10.1086/300682|arxiv=astro.ph/9809227 |s2cid=16098861}}{{cite journal|bibcode=2012ApJ...747..145M|title=New Evidence Supporting Cluster Membership for the Keystone Calibrator Delta Cephei|journal=The Astrophysical Journal|volume=747|issue=2|pages=145|last1=Majaess|first1=D.|last2=Turner|first2=D.|last3=Gieren|first3=W.|year=2012|doi=10.1088/0004-637X/747/2/145|arxiv=1201.0993|s2cid=118672744}} and the availability of precise parallaxes observed by the Hubble, Hipparcos, and Gaia space telescopes.{{cite journal|bibcode=2002AJ....124.1695B|title=Astrometry with the Hubble Space Telescope: A Parallax of the Fundamental Distance Calibrator δ Cephei|journal=The Astronomical Journal|volume=124|issue=3|pages=1695|last1=Benedict|first1=G. Fritz|last2=McArthur|first2=B. E.|last3=Fredrick|first3=L. W.|last4=Harrison|first4=T. E.|last5=Slesnick|first5=C. L.|last6=Rhee|first6=J.|last7=Patterson|first7=R. J.|last8=Skrutskie|first8=M. F.|last9=Franz|first9=O. G.|last10=Wasserman|first10=L. H.|last11=Jefferys|first11=W. H.|last12=Nelan|first12=E.|last13=Van Altena|first13=W.|last14=Shelus|first14=P. J.|last15=Hemenway|first15=P. D.|last16=Duncombe|first16=R. L.|last17=Story|first17=D.|last18=Whipple|first18=A. L.|last19=Bradley|first19=A. J.|year=2002|doi=10.1086/342014|arxiv=astro.ph/0206214 |s2cid=42655824}} The accuracy of parallax distance measurements to Cepheid variables and other bodies within 7,500 light-years is vastly improved by comparing images from Hubble taken six months apart, from opposite points in the Earth's orbit. (Between two such observations 2 AU apart, a star at a distance of 7500 light-years = 2300 parsecs would appear to move an angle of ²/₂₃₀₀ arc-seconds = 2 x 10⁻⁷ degrees, the resolution limit of the available telescopes.){{cite journal|bibcode=2014ApJ...785..161R|title=Parallax beyond a Kiloparsec from Spatially Scanning the Wide Field Camera 3 on the Hubble Space Telescope|journal=The Astrophysical Journal|volume=785|issue=2|pages=161|last1=Riess|first1=Adam G.|last2=Casertano|first2=Stefano|last3=Anderson|first3=Jay|last4=MacKenty|first4=John|last5=Filippenko|first5=Alexei V.|year=2014|doi=10.1088/0004-637X/785/2/161|arxiv=1401.0484 |s2cid=55928992}}

File:Polaris time-lapse illustrating Cepheid type variability.gif

{{See also|Kappa–mechanism}}

The accepted explanation for the pulsation of Cepheids is called the Eddington valve,{{cite journal|bibcode=1984S&T....68..519S|title=Eddington's Valve and Cepheid Pulsations|journal=Sky and Telescope|volume=68|pages=519|last1=Smith|first1=D. H.|year=1984}}{{cite book |doi=10.1888/0333750888/4130 |chapter=American Association of Variable Star Observers |title=The Encyclopedia of Astronomy and Astrophysics |year=2001 |isbn=0-333-75088-8 }} or "κ-mechanism", where the Greek letter κ (kappa) is the usual symbol for the gas opacity.

Helium is the gas thought to be most active in the process. Doubly ionized helium (helium whose atoms are missing both electrons) is more opaque than singly ionized helium. As helium is heated, its temperature rises until it reaches the point at which double ionisation spontaneously occurs and is sustained throughout the layer in much the same way a fluorescent tube 'strikes'. At the dimmest part of a Cepheid's cycle, this ionized gas in the outer layers of the star is relatively opaque, and so is heated by the star's radiation, and due to the increasing temperature, begins to expand. As it expands, it cools, but remains ionised until another threshold is reached at which point double ionization cannot be sustained and the layer becomes singly ionized hence more transparent, which allows radiation to escape. The expansion then stops, and reverses due to the star's gravitational attraction. The star's states are held to be either expanding or contracting by the hysteresis{{Cite journal |last1=Auvergne |first1=M. |last2=Baglin |first2=A. |last3=Morel |first3=P. -J. |date=1981-12-01 |title=On the existence of hysteresis effects in pulsating stars |url=https://ui.adsabs.harvard.edu/abs/1981A&A...104...47A |journal=Astronomy and Astrophysics |volume=104 |issue=1 |pages=47–56 |bibcode=1981A&A...104...47A |issn=0004-6361}} generated by the doubly ionized helium and indefinitely flip-flops between the two states reversing every time the upper or lower threshold is crossed. This process is rather analogous to the relaxation oscillator found in electronics.{{Citation needed|date=September 2024}}

