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The Effect of Mass Loss on the Tidal Evolution of Extrasolar Planet THE EFFECT OF MASS LOSS ON THE TIDAL EVOLUTION OF EXTRASOLAR PLANET The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Guo, J. H. “THE EFFECT OF MASS LOSS ON THE TIDAL EVOLUTION OF EXTRASOLAR PLANET.” The Astrophysical Journal 712, no. 2 (March 10, 2010): 1107–1115. © 2010 American Astronomical Society. As Published http://dx.doi.org/10.1088/0004-637x/712/2/1107 Publisher Institute of Physics/American Astronomical Society Version Final published version Citable link http://hdl.handle.net/1721.1/95901 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The Astrophysical Journal, 712:1107–1115, 2010 April 1 doi:10.1088/0004-637X/712/2/1107 C 2010. The American Astronomical Society. All rights reserved. Printed in the U.S.A. THE EFFECT OF MASS LOSS ON THE TIDAL EVOLUTION OF EXTRASOLAR PLANET J. H. Guo1,2 National Astronomical Observatories/Yunnan Observatory, Chinese Academy of Sciences, P.O. Box 110, Kunming 650011, China Received 2009 October 15; accepted 2010 February 9; published 2010 March 10 ABSTRACT By combining mass loss and tidal evolution of close-in planets, we present a qualitative study on their tidal migrations. We incorporate mass loss in tidal evolution for planets with different masses and find that mass loss could interfere with tidal evolution. In an upper limit case (β = 3), a significant portion of mass may be evaporated in a long evolution timescale. Evidence of greater modification of the planets with an initial separation of about 0.1 AU than those with a = 0.15 AU can be found in this model. With the assumption of a large initial eccentricity, the planets with initial mass 1 MJ and initial distance of about 0.1 AU could not survive. With the supposition of β = 1.1, we find that the loss process has an effect on the planets with low mass at a ∼ 0.05 AU. In both cases, the effect of evaporation on massive planets can be neglected. Also, heating efficiency and initial eccentricity have significant influence on tidal evolution. We find that even low heating efficiency and initial eccentricity have a significant effect on tidal evolution. Our analysis shows that evaporation on planets with different initial masses can accelerate (decelerate) the tidal evolution due to the increase (decrease) in tide of the planet (star). Consequently, the effect of evaporation cannot be neglected in evolutionary calculations of close-in planets. The physical parameters of HD 209458b can be fitted by our model. Key words: planetary systems – planets and satellites: general – stars: individual (HD 209458) 1. INTRODUCTION could lead to a correlation between the mass period and surface gravity period found by Mazeh et al. (2005) and Southworth There are considerable observations for exoplanets in the past et al. (2007). The evolutionary history of the star would be several years and over 400 planets have been discovered outside considered in order to estimate a more exact mass loss rate of the solar system. From the side of observation it is easier because a higher mass loss rate is possible when the young star to detect massive exoplanets. Jupiter-like exoplanets orbiting emits a stronger X-ray and ultraviolet (XUV) flux (Lammer et al. their host stars at a small separation of less than 0.1 AU have 2003; Penz et al. 2008). Baraffe et al. (2004, 2005) included the been detected (e.g., Table 1 of Pont 2009). To understand their influence of mass loss on evolutionary calculations, and mass −8 −1 −12 −1 interior structures and compositions, the precise masses, radii, loss rates varying from 10 MJ yr to 10 MJ yr are and orbital parameters are required. An advantage of transit in obtained from their calculations. All the above studies showed finding planets is that the masses and radii of exoplanets can be that a significant portion of mass could be evaporated in a long determined. Thus, theoretical models can be used to constrain evolution timescale. the interior properties of transiting exoplanets (Bodenheimer In order to better understand the fundamental properties of et al. 2001; Guillot & Showman 2002). The star–planet system planets, we focus on the combination between the mass loss and is often considered similarly to the binary system in which tidal the tidal action. The main aim of this paper is to upgrade the tidal interaction can result in exchange between the spin angular model with mass loss. The mass loss is coupled to the orbital momentum and orbit angular momentum so that their spin and evolution in this model; therefore, it can