Physics 321
Astrophysics II:  Lecture #8
Prof. Dale E. Gary

Stellar Interiors -- II
Stellar Energy Sources

Stellar Energy Sources

Chemical Energy Gravitational Energy
Given this onion-like structure, we can consider the shell of uniform density r, at radius r and thickness dr.  Such a shell will have a mass dm = 4pr2r dr.  Inserting this into (2), we have
dU = - GMr4pr2r dr / r
which can be integrated over the radius of the star, from the center (r = 0) to the surface (r = R) to give the total potential energy of the star:
U = - 4pG Mr rrdr         (3)
In this expression, note that Mr is itself an integral, since the total mass interior to the radius r requires integrating the density from the center to the point r:
Mr = 4p  rr2dr         (4)
so the entire expression requires the change of density with radius in the star.  However, we can make some simplifying assumptions to see if gravitational potential energy can be the energy source for the Sun.  Let's assume that the density is constant, with the value of the average density <r> = Mr / (4/3 pR3), so that (4) becomes Mr ~ 4/3 pr3<r>.  Then the potential energy, (3), becomes
U ~ - 4pG <r> 4/3 pr4dr


16p2 3GM 2
U ~
G <r>2R5  ~ 

We will discuss in a moment the virial theorem, which says that a gravitational system moving from one equilibrium state to another will release 1/2 its available energy as internal kinetic energy, and the other 1/2 is available to be radiated away.  Thus, if in the initial configuration all of the mass is at infinity (zero potential energy) then the amount that can be radiated in getting to the final (current) configuration is -U/2, or

DE = Ef-Ei = -U/2 = 3GM2 / 10R
In the case of the Sun, this is DE =3GM2 / 10R ~ 1.1 x 1041 J.  Again, since the Sun shines at the rate L = dE/dt = 3.826 x 1026 J/s, it will take only
Dt = tKH = DE / L = 1.1 x 1041 J / 3.826 x 1026 J/s ~ 107 years.
This time, tKH is called the Kelvin-Helmholtz time scale, and was calculated near the turn of the century.  At the time, this was considered an iron-clad argument that the Earth and Sun coud not be much older than this, although geologists and paleontologists were finding that the Earth must be much older.  We now know from radio-active dating techniques that the Earth and Moon are at least 4.5 billion years old.  Clearly there must be another source of energy for the Sun.  We have looked at gravitational energy in some detail, however, because for some objects and at some parts of their life-cycle gravitational energy is the dominant energy source.
Virial Theorem
As we mentioned, the virial theorem states that when a system is in one equilibrium state and changes to another equilibrium state, the difference in energy goes equally into two parts: (1) the internal energy of the system and (2) radiation or other loss mechanisms.  What do we mean by equilibrium?  An equilibrium is a situation in which the configuration of the system is not changing.  However, by the phrase "not changing" we do not mean that the system cannot evolve.  Rather, the system is allowed to change only slowly over some relevant timescale.  Consider a planet orbiting around a star.  If the planet orbit does not change appreciably over many orbits, then the system can be said to be in equilibrium.  Consider a collapsing cloud of gas.  If many collisions between particles occur before the cloud changes radius appreciably, the system can again be said to be in equilibrium.

In the collapsing cloud example, above, we used the virial theorem to find the amount of energy radiated away.  The virial theorem also says that this same amount of energy (3GM2 / 10R) goes into internal energy (kinetic energy, or heat in this case).  It the cloud starts out at temperature 0 K, what will be its temperature at radius R?  The number of particles in the cloud is N = M / mmH.  and each has thermal energy is 3/2 kT, so the total thermal energy is

3/2 NkT = 3GM2 / 10R
so the temperature change is
DT = Tf - Ti = GM 2 / 5RNk = GM mmH / 5kR ~ 2.78 x 106 K
where the numerical value is found using solar values for the mass and radius.  In fact, the internal temperature of the Sun is higher than this, as we can see by simple application of hydrostatic equilibrium and the ideal gas law: The virial theorem has a wide range of validity and usefulness. We can use it in problems concerning planetary orbits, stellar collapse, accretion disks around black holes, and motions in clusters of galaxies.
Nuclear Energy