Monte Carlo methods for electron transport

2013-11-05T09:34:56Z

134.91.214.134: corrected a tex lapse in the integral representation of the scattering matrix element <k|H'|k'>: k' ist now a subscript.

'''Classical Cepheids''' (also known as '''Population I Cepheids''', '''Type I Cepheids''', or '''[[Delta Cephei]] variables''') are a type of [[Cepheid variable]] [[star]]. They are [[population I]], [[variable stars]] that exhibit [[periodic function|pulsation periods]] in the order of a few days to months, are 4–20 times more massive than the Sun,<ref name=turner96>Turner, David G.[http://adsabs.harvard.edu/abs/1996JRASC..90...82T "The Progenitors of Classical Cepheid Variables"]. ''JRASC'' (1996)</ref> and up to 100,000 times more luminous.<ref name=turner10>Turner, David G.[http://adsabs.harvard.edu/cgi-bin/bib_query?arXiv:0912.4864 "The PL calibration for Milky Way Cepheids and its implications for the distance scale"]. ''Ap&SS'' (2010)</ref> Classical Cepheids are yellow supergiants of [[spectral classification|spectral class]] F6 – K2 and their radii change by (~25% for the longer-period [[l Car]]) millions of kilometers during a pulsation cycle.<ref>Rodgers, A. W. [http://articles.adsabs.harvard.edu//full/1957MNRAS.117...85R/0000092.000.html "Radius variation and population type of cepheid variables"]. ''Monthly Notices of the Royal Astronomical Society''. 117 (1956) 84–94</ref><ref name="strohmeier1972">W. Strohmeier, ''Variable Stars'', Pergamon (1972)</ref>

There exists a well-defined relationship between a Classical Cepheid variable's [[luminosity]] and pulsation period,<ref name=udalski99>Udalski, A.; Soszynski, I.; Szymanski, M.; Kubiak, M.; Pietrzynski, G.; Wozniak, P.; Zebrun, K.[http://adsabs.harvard.edu/abs/1999AcA....49..223U "The Optical Gravitational Lensing Experiment. Cepheids in the Magellanic Clouds. IV. Catalog of Cepheids from the Large Magellanic Cloud"]. ''Acta A.'' (1999)</ref><ref name=sos08>Soszynski, I.; Poleski, R.; Udalski, A.; Szymanski, M. K.; Kubiak, M.; Pietrzynski, G.; Wyrzykowski, L.; Szewczyk, O.; Ulaczyk, K. [http://adsabs.harvard.edu/abs/2008AcA....58..163S "The Optical Gravitational Lensing Experiment. The OGLE-III Catalog of Variable Stars. I. Classical Cepheids in the Large Magellanic Cloud"]. ''Acta A.'' (2008)</ref> 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>Freedman, Wendy L.; Madore, Barry F. [http://adsabs.harvard.edu/abs/2010ARA%26A..48..673F "The Hubble Constant"], ''Annual Review of Astronomy and Astrophysics'', 2010</ref> HST observations of Classical Cepheid variables have enabled firmer constraints on [[Hubble's law]].<ref name=freedman2001>Freedman, W. et al. [http://adsabs.harvard.edu/abs/2001ApJ...553...47F "Final Results from the Hubble Space Telescope Key Project to Measure the Hubble Constant"], ''The Astrophysical Journal'', Volume 553, Issue 1, pp. 47–72 (2001)</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 Sun's height above the galactic plane and the Galaxy's local spiral structure.<ref name=majaess2009>Majaess D. J., Turner D. G., Lane D. J. [http://adsabs.harvard.edu/abs/2009arXiv0903.4206M "Characteristics of the Galaxy according to Cepheids"], ''Monthly Notices of the Royal Astronomical Society'' (2009)</ref>

