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Detail investigation of the inclined pressure structure and gravity darkening in critical rotating star Achernar

Tai Li-Ting Song Han-Feng Wang Jiang-Tao

Detail investigation of the inclined pressure structure and gravity darkening in critical rotating star Achernar

Tai Li-Ting, Song Han-Feng, Wang Jiang-Tao
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  • Rotation and tide are two important factors that have very important impacts on the stellar structure and evolution. Based on the observational data of Achernar, we have derived the inclined pressure structure in a single rotating star or as a member in the binaries. We have given the distributions of the physical quantities on the isobaric surface and these distributions are derived from the Legendre series of expansions. We have also found the relationship between all levels of perturbation potential functions (including rotational and tidal distortions) and the distributions of density and pressure under the condition of inclined pressure structure. In particular, the gravitational darkening with the models including the effects of rotation and tide is investigated. We have found that the critical ratio of equatorial radius to the polar radius is consistent with the observations in rotating binaries better than that in single rotating model. The reason is that the tidal force can make the polar radius shortened because the tidal force exerts an inward force to the two polar points. However, the theoretical angular velocity in binaries is smaller than that observed. It is also shown that the positive shear enhances the centrifugal force and decreases the mean effective gravitational acceleration and effective temperatures whereas the negative shear plays a role to strengthen the effective gravitational acceleration. Moreover, the solid body rotation has not been supported inside Achernar because magnetic fields have not been detected through observations. Furthermore, the theoretical angular velocity in rigid rotation is higher than the angular velocity observed. Achernar has a periodic variation of light curves due to mass outburst, which also supports differential rotation. A positive shear indicates that the mass in accretion disks is falling to Achernar and the Achernar is spun up to critical rotation according to current observations. By comparing the theoretical results with observations, it can be seen that when the theoretical spin angular velocity of Achernar is 4.65 10-5 s-1 and the positive shears / s are 0.7851, the temperature of the polar points is 16041 K and that of equatorial sphere is 12073 K. Relative errors between the theoretical values and observations are less than 3% and are listed in the text. This model is the best and is the most possible one for Achernar.
      Corresponding author: Song Han-Feng, sci.hfsong@gzu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11463002), the Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Science (Grant No. OP201405), and the Graduate Innovation Fund in Guizhou University, China (Grant No. 2015055).
    [1]

    Huang R Q, Yu K N 1998 Stellar Astrophysics (New York: Springer Verlag) p313

    [2]

    Paczynski B 1971 Annu. Rev. Astron. Astrophys. 9 183

    [3]

    Kippenhahn R, Thomas H C 1970 Proceedings of IAU Colloq. 4 Columbus, USA, September 8-11, 1969 p20

    [4]

    Endal A S, Sofia S 1976 Astrophys. J. 210 184

    [5]

    Pinsonneault M H, Kawaler S D, Sofia S, Demarqure P 1989 Astrophys. J. 338 424

    [6]

    Pinsonneault M H, Kawaler S D, Demarqure P 1990 Astrophys. J. Suppl. Ser. 74 501

    [7]

    Pinsonneault M H, Deliyannis C P, Demarqure P 1991 Astrophys. J. 367 239

    [8]

    Song H F, Zhang B, Zhang J, Wu H B, Peng Q H 2003 Chin. Phys. Lett. 20 2084

    [9]

    Wen D H, Zhou Y 2013 Chin. Phys. B 22 080401

    [10]

    Zhang J, Wang B, Zhang B, Han Z W 2012 Chin. Phys. Lett. 29 019701

    [11]

    von Zeipel H 1924 Mon. Not. Roy. Astron. Soc. 84 665

    [12]

    de Souza D A, Kervella P, Jankov S, Abe L, Vakili F, di Folco E, Paresce F 2003 Astron. Astrophys. 407 L47

    [13]

    Naz Y 2009 Astron. Astrophys. 506 1055

    [14]

    Jackson S, MacGregor K B, Skumanich A 2004 Astrophys. J. 606 1196

    [15]

    Maeder A, Stahler S 2009 Physics, Formation and Evolution of Rotating Stars (Germany: Springer-Verlag) pp22-24

    [16]

    Kervella P, Domiciano de Souza A D, Bendjoya P 2008 Astron. Astrophys. 484 13

    [17]

    Zorec J, Domiciano de Souza A D, Frmat Y, Vakili F 2005 Semaine de l'Astrophysique Francaise Strasbourg, France, June 27-July 1, 2005 p363

