Hydrostatic Equilibrium

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Hydrostatic Equilibrium

• Physical Astronomy
• Professor Lee Carkner
• Lecture 10

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Questions

• How does the Voigt profile depend on wavelength?
• Increase l, increase width
• How does the Voigt profile depend on temperature?
  
• Increase T, increase width
• How does the Voigt profile depend on atomic mass?
  
• Increase m, decrease width
• How does the Voigt profile depend on number
  density?
• Increase n, increase width
• Goes as n

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Stellar Interiors

• We cannot see stellar interiors directly
• Our theories on the interior of stars come from
  stellar models
• Key variables are pressure, temperature and
  density as a function of radius

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Equilibrium

• A star is just a big ball of gas
• Since the star is static, the two forces must be
  equal
• Force of gravity on small mass dm is
• Fg -(GMrdm)/r2
• Where r is the distance from the center of the
  star and Mr is the mass interior to r

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Pressure

• Since P F/A and dm rAdr, we can write
• dP/dr -GMrr/r2 -rg
• dP/dr is the pressure gradient
• Pressure decreases as we move towards the surface

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Equation of State

• Is related to density and temperature
• Example ideal gas law
• Where N is number of particles and n is the
  number density of particles
• Remember ideal gas law is gets less accurate as
  the density increases

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Mean Weight

• n r/mmean
• We then define the mean molecular weight, m in
  terms of the mass of hydrogen
• m mmean/mH
• P rkT/mmH

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Gas Composition

• If there are heavier elements, m increases
• For example, stars are mostly hydrogen with
  significant helium and very small amounts of
  heavier elements
• If the gas is ionized, m decreases
• Ionized gas, m 0.62

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Mass Continuity

• Mass is continuous
• dMr r(4pr2 dr)
• dMr/dr 4pr2 r
• The total mass is just the integral over the
  whole star

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Particle Energy

• Kinetic energy thermal energy
• ½mv2 (3/2)kT
• Eventually fast enough to overcome Coulomb
  repulsion

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Radiation Pressure

• Radiation pressure
• Prad (4s/3c)T4
• Note that Prad is strongly temperature dependant

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Energy Transport

• Energy is transported in stars either by
  convection or radiation
• Convection dominates when
• Radiation cant get through
• Even with low opacity a lot of photons get
  absorbed

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Transport in Stars

• Low mass stars are un-ionized and have high
  opacity (convection)
• Near the cores of medium mass stars there is high
  ionization and thus low opacity (rad then conv)

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Next Time

• Read 10.3, 10.6, 11.1
• Homework 10.22, 10.23a, 11.1, 11.2a