View source for Classical Cepheid variable

[[File:HR-vartype.svg|thumb|400px|[[Hertzsprung–Russell diagram]] showing the location of  several types of [[variable star]]s superimposed on a display of the different [[Stellar classification#Yerkes spectral classification|luminosity classes]].]]

'''Classical Cepheids''' are a type of [[Cepheid variable]] [[star]]. They are young, [[population I]] [[variable stars]] that exhibit regular radial [[Stellar pulsation|pulsations]] with periods of a few days to a few weeks and visual [[amplitude]]s ranging from a few tenths of a [[Magnitude (astronomy)|magnitude]] up to about 2 magnitudes. Classical Cepheids are also known as '''Population I Cepheids''', '''Type I Cepheids''', and '''Delta Cepheid variables'''.

There exists a well-defined [[Period-luminosity relation|relationship]] between a classical Cepheid variable's [[luminosity]] and pulsation period,<ref name=OGLE_4>{{cite journal|bibcode=1999AcA....49..223U|title=The Optical Gravitational Lensing Experiment. Cepheids in the Magellanic Clouds. IV. Catalog of Cepheids from the Large Magellanic Cloud|journal=Acta Astronomica|volume=49|pages=223–317|last1=Udalski|first1=A.|last2=Soszynski|first2=I.|last3=Szymanski|first3=M.|last4=Kubiak|first4=M.|last5=Pietrzynski|first5=G.|last6=Wozniak|first6=P.|last7=Zebrun|first7=K.|year=1999|arxiv = astro-ph/9908317 }}</ref><ref name=OGLE_1/> securing Cepheids as viable [[standard candle]]s for establishing the galactic and [[extragalactic distance scale]]s.<ref name=freedman2001/><ref name=tammannsandage2008/><ref name=majaess2009/><ref name=freedman2010>{{cite journal|bibcode=2010ARA&A..48..673F|title=The Hubble Constant|journal=Annual Review of Astronomy and Astrophysics|volume=48|pages=673–710|last1=Freedman|first1=Wendy L.|last2=Madore|first2=Barry F.|year=2010|doi=10.1146/annurev-astro-082708-101829|arxiv = 1004.1856 |s2cid=13909389}}</ref> [[Hubble Space Telescope]] (HST) observations of classical Cepheid variables have enabled firmer constraints on [[Hubble's law]], which describes the expansion rate of the [[observable Universe]].<ref name=freedman2001>{{cite journal|bibcode=2001ApJ...553...47F|title=Final Results from the Hubble Space Telescope Key Project to Measure the Hubble Constant|journal=The Astrophysical Journal|volume=553|issue=1|pages=47–72|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|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.|year=2001|doi=10.1086/320638|arxiv = astro-ph/0012376 |s2cid=119097691}}</ref><ref name=tammannsandage2008/><ref name=freedman2010/><ref name=ngeow2006/><ref name=macririess2009/> Classical Cepheids have also been used to clarify many characteristics of our galaxy, such as the local [[spiral arm]] structure and the Sun's distance from the [[galactic plane]].<ref name=majaess2009>{{cite journal|bibcode=2009MNRAS.398..263M|title=Characteristics of the Galaxy according to Cepheids|journal=Monthly Notices of the Royal Astronomical Society|volume=398|issue=1|pages=263–270|last1=Majaess|first1=D. J.|last2=Turner|first2=D. G.|last3=Lane|first3=D. J.|year=2009|doi=10.1111/j.1365-2966.2009.15096.x|doi-access=free |arxiv = 0903.4206 |s2cid=14316644}}</ref>

Around 800 classical Cepheids are known in the [[Milky Way]] galaxy, out of an expected total of over 6,000.  Several thousand more are known in the [[Magellanic Clouds]], with more discovered in other galaxies;<ref>{{cite journal|bibcode=2003ASPC..298..237S|title=Cepheids: Observational properties, binarity and GAIA|journal=GAIA Spectroscopy: Science and Technology|volume=298|pages=237|last1=Szabados|first1=L.|year=2003}}</ref> the [[Hubble Space Telescope]] has identified some in [[NGC 4603]], which is 100 million [[light year]]s distant.<ref name="HST_NGC_4603_Cepheid_Distance">{{Cite journal | doi = 10.1086/307764| title = A Cepheid Distance to NGC 4603 in Centaurus| journal = The Astrophysical Journal| volume = 523| issue = 2| pages = 506| year = 1999| last1 = Newman | first1 = J. A. | last2 = Zepf | first2 = S. E. | last3 = Davis | first3 = M. | last4 = Freedman | first4 = W. L. | last5 = Madore | first5 = B. F. | last6 = Stetson | first6 = P. B. | last7 = Silbermann | first7 = N.| last8 = Phelps | first8 = R. |arxiv = astro-ph/9904368 |bibcode = 1999ApJ...523..506N | s2cid = 15343736}}</ref>

==Properties==
[[File:Evolutionary track 5m.svg|thumb|left|upright=1.0|The evolutionary track of {{solar mass|5}} star crossing the [[instability strip]] during a helium burning [[blue loop]]]]
Classical Cepheid variables are 4–20 times more massive than the Sun,<ref name=turner96>{{cite journal|bibcode=1996JRASC..90...82T|title=The Progenitors of Classical Cepheid Variables|journal=Journal of the Royal Astronomical Society of Canada|volume=90|pages=82|last1=Turner|first1=David G.|year=1996}}</ref> and around 1,000 to 50,000 (over 200,000 for the unusual [[V810 Centauri]]) times more luminous.<ref name=turner10>{{Cite journal | doi = 10.1007/s10509-009-0258-5| title = The PL calibration for Milky Way Cepheids and its implications for the distance scale| journal = Astrophysics and Space Science| volume = 326| issue = 2| pages = 219–231| year = 2010| last1 = Turner | first1 = D. G. |arxiv = 0912.4864 |bibcode = 2010Ap&SS.326..219T | s2cid = 119264970}}</ref>  Spectroscopically they are bright giants or low luminosity supergiants of [[spectral classification|spectral class]] F6 – K2.  The temperature and spectral type vary as they pulsate.  Their radii are a few tens to a few hundred times that of the sun.  More luminous Cepheids are cooler and larger and have longer periods.  Along with the temperature changes their radii also change during each pulsation (e.g. by ~25% for the longer-period [[L Carinae|l Car]]), resulting in brightness variations up to two magnitudes.  The brightness changes are more pronounced at shorter wavelengths.<ref name=rodgers>{{cite journal|bibcode=1957MNRAS.117...85R|title=Radius variation and population type of cepheid variables|journal=Monthly Notices of the Royal Astronomical Society|volume=117|pages=85–94|last1=Rodgers|first1=A. W.|year=1957|doi=10.1093/mnras/117.1.85|doi-access=free}}</ref>