    [18]

    Zhan Q, Song H F, Tai L T,Wang J T 2015 Acta Phys. Sin. 64 089701 (in Chinese) [詹琼, 宋汉峰, 邰丽婷, 王江涛 2015 物理学报 64 089701]

    [19]

    Zahn J P 2010 Astron. Astrophys. 517 A7

    [20]

    Kopal Z 1959 Close Binary Systems (1st Ed.) (New York: Wiley) p30

    [21]

    Song H F, Wang J Z, Li Y 2013 Acta Phys. Sin. 62 059701 (in Chinese) [宋汉峰, 王靖洲, 李云 2013 物理学报 62 059701]

    [22]

    Song H F, Zhong Z, Lu Y 2009 Astron. Astrophys. 504 161

    [23]

    Song H F, Lu Y, Wang J Z 2011 Publ. Astron. Soc. Jap. 63 835

    [24]

    Song H F, Maeder A, Meynet G, Huang R Q, Ekstrm S, Granada A 2013 Astron. Astrophys. 556 A100

    [25]

    Landin N R, Mendes L T S, Vaz P R 2009 Astron. Astrophys. 494 209

    [26]

    Zhou K, Yang Z Y, Zou D C, Yue R H 2012 Chin. Phys. B 21 020401

    [27]

    Maeder A 1999 Astron. Astrophys. 347 185

    [28]

    Espinosa Lara F, Rieutord M 2011 Astron. Astrophys. 533 A43

    [29]

    Claret A 2012 Astron. Astrophys. 538 A3

    [30]

    de Souza D A, Kervella P, Moser Faes D, Dalla Vedova G, Mrand A, Le Bouquin J B, Espinosa Lara F, Rieutord M, Bendjoya P, Carciofi A C, Hadjara M, Millour F, Vakili F 2014 Astron. Astrophys. 569 A10

    [31]

    Vink J S, de Koter A, Lamers H J G L M 2001 Astron. Astrophys. 369 574

    [32]

    Goss K J F, Karoff C, Chaplin W J, Elsworth Y, Stevens I R 2011 Mon. Not. Roy. Astron. Soc. 411 162

  • [1]

    Huang R Q, Yu K N 1998 Stellar Astrophysics (New York: Springer Verlag) p313

    [2]

    Paczynski B 1971 Annu. Rev. Astron. Astrophys. 9 183

    [3]

    Kippenhahn R, Thomas H C 1970 Proceedings of IAU Colloq. 4 Columbus, USA, September 8-11, 1969 p20

    [4]

    Endal A S, Sofia S 1976 Astrophys. J. 210 184

    [5]

    Pinsonneault M H, Kawaler S D, Sofia S, Demarqure P 1989 Astrophys. J. 338 424

    [6]

    Pinsonneault M H, Kawaler S D, Demarqure P 1990 Astrophys. J. Suppl. Ser. 74 501

    [7]

    Pinsonneault M H, Deliyannis C P, Demarqure P 1991 Astrophys. J. 367 239

    [8]

    Song H F, Zhang B, Zhang J, Wu H B, Peng Q H 2003 Chin. Phys. Lett. 20 2084

    [9]

    Wen D H, Zhou Y 2013 Chin. Phys. B 22 080401

    [10]

    Zhang J, Wang B, Zhang B, Han Z W 2012 Chin. Phys. Lett. 29 019701

    [11]

    von Zeipel H 1924 Mon. Not. Roy. Astron. Soc. 84 665

    [12]

    de Souza D A, Kervella P, Jankov S, Abe L, Vakili F, di Folco E, Paresce F 2003 Astron. Astrophys. 407 L47

    [13]

    Naz Y 2009 Astron. Astrophys. 506 1055

    [14]

    Jackson S, MacGregor K B, Skumanich A 2004 Astrophys. J. 606 1196

    [15]

    Maeder A, Stahler S 2009 Physics, Formation and Evolution of Rotating Stars (Germany: Springer-Verlag) pp22-24

    [16]

    Kervella P, Domiciano de Souza A D, Bendjoya P 2008 Astron. Astrophys. 484 13

    [17]

    Zorec J, Domiciano de Souza A D, Frmat Y, Vakili F 2005 Semaine de l'Astrophysique Francaise Strasbourg, France, June 27-July 1, 2005 p363

    [18]

    Zhan Q, Song H F, Tai L T,Wang J T 2015 Acta Phys. Sin. 64 089701 (in Chinese) [詹琼, 宋汉峰, 邰丽婷, 王江涛 2015 物理学报 64 089701]