In 1879, August Ritter (1826–1908) demonstrated that the adiabatic radial pulsation period for a homogeneous sphere is related to its surface gravity and radius through the relation:

$$T\; =\; k\; \backslash ,\backslash sqrt\; \backslash frac\; R\; g$$

where k is a proportionality constant. Now, since the surface gravity is related to the sphere mass and radius through the relation:

$$g\; =\; k\text{'}\; \backslash frac\; M\; \{R^2\}\; =\; k\text{'}\; \backslash frac\; \{RM\}\; \{R^3\}\; =\; k\text{'}\; R\backslash rho$$

one finally obtains:

$$T\; \backslash sqrt\; \backslash rho\; =\; Q$$

where Q is a constant, called the pulsation constant.{{cite book|author1=Maurizio Salaris|author2=Santi Cassisi|title=Evolution of Stars and Stellar Populations|url=https://books.google.com/books?id=r1dNzr8viRYC|date=13 December 2005|publisher=John Wiley & Sons|isbn=978-0-470-09222-4|page=180}}

• Classical Cepheids include: Eta Aquilae, Zeta Geminorum, Beta Doradus, RT Aurigae, Polaris, as well as Delta Cephei.
• Type II Cepheids include: W Virginis, Kappa Pavonis and BL Herculis.{{cite journal|bibcode=1996AstL...22..326G|title=A spectroscopic study of the pulsating star BL Her|journal=Astronomy Letters|volume=22|issue=3|pages=326|last1=Gorynya|first1=N. A.|last2=Samus|first2=N. N.|last3=Rastorguev|first3=A. S.|last4=Sachkov|first4=M. E.|year=1996}}
• Anomalous Cepheids include: XZ Ceti{{cite journal|bibcode=2007A&A...461..613S|arxiv=astro.ph/0609097|title=The anomalous Cepheid XZ Ceti|journal=Astronomy and Astrophysics|volume=461|issue=2|pages=613–618|last1=Szabados|first1=L.|last2=Kiss|first2=L. L.|last3=Derekas|first3=A.|year=2007|doi=10.1051/0004-6361:20065690|s2cid=18245078}} (overtone pulsation mode){{citation|arxiv=2012.09709|title=TESS observations of Cepheid stars: first light results|year=2020|bibcode=2021ApJS..253...11P|last1=Plachy|first1=E.|last2=Pál|first2=A.|last3=Bódi|first3=A.|last4=Szabó|first4=P.|last5=Molnár|first5=L.|last6=Szabados|first6=L.|last7=Benkő|first7=J. M.|last8=Anderson|first8=R. I.|last9=Bellinger|first9=E. P.|last10=Bhardwaj|first10=A.|last11=Ebadi|first11=M.|last12=Gazeas|first12=K.|last13=Hambsch|first13=F. -J.|last14=Hasanzadeh|first14=A.|last15=Jurkovic|first15=M. I.|last16=Kalaee|first16=M. J.|last17=Kervella|first17=P.|last18=Kolenberg|first18=K.|last19=Mikołajczyk|first19=P.|last20=Nardetto|first20=N.|last21=Nemec|first21=J. M.|last22=Netzel|first22=H.|last23=Ngeow|first23=C. -C.|last24=Ozuyar|first24=D.|last25=Pascual-Granado|first25=J.|last26=Pilecki|first26=B.|last27=Ripepi|first27=V.|last28=Skarka|first28=M.|last29=Smolec|first29=R.|last30=Sódor|first30=Á.|journal=The Astrophysical Journal Supplement Series|volume=253|issue=1|page=11|doi=10.3847/1538-4365/abd4e3|s2cid=229297708|display-authors=1 |doi-access=free }} and BL Boötis.

{{Reflist}}

• [https://dogwood.physics.mcmaster.ca/Cepheid/ McMaster Cepheid Photometry and Radial Velocity Data Archive]
• [http://www.aavso.org/ American Association of Variable Star Observers]
• Survey of Warsaw University at Las Campanas Observatory: OGLE-III (Optical Gravitational Lensing Experiment) [http://ogle.astrouw.edu.pl/atlas/ Variable Stars catalog website]
• [http://www.astro.utoronto.ca/DDO/research/cepheids/cepheids.html David Dunlap Observatory of Toronto University: Galactic Classical Cepheids database]

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