Today in Astronomy 241: model stellar interiors

1
Today in Astronomy 241 model stellar interiors

• Todays reading Carroll and Ostlie Ch.
  10.5-10.6, on
• The ingredients of stellar interior models
• Simple analytical models
• StatStar, a simple computer stellar-interior
  model
• Density in a portion of a model stellar interior,
  showing convective interpenetration of unstable,
  turbulent gas (left) and stable gas (right). By
  Andrea Malagoli et al. (U. Chicago).

2
The equations of stellar structure
3
Constitutive relations
Pressure equation of state
Rosseland mean opacity
4
Constitutive relations (continued)
Nuclear energy generation
5
Boundary conditions
No singularities. or match to solution for
atmosphere.
Vogt-Russell theorem the mass and composition
of a star uniquely determines its radius,
luminosity, internal structure, and subsequent
evolution.
6
A simple analytical stellar model the LSM

• Unfortunately, there are no general, analytical
  solutions to the equations of stellar structure
  usually stellar-interior models are generated by
  computer solution.
• One class of analytically tractable models
  involves cheating initially by imposing a
  solution for the density, and using this to start
  solving for all else.
• In particular, the choiceis fruitful this is
  called the linear stellar model.

7
Linear stellar model (continued)

• Apart from the cheat, most of the physics of
  stellar interiors is preserved, allowing an
  analytical understanding of many important
  consequences, e.g. the main sequence in the H-R
  diagram.
• However, and unsurprisingly, the linear stellar
  model is not very accurate.

8
A simple computer stellar model StatStar

• StatStar resembles, but is simpler than, research
  grade code used by astronomers for stellar
  structure and evolution calculations.
• It proceeds by
• division the star into many spherical zones (e.g.
  of width Dr),
• conversion of the equations into difference
  equations (e.g. dP/dr DP/Dr),
• modification of the surface boundary conditions,
  to avoid the r 0 problem,
• integration from the outside in, by adding the
  contributions of the zones,

9
A simple computer stellar model StatStar

• comparison of conditions derived for the center,
  to the boundary conditions if they dont match
  the initial parameters are modified and another
  integration is done (the general procedure called
  a shooting method).
• The simplicity of StatStar is reflected in the
  limitations to its accuracy for example, in its
  solution for and solar abundances,
• the luminosity comes out to
• the core winds up being convective, and the outer
  regions radiative, the opposite of the real Sun.
• But overall it reproduces the main sequence
  pretty well, and of course is much more accurate
  than the analytical cheats.

10
Members of binary systems vs. StatStar
11
Members of binary systems vs. StatStar (continued)
12
Members of binary systems vs. StatStar (continued)
13
In class derive adiabatic temperature gradient

• Using the ideal gas law (eqn. 10.11),
• a form of the adiabatic gas law (eqn. 10.86),
• the hydrostatic equilibrium condition (eqn.
  10.6).
• Show a full derivation of the adiabatic
  temperature gradient (eqn. 10.89), i.e. the steps
  that lead to this

14
In-class problem (linear stellar model)

• Starting withintegrate the equation of mass
  conservation, thereby obtaining expressions for
  mass as a function of radius,and the central
  density,(It will be convenient to leave in
  the equations, and to use x r/R as the
  independent variable.)
• Then integrate the equation of hydrostatic
  equilibrium, to get the pressure as a function of
  radius, and the central pressure.

15
Linear stellar model (continued)

• Then use the ideal gas law to get the temperature
  as a function of radius, and the central
  temperature. Note that
• Assume that radiative energy transport dominates,
  and that a Kramers law can be used for the
  opacity, to obtain an expression for dT/dr.
  Evaluate it at r R/2.
• Then produce an expression for dT/dr from the
  result of problem C, similarly evaluated at r
  R/2.

16
Todays in-class problems (continued)

• Set equal the results from problems D and E, and
  obtain an expression for the luminosity generated
  within r R/2 which, because of the strong
  temperature dependence of energy generation,
  should be equal to the total luminosity of the
  star.