Cepheid variables may pulsate in a [[fundamental mode]], the first [[overtone]], or rarely a mixed mode.  Pulsations in an overtone higher than first are rare but interesting.<ref name=OGLE_1/>  The majority of classical Cepheids are thought to be fundamental mode pulsators, although it is not easy to distinguish the mode from the shape of the light curve.  Stars pulsating in an overtone are more luminous and larger than a fundamental mode pulsator with the same period.<ref name=bobo>{{Cite journal | doi = 10.1086/320344| title = On the Pulsation Mode Identification of Short-Period Galactic Cepheids| journal = The Astrophysical Journal| volume = 552| issue = 2| pages = L141| year = 2001| last1 = Bono | first1 = G.| last2 = Gieren | first2 = W. P.| last3 = Marconi | first3 = M.| last4 = Fouqué | first4 = P.|arxiv = astro-ph/0103497 |bibcode = 2001ApJ...552L.141B | s2cid = 16131313}}</ref>

When an [[intermediate mass star]] (IMS) first evolves away from the [[main sequence]], it crosses the instability strip very rapidly while the hydrogen shell is still burning.  When the helium core ignites in an IMS, it may execute a [[blue loop]] and crosses the instability strip again, once while evolving to high temperatures and again evolving back towards the [[asymptotic giant branch]].  Stars more massive than about {{solar mass|8–12}} start core helium burning before reaching the [[red-giant branch]] and become [[red supergiant]]s, but may still execute a blue loop through the instability strip.  The duration and even existence of blue loops is very sensitive to the mass, [[metallicity]], and helium abundance of the star.  In some cases, stars may cross the instability strip for a fourth and fifth time when helium shell burning starts.{{citation needed|date=May 2016}}  The rate of change of the period of a Cepheid variable, along with chemical abundances detectable in the spectrum, can be used to deduce which crossing a particular star is making.<ref name=svvul>{{Cite journal | doi = 10.1051/0004-6361:20040163| title = On the crossing mode of the long-period Cepheid SV Vulpeculae| journal = Astronomy and Astrophysics| volume = 423| pages = 335–340| year = 2004| last1 = Turner | first1 = D. G.| last2 = Berdnikov | first2 = L. N.|bibcode = 2004A&A...423..335T | doi-access = free}}</ref>

Classical Cepheid variables were B type [[main sequence|main-sequence stars]] earlier than about B7, possibly late O stars, before they ran out of hydrogen in their cores.  More massive and hotter stars develop into more luminous Cepheids with longer periods, although it is expected that young stars within our own galaxy, at near solar metallicity, will generally lose sufficient mass by the time they first reach the [[instability strip]] that they will have periods of 50 days or less.  Above a certain mass, {{solar mass|20–50}} depending on metallicity, red supergiants will evolve back to blue supergiants rather than execute a blue loop, but they will do so as unstable [[yellow hypergiant]]s rather than regularly pulsating Cepheid variables.  Very massive stars never cool sufficiently to reach the instability strip and do not ever become Cepheids.  At low metallicity, for example in the Magellanic Clouds, stars can retain more mass and become more luminous Cepheids with longer periods.<ref name=turner10/>

==Light curves==
[[File:Delta Cephei lightcurve.jpg|thumb|left|[[Delta Cephei]] lightcurve]]
[[File:Delta cephei ubvri engle 2014.png|right|thumb|Phase-folded UBVRI [[light curves]] of Delta Cephei, prototype of the classical Cepheids, showing magnitude versus pulsation phase<ref name=engle2014>{{Cite journal | title = The Secret Lives of Cepheids: Evolutionary Changes and Pulsation-induced Shock Heating in the Prototype Classical Cepheid δ Cep | year = 2014 | last1 = Engle | first1 = Scott G. | last2 = Guinan | first2 = Edward F. | last3 = Harper | first3 = Graham M. | last4 = Neilson | first4 = Hilding R. | last5 = Remage Evans | first5 = Nancy | journal = [[The Astrophysical Journal]] | volume = 794 | issue = 1 | pages = 80 | doi = 10.1088/0004-637X/794/1/80 | arxiv = 1409.8628 | bibcode = 2014ApJ...794...80E| s2cid = 119189134 }}</ref>]]
A Cepheid light curve is typically asymmetric with a rapid rise to maximum light followed by a slower fall to minimum (e.g. [[Delta Cephei]]).  This is due to the phase difference between the radius and temperature variations and is considered characteristic of a fundamental mode pulsator, the most common type of type I Cepheid.  In some cases the smooth pseudo-sinusoidal light curve shows a "bump", a brief slowing of the decline or even a small rise in brightness, thought to be due to a resonance between the fundamental and second overtone.  The bump is most commonly seen on the descending branch for stars with periods around 6 days (e.g. [[Eta Aquilae]]).  As the period increases, the location of the bump moves closer to the maximum and may cause a double maximum, or become indistinguishable from the primary maximum, for stars having periods around 10 days (e.g. [[Zeta Geminorum]]).  At longer periods the bump can be seen on the ascending branch of the light curve (e.g. [[X Cygni]]),<ref name=Kovtyukh_et_al_2005>{{citation | postscript=.
 | title=Phase-dependent Variation of the Fundamental Parameters of Cepheids. II. Periods Longer than 10 Days
 | last1=Kovtyukh | first1=V. V. | last2=Andrievsky | first2=S. M.
 | last3=Belik | first3=S. I. | last4=Luck | first4=R. E.
 | display-authors=1 | journal=The Astronomical Journal
 | volume=129 | issue=1 | pages=433–453 | date=January 2005
 | doi=10.1086/426339 | bibcode=2005AJ....129..433K
| s2cid=120666782
 | doi-access=free }}</ref> but for period longer than 20 days the resonance disappears.

