﻿ 开普勒-47双星系统的宜居带 The Habitable Zone of Kepler-47 Binary System

Astronomy and Astrophysics
Vol.04 No.03(2016), Article ID:18017,12 pages
10.12677/AAS.2016.43007

The Habitable Zone of Kepler-47 Binary System

Juanxiu Hu1,2, Jianpo Guo1,2, Chaoqiong He1,2, Rongqin Cang1,2

1Department of Science and Technology, Puer University, Puer Yunnan

2Open Key Laboratory of Mechanics in Yunnan Province, Puer Yunnan

Received: Jun. 25th, 2016; accepted: Jul. 16th, 2016; published: Jul. 19th, 2016

ABSTRACT

Provided that there are lives living on an exoplanet, it must be located in the habitable zone of its host star. Considering the effects of stellar effective temperature on habitable zone, we give the distance equations on the inside and the outside boundary of binary’s habitable zone. Using the numerical methods, we obtain the habitable zone of the Kepler-47 binary system. The average distances on the inside and the outside boundary are 0.77761 and 1.55287 AU, respectively. The distance of the Kepler-47 inner planet is too small to be within the habitable zone. The orbit of the Kepler-47 outer planet may be an ellipse. Its eccentricity is 0.411; its aphelion is within the habitable zone, and its perihelion is not within the habitable zone. In an orbital period, there are 237.837 - 239.056 days that the Kepler-47 outer planet is within the habitable zone and there are 64.102 - 65.321 days that it is not within the habitable zone.

Keywords:Stellar Effective Temperature, Numerical Method, Habitable Zone, Kepler-47 Binary System, Planet

1普洱学院理工学院，云南 普洱

2云南省高校力学开放重点实验室，云南 普洱

1. 引言

2. 开普勒-47双星系统的轨道

2.1. 开普勒-47双星系统主星和次星的轨道

(1)

(2)

(3)

2.2. 开普勒-47双星系统两颗行星的轨道

(4)

3. 开普勒-47主星和次星周围的宜居带

Jones et al. (2006)给出了单星周围宜居带内、外边界的辐射流量 [32] ：

(5)

(6)

Table 1. The related data of the Kepler-47 binary system

Figure 1. The orbit diagram for the Kepler-47 secondary star, inner planet and outer planet. The orange, red, pink and blue denote the primary star, secondary star, inner planet and outer planet, respectively

(7)

(8)

0.23224 AU。图3显示的是当主星和次星都位于x轴上，并且次星在y轴侧时单独由次星产生的宜居带。以主星和次星的公共质心为坐标原点，此时次星的坐标是(0.06207 AU, 0)。则次星宜居带的内边界与x轴的两个相交点的坐标分别为(−0.05339 AU, 0)和(0.17753 AU, 0)；次星宜居带的外边界与x轴的两个交点的坐标分别为(−0.17017 AU, 0)和(0.29431 AU, 0)。

Figure 2. The habitable zone (green) of the Kepler-47 primary star

Figure 3. The habitable zone of the Kepler-47 secondary star

4. 开普勒-47双星系统周围的宜居带

4.1. 双星系统周围宜居带的内、外边界

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(10)

(11)

(12)

(13)

(14)

4.2. 开普勒-47双星系统周围宜居带的内、外边界与x轴的交点坐标

(15)

(16)

4.3. 开普勒-47双星系统周围的宜居带

(17)

(18)

(19)

(20)

4.2节已经求出双星宜居带外边界横坐标的取值范围是。在的范围，以0.001 AU的间隔选取不同的x值，根据(20)就可以解出相应的y值，把这些坐标点连起来就是双星宜居带的外边界。双星宜居带内边界和外边界之间的区域就是双星宜居带，如图4所示。

4.4. 开普勒-47外行星与双星宜居带的关系

(21)

(22)

Figure 4. The habitable zone of the Kepler-47 binary system

Figure 5. The diagram that the orbital perihelion of the Kepler-47 outer planet and the primary star are located at the same side

Figure 6. The diagram that the orbital perihelion of the Kepler-47 outer planet and the primary star are located at the two opposite sides

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5. 与前人工作对比

Orosz et al. (2012b)给出次星相对于主星公转的半长轴为0.0836 AU，内行星的轨道半长轴为0.2956 AU，外行星的轨道半长轴为0.989 AU [25] 。我们根据开普勒-47双星系统主星和次星的质量，次星的公转周期、两颗行星的公转周期；算出次星相对于主星公转的半长轴为0.08361 AU，内行星的轨道半长轴(相对于主星和次星的公共质心)为0.29559 AU，外行星的轨道半长轴(相对于主星和次星的公共质心)为0.98927AU。可见，我们求出的开普勒-47双星系统中的轨道半长轴与Orosz et al. (2012b)给出的数据几乎完全相等。Orosz et al. (2012b)指出开普勒-47内行星不在宜居带中，而外行星处在宜居带中 [25] 。我们的计算结果表明内行星轨道半长轴比双星宜居带的内边界要小很多，不在双星宜居带中。如果外行星的轨道是正圆，则外行星的整个轨道都处在宜居带中。我们求出的双星宜居带数据与Orosz et al. (2012b)的结论也是相符合的。

