14-Stellar Interiors - Physics

Equation of Mass Continuity (Conservation)If M ris the total mass inside radius r , thenr( )M r = ∫ 4πr 2 ρ( r)dr, dM0r = 4πr 2 dr ρ rSo,dL rdr = 4πr2 ρ rdM rdr = 4πr2 ρ r( )ε rvol.of sphericalshell at r withthicknessdr( )( ) = dL rdr( )( ) = dM rdrEquation of Mass ContinuityAs we will see later, most modern models will use M rinstead of r as the dependent variable in the models(integrating dM rinstead of dr ), so this equation will be used to do the conversion.Equation of Energy ConservationIn a similar manner, we can track the energy generated within a radius r due to the energy being generated per unitmass ε ( r) or per unit volume, which would be ρ( r)ε ( r).∫rL r = 4πr 20ρ( r)ε ( r)dr, dL r = 4πr 2 drρ( r)ε rdL rdr = 4πr2 ρ( r)ε r( ) = dL ( r )drEquation of Energy Conservation( )( )10