A minority of classical Cepheids show nearly symmetric sinusoidal light curves.  These are referred to as s-Cepheids, usually have lower amplitudes, and commonly have short periods.  The majority of these are thought to be first overtone (e.g. [[X Sagittarii]]), or higher, pulsators, although some unusual stars apparently pulsating in the fundamental mode also show this shape of light curve (e.g. [[S Vulpeculae]]).  Stars pulsating in the first overtone are expected to only occur with short periods in our galaxy, although they may have somewhat longer periods at lower metallicity, for example in the Magellanic Clouds.  Higher overtone pulsators and Cepheids pulsating in two overtones at the same time are also more common in the Magellanic Clouds, and they usually have low amplitude somewhat irregular light curves.<ref name=OGLE_1>{{cite journal|bibcode=2008AcA....58..163S|title=The Optical Gravitational Lensing Experiment. The OGLE-III Catalog of Variable Stars. I. Classical Cepheids in the Large Magellanic Cloud|journal=Acta Astronomica|volume=58|pages=163|last1=Soszynski|first1=I.|last2=Poleski|first2=R.|last3=Udalski|first3=A.|last4=Szymanski|first4=M. K.|last5=Kubiak|first5=M.|last6=Pietrzynski|first6=G.|last7=Wyrzykowski|first7=L.|last8=Szewczyk|first8=O.|last9=Ulaczyk|first9=K.|year=2008|arxiv = 0808.2210 }}</ref><ref name=OGLE_7>{{cite journal|bibcode=2010AcA....60...17S|title=The Optical Gravitational Lensing Experiment. The OGLE-III Catalog of Variable Stars. VII. Classical Cepheids in the Small Magellanic Cloud|journal=Acta Astronomica|volume=60|issue=1|pages=17|last1=Soszyñski|first1=I.|last2=Poleski|first2=R.|last3=Udalski|first3=A.|last4=Szymañski|first4=M. K.|last5=Kubiak|first5=M.|last6=Pietrzyñski|first6=G.|last7=Wyrzykowski|first7=Ł.|last8=Szewczyk|first8=O.|last9=Ulaczyk|first9=K.|year=2010|arxiv = 1003.4518 }}</ref>

==Discovery==
[[File:PSM V69 D184 Light curves of variable stars.png|thumb|Historical light curves of [[W Sagittarii]] and [[Eta Aquilae]]]]On September 10, 1784 [[Edward Pigott]] detected the variability of [[Eta Aquilae]], the first known representative of the class of classical Cepheid variables.  However, the namesake for classical Cepheids is the star [[Delta Cephei]], discovered to be variable by [[John Goodricke]] a month later.<ref name=hoskin>{{cite journal|bibcode=1979JHA....10...23H|title=Goodricke, Pigott and the Quest for Variable Stars|journal=Journal for the History of Astronomy|volume=10|pages=23–41|last1=Hoskin|first1=M.|year=1979|doi=10.1177/002182867901000103|s2cid=118155505}}</ref>  Delta Cephei is also of particular importance as a calibrator for the period-luminosity relation since its distance is among the most precisely established for a Cepheid, thanks in part to its membership in a [[star cluster]]<ref
name=dezeeuw1999>{{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}}</ref><ref name=majaess2012>{{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}}</ref> and the availability of precise [[Hubble Space Telescope]] and [[Hipparcos]] parallaxes.<ref name=benedict2002>{{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}}</ref>

