2024年IOAA理论第9题-吸积物理
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英文题目
T9. Physics of Accretion (35 points)
The accretion of matter onto compact objects, such as neutron stars and black holes, is one of the most efficient ways to produce radiant energy in astrophysical systems. Consider an element of gas of mass $$\Delta m$$ in a stationary and geometrically thin disc of matter with a maximum radius of $$R_{max}$$ and minimum stable orbital radius of $$R_{min}$$ (with $$R_{min}/R_{max} \ll 1$$), in rotation around a compact object of mass $$M$$ and radius $$R$$.
(a) (6 points) Assuming that an element of gas in the disc follows an approximately Keplerian circular orbit, find an expression for the total mechanical energy per unit mass $$\frac{\Delta E}{\Delta m}$$ released by this gas from when it starts orbiting at a radius $$R_{max}$$ until it reaches an orbital radius of $$r \ll R_{max}$$. This process occurs very slowly, transforming kinetic energy into internal energy of the gas disc through viscous dissipation. Note: Ignore the gravitational interaction between particles within the accretion disc and give your final answer in terms of $$G$$, $$M$$, and $$r$$.
(b) (5 points) Considering that the disc receives mass at an average rate of $$\dot{M}$$, and assuming that all the mechanical energy lost is ultimately converted into radiation, find an expression for the total luminosity of the disc.
(c) (8 points) Consider now the ring composed of all mass elements with radii between $$r$$ and $$r + \Delta r$$. In this scenario, find an expression for the luminosity generated by the disc over its small width $$\Delta r$$ at this radius; that is, find the expression for $$\frac{\Delta E}{\Delta t \Delta r}$$.
(d) (10 points) Assuming that the gravitational energy released in this ring is locally emitted by the surface of the ring in the form of black-body radiation, find an expression for the surface temperature $$T$$ of the ring.
(e) (3 points) Consider that the central object is a stellar black hole with a mass of $$3M_{\odot}$$ and a rate of accretion of $$\dot{M} = 10^{-9} M_{\odot} / \text{year}$$. Consider also that $$R_{min} = 3R_{sch}$$, where $$R_{sch}$$ is the Schwarzschild radius of the black hole. Determine the luminosity of the disc and the peak wavelength of emission of its innermost part. Ignore gravitational redshift effects and assume that the emission from the innermost part of the ring dominates the total emission.
(f) (3 points) Now, considering another accretion system with $$\dot{M} = 1 \, M_{\odot}/$$ year and a peak emission wavelength of $$\lambda = 6 \times 10^{-8}$$ m, estimate the mass of this black hole.
中文翻译
T9. 吸积物理学（35分）
物质向致密天体（如中子星和黑洞）的吸积是天体物理系统中产生辐射能最有效的方式之一。考虑一个质量为$$\Delta m$$的气体元素，其位于一个静止且几何学上薄的物质盘中，该盘的最大半径为$$R_{max}$$，最小稳定轨道半径为$$R_{min}$$（满足$$R_{min}/R_{max} \ll 1$$），围绕一个质量为$$M$$、半径为$$R$$的致密天体旋转。
(a)（6分）假设盘中的气体元素遵循近似开普勒圆形轨道，求该气体从初始轨道半径$$R_{max}$$到达到轨道半径$$r \ll R_{max}$$的过程中，单位质量释放的总机械能$$\frac{\Delta E}{\Delta m}$$的表达式。此过程进行得非常缓慢，动能通过粘滞耗散转化为气体盘的内部能量。注意：忽略吸积盘内粒子间的引力相互作用，最终答案用$$G$$、$$M$$和$$r$$表示。
(b)（5分）假设盘以平均速率$$\dot{M}$$接收质量，且所有损失的机械能最终转化为辐射，求盘的总光度表达式。
(c)（8分）考虑由半径介于$$r$$和$$r + \Delta r$$之间的所有质量元素组成的环。在此情况下，求该半径处盘在微小宽度$$\Delta r$$上产生的光度表达式，即求$$\frac{\Delta E}{\Delta t \Delta r}$$的表达式。
(d)（10分）假设此环中释放的引力能通过黑体辐射形式从环表面局部发射，求环的表面温度$$T$$的表达式。
(e)（3分）设中心天体为质量$$3M_{\odot}$$的恒星黑洞，吸积率为$$\dot{M} = 10^{-9} M_{\odot} / \text{年}$$，且$$R_{min} = 3R_{sch}$$（其中$$R_{sch}$$为黑洞的史瓦西半径）。求盘的亮度及其最内层的峰值发射波长。忽略引力红移效应，并假设环最内层的发射主导总辐射。
(f)（3分）考虑另一个吸积系统，其吸积率为$$\dot{M} = 1 \, M_{\odot}/$$年，峰值发射波长为$$\lambda = 6 \times 10^{-8}$$米，估算该黑洞的质量。
官方解答
解答：
(a) 对于开普勒轨道：
$$\frac{v_k^2}{r} = \frac{GM}{r^2} \Rightarrow v_k^2 = \frac{GM}{r}$$
总机械能：
$$E(r) = \frac{1}{2} \Delta m v_k^2 - \frac{GM \Delta m}{r} = -\frac{1}{2} \frac{GM \Delta m}{r}$$
能量差：
$$E(R_{max}) - E(r) = \frac{1}{2} GM \Delta m \left( \frac{1}{r} - \frac{1}{R_{max}} \right) \approx \frac{GM \Delta m}{2r}$$
因此：
$$\frac{\Delta E}{\Delta m} \approx \frac{GM}{2r}$$
(b) 总光度：
$$L_{Tot} = \frac{\Delta E_{Tot}}{\Delta t} = \frac{GM \dot{M}}{2R_{min}}$$
(c) 考虑微小质量元素的能量变化：
$$\frac{\Delta E}{\Delta m} \approx \frac{GM \Delta r}{2r^2}$$
乘以质量流率：
$$\frac{\Delta E}{\Delta t \Delta r} = \frac{GM \dot{M}}{2r^2}$$
(d) 使用斯特藩-玻尔兹曼定律：
$$4\pi r \Delta r \sigma T^4 = \frac{\Delta E}{\Delta t}$$
代入得：
$$T = \left( \frac{GM \dot{M}}{8\pi \sigma r^3} \right)^{1/4}$$
(e) 史瓦西半径：
$$R_{sch} = \frac{2GM}{c^2} \Rightarrow R_{min} = \frac{6GM}{c^2}$$
总光度：
$$L_{Tot} = \frac{\dot{M}c^2}{12} \approx 5 \times 10^{29} \, \text{W}$$
表面温度：
$$T = 5.5 \times 10^6 \, \text{K}$$
峰值波长（维恩定律）：
$$\lambda = \frac{b}{T} = 5 \times 10^{-10} \, \text{m}$$
(f) 由维恩定律得温度：
$$T = \frac{b}{\lambda} = 4.8 \times 10^4 \, \text{K}$$
结合吸积公式估算质量：
$$M \approx 1.3 \times 10^9 M_{\odot}$$
