1 Stellar Structure Hydrostatic Equilibrium Spherically symmetric Newtonian equation of hydrostatics dP dr Gm r 2 dm dr 4 r 2 1 m r is mass enclosed within radius r Conditions at stellar centers Q P Gm2 8 r 4 dQ dP Gm dm Gm2 Gm2 0 dr dr 4 r 4 dr 2 r 5 2 r 5 Q r 0 Pc 2 Q r R GM 2 8 R4 M and R are total mass and radius Central pressure Pc Milne inequality 4 2 2 M R GM 14 dynes cm 2 4 10 Pc 4 M R 8 R 3 Average density is 3M M R 3 1 4 g cm 3 3 M R 4 R Estimate of Tc from perfect gas law M R Pc Tc 2 1 106 K No M R is mean molecular weight Tc too low by factor of 7 Better Estimate i h 2 c 1 r R M 8 15 cR3 1 r 6 4 2 2 1 r 2 2 r 4 P Pc G c R 3 2 R 5 R 10 R 4 5 6 2 P R 0 15GM 2 2 4 M R 15 3 5 10 dynes cm 2 7 4 M R 16 R 2 r 2 2 2 1 r Pc P Pc 1 1 1 R 2 R 2 c c 8 The central density is 5 M R 3 15M 3 6 g cm 3 9 c 3 2 M R 8 R and the central temperature becomes M R Pc K 10 7 0 106 Tc cNo M R Mean molecular weight Perfect ionized gas kB 1 X P T 1 Zi ni NoT N T 11 Pc i Zi is charge of ith isotope Abundance byP mass of H He and everything else denoted by X Y and Z i He niAi No Assuming 1 Zi Ai 2 for i He 1 X ni 1 Zi 3 4 4 2X Y 4 No 2 6X Y 3 5X Z i He 12 Solar gas X 0 75 Y 0 22 Z 0 03 has 0 6 Number of electrons per baryon for completely ionized gas Zi He Ai 2 X ni Zi Z 1 Y Y X 1 X 13 Ye X 2 No 2 2 2 i He 3 The Virial Theorem Position momentum mass of ith particle ri p i mi Newton s Law F i p i with p i mi r i 2I X X d X 1 d d p i ri mi r i ri p i ri p i r i dt dt 2 d2 t 14 P Moment of inertia I mi ri2 Static situation d2I dt2 0 P 2 P Non relativistic gas mi ri p i r i 2K Total kinetic energy 1X 1X 1X K p i ri p i ri Fi ri 1 2 2 2 2 15 Sum is virial of Clausius For perfect gas only gravitational forces contribute since forces involved in collisions cancel X Gmimj X X G G Fij ri rj Fi ri 16 rij pairs pairs is gravitational potential energy rij ri rj Perfect gas with constant ratio of specific heats cp cv K 3 2 N T U 1 1 N T U is internal energy E is total energy E U U 4 3 E U U 2K 3 4 3 1 17 For 4 3 E 0 4 3 E 0 configuration unstable 4 3 E 0 configuration stable and bound by energy E Application contraction of self gravitating mass 0 If 4 3 E 0 so energy is radiated However U 0 so star grows hotter 4 P Relativistic gas p i ri c p i K Another derivation P V dP 1 Gm 1 dm d 3 r 3 18 where V 4 r 3 3 Its integral is Z V r dP P V R 0 Z 1 P r dV 3 19 R from Eq 18 Thus 3 P r dV Non relativistic case P 2 3 2K E K 2 Relativistic case similar to non relativistic case with 4 3 P 3 K E 0 The critical nature of 4 3 is important in stellar evolution Regions of a star which through ionization or pair production maintain 4 3 will be unstable and will lead to instabilities or oscillations Entire stars can become unstable if the average adiabatic index drops close to 4 3 and this actually sets an upper limit to the masses of stars As we will see the proportion of pressure contributed by radiation is a steeply increasing function of mass and radiation has an effective of 4 3 We now turn our attention to obtaining more accurate estimates of the conditions inside stars 5 Polytropic Equations of State The polytropic equation of state common in nature satisfies n 1 n P K K 20 n is the polytropic index and is the polytropic exponent 1 Non degenerate gas nuclei electrons and radiation pressure If Pgas Ptotal is fixed throughout a star 1 3 No 3No P 1 4 3 21a a 1 3 3No T 1 1 3 21b a Here and a are the mean molecular weight of the gas and the radiation constant respectively Thus n 3 2 A star in convective equilibrium Entropy is constant If radiation pressure is ignored then n 3 2 3 2 2 5 h s ln No constant 22a 2 2mT 2 5 h 2 No 5 3 exp s K 5 3 22b P 2m 3 3 3 An isothermal non degenerate perfect gas with pairs radiation and electrostatic interactions negligible n Could apply to a dense molecular cloud core in initial collapse and star formation 4 An incompressible fluid n 0 This case can be roughly applicable to neutron stars 5 Non relativistic degenerate fermions n 3 2 Lowdensity white dwarfs cores of evolved stars 6 6 Relativistic degenerate fermions n 3 High density white dwarfs 7 Cold matter at very low densities below 1 g cm 3 with Coulomb interactions resulting in a pressure density law of the form P 10 3 i e n 3 7 Don t confuse polytropic with adiabatic indices A polytropic change has c dQ dT is constant where dQ T dS An adiabatic change is a specific case c 0 c c c ln P p v ln V cp c cv where the adiabatic exponent ln P ln V s If cp cv as for a perfect gas c cp c cv In the adiabatic case c 0 and regardless of s value Polytropes Self gravitating fluid with a polytropic equation of state is a polytrope with Z Z 3 GM 2 Gm r dm r 3 P dV 23 r 5 n R For a perfect gas with constant specific heats 3 4 1 GM 2 E 1 5 n R For the adiabatic case n 1 1 24 n 3 GM 2 E 25 5 n R For a mixture of a perfect gas and radiation Z Z 4 3 U 3 1 P dV P dV 26 …