Stellar Interior

Solar Facts Radius: –R  = 7  10 5 km = 109 R E Mass : –M  = 2  kg –M  = 333,000 M E Density: –   = 1.4 g/cm 3 –(water is 1.0 g/cm 3, Earth is 5.6 g/cm 3 ) Composition: –Mostly H and He Temperature: –Surface is 5,770 K –Core is 15,600,000 K Power: –4  W

Solar Layers Core –0 to 0.25 R  –Nuclear fusion region Radiative Zone –0.25 to 0.70 R  –Photon transport region Convective Zone –0.70 to 1 R  –Fluid flow region

Equilibrium A static model of a star can be made by balancing gravity against pressure. –Need mass density and pressure FgFg FbFb FtFt

Particles and States The particles in a star form a nearly ideal fluid. –Classical ideal gas –Quantum fluid The particles quantum states can be found by considering the particle in a box. –Dimension L –Wave vector (k x, k y, k z ) note:

Internal Energy The internal energy depends on the quantum states. –Density of states g(p)dp –Energy of each state  p –Number in each state f(  p ) The distribution depends on the type of particle –Fermion or boson –Reduces to Maxwell

Pressure The energy is related to the thermodynamic properties. –Temperature T –Pressure P –Chemical potential  The pressure comes from the energy. –Related to kinetic energy density

Relativity Effects The calculation for the ideal gas applied to both non-relativistic and relativistic particles. For non-relativistic particles For ultra-relativistic particles

Ideal Gas A classical gas assumes that the average occupation of any quantum state is small. –States are g(p)dp –State occupancy g s –Maxwellian f(  p ) The number N can be similarly integrated. –Compare to pressure –Equation of state –True for relativistic, also

Particle Density The equation of state is the same for both non-relativistic and relativistic particles. –Derived quantities differ For non-relativistic particles For ultra-relativistic particles

Electron Gas Electrons are fermions. –Non-relativistic –Fill lowest energy states The Fermi momentum is used for the highest filled state. This leads to an equation of state.

Relativistic Electron Gas Relativistic electrons are also fermions. –Fill lowest energy states –Neglect rest mass The equation of state is not the same as for non-relativistic electrons.

Electron Regimes Region A: classical, non- relativistic –Ideal gases, P = nkT Region B: classical, ultra- relativistic –P = nkT Region C: degenerate, non- relativistic –Metals, P = K NR n 5/3 Region D: degenerate, ultra- relativistic –P = K UR n 4/ T(K) n(m 3 ) A B CD

Hydrogen Ionization Particle equilibrium is dominated by ionized hydrogen. Equilibrium is a balance of chemical potentials. n = 1 n = 2 n = 3  p = p 2 /2m

Saha Equation The masses in H are related. –Small amount  n for degeneracy Protons and electrons each have half spin, g s = 2. –H has multiple states. The concntration relation is the Saha equation.