Next: One-Dimensional Compressible Inviscid Flow Up: Equilibrium of Compressible Fluids Previous: Eddington Solar Model

# 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 is

Evaluate this fraction for , , , respectively.

2. If the absolute temperature in the atmosphere diminishes upwards according to the law

where is a constant, show that the pressure varies as

3. If the absolute temperature in the atmosphere diminishes upward according to the law

where is a constant, show that the pressure varies as

4. Show that if the absolute temperature, , in the atmosphere is any given function of the altitude, , then the vertical distribution of pressure in the atmosphere is given by

5. Show that if the Earth were surrounded by an atmosphere of uniform temperature then the pressure a distance from the Earth's center would be

6. Show that if the whole of space were occupied by air at the uniform temperature then the densities at the surfaces of the various planets would be proportional to the corresponding values of

where is the radius of the planet, and 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

where the suffixes refer to the initial states of the two strata. Hence, show that for stability the ratio must increase upwards.
8. A spherically symmetric star is such that is the mass contained within radius . Show that the star's total gravitational potential energy can be written in the following three alternative forms:

Here, is the total mass, the radius, the gravitational potential per unit mass (defined such that as ), the pressure, and .

9. Suppose that the pressure and density inside a spherically symmetric star are related according to the polytropic gas law,

where is termed the polytropic index. Let , where is the central mass density. Demonstrate that satisfies the Lane-Emden equation

where , and

Show that the physical solution to the Lane-Emden equation, which is such that and , for some , is

for ,

for , and

for . 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

where is the total mass, and the radius.
10. A spherically symmetric star of radius has a mass density of the form

Show that the central mass density is four times the mean density. Demonstrate that the central pressure is

where is the mass of the star. Finally, show that the total gravitational potential energy of the star can be written

Next: One-Dimensional Compressible Inviscid Flow Up: Equilibrium of Compressible Fluids Previous: Eddington Solar Model
Richard Fitzpatrick 2016-01-22