1B11 Foundations of Astronomy Sun (and stellar) Models Silvia Zane, Liz Puchnarewicz

1B11 Physical State of the stellar (Sun) interior  Although stars evolve, their properties change so slowly that at any time it is a good approximation to neglect the rate of change of these properties.  Stars are spherical and symmetrical about their centre; all physical quantities depend just on r, the distance from the centre.. Fundamental assumptions:

1B11 1) Equation of hydrostatic equilibrium.  Consider a small volume element at distance r from the centre, cross section S=2  r, thickness dr Concept 1): Stars are self-gravitating bodies in dynamical equilibrium  balance of gravity and internal pressure forces. (1)

1B11 2) Equation of distribution of mass.  Consider the same small volume element at distance r from the centre, cross section S=2  r, thickness dr (2)

1B11 First consequence: upper limit on central P  From (1) and (2): At all points within the star r 1/R S 4 : For the Sun: P c > Nm -2 = atm

1B11 Toward the E-balance equation: The virial theorem Thermal energy/unit volume u=nfkT/2=(  /  m H )fkT/2 Ratio of specific heats  =c P /c V =(f+2)/f (f=3:  =5/3) U= total thermal Energy;  = total gravitational energy

1B11 Toward the E-balance equation: The virial theorem E is negative and equal to  /2 or –U A decrease in E leads to a decrease in  but an increase in U and hence T. A star, with no hidden energy sources, c.omposed of a perfect gas, contracts and heat up as it radiates energy Stars have a negative “heat capacity” = they heat up when their total energy decreases. For a fully ionized gas  =5/3 and 2U+  =0 Total Energy of the star: E=U+ 

1B11 Toward the E-balance equation Energy loss at stellar surface as measured by stellar luminosity is compensated by energy release from nuclear reactions through the stellar interior. Sources of stellar energy: since stars lose energy by radiation, stars supported by thermal pressure require an energy source to avoid collapse.  r =nuclear energy released per unit mass per s. Depends on T,  and chemical composition During rapid evolutionary phases (contraction/expansion): TdS/dt accounts for the gravitational energy term

1B11 The equations of Stellar structure P,k,  r are functions of ,T, chemical composition (basic physics provides these expressions) In total: 4, coupled, non-linear partial differential equations (+ 3 constitutive relations) for 7 unknowns: P, ,T, M, L, k,  r as a function of r. These completely determine the structure of a star of given composition, subject to suitable boundary conditions. in general, only numerical solutions can be obtained (=computer). Summary:

1B11 The equations of Stellar structure Boundary conditions: M=0, L=0 and r=0; M=M s L=4R S 2  T eff 4 and P=2/3g/k These equations must be solved for specified M s and composition. Using mass as independent variable (better from a theoretical point of view): “For a given chemical composition, only a single equilibrium configuration exists for each mass; thus the internal structure is fixed”. This “theorem” has not been proven and is not rigorously true; there are unknown exceptions (for very special cases) Uniqueness of solution: the Vogt Russel “theorem”:

1B11 Last ingredient: Equation of State N=number density of particles;  =mean particle mass in units of m H. Define: Perfect gas: X= mass fraction of H (Sun=0.70) Y= mass fraction of He (Sun =0.28) Z= mass fraction of heavy elements (metals) (Sun=0.02) X+Y+Z=1

1B11 Last ingredient: Equation of State A=average atomic weight of heavier elements; each metal atom contributes ~A/2 electrons Total number of particles: If the material is assumed to be fully ionized: N=(2X+3Y/4+Z/2)  /m H (1/  ) = 2X+3Y/4+Z/2) Very good approximation is “standard” conditions!

1B11 Deviations from a perfect gas 1) When radiation pressure is important (very massive stars): The most important situations in which a perfect gas approximation breaks down are: 2) In stellar interiors where electrons becomes degenerate (very compact stars, with extremely high density): here the number density of electrons is limited by the Pauli exclusion principle)