Over 700 classical Cepheids are known in the [[Milky Way]] Galaxy,<ref name="cox1980">Cox, John P., ''Theory of Stellar Pulsations'', Princeton (1980)</ref> and several thousand extragalactic Cepheids have been discovered. The [[Hubble Space Telescope]] has identified classical Cepheids in [[NGC 4603]], which is 100 million [[light year]]s distant.<ref name="HST_NGC_4603_Cepheid_Distance">Jeffrey A. Newman, Stephen E. Zepf, Marc Davis, Wendy L. Freedman, Barry F. Madore, Peter B. Stetson, N. Silbermann and Randy Phelps [http://iopscience.iop.org/0004-637X/523/2/506/ "A Cepheid Distance to NGC 4603 in Centaurus"]. ''The Astrophysical Journal''. 523 (1999) 506–520</ref>

==Discovery==
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 few months later. [[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>de Zeeuw, P. T.; Hoogerwerf, R.; de Bruijne, J. H. J.; Brown, A. G. A.; Blaauw, A. (1999). [http://adsabs.harvard.edu/abs/1999AJ....117..354D ''A HIPPARCOS Census of the Nearby OB Associations''], AJ</ref><ref name=majaess2012>Majaess, D.; Turner, D.; Gieren, W. (2012). [http://adsabs.harvard.edu/abs/2012ApJ...747..145M ''New Evidence Supporting Cluster Membership for the Keystone Calibrator Delta Cephei''], ApJ</ref> and the availability of precise [[Hubble Space Telescope]]/[[Hipparcos]] parallaxes.<ref name=benedict2002/>

==Period-luminosity relation==
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>Leavitt, Henrietta S. [http://www.adsabs.harvard.edu/abs/1908AnHar..60...87L "1777 Variables in the Magellanic Clouds"]. ''Annals of Harvard College Observatory''. LX(IV) (1908) 87–110</ref> She published it in 1912<ref>Miss Leavitt in Pickering, Edward C. [http://adsabs.harvard.edu/abs/1912HarCi.173....1L "Periods of 25 Variable Stars in the Small Magellanic Cloud"] ''Harvard College Observatory Circular'' 173 (1912) 1–3.</ref> with further evidence. Once the [[period-luminosity relationship]] 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 relationship has been calibrated by many astronomers throughout the twentieth century, beginning with [[Ejnar Hertzsprung|Hertzsprung]].<ref>Hertzsprung, E., "Über die räumliche Verteilung der Veränderlichen vom δ Cephei-Typus." ''Astronomischen Nachrichten'', 196 p. 201–210 (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>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"], ''The Astronomical Journal'', Volume 133, Issue 4, pp. 1810-1827 (2007)</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>Kervella, Pierre: [http://www.eso.org/public/outreach/press-rel/pr-2008/pr-05-08.html ''Light echoes whisper the distance to a star'']</ref> However, that latter finding has been actively debated in the literature.<ref name=benedict2007>Bond, H. E., Sparks, W. B. [http://adsabs.harvard.edu/abs/2009A%26A...495..371B "On geometric distance determination to the Cepheid RS Puppis from its light echoes"], ''Astronomy and Astrophysics'', Volume 495, Issue 2, 2009, pp.371-377</ref>

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

: <math> M_v = (-2.43\pm0.12) (\log_{10}(P) - 1) - (4.05 \pm 0.02) \, </math>

with <math>P</math> measured in days.
<ref
name=benedict2002>Benedict, G. Fritz; McArthur, B. E.; Fredrick, L. W.; Harrison, T. E.; Slesnick, C. L.; Rhee, J.; Patterson, R. J.; Skrutskie, M. F.; Franz, O. G.; Wasserman, L. H.; Jefferys, W. H.; Nelan, E.; van Altena, W.; Shelus, P. J.; Hemenway, P. D.; Duncombe, R. L.; Story, D.; Whipple, A. L.; Bradley, A. J. (2002). [http://adsabs.harvard.edu/abs/2002AJ....124.1695B ''Astrometry with the Hubble Space Telescope: A Parallax of the Fundamental Distance Calibrator δ Cephei''], AJ</ref><ref
name=benedict2007>Benedict, G. Fritz; McArthur, Barbara E.; Feast, Michael W.; Barnes, Thomas G.; Harrison, Thomas E.; Patterson, Richard J.; Menzies, John W.; Bean, Jacob L.; Freedman, Wendy L. (2007). [http://adsabs.harvard.edu/abs/2007AJ....133.1810B ''Hubble Space Telescope Fine Guidance Sensor Parallaxes of Galactic Cepheid Variable Stars: Period-Luminosity Relations''], AJ</ref> The following relations can also be used to calculate the distance <math>d</math> to classical Cepheids:
: <math> 5\log_{10}{d}=V+ (3.34) \log_{10}{P} - (2.45) (V-I) + 7.52 \,. </math><ref name=benedict2007/>

or

: <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"], ApJ Letters'', Volume 741, Issue 2, article id. L2 (2011)</ref>
<math>I</math> and <math>V</math> represent near infrared and visual apparent mean magnitudes, respectively.