Haghighipour与Kaltenegger (2013)计算了开普勒-47双星系统的宜居带；分狭窄宜居带和经验宜居带两组数据。狭窄宜居带的内、外边界半径分别为0.904和1.569 AU；而经验宜居带的内、外边界半径分别为0.697和1.664 AU [20] 。在本文4.2节中，我们求出的开普勒-47双星系统宜居带的内边界与x轴的两个交点坐标为(−0.79726 AU, 0)和(0.75795 AU, 0)；宜居带外边界与x轴的两个交点坐标为(−1.57253AU, 0)和(1.53321 AU, 0)。可以求出双星宜居带内、外边界的平均半径分别为0.77761和1.55287 AU。可见，我们求出的双星宜居带正好介于Haghighipour与Kaltenegger (2013)求出的狭窄宜居带和经验宜居带之间。Haghighipour与Kaltenegger (2013)在考虑外行星椭率的情况下，对比了外行星轨道与宜居带的关系，外行星整体上处于宜居带中，可是在近日点附近并不在宜居带中 [20] 。本文4.4节的计算结果不但给出了相同的结论，还计算出外行星在一个公转周期内，有237.837至239.056天处在宜居带中，而有64.102至65.321天不在宜居带中。

6. 总结

The Habitable Zone of Kepler-47 Binary System[J]. 天文与天体物理, 2016, 04(03): 56-67. http://dx.doi.org/10.12677/AAS.2016.43007

1. 1. Marois C, Macintosh B, Barman T, et al. Sci., 2008, 322: 1348 http://dx.doi.org/10.1126/science.1166585

2. 2. Borucki W J, Koch D G, Batalha N, et al. ApJ, 2012, 745: 120 http://dx.doi.org/10.1088/0004-637X/745/2/120

3. 3. Borucki WJ, Agol E, Fressin F, et al. Sci., 2013, 340: 587 http://dx.doi.org/10.1126/science.1234702

4. 4. Pepe F, Cameron AC, Latham D W, et al. Natur., 2013, 503: 377 http://dx.doi.org/10.1038/nature12768

5. 5. Jenkins J M, Twicken J D, Batalha N M, et al. AJ, 2015, 150: 56

6. 6. Williams D M, Kasting J F. Icarus, 1997, 129: 254 http://dx.doi.org/10.1006/icar.1997.5759

7. 7. Menou K, Tabachnik S. ApJ, 2003, 583: 473 http://dx.doi.org/10.1086/345359

8. 8. Valencia D, O’Connell R J, Sasselov D. Icar., 2006, 181: 545 http://dx.doi.org/10.1016/j.icarus.2005.11.021

9. 9. Kasting J F, Whitmire D P, Reynolds R T. Icar., 1993, 101: 108 http://dx.doi.org/10.1006/icar.1993.1010

10. 10. Hart M H. Icar., 1978, 33: 23 http://dx.doi.org/10.1016/0019-1035(78)90021-0

11. 11. Franck S, von Bloh W, Bounama C, et al. JGR, 2000, 105: 1651

12. 12. Noble M, Musielak Z E, Cuntz M. ApJ, 2002, 572: 1024 http://dx.doi.org/10.1086/340430

13. 13. von Bloh W., Bounama C, Cuntz M, Franck S. A&A, 2007, 476: 1365

14. 14. Vladilo G, Murante G, Silva L, et al. ApJ, 2013, 767: 65 http://dx.doi.org/10.1088/0004-637X/767/1/65

15. 15. Kopparapu R K. ApJL, 2013, 767: L8 http://dx.doi.org/10.1088/2041-8205/767/1/L8

16. 16. Williams D M, Kasting J F, Wade R A. Natur., 1997, 385: 234 http://dx.doi.org/10.1038/385234a0

17. 17. Duquennoy A, Mayor M. A&A, 1991, 248: 485

18. 18. Richichi A, Leinert Ch, Jameson R, Zinnecker H. A&A, 1994, 287: 145

19. 19. Kaltenegger L, Haghighipour N. ApJ, 2013, 777: 165 http://dx.doi.org/10.1088/0004-637X/777/2/165

20. 20. Haghighipour N, Kaltenegger L. ApJ, 2013, 777: 166 http://dx.doi.org/10.1088/0004-637X/777/2/166

21. 21. Chauvin G, Beust H, Lagrange A M, Eggenberger A. A&A, 2011, 528: 8

22. 22. Doyle L R, Carter J A, Fabrycky D C, et al. Sci., 2011, 333: 1602 http://dx.doi.org/10.1126/science.1210923

23. 23. Welsh W F, Orosz J A, Carter, J A, et al. Natur., 2012, 481: 475 http://dx.doi.org/10.1038/nature10768

24. 24. Orosz J A, Welsh W F, Carter J A, et al. ApJ, 2012, 758: 87 http://dx.doi.org/10.1088/0004-637X/758/2/87

25. 25. Orosz J A, Welsh W F, Carter J A, et al. Sci., 2012, 337: 1511

26. 26. Schwamb M E, Orosz J A, Carter J A, et al. ApJ, 2013, 768: 127 http://dx.doi.org/10.1088/0004-637X/768/2/127

27. 27. Forget F, Pierrehumbert R T. Sci., 1997, 278: 1273 http://dx.doi.org/10.1126/science.278.5341.1273

28. 28. Williams D M, Kasting J F. Icar., 1997, 129: 254 http://dx.doi.org/10.1006/icar.1997.5759

29. 29. Mischna M A, Kasting J F, Pavlov A, Freedman R. Icar., 2000, 145: 546 http://dx.doi.org/10.1006/icar.2000.6380

30. 30. Jones B W. Life in the Solar System and beyond. London: Springer, 2004 http://dx.doi.org/10.1007/978-1-85233-897-8

31. 31. Clube S V M, Hoyle F, Napier W M, Wickramasinghe N C. Ap&SS, 1996, 245: 43 http://dx.doi.org/10.1007/BF00637802

32. 32. Jones B W, Sleep P N, Underwood D R. ApJ, 2006, 649: 1010 http://dx.doi.org/10.1086/506557

33. 33. Guo J P, Zhang F H, Chen X F, Han Z W. Ap&SS, 2009, 323: 367 http://dx.doi.org/10.1007/s10509-009-0081-z