系外行星系統(tǒng)中潮汐和共振的動力學(xué)研究

陳媛媛

(南京大學(xué)天文與空間科學(xué)學(xué)院南京210093)

系外行星系統(tǒng)中潮汐和共振的動力學(xué)研究

陳媛媛?

(南京大學(xué)天文與空間科學(xué)學(xué)院南京210093)

近年來,行星形成理論與行星系統(tǒng)動力學(xué)已經(jīng)成為天文學(xué)研究的一個重要領(lǐng)域.隨著系外行星探測的不斷深入,各種與太陽系相比特征迥異的系外行星系統(tǒng)被發(fā)現(xiàn).大批離恒星極近的行星被發(fā)現(xiàn),它們的周期只有幾天,從而會受到強(qiáng)烈的潮汐耗散作用.很多多行星系統(tǒng)中相鄰行星的周期比都接近簡單整數(shù)比,這預(yù)示著它們很可能處在平運(yùn)動共振.行星的軌道面與恒星的赤道面夾角的范圍也從太陽系內(nèi)的行星的?7°擴(kuò)展到0°～180°的整個有效范圍,出現(xiàn)了不少逆行的熱木星.這些新現(xiàn)象在挑戰(zhàn)傳統(tǒng)的行星形成理論與行星系統(tǒng)動力學(xué)的同時,也為其進(jìn)一步的完善和發(fā)展提供了前所未有的機(jī)遇.本文將基于最新的觀測數(shù)據(jù)和統(tǒng)計特征,從行星系統(tǒng)動力學(xué)角度出發(fā),將潮汐作用與諸共振相結(jié)合,研究行星演化過程中的不同構(gòu)型.

本文首先回顧了與潮汐力和共振相關(guān)的系外行星方面的主要應(yīng)用和最新進(jìn)展.然后分別給出了最經(jīng)典的和當(dāng)前最常用的潮汐模型的推導(dǎo)和各根數(shù)的平均變化率,近距離接觸了平衡潮模型的簡化假設(shè)和建模過程.之后從動力學(xué)角度出發(fā),利用數(shù)值模擬和理論分析相結(jié)合的方法,研究了以下3個問題:行星的自轉(zhuǎn)-軌道共振對其軌道偏心率的影響;潮汐作用下近2:1平運(yùn)動共振和Laplace共振的演化特點(diǎn);外氣體盤引力對空洞內(nèi)行星軌道激發(fā)的促進(jìn)作用.同時考慮潮汐耗散和行星形變產(chǎn)生的引力,本文第3章得出結(jié)論,處于非同步自轉(zhuǎn)-軌道共振比處于半平衡狀態(tài)下的行星軌道耗散速率更大,從而偏心率也被圓化得更快.為解釋HD40307系統(tǒng)中三行星近2:1的兩個周期比的形成,本文第4章分不同情況模擬它們的演化路徑.如果行星在氣體盤消散后的演化很穩(wěn)定,由行星間相互作用產(chǎn)生的偏心率很小(～10?4),導(dǎo)致周期比的變化時標(biāo)遠(yuǎn)大于系統(tǒng)的年齡.而如果行星經(jīng)歷過不穩(wěn)定階段,在期間產(chǎn)生的自由偏心率便可以有效地加速周期比的演化.在這種情況下,存在3條路徑可以達(dá)到當(dāng)前構(gòu)型,3條路徑的半長徑初值分別對應(yīng)周期比平面上的3個不同區(qū)域.由此可推斷,氣體盤耗散后的不穩(wěn)定階段是系統(tǒng)在潮汐作用下從2:1共振演化到當(dāng)前構(gòu)型的必要條件.

本文第5章針對最新觀測到的逆行熱木星,提出一種可以減小軌道激發(fā)的臨界傾角的機(jī)制.考慮外氣體盤的引力,空洞內(nèi)的行星在合適位置上會發(fā)生長期共振,長期共振激發(fā)的軌道傾角又有可能引發(fā)行星之間的Kozai共振,從而激發(fā)內(nèi)行星的偏心率和傾角.我們發(fā)展了長期攝動下三體問題的根數(shù)變化率方程(從相對于不變平面的形式擴(kuò)展到相對于任意平面的形式),并給出了二維盤引力下各根數(shù)的變化率,把這兩部分線性疊加而得到的演化方程可以很好地近似N體模擬的結(jié)果.利用演化方程對參數(shù)空間的掃描,我們初步給出了可以形成逆行熱木星的臨界條件,并較完整地討論了各個相關(guān)參數(shù)的影響.

