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2 equations of stellar structure Stellar Structure: TCD 2006: 2.1 a stellar interior Stellar Structure: TCD 2006: 2.2 assumptions Isolated body – only forces are Surface: r=R self-gravity internal pressure Spherical symmetry Neglect: rotation X,Y,Z magnetic fields r Consider spherical system of mass M and radius R Internal structure described by: r radius m,l,P,T m(r) mass within r Centre: r=0 l(r) flux through r 0,0,Pc,Tc T(r) temperature at r P(r) pressure at r [ (r) density at r ] Stellar Structure: TCD 2006: 2.3 M,L,0,Teff mass continuity Consider a spherical shell of radius r thickness r (r <<r) density Its mass (volume x density): m = 4 r2 r As r0: dm/dr = 4 r2 r r dm 4r 2 dr 2.1 Also: m= 4 r2 dr Stellar Structure: TCD 2006: 2.4 hydrostatic equilibrium Consider forces at any point. A sphere of radius r acts as a gravitational mass situated at the centre, giving rise to a force: g = Gm/r2 g If a pressure gradient (dP/dr) exists, there will be a nett inward force acting dm on 4r r an element of thickness r and areadA: dP/dr r A m / dP/dr (element mass is m = r A) zP 2 The sum of inward forces is then m ( g +1/ dP/dr ) = - m d2r/dt2 P+P r r In order to oppose gravity, pressure must increase towards the centre. For hydrostatice equilibrium, forces must balance: dP/dr = - Gm / r2 Stellar Structure: TCD 2006: 2.5 2.5 Virial theorem The whole system is in equilibrium if 2.5 is satisfied at all r, whence it is possible to derive a simple relation between average internal pressure and the gravitational potential energy of the system. Multiplying both sides by 4r3 and integrate from r=0 to r=R: R 0 R Gm dP dr 4r 2 dr 0 dr r 4r 3 Integrate lhs by parts (du=dP/dr.dr, dv=4r3), and substitute dm = 4r2dr 4r P 3 R 0 R 3 4r 2 Pdr 0 M 0 Gm dm r Since P(R)=0, the first term iz zero. Substituting 4r2dr=dv V 3 Pdv 0 M 0 Gm dm r Stellar Structure: TCD 2006: 2.6 Virial theorem (2) V 3 Pdv 0 M 0 Gm dm r lhs: simply 3<P>V, where <P> is the volume-averaged pressure, rhs: is the gravitational potential energy of the system Egrav. Thus the average pressure needed to support a system with gravitational energy Egrav and volume V is given by 1 E grav P 3 V 2.6 This is the Virial Theorem. The physical meaning of pressure depends on the system itself, but it can be applied to clusters of galaxies, cooling flows, globular cluster as well as to individual stars. Stellar Structure: TCD 2006: 2.7 Virial theorem: non-relativistic gas In a star, an equation of state relates the gas pressure to the translational kinetic energy of the gas particles. For nonrelativistic particles: P = nkT = kT/V and Ekin = 3/2 kT and hence P = 2/3 Ekin/V 2.7 Applying the Viral theorem: for a self-gravitating system of volume V and gravitational energy Egrav, the gravitational and kinetic energies are related by 2Ekin + Egrav = 0 2.8 Then the total energy of the system, Etot = Ekin + Egrav = –Ekin = 1/2 Egrav 2.9 These equations are fundamental. If a system is in h-s equilibrium and tightly bound, the gas is HOT. If the system evolves slowly, close to h-s equilibrium, changes in Ekin and Egrav are simply related to changes in Etot. Stellar Structure: TCD 2006: 2.8 Virial theorem: ultra-relativistic gas For ultra-relativistic particles: Ekin = 3 kT and hence P = 1/3 Ekin/V 2.10 Applying the Viral theorem: Ekin + Egrav = 0 2.11 Thus h-s equilibrium is only possible if Etot = 0. As the u-r limit is approached, ie the gas temperature increases, the binding energy decreases and the system is easily disrupted. Occurs in supermassive stars (photons provide pressure) or in massive white dwarfs (rel. electrons provide pressure). Stellar Structure: TCD 2006: 2.9 conservation of energy Consider a spherical volume element dv=4 r2 dr Conservation of energy demands that energy out must equal energy in + energy produced or lost within the element If is the energy produced per unit mass, then l+l = l + m dl/dm = Since dm = 4r2 dr, dl/dr = 4r2 2.12 We will consider the nature of energy sources, , later. Stellar Structure: TCD 2006: 2.10 l+l r l r m radiative energy transport A temperature difference between the centre and surface of a star implies there must be a temperature gradient, and hence a flux of energy. If transported by radiation, then this flux obeys Flick’s law of diffusion: F = -D d(aT4)/dr where aT4 is the radiation energy density and D is a diffusion coefficient. We state (for now) that D is related to the “opacity” (actually: D = c/) The flux must be multiplied by 4r2 to obtain a luminosity l, whence L = - (4r2c / 3) d(aT4)/dr dT/dr = 3/4acT3 l/4r2 Stellar Structure: TCD 2006: 2.11 2.16a radiative energy transport (2) Writing the temperature gradient as d ln T : d ln P dT GmT 2 dr r P 2.16 where, in radiative