==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=turner10/><ref name=tammannsandage2008>Tammann, G. A.; Sandage, A.; Reindl, B. [http://adsabs.harvard.edu/abs/2008A%26ARv..15..289T "The expansion field: the value of H 0"], ''Annual Review of Astronomy and Astrophysics'', 2008</ref><ref name=ngeow2006>Ngeow, C., Kanbur, S. M. [http://adsabs.harvard.edu/abs/2006ApJ...642L..29N "The Hubble Constant from Type Ia Supernovae Calibrated with the Linear and Nonlinear Cepheid Period-Luminosity Relations"], ''The Astrophysical Journal'', 2006</ref><ref>Feast, Michael W. & Robin M. Catchpole. [http://adsabs.harvard.edu/abs/1997MNRAS.286L...1F "The Cepheid period-luminosity zero-point from Hipparcos trigonometrical parallaxes"]. ''[[Monthly Notices of the Royal Astronomical Society]]''. 286 (1997) L 1–5.</ref><ref name=stanekudalski1999>Stanek, K. Z., Udalski, A. [http://adsabs.harvard.edu/abs/1999astro.ph..9346S "The Optical Gravitational Lensing Experiment. Investigating the Influence of Blending on the Cepheid Distance Scale with Cepheids in the Large Magellanic Cloud"], ''ApJ'', 1999</ref><ref name=udalski2001>Udalski, A. et al. [http://adsabs.harvard.edu/abs/2001AcA....51..221U "The Optical Gravitational Lensing Experiment. Cepheids in the Galaxy IC1613: No Dependence of the Period-Luminosity Relation on Metallicity"], ''Acta Astronomica'', 2001</ref><ref name=macri2006>Macri, L. M.; Stanek, K. Z.; Bersier, D.; Greenhill, L. J.; Reid, M. J. [http://adsabs.harvard.edu/abs/2006ApJ...652.1133M "A New Cepheid Distance to the Maser-Host Galaxy NGC 4258 and Its Implications for the Hubble Constant"], ''The Astrophysical Journal'', 2006</ref><ref name=bono2008>Bono, G.; Caputo, F.; Fiorentino, G.; Marconi, M.; Musella, I.[http://adsabs.harvard.edu/abs/2008ApJ...684..102B "Cepheids in External Galaxies. I. The Maser-Host Galaxy NGC 4258 and the Metallicity Dependence of Period-Luminosity and Period-Wesenheit Relations"], ''The Astrophysical Journal'', 2008</ref><ref name=majaess2009b>Majaess, D.; Turner, D.; Lane, D. [http://adsabs.harvard.edu/abs/2009AcA....59..403M "Type II Cepheids as Extragalactic Distance Candles"], ''Acta Astronomica'', 2009</ref><ref name=madore2009>Madore, Barry F., Freedman, Wendy L. [http://adsabs.harvard.edu/abs/2009ApJ...696.1498M " Concerning the Slope of the Cepheid Period-Luminosity Relation"], ''The Astrophysical Journal'', 2009</ref><ref name=scowcroft2009>Scowcroft, V.; Bersier, D.; Mould, J. R.; Wood, P. R. [http://adsabs.harvard.edu/abs/2009MNRAS.396.1287S "The effect of metallicity on Cepheid magnitudes and the distance to M33"], ''Monthly Notices of the Royal Astronomical Society'', Volume 396, Issue 3, pp. 1287-1296, 2009</ref><ref name=majaess2010>Majaess, D. [http://adsabs.harvard.edu/abs/2010AcA....60..121M "The Cepheids of Centaurus A (NGC 5128) and Implications for H0"], ''Acta Astronomica'', 2010</ref>