In recent years,the planet formation theory and planetary system dynamics have become an important area of astronomy.With more details of exoplanets being found,many characteristics quite di ff erent from the solar system have been found in the exoplanetary systems.A large number of planets are found to be very close to their host star,and their periods are only a few days,which brings strong tidal dissipation with the star.Many period ratios of adjacent planets in multi-planetary systems are close to the simple integer ratios, which indicates that the planets are likely in the mean motion resonances(MMRs).The range of the angles between the orbital plane of the planets and the equatorial plane of their hosts expands from ?7°for the planets in the solar system to 0°～180°,and some retrograde hot Jupiters exist.These new phenomena are testing out the traditional planetary formation theory and planetary system dynamics,but also provide an unprecedented opportunity for their further improvement and development.Based on these latest observational data and statistical features,the thesis investigates some special con fi gurations combining the resonances and tidal dissipation by the way of planetary system dynamics.

The thesis fi rst reviews the primary applications and the latest progress in the tide as well as various resonances of exoplanets.Then it gives some tidal model derivations, including the classic one and most popular one,in order to understand the assumptions of the equilibrium tide.Meanwhile,the average rates of change of orbital elements under tidal dissipation are exhibited.By both numerical simulation and theoretical analysis,the following three questions are investigated:the evolution of the eccentricity of planets in the non-synchronous spin-orbit resonances,the characteristics of nearly 2:1 MMR and Laplace resonance under tidal dissipation,and the promoting role of the gravity of outer gas disk for exciting the planets in its inner cavity.Chapter 3 takes into account the tidal dissipation and the gravity from planet deformation,and concludes that,the tidal dissipation rates in all the non-synchronous spin-orbit resonances are greater than that in the quasi-stationary state of the spin of the planet,so the eccentricity is also damped within a shorter duration.In order to explain the formation of two period ratios nearly 2:1 in the three planet HD40307 system, Chapter 4 simulates the evolution paths of planets in two di ff erent situations.If the planets are always stable during and after the dissipation of gas disk,their eccentricities directly from the interaction among planets are very small(～10?4).So the changing timescale of period ratios is much larger than the age of the system.On the contrary,if the planets have experienced unstable phases,their eccentricities would be excited,which can accelerate the evolution of the period ratios e ff ectively.In this situation,three paths exist to achieve the current con fi guration,whose initial semi-major axes respectively correspond to three di ff erent regions on the plane of two period ratios.It can be inferred that the instability stage after the dissipation of the gas disk is a necessary condition for the system to achieve the evolution from 2:1 MMRs to the current con fi guration under tidal dissipation with thestar.

Chapter 5 proposes a mechanism to reduce the critical inclination of orbital pumping,in order to explain the retrograde hot Jupiters in latest observations.Considering the gravity of the outer gas disk,a secular resonance would occur between the planets in the inner cavity if they are in appropriate positions,which pumps the mutual inclination of the planets and induces the Kozai resonance between them in some situations.Then the eccentricity and inclination of the inner planet will be excited eventually.We develop the equation of the rates of change of orbital elements under the secular perturbation in hierarchical threebody system(with respect to an arbitrary plane rather than the invariant plane of the two orbits),as well as give the rates of change of orbital elements under a 2-D disk gravity equations.Combining the two parts,the results by integrating the evolution equations are good approximation of the N-body simulations.By scanning the parameter spaces using the evolution equations,we get the preliminary critical condition for the retrograde hot Jupiters formed,and give a complete discussion of the impact of relevant parameters.

The Dynamics of Tide and Resonances in Exoplanetary Systems

CHEN Yuan-yuan
(School of Astronomy and Space Science,Nanjing University,Nanjing 210093)

10.15940/j.cnki.0001-5245.2015.03.011

?2013-06-13獲得博士學(xué)位,導(dǎo)師:南京大學(xué)天文與空間科學(xué)學(xué)院周濟(jì)林教授;chenyy@pmo.ac.cn