==Period-luminosity relation==
{{main|Period-luminosity relation}} <!-- largely duplicated from that article, but kept here in full for now, pending clarification on whether the main article is about all period-luminosity relations or just the Classical Ceoheid one -->
[[File:Period-Luminosity Relation for Cepheids.png|thumb|left|The two Period-Luminosity Characteristics of Classic and Type II Cepheids]]
A classical Cepheid's luminosity is directly related to its period of variation.  The longer the pulsation period, the more luminous the star.  The period-luminosity relation for classical Cepheids was discovered in 1908 by [[Henrietta Swan Leavitt]] in an investigation of thousands of variable stars in the [[Magellanic Clouds]].<ref name=leavitt>{{cite journal|bibcode=1907AnHar..60...87L|title=1777 variables in the Magellanic Clouds|journal=Annals of Harvard College Observatory|volume=60|pages=87|last1=Leavitt|first1=Henrietta S.|year=1908}}</ref> She published it in 1912<ref>{{cite journal|bibcode=1912HarCi.173....1L|title=Periods of 25 Variable Stars in the Small Magellanic Cloud|journal=Harvard College Observatory Circular|volume=173|pages=1|last1=Leavitt|first1=Henrietta S.|last2=Pickering|first2=Edward C.|year=1912}}</ref> with further evidence.  Once the period-luminosity relation is calibrated, the luminosity of a given Cepheid whose period is known can be established.  Their distance is then found from their apparent brightness.  The period-luminosity relation has been calibrated by many astronomers throughout the twentieth century, beginning with [[Ejnar Hertzsprung|Hertzsprung]].<ref>{{cite journal|bibcode=1913AN....196..201H|title=Über die räumliche Verteilung der Veränderlichen vom δ Cephei-Typus|journal=Astronomische Nachrichten|volume=196|pages=201|last1=Hertzsprung|first1=Ejnar|year=1913}}</ref>  Calibrating the period-luminosity relation has been problematic; however, a firm Galactic calibration was established by Benedict et al. 2007 using precise HST parallaxes for 10 nearby classical Cepheids.<ref name=benedict2007>{{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 }}</ref> Also, in 2008, [[European Southern Observatory|ESO]] astronomers estimated with a precision within 1% the distance to the Cepheid [[RS Puppis]], using [[light echo]]s from a nebula in which it is embedded.<ref>{{Cite journal | doi = 10.1051/0004-6361:20078961| title = The long-period Galactic Cepheid RS Puppis| journal = Astronomy and Astrophysics| volume = 480| pages = 167–178| year = 2008| last1 = Kervella | first1 = P.| last2 = Mérand | first2 = A.| last3 = Szabados | first3 = L.| last4 = Fouqué | first4 = P.| last5 = Bersier | first5 = D.| last6 = Pompei | first6 = E.| last7 = Perrin | first7 = G.| issue = 1|arxiv = 0802.1501 |bibcode = 2008A&A...480..167K | s2cid = 14865683| url = https://hal.archives-ouvertes.fr/hal-00250342}}</ref>  However, that latter finding has been actively debated in the literature.<ref name=Bond_2009>{{cite journal|bibcode=2009A&A...495..371B|title=On geometric distance determination to the Cepheid RS Puppis from its light echoes|journal=Astronomy and Astrophysics|volume=495|issue=2|pages=371|last1=Bond|first1=H. E.|last2=Sparks|first2=W. B.|year=2009|doi=10.1051/0004-6361:200810280|arxiv = 0811.2943 }}</ref>

The following experimental [[correlation]]s between a Population I Cepheid's period ''P'' and its mean [[absolute magnitude]] ''M''<sub>v</sub> was established from [[Hubble Space Telescope]] [[trigonometric parallax]]es for 10 nearby Cepheids:

: <math> M_\mathrm{v} = (-2.43\pm0.12) \left(\log_{10}P - 1\right) - (4.05 \pm 0.02)\,</math><ref name=benedict2007/>

with ''P'' measured in days.

The following relations can also be used to calculate the distance ''d'' to classical Cepheids:
: <math> 5\log_{10}{d}=V+ 3.34 \log_{10}{P} - 2.45 (V-I) + 10.52\,</math><ref name=benedict2007/>

or

: <math> 5\log_{10}{d}=V+ 3.37 \log_{10}{P} - 2.55 (V-I) + 10.48\,</math><ref name=majaess2011>{{cite journal|bibcode=2011ApJ...741L..27M|title=New Evidence Supporting Membership for TW Nor in Lyngå 6 and the Centaurus Spiral Arm|journal=The Astrophysical Journal Letters|volume=741|issue=2|pages=L27|last1=Majaess|first1=Daniel|last2=Turner|first2=David|last3=Moni Bidin|first3=Christian|last4=Mauro|first4=Francesco|last5=Geisler|first5=Douglas|last6=Gieren|first6=Wolfgang|last7=Minniti|first7=Dante|last8=Chené|first8=André-Nicolas|last9=Lucas|first9=Philip|last10=Borissova|first10=Jura|last11=Kurtev|first11=Radostn|last12=Dékány|first12=Istvan|last13=Saito|first13=Roberto K.|year=2011|doi=10.1088/2041-8205/741/2/L27|arxiv = 1110.0830 |s2cid=12220317}}</ref>
''I'' and ''V'' represent near infrared and visual apparent mean magnitudes, respectively. The distance ''d'' is in [[parsec]]s.

==Small amplitude Cepheids==
Classical Cepheid variables with visual amplitudes below 0.5 magnitudes, almost symmetrical sinusoidal light curves, and short periods, have been defined as a separate group called small amplitude Cepheids.  They receive the acronym DCEPS in the GCVS.  Periods are generally less than 7 days, although the exact cutoff is still debated.<ref name=gcvs>{{cite journal|bibcode=2009yCat....102025S|title=VizieR Online Data Catalog: General Catalogue of Variable Stars (Samus+ 2007–2013)|journal=VizieR On-Line Data Catalog: B/GCVS. Originally Published in: 2009yCat....102025S|volume=1|display-authors=etal|last1=Samus|first1=N. N.|last2=Durlevich|first2=O. V.|year=2009}}</ref>
The term s-Cepheid is used for short period small amplitude Cepheids with sinusoidal light curves that are considered to be first overtone pulsators.  They are found near the red edge of the instability strip.  Some authors use s-Cepheid as a synonym for the small amplitude DCEPS stars, while others prefer to restrict it only to first overtone stars.<ref name=turner>{{cite journal|bibcode=2013ApJ...772L..10T|arxiv=1306.1228|title=The Pulsation Mode and Distance of the Cepheid FF Aquilae|journal=The Astrophysical Journal Letters|volume=772|issue=1|pages=L10|last1=Turner|first1=D. G.|last2=Kovtyukh|first2=V. V.|last3=Luck|first3=R. E.|last4=Berdnikov|first4=L. N.|year=2013|doi=10.1088/2041-8205/772/1/L10|s2cid=54710833}}</ref><ref name=antonello>{{cite journal|bibcode=1990A&A...236..138A|title=The separation of S-Cepheids from classical Cepheids and a new definition of the class|journal=Astronomy and Astrophysics|volume=236|pages=138|last1=Antonello|first1=E.|last2=Poretti|first2=E.|last3=Reduzzi|first3=L.|year=1990}}</ref>

