d r v >
For an ideal gas, the adiabatic temperature gradient is
'dT x d r adiabatic cP - cV T dP c P P dr where cV is the specific heat at fixed volume. For a monatomic gas (complete ionisation), (cP - cV)/cP = 2/5.
The final equilibrium condition to be considered is that of energy balance, at thermal equilibrium, dL(r)/dr = 4n r2p (r) e (p,T,X,...),
stating that the radiative energy loss must be made up for by an energy source term, e , which describes the energy production as function of density, temperature, composition (e.g. hydrogen abundance X), etc. Since the energy production processes in the Sun are associated with nuclear reactions, notably the thermonuclear fusion of hydrogen nuclei into helium, then the evaluation of e requires a detailed knowledge of cross sections for the relevant nuclear reactions.
Before touching on the nuclear reaction rates, it should be mentioned that the condition of thermal equilibrium is not nearly as critical as that of hydrostatic equilibrium. Should the nuclear processes suddenly cease in the Sun's interior, the luminosity would, for some time, remain unchanged at the value implied by the temperature gradient (2.22). The stores of thermal and gravitational energy would ensure an apparent stability during part of the time (called the "Kelvin time") needed for gravitational collapse. This time may be estimated from the virial theorem, since the energy available for radiation (when there is no new energy generation) is the difference between the total gravitational energy and the kinetic energy bound in the thermal motion (i.e. the temperature increase associated with contraction), tK
, = (e gravit e kJ/L(R) = - (e pot + e Un)/L(R) = e Ur/L(R) = 3 x 107 y.
On the other hand, this time span is short enough to prove that energy production takes place in the Sun's interior.
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