Exercises

1. Prove that the fraction of the whole mass of an isothermal atmosphere that lies between the ground and a horizontal plane of height $z$ is
   $$1 - {\rm e}^{-z/H}.$$
   Evaluate this fraction for $z=H$ , $2\,H$ , $3\,H$ , respectively.

2. If the absolute temperature in the atmosphere diminishes upwards according to the law
   $$\frac{T}{T_0} = 1 - \frac{z}{c},$$
   where $c$ is a constant, show that the pressure varies as
   $$\frac{p}{p_0} = \left(1-\frac{z}{c}\right)^{c/H}.$$

3. If the absolute temperature in the atmosphere diminishes upward according to the law
   $$\frac{T}{T_0} = \frac{1}{1+\beta\,z},$$
   where $\beta$ is a constant, show that the pressure varies as
   $$\frac{p}{p_0} = \exp\!\left(-\frac{z}{H}-\frac{1}{2}\,\frac{\beta\,z^{\,2}}{H}\right).$$

4. Show that if the absolute temperature, $T$ , in the atmosphere is any given function of the altitude, $z$ , then the vertical distribution of pressure in the atmosphere is given by
   $$\ln\frac{p}{p_0} = -\frac{T_0}{H}\int_0^z \frac{dz}{T}.$$

5. Show that if the Earth were surrounded by an atmosphere of uniform temperature then the pressure a distance $r$ from the Earth's center would be
   $$\frac{p}{p_0} = \exp\left[\frac{a^{\,2}}{H}\left(\frac{1}{r}-\frac{1}{a}\right)\right],$$
   where $a$ is the Earth's radius.

6. Show that if the whole of space were occupied by air at the uniform temperature $T$ then the densities at the surfaces of the various planets would be proportional to the corresponding values of
   $$\exp\left(\frac{g\,M\,a}{R\,T}\right),$$
   where $a$ is the radius of the planet, and $g$ its surface gravitational acceleration.

7. Prove that in an atmosphere arranged in horizontal strata the work (per unit mass) required to interchange two thin strata of equal mass without disturbance of the remaining strata is
   $$\frac{1}{\gamma-1}\,\left(\rho_2^{\,\gamma-1}-\rho_1^{\,\gamma-1}\right)\left(\frac{p_1}{\rho_1^{\,\gamma}}-\frac{p_2}{\rho_2^{\,\gamma}}\right),$$
   where the suffixes refer to the initial states of the two strata. Hence, show that for stability the ratio $p/\rho^{\,\gamma}$ must increase upwards.
8. A spherically symmetric star is such that $m(r)$ is the mass contained within radius $r$ . Show that the star's total gravitational potential energy can be written in the following three alternative forms:
   $$U= - \int_0^{M_\ast}\frac{G\,m}{r}\,dm = \frac{1}{2}\int_0^{M_\ast}{\mit\Psi}\,dm=-3\int_0^R p\,dV$$
   Here, ${M_\ast}$ is the total mass, $R$ the radius, ${\mit\Psi}(r)$ the gravitational potential per unit mass (defined such that ${\mit\Psi}\rightarrow 0$ as $r\rightarrow \infty$ ), $p(r)$ the pressure, and $dV = 4\pi\,r^{\,2}\,dr$ .

9. Suppose that the pressure and density inside a spherically symmetric star are related according to the polytropic gas law,
   $$p = K\,\rho^{\,(1+n)/n},$$
   where $n$ is termed the polytropic index. Let $\rho=\rho_c\,\theta^{\,n}$ , where $\rho_c$ is the central mass density. Demonstrate that $\theta$ satisfies the Lane-Emden equation
   $$\frac{1}{\xi^{\,2}}\frac{d}{d\xi}\!\left(\xi^{\,2}\,\frac{d\theta}{d\xi}\right)=-\theta^{\,n},$$
   where $r=a\,\xi$ , and
   $$\xi=\left[\frac{(n+1)\,K}{4\pi\,G}\,\rho_c^{\,(n-1)/n}\right]^{1/2}.$$
   Show that the physical solution to the Lane-Emden equation, which is such that $\theta(0)=1$ and $\theta(\xi_1)=0$ , for some $\xi_1>0$ , is
   $$\theta=1-\frac{\xi^{\,2}}{6}$$
   for $n=0$ ,
   $$\theta=\frac{\sin\xi}{\xi}$$
   for $n=1$ , and
   $$\theta = \frac{1}{(1+\xi^{\,2}/3)^{1/2}}$$
   for $n=5$ . Determine the ratio of the central density to the mean density in all three cases. Finally, demonstrate that, in the general case, the total gravitational potential energy can be written
   $$U= - \frac{3}{5-n}\,\frac{G\,M_\ast^{\,2}}{R},$$
   where $M_\ast$ is the total mass, and $R=a\,\xi_1$ the radius.
10. A spherically symmetric star of radius $R$ has a mass density of the form
    $$\rho(r)=\rho_c\,(1-r/R).$$
    Show that the central mass density is four times the mean density. Demonstrate that the central pressure is
    $$p_c = \frac{5}{4\pi}\,\frac{G\,M_\ast^{\,2}}{R},$$
    where $M_\ast$ is the mass of the star. Finally, show that the total gravitational potential energy of the star can be written
    $$U = - \frac{26}{35}\,\frac{G\,M_\ast^{\,2}}{R}.$$