2 equations of stellar structure

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2 equations of stellar structure
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a stellar interior
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assumptions

• 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

Surface rR
X,Y,Z
r
M,L,0,Teff
m,l,P,T
Centre r0
0,0,Pc,Tc
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mass continuity

• Consider a spherical shell of radius rthickness
  ?r (?r ltltr)density ?
• Its mass (volume x density)
• ?m 4 ? r2 ? ?r
• As ?r?0
• dm/dr 4 ? r2 ? 2.1
• Also
• m 4 ? ? r2 ? dr

?r
?
r
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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
• 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

In order to oppose gravity, pressure must
increase towards the centre. For hydrostatice
equilibrium, forces must balance dP/dr - Gm ?
/ r2 2.5
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Virial theorem
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Virial theorem (2)
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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 Egrav are simply
  related to changes in Etot.

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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).

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conservation of energy

• Consider a spherical volume element dv4 ? 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 4?r2 ? dr,
• dl/dr 4?r2 ? ? 2.12
• We will consider the nature of energy sources,
  ?, later.

l?l
?r
??m
l
r
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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
  Flicks 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?r2 2.16a

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radiative energy transport (2)
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convective energy transport
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convective energy transport (2)
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convective energy transport (3)
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energy transport
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equations of stellar structure
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odes Lagrangian form
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boundary conditions
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constitutive relations
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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