Small amplitude Cepheids (DCEPS) include [[Polaris]] and [[FF Aquilae]], although both may be pulsating in the fundamental mode.  Confirmed first overtone pulsators include [[BG Crucis]] and [[BP Circini]].<ref name=usenko>{{cite journal|bibcode=2014AstL...40..800U|title=Spectroscopic studies of Cepheids in Circinus (AV Cir, BP Cir) and Triangulum Australe (R TrA, S TrA, U TrA, LR TrA)|journal=Astronomy Letters|volume=40|issue=12|pages=800|last1=Usenko|first1=I. A.|last2=Kniazev|first2=A. Yu.|last3=Berdnikov|first3=L. N.|last4=Kravtsov|first4=V. V.|year=2014|doi=10.1134/S1063773714110061|s2cid=122745580}}</ref><ref name=evans>{{cite journal|bibcode=2015MNRAS.446.4008E|arxiv=1411.1730|title=Observations of Cepheids with the MOST satellite: Contrast between pulsation modes|journal=Monthly Notices of the Royal Astronomical Society|volume=446|issue=4|pages=4008|last1=Evans|first1=N. R.|last2=Szabó|first2=R.|last3=Derekas|first3=A.|last4=Szabados|first4=L.|last5=Cameron|first5=C.|last6=Matthews|first6=J. M.|last7=Sasselov|first7=D.|last8=Kuschnig|first8=R.|last9=Rowe|first9=J. F.|last10=Guenther|first10=D. B.|last11=Moffat|first11=A. F. J.|last12=Rucinski|first12=S. M.|last13=Weiss|first13=W. W.|year=2015|doi=10.1093/mnras/stu2371|doi-access=free }}</ref>

==Uncertainties in Cepheid determined distances==
Chief among the uncertainties tied to the 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) and a changing (typically unknown) extinction law on classical Cepheid distances.  All these topics are actively debated in the literature.<ref name=tammannsandage2008>{{cite journal|bibcode=2008A&ARv..15..289T|title=The expansion field: The value of H 0|journal=The Astronomy and Astrophysics Review|volume=15|issue=4|pages=289|last1=Tammann|first1=G. A.|last2=Sandage|first2=A.|last3=Reindl|first3=B.|year=2008|doi=10.1007/s00159-008-0012-y|arxiv = 0806.3018 |s2cid=18463474}}</ref><ref name=ngeow2006>{{cite journal|bibcode=2006ApJ...642L..29N|title=The Hubble Constant from Type Ia Supernovae Calibrated with the Linear and Nonlinear Cepheid Period-Luminosity Relations|journal=The Astrophysical Journal|volume=642|issue=1|pages=L29–L32|last1=Ngeow|first1=C.|last2=Kanbur|first2=S. M.|year=2006|doi=10.1086/504478|arxiv = astro-ph/0603643 |s2cid=17860528}}</ref><ref name=turner10/><ref>{{cite journal|bibcode=1997MNRAS.286L...1F|title=The Cepheid period-luminosity zero-point from HIPPARCOS trigonometrical parallaxes|journal=Monthly Notices of the Royal Astronomical Society|volume=286|issue=1|pages=L1–L5|last1=Feast|first1=M. W.|last2=Catchpole|first2=R. M.|year=1997|doi=10.1093/mnras/286.1.l1|doi-access=free}}</ref><ref name=stanekudalski1999>{{cite arXiv |title=The Optical Gravitational Lensing Experiment. Investigating the Influence of Blending on the Cepheid Distance Scale with Cepheids in the Large Magellanic Cloud|eprint=astro-ph/9909346|last1=Stanek|first1=K. Z.|last2=Udalski|first2=A.|year=1999}}</ref><ref name=udalski2001>{{cite journal|bibcode=2001AcA....51..221U|title=The Optical Gravitational Lensing Experiment. Cepheids in the Galaxy IC1613: No Dependence of the Period-Luminosity Relation on Metallicity|journal=Acta Astronomica|volume=51|pages=221|last1=Udalski|first1=A.|last2=Wyrzykowski|first2=L.|last3=Pietrzynski|first3=G.|last4=Szewczyk|first4=O.|last5=Szymanski|first5=M.|last6=Kubiak|first6=M.|last7=Soszynski|first7=I.|last8=Zebrun|first8=K.|year=2001|arxiv = astro-ph/0109446 }}</ref><ref name=macri2006>{{cite journal|bibcode=2006ApJ...652.1133M|title=A New Cepheid Distance to the Maser-Host Galaxy NGC 4258 and Its Implications for the Hubble Constant|journal=The Astrophysical Journal|volume=652|issue=2|pages=1133–1149|last1=Macri|first1=L. M.|last2=Stanek|first2=K. Z.|last3=Bersier|first3=D.|last4=Greenhill|first4=L. J.|last5=Reid|first5=M. J.|year=2006|doi=10.1086/508530|arxiv = astro-ph/0608211 |s2cid=15728812}}</ref><ref name=bono2008>{{cite journal|bibcode=2008ApJ...684..102B|title=Cepheids in External Galaxies. I. The Maser-Host Galaxy NGC 4258 and the Metallicity Dependence of Period-Luminosity and Period-Wesenheit Relations|journal=The Astrophysical Journal|volume=684|issue=1|pages=102–117|last1=Bono|first1=G.|last2=Caputo|first2=F.|last3=Fiorentino|first3=G.|last4=Marconi|first4=M.|last5=Musella|first5=I.|year=2008|doi=10.1086/589965|arxiv = 0805.1592 |s2cid=6275274}}</ref><ref name=majaess2009b>{{cite journal|bibcode=2009AcA....59..403M|title=Type II Cepheids as Extragalactic Distance Candles|journal=Acta Astronomica|volume=59|issue=4|pages=403|last1=Majaess|first1=D.|last2=Turner|first2=D.|last3=Lane|first3=D.|year=2009|arxiv = 0909.0181 }}</ref><ref name=madore2009>{{cite journal|bibcode=2009ApJ...696.1498M|title=Concerning the Slope of the Cepheid Period-Luminosity Relation|journal=The Astrophysical Journal|volume=696|issue=2|pages=1498–1501|last1=Madore|first1=Barry F.|last2=Freedman|first2=Wendy L.|year=2009|doi=10.1088/0004-637X/696/2/1498|arxiv = 0902.3747 |s2cid=16325249}}</ref><ref name=scowcroft2009>{{cite journal|bibcode=2009MNRAS.396.1287S|title=The effect of metallicity on Cepheid magnitudes and the distance to M33|journal=Monthly Notices of the Royal Astronomical Society|volume=396|issue=3|pages=43–47|last1=Scowcroft|first1=V.|last2=Bersier|first2=D.|last3=Mould|first3=J. R.|last4=Wood|first4=P. R.|year=2009|doi=10.1111/j.1365-2966.2009.14822.x|doi-access=free |arxiv=0903.4088}}</ref><ref name=majaess2010>{{cite journal|bibcode=2010AcA....60..121M|title=The Cepheids of Centaurus A (NGC 5128) and Implications for H0|journal=Acta Astronomica|volume=60|issue=2|pages=121|last1=Majaess|first1=D.|year=2010|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.<ref name=freedman2001/><ref name=tammannsandage2008/><ref name=freedman2010/><ref name=ngeow2006/><ref name=macririess2009/> 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.<ref name="freedman2010"/><ref name=macririess2009>{{cite conference|bibcode=2009AIPC.1170...23M|title=The SH0ES Project: Observations of Cepheids in NGC 4258 and Type Ia SN Hosts|conference=Stellar Pulsation: Challenges for Theory and Observation: Proceedings of the International Conference. AIP Conference Proceedings|volume=1170|pages=23–25|last1=Macri|first1=Lucas M.|last2=Riess|first2=Adam G.|last3=Guzik|first3=Joyce Ann|last4=Bradley|first4=Paul A.|year=2009|doi=10.1063/1.3246452}}</ref>