    [19]

    Zahn J P 2010 Astron. Astrophys. 517 A7

    [20]

    Kopal Z 1959 Close Binary Systems (1st Ed.) (New York: Wiley) p30

    [21]

    Song H F, Wang J Z, Li Y 2013 Acta Phys. Sin. 62 059701 (in Chinese) [宋汉峰, 王靖洲, 李云 2013 物理学报 62 059701]

    [22]

    Song H F, Zhong Z, Lu Y 2009 Astron. Astrophys. 504 161

    [23]

    Song H F, Lu Y, Wang J Z 2011 Publ. Astron. Soc. Jap. 63 835

    [24]

    Song H F, Maeder A, Meynet G, Huang R Q, Ekstrm S, Granada A 2013 Astron. Astrophys. 556 A100

    [25]

    Landin N R, Mendes L T S, Vaz P R 2009 Astron. Astrophys. 494 209

    [26]

    Zhou K, Yang Z Y, Zou D C, Yue R H 2012 Chin. Phys. B 21 020401

    [27]

    Maeder A 1999 Astron. Astrophys. 347 185

    [28]

    Espinosa Lara F, Rieutord M 2011 Astron. Astrophys. 533 A43

    [29]

    Claret A 2012 Astron. Astrophys. 538 A3

    [30]

    de Souza D A, Kervella P, Moser Faes D, Dalla Vedova G, Mrand A, Le Bouquin J B, Espinosa Lara F, Rieutord M, Bendjoya P, Carciofi A C, Hadjara M, Millour F, Vakili F 2014 Astron. Astrophys. 569 A10

    [31]

    Vink J S, de Koter A, Lamers H J G L M 2001 Astron. Astrophys. 369 574

    [32]

    Goss K J F, Karoff C, Chaplin W J, Elsworth Y, Stevens I R 2011 Mon. Not. Roy. Astron. Soc. 411 162

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  • Received Date:  27 August 2015
  • Accepted Date:  06 December 2015
  • Published Online:  05 February 2016

Detail investigation of the inclined pressure structure and gravity darkening in critical rotating star Achernar

    Corresponding author: Song Han-Feng, sci.hfsong@gzu.edu.cn
  • 1. Department of Physics, College of Science, Guizhou University, Guiyang 550025, China;
  • 2. Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming 650011, China;
  • 3. Joint Research Centre for Astronomy, National Astronomical Observatory-Guizhou University, Guiyang 550025, China
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant No. 11463002), the Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Science (Grant No. OP201405), and the Graduate Innovation Fund in Guizhou University, China (Grant No. 2015055).

Abstract: Rotation and tide are two important factors that have very important impacts on the stellar structure and evolution. Based on the observational data of Achernar, we have derived the inclined pressure structure in a single rotating star or as a member in the binaries. We have given the distributions of the physical quantities on the isobaric surface and these distributions are derived from the Legendre series of expansions. We have also found the relationship between all levels of perturbation potential functions (including rotational and tidal distortions) and the distributions of density and pressure under the condition of inclined pressure structure. In particular, the gravitational darkening with the models including the effects of rotation and tide is investigated. We have found that the critical ratio of equatorial radius to the polar radius is consistent with the observations in rotating binaries better than that in single rotating model. The reason is that the tidal force can make the polar radius shortened because the tidal force exerts an inward force to the two polar points. However, the theoretical angular velocity in binaries is smaller than that observed. It is also shown that the positive shear enhances the centrifugal force and decreases the mean effective gravitational acceleration and effective temperatures whereas the negative shear plays a role to strengthen the effective gravitational acceleration. Moreover, the solid body rotation has not been supported inside Achernar because magnetic fields have not been detected through observations. Furthermore, the theoretical angular velocity in rigid rotation is higher than the angular velocity observed. Achernar has a periodic variation of light curves due to mass outburst, which also supports differential rotation. A positive shear indicates that the mass in accretion disks is falling to Achernar and the Achernar is spun up to critical rotation according to current observations. By comparing the theoretical results with observations, it can be seen that when the theoretical spin angular velocity of Achernar is 4.65 10-5 s-1 and the positive shears / s are 0.7851, the temperature of the polar points is 16041 K and that of equatorial sphere is 12073 K. Relative errors between the theoretical values and observations are less than 3% and are listed in the text. This model is the best and is the most possible one for Achernar.

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