Балкански 2010, Теория — Задача 2. Stability of a spherical star

Concours Général de Physique “Minko Balkanski” · 8 май 2010 г. · 8 т.

Условие

1. Thermal stability.

We are going to use a model of a spherical homogeneous star with radius $R$ and mass $M$ made by hydrogen atoms with molar mass $M_H$, making the hypothesis of a perfect gaz in equilibrium with temperature $T$ uniformly distributed. Every atom has a kinetic energy $e_c = \frac{3}{2}kT$, where $k = \frac{\Re}{Na}$.

1. (a) Supposing that the Sun is an isolated system, give its total energy $E_S$.

1. (b) What is the condition for having a constant radius of the Sun, ie. the Sun is not in infinite expansion?

1. (c) Give the maximum temperature $T$ corresponding to the previous condition.

1. (d) Numerical application: calculate this temperature.

2. Dynamical stability.

We are going to use a model of a spherical homogeneous star with radius $R$ and mass $M$, rotating with a constant period around one of its diameters and made by a gaz of particles moving with the same angular speed as the star.

2. (a) Give the speed of liberation $v_i$ at the surface of the star.

2. (b) Write a condition for the dynamical stability of the star.

2. (c) Numerical application: does the Sun verify this dynamical stability? Justify your answer with numerical values.

2. (d) What happens to a star which has a liberation speed greater than the speed of light? What is the condition for the fraction $\frac{M}{R}$ for these objects? Give the same condition using the density of the star.

2. (e) Numerical application: calculate for a star with radius $R_S$, the minimal density that gives a liberation speed greater than the speed of light. Comment this numerical value.

3. Hydrostatic aspects.

We are going to use a model of a spherical homogeneous star with radius $R$ and mass $M$, without rotation, whose density $\rho(r)$, internal pressure $P(r)$ and temperature $T(r)$ depend only of the distance $r$ from the center of the star. Furthermore, the star is modelled by perfect gaz of hydrogen atoms in hydrostatic equlibrium, ie. every element of volume $dV$ is equilibrated by the gravitational and pressure forces.

3. (a) Remind the fundamental principe of the fluid statics, writing $\overrightarrow{grad}P$ as a function of $\rho(r)$ and $\vec{g}$.

3. (b) Give $\frac{dP}{dr}$ in the case of spherical star as a function of $G$, $r$, $\rho(r)$ and $M(r)$, the mass contained in the ball with radius $r$.

3. (c) We want to estimate the temperature $T_C$ in the center of the Sun. In order to do that, we suppose that the function $P(r)$ is linear and zero at the surface of the Sun. We will substitute $\rho(r)$ by the average density of the Sun and the term $\frac{M(r)}{r^2}$ by the “average“ value $\frac{M_S/2}{(R/2)^2}$. Give $P_C$ and then find $T_C$, as a function of $G$, $M_S$, $R_S$, $\Re$ and $M_H$.

3. (d) Numerical application: calculate $T_C$.

Решение

Оригинал в Архива: CGP_2010_en.pdf