[论文解读] The equilibrium tide in stars and giant planets

F. Remus, S. Mathis+1 more|arXiv (Cornell University)|May 15, 2012

Stellar, planetary, and galactic studies参考文献 30被引用 46

一句话总结

本文通过在每个潮汐势的傅里叶分量参考系中建立旋转参考系中的潮汐动力学，重新定义了流体恒星和类木行星中的平衡潮。该方法消除了先前惯性参考系模型中存在的虚假‘伪共振’现象。基于湍流粘性建立了一个物理解释的潮汐耗散模型，表明品质因数Q依赖于潮汐频率和对流周转时间，从而揭示两种不同的耗散区：低频时为恒定时间延迟，高频时为恒定相位滞后角，对轨道圆化和自转同步的时间尺度具有重要意义。

ABSTRACT

Context. Since 1995, more than 500 extrasolar planets have been discovered orbiting very close to their parent star, where they experience strong tidal interactions. Their orbital evolution depends on the physical mechanisms that cause tidal dissipation, which remain poorly understood. Aims. We refine the theory of the equilibrium tide in fluid bodies that are partly or entirely convective, to predict the dynamical evolution of the systems. In particular, we examine the validity of modeling the tidal dissipation using the quality factor Q, which is commonly done. We consider here the simplest case where the considered star or planet rotates uniformly, all spins are aligned, and the companion is reduced to a point mass. Methods. We expand the tidal potential as a Fourier series, and express the hydrodynamical equations in the reference frame, which rotates with the corresponding Fourier component. The results are cast in the form of a complex disturbing function, which may be implemented directly in the equations governing the dynamical evolution of the system. Results. The first manifestation of the tide is to distort the shape of the star or planet adiabatically along the line of centers. This generates the divergence-free velocity field of the adiabatic equilibrium tide, which is stationary in the frame rotating with the considered Fourier component of the tidal potential; this large-scale velocity field is decoupled from the dynamical tide. The tidal kinetic energy is dissipated into heat by means of turbulent friction, which is modeled here as an eddy-viscosity acting on the adiabatic tidal flow. This dissipation induces a second velocity field, the dissipative equilibrium tide, which is in quadrature with the exciting potential; this field is responsible for the imaginary part of the disturbing function, which is implemented in the dynamical evolution equations, from which one derives the characteristic evolutionary times. Conclusions. The rate at which the system evolves depends on the physical properties of the tidal dissipation, and specifically on both how the eddy viscosity varies with tidal frequency and the thickness of the convective envelope for the fluid equilibrium tide. At low frequency, this tide is retarded by a constant time delay, whereas it lags behind by a constant angle when the tidal frequency exceeds the convective turnover rate.

研究动机与目标

• 在旋转参考系中重新表达平衡潮，以正确将其与动力潮分离，并消除早期惯性参考系模型中存在的虚假共振。
• 将经验性的品质因数Q与对流包层中的物理湍流耗散过程相协调，超越行星科学中常用的恒定Q近似。
• 建立一个考虑潮汐耗散对潮汐频率和对流周转 timescale 依赖性的形式化方法，实现对轨道和自转演化的精确预测。
• 提供一个计算密近双星系统和系外行星系统中轨道圆化、自转同步和自转轴对齐等特征 timescale 的框架。
• 为未来工作奠定基础，以将模型扩展至非共面轨道、倾斜自转、差速旋转以及弹性的核心。

提出的方法

• 将潮汐势展开为傅里叶级数，并在每个分量的旋转参考系中分析其谐波分量。
• 在旋转参考系中表述流体动力学方程，将绝热平衡潮识别为在该参考系中静止的无散度、无旋速度场。
• 通过作用于绝热潮汐流的涡流粘性来建模湍流耗散，生成与潮汐势正交的耗散性平衡潮。
• 推导复数扰动函数，其实部和虚部分别对应于绝热和耗散分量。
• 将潮汐耗散率表达为k²/Q，其中Q依赖于潮汐频率σₗ和对流周转时间，采用物理解释的粘性模型。
• 应用拉格朗日运动方程，推导轨道元素（半长轴、偏心率、平近点角）和角动量的长期演化方程，纳入频率依赖的耗散效应。

实验结果

研究问题

• RQ1 为何对每个潮汐势傅里叶分量采用旋转参考系能更准确地分离平衡潮与动力潮，并消除早期惯性参考系模型中的虚假共振？
• RQ2 在对流流体天体中，潮汐品质因数Q的频率依赖性的物理解释是什么？它与湍流粘性有何关联？
• RQ3 当潮汐周期短于局部对流周转时间时，潮汐耗散区会发生怎样的变化？
• RQ4 这种频率依赖的耗散对轨道圆化和自转同步的特征 timescale 有何影响？
• RQ5 该新形式化方法与经典恒定Q或恒定相位滞后近似相比，在预测密近恒星与行星系统动力演化方面有何差异？

主要发现

• 平衡潮在旋转参考系中被正确描述为无散度、无旋速度场，证实了其物理一致性，并解决了此前对其无旋性质的疑虑。
• 通过在每个潮汐谐波的旋转参考系中分析，该方法成功滤除了动力潮，并消除了早期惯性参考系模型中困扰研究者的‘伪共振’现象。
• 出现两种不同的潮汐耗散区：在低频时（σₗ ≪ ω_conv），系统表现出恒定时间延迟；在高频时（σₗ ≫ ω_conv），表现出恒定相位滞后角。
• 当对流周转时间短于潮汐周期时，逆品质因数k²/Q与潮汐频率σₗ成正比；而当潮汐周期短于周转时间时，k²/Q与频率无关。
• 推导出的轨道和自转元素演化方程包含了频率依赖的耗散效应，使得对圆化和同步化 timescale 的预测比恒定Q模型更为精确。
• 对潮汐耗散率的定量估计表明，对流包层的厚度和潮汐频率σₗ是关键参数，且在高频时湍流粘性按νₜ ∝ σₗ⁻¹规律变化。