Today in Astronomy 241: nuclear fusion in stars

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Today in Astronomy 241 nuclear fusion in stars

• Todays reading Carroll and Ostlie Chap. 10.3,
  on
• Nuclear fusion
• Nuclear energy generation in stars
• The luminosity equation

(NASA/GSFC)
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The Sun is nuclear powered.

• As you are presumed to know, from AST 111 and
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• The solar system is known to be about 4.5?109
  years old, from isotope dating of meteorites and
  of the oldest rocks found on the Moon. The Sun is
  at least as old as the solar system.
• Thus the Sun cannot be gravity powered this
  would only last of order 107 years (example
  10.3).
• Thus the Sun cannot be burning in the normal
  sense (chemical-energy powered) this would only
  last of order 105 years (problem 10.3).
• Thus the Sun is nuclear powered, by fusion of the
  nuclei of light elements (mostly hydrogen) this
  can last at least 1010 years (example 10.4).

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Nuclear reaction cross sections

• The quantity that is needed in nuclear reaction
  calculations, and is measurable in the
  laboratory
• Cross section for fusion of two nuclei (charges
  Z1 and Z2)

where S(E) varies little with varying energy, and
where
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Nuclear reaction rates

• Reaction rate coefficient (reactions per unit
  time per unit volume) for particles i and target
  x
• The integrand, apart from S(E), is sharply peaked
  at energy(the Gamow peak see problem 10.8)

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MB distrib. X Penetration prob. gt Gamow peak
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Nuclear energy generation in stars

• S(E) usually expanded in a power series result
  of integral parameterized aswhere a is
  usually 2 (two-body collisions) and b can vary
  over a wide range (1 to 40!). The energy
  generation rate for a given reaction is where
  a a-1, and the total energy generation rate is
  the sum of these expressions for all reactions.

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The luminosity equation

• where Lr is the luminosity generated within the
  star between the center and radius r, and r and e
  are the mass density and total energy generation
  rate at radius r.
• This is the fourth of the important equations
  that must be solved simultaneously to determine
  the interior structure of a star, along with
  hydrostatic equilibrium, mass conservation, and
  the equation of state.

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The proton-proton chains

• PP I for example (70 of pp chain reactions)

From Chaisson and McMillan, Astronomy Today
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Leftover problem from last class

• Recall that for
  photons. Show that the momentum distribution
  function for photons emitted by a blackbody
  isand from this and the pressure integral
  re-derive the radiation pressure from a
  blackbody.

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Todays in-class problems

• Problem 10.7.
• Show also that apart from the factor S(E), the
  integrand in the equation for nuclear reaction
  rates can be written as
• Calculate the ratio of the values of
  at 107 and 108 K, to gain an impression of
  the extreme temperature sensitivity of nuclear
  reactions in stars.

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Todays in-class problems (continued)

• Answers and/or secrets of the problems we did
  last class
• 1. Were supposed to derive the ideal gas law
  from the pressure integral and the
  Maxwell-Boltzmann distribution. Ignoring
  relativity, we write the pressure integral as
• Abbreviate and instead of
  considering the integral directly, try instead
  to calculate

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Todays in-class problems (continued)

• Substitute polar coordinates
• and then make a few more substitutions

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Todays in-class problems (continued)

• You have done the u integral many times
  integrate it by parts four times, and you get

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Todays in-class problems (continued)

• Now, note that ? is half a cycle for both sin and
  cos, so

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Todays in-class problems (continued)

• Thus,

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Todays in-class problems (continued)

• One can use the same basic integration of
  Gaussians process, or a combination of this
  basic process and integration by parts, to
  integrate the more general formBut that
  would take a few more pages, and I think its
  safe to leave to the imagination

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Todays in-class problems (continued)

• 2. To derive blackbody radiation pressure from
  the pressure integral and the momentum
  distribution of photons in blackbody radiation,
  start with the fully-relativistic form of the
  integral and take the speed of the particle to be
  c