==Examples==
Several classical Cepheids have variations that can be recorded with night-by-night, trained [[Bright Star Catalogue|naked eye]] observation, including the prototype [[Delta Cephei]] in the far north, [[Zeta Geminorum]] and [[Eta Aquilae]] ideal for observation around the tropics (near the ecliptic and thus zodiac) and in the far south [[Beta Doradus]].  The closest class member is the North Star ([[Polaris]]) whose distance is debated and whose present variability is approximately 0.05 of a magnitude.<ref name=freedman2010/>

<!-- placing a zero before the period if the range is less than ten days allows us to accurately list by period length  -->
{| class="wikitable sortable" style="font-size:90%;"
|- style="background:#efefef;"
! style="width:10%;"|'''Designation (name)'''
! style="width:10%;"|'''Constellation'''
! style="width:15%;"|'''Discovery'''
! style="width:10%;"|''Maximum [[Apparent magnitude]]'' (m<sub>V</sub>)<ref name=berdnikov>{{cite journal|bibcode=2008yCat.2285....0B|title=VizieR Online Data Catalog: Photoelectric observations of Cepheids in UBV(RI)c (Berdnikov, 2008)|journal=VizieR On-Line Data Catalog: II/285. Originally Published in: 2008yCat.2285....0B|volume=2285|pages=0|last1=Berdnikov|first1=L. N.|year=2008}}</ref>
! style="width:10%;"|''Minimum [[Apparent magnitude]]'' (m<sub>V</sub>)<ref name=berdnikov/>
! style="width:15%;"|'''Period''' (days)<ref name=berdnikov/>
! style="width:15%;"|'''Spectral class'''
! style="width:15%;"|'''Comment'''
|-
| [[Eta Aquilae|η Aql]]
| [[Aquila (constellation)|Aquila]]
| [[Edward Pigott]], 1784
| 3<sup>m</sup>.48
| 4<sup>m</sup>.39
| 07.17664
| F6 Ibv
| &nbsp;
|-
| [[FF Aquilae|FF Aql]]
| [[Aquila (constellation)|Aquila]]
| [[Charles Morse Huffer]], 1927
| 5<sup>m</sup>.18
| 5<sup>m</sup>.68
| 04.47
| F5Ia-F8Ia
| &nbsp;
|-
| [[TT Aquilae|TT Aql]]
| [[Aquila (constellation)|Aquila]]
|
| 6<sup>m</sup>.46
| 7<sup>m</sup>.7
| 13.7546
| F6-G5
| &nbsp;
|-
| [[U Aquilae|U Aql]]
| [[Aquila (constellation)|Aquila]]
|
| 6<sup>m</sup>.08
| 6<sup>m</sup>.86
| 07.02393
| F5I-II-G1
| &nbsp;
|-
| [[T Antliae|T Ant]]
| [[Antlia]]
|
| 5<sup>m</sup>.00
| 5<sup>m</sup>.82
| 05.898
| G5
| possibly has unseen companion. Previously thought to be a type II Cepheid<ref>{{cite journal|doi=10.1051/0004-6361:20030835|title=The nature of the Cepheid T Antliae|journal=Astronomy and Astrophysics|volume=407|pages=325–334|year=2003|last1=Turner|first1=D. G.|last2=Berdnikov|first2=L. N.|bibcode=2003A&A...407..325T|doi-access=free}}</ref>
|-
| [[RT Aurigae|RT Aur]]
| [[Auriga (constellation)|Auriga]]
|
| 5<sup>m</sup>.00
| 5<sup>m</sup>.82
| 03.73
| F8Ibv
| &nbsp;
|-
| [[HD 84810|l Car]]
| [[Carina (constellation)|Carina]]
| &nbsp;
| 3<sup>m</sup>.28
| 4<sup>m</sup>.18
| 35.53584
| G5 Iab/Ib
| &nbsp;
|-
| [[Delta Cephei|δ Cep]]
| [[Cepheus (constellation)|Cepheus]]
| [[John Goodricke]], 1784
| 3<sup>m</sup>.48
| 4<sup>m</sup>.37
| 05.36634
| F5Ib-G2Ib
| double star, visible in binoculars
|-
| [[AX Circini|AX Cir]]
| [[Circinus]]
| &nbsp;
| 5<sup>m</sup>.65
| 6<sup>m</sup>.09
| 05.273268
| F2-G2II
| spectroscopic binary with {{Solar mass|5|link=y}} B6 companion
|-
| [[BP Circini|BP Cir]]
| [[Circinus]]
| &nbsp;
| 7<sup>m</sup>.31
| 7<sup>m</sup>.71
| 02.39810
| F2/3II-F6
| spectroscopic binary with {{Solar mass|4.7|link=y}} B6 companion
|-