fit the semimajor axis, orbit periods are synchronized (Zahn 1977;Hut1980; Eggleton eccentricity, radius, and mass simultaneously. We present our & Kiseleva 1998). The tidal effects on planets have been thought models in Section 2. The results and applications are given in to explain the low e value by Rasio et al. (1996). Current Section 3. In Section 4, we discuss and summarize our results. researches show that close-in planets may be the consequence of tidal interaction, and the tidal action will fall off very rapidly 2. THE MODEL with the separation (Jackson et al 2008, 2009). Moreover, the In this paper, we want to test the potential role of mass loss in tidal heating has been calculated for the evolution in several the tidal evolution of a planet. The equations of tidal evolution studies (Gu et al. 2004; Miller et al. 2009; Ibgui & Burrows on the semimajor axis and eccentricity are (Jackson et al. 2009) 2009). These authors found that the tidal heating is an important √ energy source of evolution by coupling the tidal effect to thermal 63 GM3R5 e2 5 1 da =− s p 9 G/Ms Rs Mp evolution, but a simultaneous fit for all parameters (radius, + a dt 2Q Mp 2Q semimajor axis, eccentricity, and age) seems to be difficult. p s Vidal-Madjar et al. (2003) found evidence for atmospheric 57 × 2 −13/2 evaporation in transiting planet HD 209458b, and the estimated 1+ e a (1) − 4 lower limit of the mass loss rate is 1010 gs 1. It means that 26 9 √ the planet could lose 10 gin10 yr at least. The evaporation 1 de 63 GM3R5 e2 225 G/M R5M =− s p s s p −13/2 + a . e dt 4QpMp 16Qs 1 Also at Department of Earth, Atmospheric, and Planetary Science, Massachusetts Institute of Technology, Cambridge, MA 01742, USA. (2) 2 Also at Key Laboratory for the Structure and Evolution of Celestial Objects, The first term in Equation (1) represents the tidal effect of the Chinese Academy of Sciences, Kunming 650011, China. planet while the tidal effect of the star is expressed in the second 1107 1108 GUO Vol. 712 1 = − 3 1 term. It is worth noting that the first term is proportional to K(ξ) 1 + 2 < 1 is a nonlinear potential energy reduction Mp 2ξ 2ξ 4πρ while the second term is proportional to Mp. This implies that the factor due to the stellar tidal forces, and ξ = d( )1/3 is the 9M∗ mass loss in planet leads to an increase in the planetary tide, but Roche lobe boundary distance, where M∗ is the mass of star leads to a decrease in the stellar tide. The final effect is a subtle and d is the orbital distance. Erkaev et al. (2007) assumed a balance between the planetary tide and stellar tide. The model circular orbit in calculating the Roche lobe enhancement factor. assumes that the planetary orbital period is shorter than the star’s To account for the effect of a non-circular orbit, we replace d rotation period. Equations (1) and (2) also show that eccentricity with the average orbital distance a in Equation (5). and semimajor axis decrease with time. In fact, Dobbs-Dixon The variation of radius must be considered in this model et al. (2004) indicated that the variation of the semimajor axis because the mass loss rate is tightly related to the mean density depends on the ratio of the orbital angular velocity of the planet shown in Equation (5). The radii of planets vary due to the to the spin angular velocity of the parent star. The semimajor axis effects of cooling, irradiation, and tidal heating. However, the could increase if the orbital period of the planet is longer than irradiation of host star does not have an evident impact on the rotation period of the parent star, or vice versa. This hints that planets at a large separation. Cooling dominates the variation the spin angular momentum of the star can be transferred to the of planetary radius, while the irradiation of the parent star can orbit of the planet if the rotation period of the star is shorter than slightly change the structure of close-in planets (Fortney et al. the orbital period of the planet. A good example for the exchange 2007). In general, for a given mass R∞ decreases with the of angular momentum is Bootis which is almost synchronized increase of separation if tidal heating is neglected (Fortney et al. with the orbital motion of its planet (Donati et al. 2008). As 2007). However, a Jupiter-like planet only varies ∼0.1 RJ from a star is younger, the rotation period could be short. However, 0.02 to 0.06 AU. The variation in radius of the planet at the stars lose their spin angular momentum through stellar winds distance larger than 0.06 AU is negligible (Burrows et al.
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