2024年IOAA理论第8题-双星硬化
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英文题目
T8. Binary Hardening (25 points)
Consider a binary system of black holes, both of equal mass \(M\), separated by a distance \(a\), and revolving around their common centre of mass (CM) in circular orbits. This binary system moves against, and interacts with, a very large, uniform field of stars (each of mass \(m \ll M\)) with number density \(n\). Consider a star that approaches the system from infinity with speed \(v\) and impact parameter \(b\), in the reference frame of the CM (as shown in the figure below). Its closest approach distance to the CM is \(r_p \approx \frac{1}{2} a\). For tasks (a) and (c), you should make use of the fact that \(v^2 \ll \frac{GM}{a}\).
(a) (5 points) Obtain an expression for \(b\), in terms of \(M\), \(a\), \(v\), and physical constants. In this task, assume that the star interacts with the binary as if its total mass was fixed at the CM.
(b) (6 points) The star approaches the component with an initial speed negligible compared to the component's orbital speed, and both are moving directly towards each other. After interacting with the system, when the star is again far away from the black hole, we find that the direction of its velocity vector is reversed and the final speed is \(v_f\). Determine \(v_f\), in terms of \(M\), \(a\), and physical constants. Assume that linear momentum and mechanical energy are conserved during this interaction and that it takes place in a timescale much smaller than the binary's period. Recall that \(m \ll M\).
(c) (14 points) Upon each encounter, part of the total energy of the binary is transferred to the kinetic energy of the star. Assume that the binary orbit remains circular. Knowing this, using your results from previous tasks, and taking into account only encounters with the stars within the specified range of impact parameters, show that the reciprocal of the binary's separation increases at a constant rate:
$$ \frac{d}{dt} \left( \frac{1}{a} \right) = H \frac{G \rho}{v_0} $$
Here, \(\rho = nm\) is the mass density of the star field, and \(G\) is the universal gravitational constant. Find the dimensionless constant \(H\), which refers to hardening.
中文翻译
T8. 双星硬化（25分）
考虑一个由两个质量均为\(M\)的黑洞组成的双星系统，间距为\(a\)，绕其共同质心（CM）做圆周运动。该系统在大量均匀分布的恒星场（每颗恒星质量\(m \ll M\)，数密度\(n\)）中运动并发生相互作用。假设一颗恒星从无穷远处以速度\(v\)和撞击参数\(b\)接近系统（参考系为CM），其最接近CM的距离为\(r_p \approx \frac{1}{2}a\)。在问题(a)和(c)中，需利用\(v^2 \ll \frac{GM}{a}\)的条件。
(a) (5分) 推导撞击参数\(b\)的表达式，用\(M\)、\(a\)、\(v\)和物理常数表示。本题中假设恒星与双星系统的相互作用等效于其总质量固定在CM处。
(b) (6分) 恒星以初速度（相对于黑洞轨道速度可忽略）直接朝向其中一个成员运动。相互作用后，当恒星远离黑洞时，其速度方向反转且最终速度为\(v_f\)。确定\(v_f\)（用\(M\)、\(a\)和物理常数表示）。假设相互作用过程中线动量与机械能守恒，且作用时间远小于双星轨道周期。
(c) (14分) 每次相遇时，双星系统的部分能量转化为恒星的动能。假设双星轨道保持圆形，利用先前结果并仅考虑指定撞击参数范围内的恒星相遇，证明双星间距倒数的变化率为常数：
$$ \frac{d}{dt} \left( \frac{1}{a} \right) = H \frac{G \rho}{v_0} $$
其中\(\rho = nm\)为恒星场质量密度，求无量纲常数\(H\)（硬化常数）。
官方解答
解答：
(a) 通过机械能守恒和角动量守恒：
$$ b = \frac{\sqrt{2GMa}}{v} $$
推导过程： 最接近点速度\(v_p = \frac{b}{r_p}v\)，结合能量守恒方程：
$$ -\frac{GM(2M)m}{a/2} + \frac{1}{2}m\left(\frac{b}{a/2}v\right)^2 = \frac{1}{2}mv^2 $$
忽略\(v^2\)项后解得：
$$ \boxed{b \approx \frac{\sqrt{2GMa}}{v} $$
(b) 考虑黑洞轨道速度\(V = \sqrt{\frac{GM}{2a}}\)，通过动量与能量守恒：
$$ v_f = 2V = \sqrt{\frac{2GM}{a}} $$
推导细节： 在黑洞参考系中，恒星速度反向且大小不变，转换回CM系得：
$$ \boxed{v_f = \sqrt{\frac{2GM}{a}} $$
(c) 计算能量转移率与轨道收缩率：
$$ H = 8\pi $$
关键步骤： 1. 单次相遇能量损失：\(\Delta E = \frac{GMm}{a}\)
2. 相遇率计算：\(d\phi = 2\pi n b v db\)
3. 能量变化方程：\(\frac{dE}{dt} = -\frac{4\pi G^2 M^2 nm}{v_0}\)
4. 关联轨道能量\(E = -\frac{GM^2}{2a}\)的时间导数，最终得到：
$$ \boxed{H = 8\pi} $$