| [[BG Crucis|BG Cru]]
| [[Crux]]
| &nbsp;
| 5<sup>m</sup>.34
| 5<sup>m</sup>.58
| 03.3428
| F5Ib-G0p
| &nbsp;
|-
| [[R Crucis|R Cru]]
| [[Crux]]
| &nbsp;
| 6<sup>m</sup>.40
| 7<sup>m</sup>.23
| 05.82575
| F7Ib/II
| &nbsp;
|-
| [[S Crucis|S Cru]]
| [[Crux]]
| &nbsp;
| 6<sup>m</sup>.22
| 6<sup>m</sup>.92
| 04.68997
| F6-G1Ib-II
| &nbsp;
|-
| [[T Crucis|T Cru]]
| [[Crux]]
| &nbsp;
| 6<sup>m</sup>.32
| 6<sup>m</sup>.83
| 06.73331
| F6-G2Ib
| &nbsp;
|-
| [[X Cygni|X Cyg]]
| [[Cygnus (constellation)|Cygnus]]
| &nbsp;
| 5<sup>m</sup>.85
| 6<sup>m</sup>.91
| 16.38633
| G8Ib<ref name=tomasella>{{cite journal|bibcode=2010AJ....140.1758T|title=A High-resolution, Multi-epoch Spectral Atlas of Peculiar Stars Including RAVE, GAIA, and HERMES Wavelength Ranges|journal=The Astronomical Journal|volume=140|issue=6|pages=1758|last1=Tomasella|first1=Lina|last2=Munari|first2=Ulisse|last3=Zwitter|first3=Tomaž|year=2010|doi=10.1088/0004-6256/140/6/1758|arxiv = 1009.5566 |s2cid=119188449}}</ref>
| &nbsp;
|-
| [[SU Cygni|SU Cyg]]
| [[Cygnus (constellation)|Cygnus]]
| &nbsp;
| 6<sup>m</sup>.44
| 7<sup>m</sup>.22
| 03.84555
| F2-G0I-II<ref name=andrievsky>{{cite journal|bibcode=2005AJ....130.1880A|title=Phase-dependent Variation of the Fundamental Parameters of Cepheids. III. Periods between 3 and 6 Days|journal=The Astronomical Journal|volume=130|issue=4|pages=1880|last1=Andrievsky|first1=S. M.|last2=Luck|first2=R. E.|last3=Kovtyukh|first3=V. V.|year=2005|doi=10.1086/444541|doi-access=free}}</ref>
| &nbsp;
|-
| [[Beta Doradus|β Dor]]
| [[Dorado]]
| &nbsp;
| 3<sup>m</sup>.46
| 4<sup>m</sup>.08
| 09.8426
| F4-G4Ia-II
| &nbsp;
|-
| [[Zeta Geminorum|ζ Gem]]
| [[Gemini (constellation)|Gemini]]
| [[Johann Friedrich Julius Schmidt|Julius Schmidt]], 1825
| 3<sup>m</sup>.62
| 4<sup>m</sup>.18
| 10.15073
| F7Ib to G3Ib
| &nbsp;
|-
| [[V473 Lyrae|V473 Lyr]]
| [[Lyra]]
| &nbsp;
| 5<sup>m</sup>.99
| 6<sup>m</sup>.35
| 01.49078
| F6Ib-II
| &nbsp;
|-
| [[R Muscae|R Mus]]
| [[Musca]]
| &nbsp;
| 5<sup>m</sup>.93
| 6<sup>m</sup>.73
| 07.51
| F7Ib-G2
| &nbsp;
|-
| [[S Muscae|S Mus]]
| [[Musca]]
| &nbsp;
| 5<sup>m</sup>.89
| 6<sup>m</sup>.49
| 09.66007
| F6Ib-G0
| &nbsp;
|-
| [[S Normae|S Nor]]
| [[Norma (constellation)|Norma]]
| &nbsp;
| 6<sup>m</sup>.12
| 6<sup>m</sup>.77
| 09.75411
| F8-G0Ib
| brightest member of  open cluster [[NGC 6087]]
|-
| [[QZ Normae|QZ Nor]]
| [[Norma (constellation)|Norma]]
| &nbsp;
| 8<sup>m</sup>.71
| 9<sup>m</sup>.03
| 03.786008
| F6I
| member of  open cluster [[NGC 6067]]
|-
| [[V340 Normae|V340 Nor]]
| [[Norma (constellation)|Norma]]
| &nbsp;
| 8<sup>m</sup>.26
| 8<sup>m</sup>.60
| 11.2888
| G0Ib
| member of  open cluster [[NGC 6067]]
|-
| [[V378 Normae|V378 Nor]]
| [[Norma (constellation)|Norma]]
| &nbsp;
| 6<sup>m</sup>.21
| 6<sup>m</sup>.23
| 03.5850
| G8Ib
| &nbsp;
|-
| [[BF Ophiuchi|BF Oph]]
| [[Ophiuchus]]
| &nbsp;
| 6<sup>m</sup>.93
| 7<sup>m</sup>.71
| 04.06775
| F8-K2<ref name=densities>{{cite journal|bibcode=1953ZA.....32..125K|title=The Density of Stars of Different Spectral Types. With 1 figure|journal=Zeitschrift für Astrophysik|volume=32|pages=125|last1=Kreiken|first1=E. A.|year=1953}}</ref>
| &nbsp;
|-
| [[RS Puppis|RS Pup]]
| [[Puppis]]
| &nbsp;
| 6<sup>m</sup>.52
| 7<sup>m</sup>.67
| 41.3876
| F8Iab
| &nbsp;
|-
| [[S Sagittae|S Sge]]
| [[Sagitta]]
 | [[John Ellard Gore]], 1885