equilibrum d ln T 3 lP rad d ln P 16ac GmT 4 2.17 These equations are obtained by combining 2.16a with h-s equilibrium and taking logs. Stellar Structure: TCD 2006: 2.12 convective energy transport An element of gas is at some radius r. Consider its upward displacement by a * distance r, allowing it to expand * P2 ,2 P2,2 adiabatically until the pressure within is equal to the pressure outside. Release r the element. If it continues to move upwards, the layer in question is * * P1 ,1 P1,1 convectively unstable. Let the pressures and densities be denoted by P*, *, and P, respectively. Initially P*1=P1 and *1=1. After the perturbation, P*2=P2 and *2=*1(P*2/P*1)1/, where PV=c and =5/3 for a highly-ionized gas. For radiative equilibrium, we require *2>2 so that the net force (bouyancy+gravitation) is downwards and the element will return to its starting position. Stellar Structure: TCD 2006: 2.13 convective energy transport (2) Eliminating asterisks and writing * P1=P2+dP, we obtain P d 1 dP * P2 ,2 2.18 P2,2 r for radiative equilibrium. * * P1 ,1 P1,1 This condition is related to the temperature gradient assuming some equation of state (e.g. P=kT/m) so that 2.18 becomes d ln T 1 ad d ln P Stellar Structure: TCD 2006: 2.14 2.19 convective energy transport (3) There are two main circumstances under which 2.19 will fail. 1.In the centre of main-sequence stars, the radiation flux l/4r2 can become very large, whilst remains small. Thus the temperature gradient dlnT/dlnP required for radiative equilibrium becomes large, and the material becomes convectively unstable. This gives rise to convective cores in massive stars. 2.In ionisation zones, a) the adiabatic exponent becomes close to unity, and b) the opacity may become very large. Hence radiative equilibrium may be violated for small values of the temperature gradient. This gives rise to convective envelopes in cool stars. Stellar Structure: TCD 2006: 2.15 energy transport From 2.17 we have in radiative equilibrium rad d ln T 3 lP d ln P 16ac GmT 4 2.20 and from 2.19 we have in adiabatic convective equilibrium 1 ad 2.21 In formulating the stellar structure problem we often require a single expression for the temperature gradient and write d ln T 1 rad ad d ln P 2.22 where is represents a convective efficiency such that 1.=0: radiative equilibrium 2.=1: adiabatic convection 3.0<<1: non-adiabiatic convection - must be determined from convection theory Stellar Structure: TCD 2006: 2.16 equations of stellar structure We have derived four time-independent equations of stellar structure. These form a set of coupled first order ode’s in one independent variable, r, and four dependent variables, m,l,P,T, which describe the structure of the star dm 4r 2 . dr 2.1 dP Gm 2 dr r 2.5 dl 4r 2 . dr 2.12 dT GmT 2 dr r P 2.16 Stellar Structure: TCD 2006: 2.17 ode’s: Lagrangian form Note that any variable could be used as the independent variable. In an Eulerian frame, r, the spatial coordinate is the independent variable. However, in dealing with most problems in stellar structure and evolution it is more appropriate to work in a Lagrangian frame, with mass as the independent variable. dr 1 dm 4r 2 2.23 dl dm 2.24 dP Gm dm 4r 4 2.25 dT Gm T 4 dm 4r P 2.26 Stellar Structure: TCD 2006: 2.18 boundary conditions To solve a set of odes, boundary conditions are required. For this 1d description of a star, the boundaries are at the centre and the surface. In the centre, the enclosed mass and luminosity are defined r(m=0) = 0 2.27 l(m=0) = 0 2.28 At the surface the temperature and pressure can be defined, to first approximation, by T(m=M) = Teff 2.29 Pgas(m=M) = 0 2.30 Teff is related to the stellar luminosity and radius by L=4R2Teff4. Thus we have four first order odes (2.24-2.26) and four bcs (2.27-2.30). Stellar Structure: TCD 2006: 2.19 constitutive relations In addition, , , refer to energy generation, density and energy transport, the last depending on and , the convective efficiency and opacity. These quantities describe the physics of the stellar material and may be expressed in terms of the state variables (P and T) and of the composition of the stellar material (X,Y,Z or Xi). These constitutive relations are required to close the system of ode’s: = (P,T,Xi) (equation of state) 2.31 = (,T,Xi) (nuclear energy generation rate) 2.32 = (,T,Xi) (opacity) 2.33 = (,T,Xi) (convective efficiency) 2.34 = (,T,,,Xi) (energy transport) Stellar Structure: TCD 2006: 2.20 2.35 2 equations of stellar structure -- review Assumptions: spherical symmetry, … Mass continuity Hydrostatic equilibrium Conservation of energy Radiative energy transport Convective energy transport The Virial theorem Eulerian and Lagrangian forms Boundary Conditions Constitutive equations Stellar Structure: TCD 2006: 2.21