Stars as Laboratories for Fundamental Physics - MPP Theory Group

6 Chapter 1
impact on the overall picture of stellar structure and evolution, with
the possible exception of supernova physics where large-scale convective
overturns may be crucial for the explosion mechanism (Chapter 11).
Second, one usually assumes hydrostatic equilibrium, i.e. one ignores
the macroscopic kinetic energy of the stellar medium. This approximation
is inadequate for a study of stellar pulsation where the inertia of
the material is obviously important, and also inadequate for “hydrodynamic
events” such as a supernova explosion (Sect. 2.1.8), and perhaps
the helium flash (Sect. 2.1.3). For most purposes, however, the changes
of the stellar structure are so slow that neglecting the kinetic energy is
an excellent approximation.
Therefore, as a first equation one uses the condition of hydrostatic
equilibrium that a spherical shell of the stellar material is held in place
by the opposing forces of gravity and pressure,
dp
dr = −G NM r ρ
r 2 . (1.1)
Here, G N is Newton’s constant, p and ρ are the pressure and mass density
at the radial position r, and M r = 4π ∫ r
0 dr′ ρ r ′2 is the integrated
mass up to the radius r.
With apologies to astrophysicists I will usually employ natural units
where ¯h = c = k B = 1. In Appendix A conversion factors are given
between various units of mass, energy, inverse length and time, temperature,
and so forth. Newton’s constant is then G N = m −2
Pl with the
Planck mass m Pl = 1.221×10 19 GeV = 2.177×10 −5 g. Stellar masses are
always denoted with the letter M to avoid confusion with an absolute
bolometric brightness which is traditionally denoted by M.
In general, the pressure is given in terms of the density, temperature,
and chemical composition by virtue of an equation of state. For a
classical monatomic gas p = 2 u with u the density of internal energy.
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One may multiply Eq. (1.1) on both sides with 4πr 3 and integrate
from the center (r = 0) to the surface (r = R). The r.h.s. gives the
total gravitational energy while the l.h.s. yields −12π ∫ R
0 dr p r2 after a
partial integration with the boundary condition p = 0 at the surface.
With p = 2 u this is −2U with U the total internal energy of the star.
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Because for a monatomic gas U is the sum of the kinetic energies of the
atoms one finds that on average for every atom
⟨E kin ⟩ = − 1 2 ⟨E grav⟩. (1.2)
This is the virial theorem which is the most important tool to understand
the behavior of self-gravitating systems.