| 5<sup>m</sup>.24
| 6<sup>m</sup>.04
| 08.382086<ref name=AAVSOS>{{cite web|url=http://www.aavso.org/vsx/index.php?view=detail.top&oid=27343 |title=S Sagittae |author =Watson, Christopher |date=4 January 2010|work=AAVSO Website|publisher=American Association of Variable Star Observers|access-date=22 May 2015}}</ref>
| F6Ib-G5Ib
| &nbsp;
|-
| [[U Sagittarii|U Sgr]]
| [[Sagittarius (constellation)|Sagittarius]] (in [[Messier 25|M25]])
| &nbsp;
| 6<sup>m</sup>.28
| 7<sup>m</sup>.15
| 06.74523
| G1Ib<ref name=2d/>
| &nbsp;
|-
| [[W Sagittarii|W Sgr]]
| [[Sagittarius (constellation)|Sagittarius]]
| &nbsp;
| 4<sup>m</sup>.29
| 5<sup>m</sup>.14
| 07.59503
| F4-G2Ib
| Optical double with [[Gamma2 Sagittarii|γ<sup>2</sup> Sgr]]
|-
| [[X Sagittarii|X Sgr]]
| [[Sagittarius (constellation)|Sagittarius]]
| &nbsp;
| 4<sup>m</sup>.20
| 4<sup>m</sup>.90
| 07.01283
| F5-G2II
|
|-
| [[V636 Scorpii|V636 Sco]]
| [[Scorpius]]
| &nbsp;
| 6<sup>m</sup>.40
| 6<sup>m</sup>.92
| 06.79671
| F7/8Ib/II-G5
| &nbsp;
|-
| [[R Trianguli Australis|R TrA]]
| [[Triangulum Australe]]
| &nbsp;
| 6<sup>m</sup>.4
| 6<sup>m</sup>.9
| 03.389
| F7Ib/II<ref name=2d>{{cite book|bibcode=1975mcts.book.....H|title=University of Michigan Catalogue of two-dimensional spectral types for the HD stars. Volume I. Declinations −90° to −53.0°|last1=Houk|first1=N.|last2=Cowley|first2=A. P.|year=1975}}</ref>
| &nbsp;
|-
| [[S Trianguli Australis|S TrA]]
| [[Triangulum Australe]]
| &nbsp;
| 6<sup>m</sup>.1
| 6<sup>m</sup>.8
| 06.323
| F6II-G2
| &nbsp;
|-
| α UMi ([[Polaris]])
| [[Ursa Minor]]
| [[Ejnar Hertzsprung]], 1911
| 1<sup>m</sup>.86
| 2<sup>m</sup>.13
| 03.9696
|  F8Ib or F8II
| &nbsp;
|-
| [[AH Velorum|AH Vel]]
| [[Vela (constellation)|Vela]]
| &nbsp;
| 5<sup>m</sup>.5
| 5<sup>m</sup>.89
| 04.227171
|  F7Ib-II
| &nbsp;
|-
| [[S Vulpeculae|S Vul]]
| [[Vulpecula]]
| &nbsp;
| 8<sup>m</sup>.69
| 9<sup>m</sup>.42
| 68.464
| G0-K2(M1)
| &nbsp;
|-
| [[T Vulpeculae|T Vul]]
| [[Vulpecula]]
| &nbsp;
| 5<sup>m</sup>.41
| 6<sup>m</sup>.09
| 04.435462
|  F5Ib-G0Ib
| &nbsp;
|-
| [[U Vulpeculae|U Vul]]
| [[Vulpecula]]
| &nbsp;
| 6<sup>m</sup>.73
| 7<sup>m</sup>.54
| 07.990676
| F6Iab-G2
| &nbsp;
|-
| [[SV Vulpeculae|SV Vul]]
| [[Vulpecula]]
| &nbsp;
| 6<sup>m</sup>.72
| 7<sup>m</sup>.79
| 44.993
| F7Iab-K0Iab
| &nbsp;
|-
| [[SU Cas]]
| [[Cassiopeia (constellation)|Cassiopeia]]
| &nbsp;
| 5<sup>m</sup>.88
| 6<sup>m</sup>.30
| 01.9
| F5II
| &nbsp;
|}

==See also==
*[[RR Lyrae variable]]
*[[Stellar pulsation]]
*[[Type II Cepheid]]

==References==
{{Reflist|2}}

==External links==
*[https://web.archive.org/web/20101009205927/http://www.institute-of-brilliant-failures.com/ The Cepheid Distance Scale: A History, by Nick Allen]
*[https://crocus.physics.mcmaster.ca/Cepheid/Classical.html List of classical Cepheids on McMaster Cepheid Photometry and Radial Velocity Data Archive] {{Webarchive|url=https://web.archive.org/web/20211008185122/https://crocus.physics.mcmaster.ca/Cepheid/Classical.html |date=2021-10-08 }}
*[http://www.aavso.org/ American Association of Variable Star Observers]
*[http://ogle.astrouw.edu.pl/atlas/classical_Cepheids.html OGLE Atlas of Variable Star Light Curves – Classical Cepheids]

{{Variable star topics}}

{{DEFAULTSORT:Cepheid Variable}}
[[Category:Classical Cepheid variables]]
[[Category:Astrometry]]
[[Category:Standard candles]]