These unresolved matters have resulted in cited values for the Hubble constant ranging between 60 km/s/Mpc and 80 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>Macri, Lucas M.; Riess, Adam G. [http://adsabs.harvard.edu/abs/2009AIPC.1170...23M "The SH0ES Project: Observations of Cepheids in NGC 4258 and Type Ia SN Hosts"], ''STELLAR PULSATION: CHALLENGES FOR THEORY AND OBSERVATION'', 2009</ref>

==Examples==
Some fairly bright Classical Cepheids which exhibit variations discernable with the naked eye include: [[Eta Aquilae]], [[Zeta Geminorum]], [[Beta Doradus]], as well as the prototype [[Delta Cephei]]. The closest Classical Cepheid is the North Star ([[Polaris]]), although the star exhibits many peculiarities and its distance is a topic of active debate.<ref name=turner2009>Turner, D. [http://adsabs.harvard.edu/abs/2010ARA%26A..48..673F "Polaris and its Kin"], ''STELLAR PULSATION: CHALLENGES FOR THEORY AND OBSERVATION'', 2009</ref>

{| 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%;"|''[[Apparent magnitude]]'' (Maximum)<ref name="autogenerated1">''(visual magnitude, unless marked'' (B) ''(= blue) or'' (p) ''(= photographic))''</ref>
! style="width:10%;"|''[[Apparent magnitude]]'' (Minimum)<ref name="autogenerated1"/>
! style="width:15%;"|'''Period'''
! style="width:15%;"|'''Spectral class'''
! style="width:15%;"|'''Comment'''
|-
| [[Eta Aquilae|η Aql]]
| [[Aquila (constellation)|Aquila]]
| [[Nathaniel Pigott|Pigott]], 1784
| 3m.48
| 4m.39
| 7.17664 d
| F6 Ibv
|  
|-
| [[FF Aquilae|FF Aql]]
| [[Aquila (constellation)|Aquila]]
|
| 5m.18
| 5m.68
| 4.47 d
| F5Ia-F8Ia
|  
|-
| [[TT Aquilae|TT Aql]]
| [[Aquila (constellation)|Aquila]]
|
| 6m.46
| 7m.7
| 13.7546 d
| F6-G5
|  
|-
| [[U Aquilae|U Aql]]
| [[Aquila (constellation)|Aquila]]
|
| 6m.08
| 6m.86
| 7.02393 d
| F5I-II-G1
|  
|-
| [[T Antliae|T Ant]]
| [[Antlia]]
|
| 5m.00
| 5m.82
| 5.898 d
| G5
| has unseen companion, possibly associated with open cluster [[NGC 6649]]
|-
| [[RT Aurigae|RT Aur]]
| [[Auriga (constellation)|Auriga]]
|
| 5m.00
| 5m.82
| 3.73 d
| F8Ibv
|  
|-
| [[HD 84810|l Car]]
| [[Carina (constellation)|Carina]]
|  
| 3m.28
| 4m.18
| 35.53584 d
| G5 Iab/Ib
|  
|-
| [[Delta Cephei|δ Cep]]
| [[Cepheus (constellation)|Cepheus]]
| [[John Goodricke]], 1784
| 3m.48
| 4m.37
| 5.36634 d
| F5Ib-G2Ib
| double star, visible in binoculars
|-
| [[AX Circini|AX Cir]]
| [[Circinus]]
|  
| 5m.65
| 6m.09
| 5.273268 d
| F2-G2II
| spectroscopic binary with 5 solar mass B6 companion
|-
| [[BP Circini|BP Cir]]
| [[Circinus]]
|  
| 7m.31
| 7m.71
| 2.39810 d
| F2/3II-F6
| spectroscopic binary with 4.7 solar mass B6 companion
|-
| [[BG Crucis|BG Cru]]
| [[Crux]]
|  
| 5m.34
| 5m.58
| 3.3428 d
