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2 equations of stellar structure

2 equations of stellar structure. a stellar interior. Isolated body – only forces are self-gravity internal pressure Spherical symmetry Neglect: rotation magnetic fields Consider spherical system of mass M and radius R Internal structure described by: r radius m(r) mass within r

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2 equations of stellar structure

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  1. 2 equations of stellar structure Stellar Structure: TCD 2006: 2.1

  2. a stellar interior Stellar Structure: TCD 2006: 2.2

  3. Isolated body – only forces areself-gravity internal pressure Spherical symmetry Neglect: rotation magnetic fields Consider spherical system of mass M and radius R Internal structure described by: r radius m(r) mass within r l(r) flux through r T(r) temperature at r P(r) pressure at r [ (r) density at r ] assumptions Surface: r=R X,Y,Z r M,L,0,Teff m,l,P,T Centre: r=0 0,0,Pc,Tc Stellar Structure: TCD 2006: 2.3

  4. Consider a spherical shell of radius rthickness r (r <<r)density  Its mass (volume x density): m = 4  r2  r As r0: dm/dr = 4  r2  2.1 Also: m= 4  r2 dr mass continuity r  r Stellar Structure: TCD 2006: 2.4

  5. 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 If a pressure gradient (dP/dr) exists, there will be a nett inward force acting on an element of thickness r and area A: dP/dr r A  m /  dP/dr (element mass is m =  r A) The sum of inward forces is then m ( g +1/  dP/dr ) = - m d2r/dt2 P+P g z  P r r hydrostatic equilibrium In order to oppose gravity, pressure must increase towards the centre. For hydrostatice equilibrium, forces must balance: dP/dr = - Gm  / r2 2.5 Stellar Structure: TCD 2006: 2.5

  6. Virial theorem Stellar Structure: TCD 2006: 2.6

  7. Virial theorem (2) Stellar Structure: TCD 2006: 2.7

  8. 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 non-relativistic 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 Egravare simply related to changes in Etot. Stellar Structure: TCD 2006: 2.8

  9. 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

  10. conservation of energy • Consider a spherical volume element dv=4  r2dr • 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 = 4r2 dr, • dl/dr = 4r2  2.12 • We will consider the nature of energy sources, , later. l+l r l m r Stellar Structure: TCD 2006: 2.10

  11. 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 4r2 to obtain a luminosity l, whence • L = - (4r2c / 3) d(aT4)/dr •  dT/dr = 3/4acT3 l/4r22.16a Stellar Structure: TCD 2006: 2.11

  12. radiative energy transport (2) Stellar Structure: TCD 2006: 2.12

  13. convective energy transport Stellar Structure: TCD 2006: 2.13

  14. convective energy transport (2) Stellar Structure: TCD 2006: 2.14

  15. convective energy transport (3) Stellar Structure: TCD 2006: 2.15

  16. energy transport Stellar Structure: TCD 2006: 2.16

  17. equations of stellar structure Stellar Structure: TCD 2006: 2.17

  18. ode’s: Lagrangian form Stellar Structure: TCD 2006: 2.18

  19. boundary conditions Stellar Structure: TCD 2006: 2.19

  20. constitutive relations Stellar Structure: TCD 2006: 2.20

  21. 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

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