For a system to achieve a mechanical equilibrium,
a relation must be satisfied
between energies such as potential, thermal and kinetic energies.
This is called the Virial relation.
For example, a harmonic oscillator
a potential energy of
and a kinetic energy
Averaging these two energies over one period,
both energies give the same absolute value proportional to the oscillation
amplitude squared as
Another example is a Kepler problem.
For simplicity, consider mass running
on a circular orbit with a radius from a body with a mass .
The gravitational and kinetic energies are equal to and
, where we used the centrifugal balance
As a result,
for the harmonic oscillator
while for the circular
Kepler problem .
This ratio is known to be related to the power of
the potential as
Important nature of the self-gravity is understood only with this relation
without solving the hydrostatic balance equations.
In the following, we describe the Virial relation satisfied with isolated
systems such as stars.
Hydrodynamic equation of motion using the Lagrangean time derivative [eq.(A.3)] is
For simplicity, consider a spherical symmetric configuration.
The equation of motion is expressed as
Multiplying radius to the equation
and integrating by the volume over a volume
from to , we obtain the Virial relation as
is an inertia of this body,
and and are, respectively, the kinetic and thermal energies as
is a gravitational energy, where
the lefthand-side of equation (2.105) is rewritten as
using equations (2.107) and (2.108).
On the other hand, the first term of the rhs of equation (2.105)
To derive this equation, we have assumed the pressure diminishes
at a radius and the surface pressure term does not appear in the
This is valid for an isolated system such as a star.
The last term of the rhs of equation (2.105) is written as
where we used equation (2.13).
This is rewritten as
The energy per unit mass is necessary for a gas element
to move from the radius , inside which mass is contained, to the
Adding the energy for all the gas,
the potential energy is obtained.
In the case of a star composed of uniform density ,
To obtain a condition for the mechanical equilibrium, we assume .
Equation (2.106) becomes
Assuming the system is static , the above equation reduces to
In the case of this reduces to .
The total energy is expressed as
For the gas with , equation (2.119) gives
a negative total energy and the system is in a confined state.
However, if , the gravity can not confine the gas.
, equation (2.118) gives
This shows us a strange nature of the self-gravitating gas (see Fig. 2.10).
That is, if the heat flux flows outward, the total energy decreases
The system must contract and equation (2.119)
(the gravitational energy decreases:
the absolute value of the gravitational energy increases).
However, for the thermal energy,
equation (2.120) indicates that
for this system to be static.
This shows that if the heat flux flows out from the system
the thermal energy increases in the self-gravitating system.
This comes from the contraction due to the gravity.
Virial relation for stars composed of gas.