Stars
II. Stellar Physics
1. Overview of the structure of stars
Still, First the Sun as an example
Overview of basic processes
Nuclear energy production  energy transport
(radiation, convection) ;
for a system in a long-time equilibrium, mass
conservation, energy conservation, force balance.
Mathematically the conditions for the internal equilibrium of a star can be
expressed as four differential equations governing the distribution of
mass, gas pressure and energy production and transport in the star.
Plus the Equation of State (Physical State of the Gas) and Boundary
Conditions.
These equations will be derived soon.
Connections with observations
Maoz, fig2.4
Energy transport I: radiation
Energy transport II:
convection, sound waves and helioseismology
The solar convection is visible on the surface as the
granulation. At the bright (high-T) center of each
granule, gas is rising upward, and at the darker (lowerT) granule boundaries, it is sinking down again. The
size of a granule seen from the Earth is typically 1”,
corresponding to about 1000 km on the solar surface.
There is also a larger scale convection called
supergranulation in the photosphere. The cells of the
supergranulation may be about 1’ in diameter.
The oscillations we see on the surface are due to
sound waves generated and trapped inside the sun.
Sound waves are produced by pressure fluctuations
in the turbulent convective motions of the sun's
interior. Since sound is produced by pressure, these
modes of vibration are called p-modes.
Helioseismology
These sound waves, and the
modes of vibration they produce,
can be used to probe the interior of
the sun the same way that
geologists uses seismic waves
from earthquakes to probe the
inside of the earth. Some of these
waves travel right through the
center of the sun. Others are bent
back toward the surface at shallow
depths. Helioseismologists can use
the properties of these waves to
determine the temperature, density,
composition, and motion of the
interior of the sun.
http://solarscience.msfc.nasa.gov/Helioseismology.shtml
The sound waves set the sun vibrating in millions of different patterns or modes:
One mode of vibration is shown
in the preceding image
as a pattern of surface
displacements exaggerated by
over 1000 times:
A cylindrical mass element located (r, r+dr)
from center of the star
2. Our goal of this Section
The goal is to understand the inner structure of stars, their equilibrium
configurations, nuclear burning, energy transport and evolution. In the
standard model, we assume that a star can be treated as a spherically
symmetric gas sphere (no non-radial motions, no magnetic fields).
This is a reasonably good approximation for most stars. The task is solved
within the approximation when we have determined:
Which equations constrain these parameters?
 (1) Hydrostatic equilibrium:
 (2) Mass-density relation:
 (3) Radial luminosity profile, i.e. the luminosity produced by nuclear burning
in a shell of radius r and thickness dr
εv(ρ, T) denotes the energy production rate per volume, whereas ε(ρ,T) is the energy
production rate per mass. They can be determined by considering all nuclear reaction
rates at a given temperature and density (only valid in inner nuclear reaction core).
 (4) Equation of energy transport (important! e.g.: inhibiting the energy
transport would imply zero luminosity and the explosion of the star).
e.g.
 (5) Equation of state (normal stars) which links ρ, P and T:
So, we finally have 5 equations for 5 unknowns and we
should be able to solve the problem in a unique way. The
uniqueness of the solution is claimed in the Russel-VogtTheorem: For a star of given chemical composition and
mass there exists only one equilibrium configuration
which solves the boundary problem of stellar structure.
[In this generality, the theorem is not proven. Local uniqueness can be
shown, however. Also, the theorem is based on the wrong assumption
that the chemical composition of a star is homogenous.]
That is, all the parameters of a star (its spectral type,
luminosity, size, radius and temperature) are determined
primarily by its mass. The emphasis on `primarily' is
important since this only applies during the `normal' or
hydrogen burning phase of a star's life.
后续课程《恒星结构和演化 》(AY15204）
Four boundary conditions
•
•
•
•
M(r=0) = 0
L(r=0) = 0
P(r=r*) = 0
M(r=r*) = M*
理论上，有唯一解 （Vogt-Russell theorem）；
但是无解析解。数值求解。
Standard Solar Model [Bahcall & Pinsonneault, Phys. Lett. B, 433 (1998) 1-8]
Nuclear fusions: Summary
• Proton-proton chain
(T<2x107 K; stars with M≤Msun)
• CNO cycle
(stars with mass≥1.2Msun)
• Triple-alpha reaction
(T> 108K)
• More reactions at higher temperatures
Carbon burning; Oxygen burning; Silicon burning
• (questions: how elements heavier than Iron are
produced? [neutron capture; see Star III.])
p-p chain
CNO cycle
C, N, O as catalysts
• Triple-alpha
(He burning)
• Carbon burning
• Oxygen burning
• Silicon burning
Onion Structure of the Massive Star
Homework: Textbook of D. Maoz, p61, 2, 6,9