| F5Ib-G0p
|  
|-
| [[R Crucis|R Cru]]
| [[Crux]]
|  
| 6m.40
| 7m.23
| 5.82575 d
| F7Ib/II
|  
|-
| [[X Cygni|X Cyg]]
| [[Cygnus (constellation)|Cygnus]]
|  
| 5m.85
| 6m.91
| 16.38633 d
| F7Ib
|  
|-
| [[SU Cygni|SU Cyg]]
| [[Cygnus (constellation)|Cygnus]]
|  
| 6m.44
| 7m.22
| 3.84555 d
| F2Iab
|  
|-
| [[Beta Doradus|β Dor]]
| [[Dorado]]
|  
| 3m.46
| 4m.08
| 9.8426 d
| F4-G4Ia-II
|  
|-
| [[Zeta Geminorum|ζ Gem]]
| [[Gemini (constellation)|Gemini]]
|  
| 3m.62
| 4m.18
| 10.15073 d
| F7Ib to G3Ib
|  
|-
| [[R Muscae|R Mus]]
| [[Musca]]
|  
| 5m.93
| 6m.73
| 7.51 d
| F7Ib-G2
|  
|-
| [[S Muscae|S Mus]]
| [[Musca]]
|  
| 5m.89
| 6m.49
| 9.66007 d
| F6Ib-G0
|  
|-
| [[BF Ophiuchi|BF Oph]]
| [[Ophiuchus]]
|  
| 6m.93
| 7m.71
| 4.06775 d
| G0II
|  
|-
| [[Kappa Pavonis|κ Pav]]
| [[Pavo (constellation)|Pavo]]
|  
| 3m.91
| 4m.78
| 9.09423 d
| F5Ib-II
|  
|-
| [[RS Puppis|RS Pup]]
| [[Puppis]]
|  
| 6m.52
| 7m.67
| 41.3876 d
| F8Iab
|  
|-
| [[U Sagittarii|U Sgr]]
| [[Sagittarius (constellation)|Sagittarius]] (in [[Messier 25|M25]])
|  
| 6m.28
| 7m.15
| 6.74523 d
| G1Ib
|  
|-
| [[W Sagittarii|W Sgr]]
| [[Sagittarius (constellation)|Sagittarius]]
|  
| 4m.29
| 5m.14
| 7.59503 d
| F4-G2Ib
| Optical double with [[Gamma Sagittarii|γ Sgr]]
|-
| [[X Sagittarii|X Sgr]]
| [[Sagittarius (constellation)|Sagittarius]]
|  
| 4m.20
| 4m.90
| 7.01283 d
| F5-G2II
|
|-
| [[V636 Scorpii|V636 Sco]]
| [[Scorpius]]
|  
| 6m.40
| 6m.92
| 6.79671 d
| F7/8Ib/II-G5
|  
|-
| [[R Trianguli Australis|R TrA]]
| [[Triangulum Australe]]
|  
| 6m.4
| 6m.9
| 3.389 d
| F7Ib/II
|  
|-
| [[S Trianguli Australis|S TrA]]
| [[Triangulum Australe]]
|  
| 6m.1
| 6m.8
| 6.323 d
| F8II
|  
|-
| α UMi ([[Polaris]])
| [[Ursa Minor]]
|  
| 1m.86
| 2m.13
| 3.9696 d
| F8Ib or F8II
|  
|-
| [[AH Velorum|AH Vel]]
| [[Vela (constellation)|Vela]]
|  
| 5m.5
| 5m.89
| 4.227171 d
| F7Ib-II
|  
|-
| [[T Vulpeculae|T Vul]]
| [[Vulpecula]]
|  
| 5m.41
| 6m.09
| 4.435462 d
| F5Ib-G0Ib
|  
|}

==See also==
*[[Type II Cepheids]]
*[[RR Lyrae variables]]
*[[Stellar pulsation theory - Regular versus irregular variability]]

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

==External links==
*[http://www.institute-of-brilliant-failures.com The Cepheid Distance Scale: A History, by Nick Allen]
*[http://www.seds.org/~spider/spider/ScholarX/variables.html SEDS: Variable Stars]
*[http://crocus.physics.mcmaster.ca/Cepheid/HomePage.html McMaster Cepheid Photometry and Radial Velocity Data Archive]
*[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:Cepheid variables]]
[[Category:Astrometry]]
[[Category:Standard candles]]
[[Category:Variable stars]]

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