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Stars II. Stellar Physics

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StarsII. Stellar Physics

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.

Maoz, fig2.4

Energy transport I: radiation

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 (lower-T) 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.

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

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-Vogt-Theorem: 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）

- 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]

- 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.])

C, N, O as catalysts

(He burning)

- Triple-alpha
- Carbon burning
- Oxygen burning
- Silicon burning

Onion Structure of the Massive Star

Homework: Textbook of D. Maoz, p61, 2, 6,9