磁化介质中相对论性瑞利-泰勒不稳定性线性理论的色散关系再探讨
Dispersion relation for the linear theory of relativistic Rayleigh Taylor instability in magnetized medium revisited
AI总结：
本文重新推导了磁化介质中相对论性瑞利-泰勒不稳定性的线性色散关系，通过选取对称的中间参考系揭示其激发判据不含相对论修正，并阐明相对论效应主要导致不稳定性增长显著变慢，为解释喷流稳定性提供理论依据。
摘要：
瑞利-泰勒不稳定性（RTI）出现在两种不同密度流体的界面处，尤其是当较重流体在有效引力场中位于较轻流体之上时。在具有高速度的天体物理系统中，相对论修正是必要的。我们研究了磁化介质中相对论性瑞利-泰勒不稳定性（R-RTI）的线性理论，其中流体可以以相对论速度平行于界面运动。我们选择了一个“中间参考系”，在该参考系中，界面两侧的流体以相同洛伦兹因子 gamma_* 沿相反方向运动。这种对称性有助于解析推导以及对不稳定性动力学的相对论效应研究。我们推导出了 R-RTI 的正确版本。我们发现，当阿特伍德数 A = (rho1 h1 - rho2 h2) / (rho1 h1 + rho2 h2) > 0 时，不稳定性被激发，其中 rho1 和 rho2 是在流体静止参考系中测量的密度，并且该判据不包含相对论修正。相对论效应主要体现在引力加速度的洛伦兹变换 g' = g / (gamma_*)^2 中，它与时间膨胀相结合，导致不稳定性的增长显著变慢（omega' = omega0 / gamma_*），并且由于中间参考系中较弱的 g，不稳定区域的长度略微拉长。考虑到时间膨胀，当在介质静止参考系中观察时，我们预期不稳定性的增长速率将大幅降低。这些解析结果应能指导对微类星体（muQSOs）、活动星系核（AGNs）、伽马射线暴（GRBs）和射电脉冲星（PSRs）等系统中不稳定性的进一步探索，在这些系统中，喷流的表观稳定性可归因于其固有稳定性（例如阿特伍德数）或 R-RTI 得以增长的持续时间被大幅延长。
英文摘要：
The Rayleigh-Taylor instability (RTI) arises at the interface between two fluids of different densities, notably when a heavier fluid lies above a lighter one in an effective gravitational field. In astrophysical systems with high velocities, relativistic corrections are necessary. We investigate the linear theory of relativistic Rayleigh-Taylor instability (R-RTI) in a magnetized medium, where fluids can move parallel to the interface at relativistic velocities. We chose an "intermediate frame" where fluids on each side of the interface move in opposite directions with identical Lorentz factors gamma_*. This symmetry facilitates analytical derivations and the study of relativistic effects on the instability's dynamics. We derive the correct version of the R-RTI. We find that the instability is activated when the Atwood number A = (rho1 h1 - rho2 h2) / (rho1 h1 + rho2 h2) > 0, where rho1 and rho2 are densities measured in the rest frame of the fluids, and that this criterion does not contain relativistic corrections. The relativistic effect is mostly contained in the Lorentz transformation of the gravitational acceleration g' = g / (gamma_*)^2, which, combined with time dilation, leads to a much slower growth of instability (omega' = omega0 / gamma_*), and a slightly elongated length of the unstable patch, due to weaker g in the intermediate frame. Taking time dilation into account, when viewed in the rest frame of the medium, we expect the instability to grow at a much reduced rate. The analytical results should guide further explorations of instability in systems such as microquasars (muQSOs), active galactic nuclei (AGNs), gamma-ray bursts (GRBs), and radio pulsars (PSRs), where the apparent stability of the jet can be attributed to either the intrinsic stability (e.g. the Atwood number) or the much prolonged duration through which R-RTI can grow.
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