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- 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.

- If the absolute temperature in the atmosphere diminishes upwards according to the law

where
is a constant,
show that the pressure varies as

- If the absolute temperature in the atmosphere diminishes upward according to the law

where
is a constant,
show that the pressure varies as

- 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

- 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

where
is the Earth's radius.

- 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.

- 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.
- 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
.

- 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.
- 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