2025年3月6日 (四) 17:07 Quan787

2025-03-06T17:07:52Z

Quan787：创建页面，内容为“{{由ai生成}} ==英文题目== '''T4. White Dwarf (10 points)''' The structure of a white dwarf is sustained against gravitational collapse by the pressure of de…”

2025-03-06T14:56:17Z

创建页面，内容为“{{由ai生成}} ==英文题目== '''T4. White Dwarf (10 points)''' The structure of a white dwarf is sustained against gravitational collapse by the pressure of de…”

新页面

{{由ai生成}}
==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. The equation of state of a gas made of non-relativistic degenerate electrons is:

$$P = \left( \frac{3}{8\pi} \right)^{2/3} \frac{h^2}{5m_e} n_e^{5/3},$$

where $n_e$ is the number of electrons per unit volume. In hydrostatic equilibrium:

$$\frac{dP}{dr} = -\frac{GM(r)\rho(r)}{r^2}.$$

Assume the density is roughly uniform and the approximation at the surface:

$$\frac{dP}{dr} \bigg|_{r=R} \approx -\frac{P_c}{R}.$$

(a) (6 points) Derive the mass-radius relationship $R = a \cdot M^b$. Find exponent $b$ and coefficient $a$ in terms of physical constants and $\mu_e$.

(b) (4 points) Estimate the radius of a carbon white dwarf ($^{12}C$) with $M = 1.0M_\odot$.

==中文翻译==

'''T4. 白矮星（10分）'''

白矮星通过简并电子压抵抗引力坍缩。非相对论性简并电子气体的状态方程为：

$$P = \left( \frac{3}{8\pi} \right)^{2/3} \frac{h^2}{5m_e} n_e^{5/3},$$

其中$n_e$为电子数密度。在流体静力平衡条件下：

$$\frac{dP}{dr} = -\frac{GM(r)\rho(r)}{r^2}.$$

假设密度均匀，并在表面采用近似：

$$\frac{dP}{dr} \bigg|_{r=R} \approx -\frac{P_c}{R}.$$

(a) (6分) 推导质量-半径关系$R = a \cdot M^b$，求指数$b$和系数$a$（用物理常数和$\mu_e$表示）。

(b) (4分) 估算碳白矮星($^{12}C$)在$M = 1.0M_\odot$时的半径。

== 官方解答 ==

(a) 结合状态方程与流体静力平衡条件：

$$R = \left( \frac{4\pi}{3} \right) \left( \frac{3}{8\pi} \right)^{2/3} \frac{h^2}{5Gm_e} \left( \frac{3}{4\pi \mu_e m_p} \right)^{5/3} M^{-1/3}$$

得：

$$b = -\frac{1}{3}, \quad a \propto \mu_e^{-5/3}$$

(b) 碳白矮星$\mu_e = 2$，代入常数计算得：

$$R \approx 1.44 \times 10^6 \ \text{m}$$

[[分类:恒星结构]]
[[分类:量子物理]]
[[分类:由ai生成]]

@@ 第4行： / 第4行： @@
 '''T4. White Dwarf (10 points)'''
-The structure of a white dwarf is sustained against gravitational collapse by the pressure of degenerate electrons. The equation of state of a gas made of non-relativistic degenerate electrons is:
+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. In hydrostatic equilibrium:
+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.
-$$\frac{dP}{dr} = -\frac{GM(r)\rho(r)}{r^2}.$$
+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:
-Assume the density is roughly uniform and the approximation at the surface:
+$$\frac{dP}{dr} = -\frac{GM(r)\rho(r)}{r^2},$$
-$$\frac{dP}{dr} \bigg|_{r=R} \approx -\frac{P_c}{R}.$$
+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 $$r$$.
-(a) (6 points) Derive the mass-radius relationship \(R = a \cdot M^b\). Find exponent \(b\) and coefficient \(a\) in terms of physical constants and \(\mu_e\).
+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:
-(b) (4 points) Estimate the radius of a carbon white dwarf (\(^{12}C\)) with \(M = 1.0M_\odot\).
+$$\frac{dP}{dr} \bigg|_{r=R} \approx -\frac{P_c}{R},$$
 ==中文翻译==
@@ 第24行： / 第34行： @@
 '''T4. 白矮星（10分）'''
-白矮星通过简并电子压抵抗引力坍缩。非相对论性简并电子气体的状态方程为：
+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\(n_e\)为电子数密度。在流体静力平衡条件下：
+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。
-$$\frac{dP}{dr} = -\frac{GM(r)\rho(r)}{r^2}.$$
+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：
-假设密度均匀，并在表面采用近似：
+$$\frac{dP}{dr} = -\frac{GM(r)\rho(r)}{r^2},$$
-$$\frac{dP}{dr} \bigg|_{r=R} \approx -\frac{P_c}{R}.$$
+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。
-(a) (6分) 推导质量-半径关系\(R = a \cdot M^b\)，求指数\(b\)和系数\(a\)（用物理常数和\(\mu_e\)表示）。
+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：
-(b) (4分) 估算碳白矮星(\(^{12}C\))在\(M = 1.0M_\odot\)时的半径。
+$$\frac{dP}{dr} \bigg|_{r=R} \approx -\frac{P_c}{R},$$
 == 官方解答 ==
-(a) 结合状态方程与流体静力平衡条件：
+(a) **解答：**
 $$R = \left( \frac{4\pi}{3} \right) \left( \frac{3}{8\pi} \right)^{2/3} \frac{h^2}{5Gm_e} \left( \frac{3}{4\pi \mu_e m_p} \right)^{5/3} M^{-1/3}$$
-得：
+因此，
-$$b = -\frac{1}{3}, \quad a \propto \mu_e^{-5/3}$$
+$$R \approx \frac{1.866 \times 10^{-2} \times \left(6.626 \times 10^{-34}\right)^2}{6.67 \times 10^{-11} \times 9.11 \times 10^{-31} \times \left(2 \times 1.67 \times 10^{-27}\right)^{5/3} \times \left(1.988 \times 10^{30}\right)^{1/3}}$$
-(b) 碳白矮星\(\mu_e = 2\)，代入常数计算得：
+计算结果为：
-$$R \approx 1.44 \times 10^6 \ \text{m}$$
+$$R \approx 1.44 \times 10^6 \, \text{m}.$$
 [[分类:恒星结构]]
 [[分类:量子物理]]
 [[分类:由ai生成]]

2024年IOAA理论第4题-白矮星 - 版本历史

2025年3月6日 (四) 17:07 Quan787

Quan787：创建页面，内容为“{{由ai生成}} ==英文题目== '''T4. White Dwarf (10 points)''' The structure of a white dwarf is sustained against gravitational collapse by the pressure of de…”