2024年IOAA理论第4题-白矮星
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英文题目
T4. White Dwarf (10 points)
The structure of a white dwarf is sustained against gravitational collapse by the pressure of degenerate electrons, a phenomenon explained by quantum physics and related to the Pauli Exclusion Principle for electrons. The equation of state of a gas made of non-relativistic degenerate electrons is the following:
$$P = \left( \frac{3}{8\pi} \right)^{2/3} \frac{h^2}{5m_e} n_e^{5/3},$$
where $$n_e$$ is the number of electrons per unit volume, which can be expressed in terms of the mass density $$\rho$$ using the dimensionless factor $$\mu_e$$, the number of nucleons (protons and neutrons) per unit electron. Also consider that the central pressure can be described by this equation of state.
In the condition of hydrostatic equilibrium, the pressure and gravitational forces balance each other at any distance $$r$$ from the centre of the star. This condition can be expressed by:
$$\frac{dP}{dr} = -\frac{GM(r)\rho(r)}{r^2},$$
where $$M(r)$$ is the mass contained in the sphere of radius $$r$$, and $$\rho(r)$$ is the mass density of the star at a radius $$r$$.
Assume that $$m_p = m_n$$, the density of a white dwarf is roughly uniform, and the following approximation is valid at the surface of the star:
$$\frac{dP}{dr} \bigg|_{r=R} \approx -\frac{P_c}{R},$$
where $$P_c$$ is the pressure at the centre of the star, and $$R$$ the star radius.
(a) (6 points) The relationship between the mass $$M$$ and the radius $$R$$ of a white dwarf can be written in the form:
$$R = a \cdot M^b$$
Find the exponent $$b$$ and determine the coefficient $$a$$ in terms of physical constants and $$\mu_e$$.
(b) (4 points) Using the relationship found in the previous part, estimate the radius of a white dwarf made of fully ionised carbon ($$^{12}C$$) with a mass of $$M = 1.0M_\odot$$.
中文翻译
T4. 白矮星（10分）
白矮星的结构由简并电子的压力维持以抵抗引力坍缩，这一现象可由量子物理和电子泡利不相容原理解释。由非相对论性简并电子组成的气体的状态方程如下：
$$P = \left( \frac{3}{8\pi} \right)^{2/3} \frac{h^2}{5m_e} n_e^{5/3},$$
其中$$n_e$$为单位体积的电子数，可通过质量密度$$\rho$$和无量纲因子$$\mu_e$$（每个电子对应的核子（质子和中子）数）表示。同时假设中心压力可用此状态方程描述。
在流体静力平衡条件下，压力与引力在距离星体中心$$r$$处的任意位置相互平衡。该条件可表示为：
$$\frac{dP}{dr} = -\frac{GM(r)\rho(r)}{r^2},$$
其中$$M(r)$$为半径$$r$$的球体内包含的质量，$$\rho(r)$$为星体在半径$$r$$处的质量密度。
假设$$m_p = m_n$$，白矮星的密度大致均匀，且在星体表面满足以下近似：
$$\frac{dP}{dr} \bigg|_{r=R} \approx -\frac{P_c}{R},$$
其中$$P_c$$为星体中心压力，$$R$$为星体半径。
(a) (6分) 白矮星质量$$M$$与半径$$R$$的关系可写为：
$$R = a \cdot M^b$$
求指数$$b$$，并用物理常数和$$\mu_e$$确定系数$$a$$。
(b) (4分) 利用上一部分得到的关系，估算由完全电离的碳（$$^{12}C$$）构成、质量为$$M = 1.0M_\odot$$的白矮星的半径。
官方解答
(a) **解答：** 在$$r = R$$处，利用题目中提供的近似条件和密度表达式$$\rho = 3M/(4\pi R^3)$$：
$$-\frac{P_c}{R} = -\frac{GM(3M/(4\pi R^3))}{R^2}$$
$$P_c = \frac{3GM^2}{4\pi R^4}$$
电子密度$$n_e$$与质量密度的关系为：
$$\rho = \mu_e m_p n_e$$
使用状态方程：
$$P_c = \left( \frac{3}{8\pi} \right)^{2/3} \frac{h^2}{5m_e} \left( \frac{3M}{4\pi R^3 \mu_e m_p} \right)^{5/3}$$
联立得：
$$\frac{3GM^2}{4\pi R^4} = \left( \frac{3}{8\pi} \right)^{2/3} \frac{h^2}{5m_e} \left( \frac{3}{4\pi \mu_e m_p} \right)^{5/3} \frac{M^{5/3}}{R^5}$$
解得：
$$R = \left( \frac{4\pi}{3} \right) \left( \frac{3}{8\pi} \right)^{2/3} \frac{h^2}{5Gm_e} \left( \frac{3}{4\pi \mu_e m_p} \right)^{5/3} M^{-1/3}$$
因此，
$$b = -\frac{1}{3}, \quad a = \left( \frac{4\pi}{3} \right) \left( \frac{3}{8\pi} \right)^{2/3} \frac{h^2}{5Gm_e} \left( \frac{3}{4\pi \mu_e m_p} \right)^{5/3}.$$
(b) **解答：** 对于完全电离的碳（$$^{12}C$$），每个电子对应2个核子，即$$\mu_e = 2$$。代入上述表达式：
$$R \approx \frac{1.866 \times 10^{-2} \times \left(6.626 \times 10^{-34}\right)^2}{6.67 \times 10^{-11} \times 9.11 \times 10^{-31} \times \left(2 \times 1.67 \times 10^{-27}\right)^{5/3} \times \left(1.988 \times 10^{30}\right)^{1/3}}$$
计算结果为：
$$R \approx 1.44 \times 10^6 \, \text{m}.$$
