Nuclear reactions and stellar Luminosity

1
Nuclear reactions and stellar Luminosity

• We have shown that to provide the observed
  luminosity of the sun for billions of years
  requires nuclear fusion reactions (burning).
  However, it is improper to say the nuclear
  reaction rate determines the luminosity. The
  luminosity is determined by the temperature
  structure within the star
• which we know fairly well from the virial theorem
• and the rate of flow of energy
• heat energy flows from hotter region to cooler
  region by
• conduction
• convection
• radiation
• So the nuclear reactions are important in
  determining
• The position on the evolutionary track for the
  main sequence phase
• the lifetime on the main sequence
• But not the luminosity of the sun at its current
  radius

2
Stellar evolution to main sequence

• Another way of thinking about this is to imagine
  the evolutionary track of the sun on the H-R
  diagram
• because of this, we present only brief comments
  on the details of the nuclear reactions. As long
  as there are more tightly bound nuclei, the star
  can contract, thus raising the temperature, until
  the nuclear reaction rate equals the luminosity
  from the star
• This party continues till we hit Fe (iron)
• Most tightly bound nucleus
• End of the line for contraction-induced
  luminosity
• Leads to supernovae

3
Nuclear reactions

• Speaking practically, all we need is the nuclear
  reaction energy generation rate e(r,T). This
  will depend on the composition, T, and density of
  the gas.
• For stars, the abundance of an element is
  described by Xatom, the fraction of the total
  mass from that element.
• X, Y, and Z are the mass fractions of Hydrogen,
  Helium, and everything else (sometimes referred
  to as metals)
• The initial sun had X 0.7, Y 0.28, and Z
  0.02
• Xoxygen 0.2
• For a human being, Xoxygen 60 and X 10
• Because of the enormous coulomb barrier, by far
  the easiest to burn (i.e. burns at the lowest
  temperatures) of abundant atoms is H.
• substantial H burning requires 3 million K
• sets limit of lowest mass star

4
Reaction rate, energy generation

• The rate of nuclear fusion energy generation is
  given by the energy released times the reaction
  rate
• Reaction rates are a generic problem in
  astropysics. They depend on
• the density of each reactant n1, n2
• the relative speed of the reactants, v
• the cross-section s (probability of a fusion
  reaction)
• consider the case where reactant-2 is stationary
  and reactant-1 moves with velocity v
• this would be the situation for electrons
  colliding with atoms, why?
• The volume swept out in time dt by one particle
  of reactant-1 is
• dV svdt
• The number of reactions is
• dN n2 dV
• the reaction rate per unit volume per unit time
  is
• r n1 dN/dt n1 n2sv

5
Energy generation rate

• If the energy liberated per reaction is Q, the
  energy generated per unit volume per unit time is
  just rQ.
• So e rQ/r
• recall definition of e
• For T.E., the velocities will be Maxwellian, and
  we have to take the average over the velocity
  distribution
• where normalized M-Bmann distribution
• and E (1/2)mv2
• Using the measured nuclear cross sections, and
  mass deficits (accounting for neutrinos, which
  simply leave the star without heating), the
  energy generation is calculated as a function of
  T. For the purposes of constructing a stellar
  model, the T dependence is fit with a power law
  near the temperature of interest.
• e e 0raTb
• a 1 for 2-body reaction (net Luminosity r2)
• b 4 to 10 (4 for p-p chain, 10 for CNO bi-cycle)

6
Coulomb barrier, tunneling, Gamow peak

• Since the interior is ionized, the nuclei are
  positively charged, and the repulsive potential
  is enormous slide 7
• Electric potential energy for 2 protons separated
  by 1 fermi is about U 1.4 MeV (see MathCAD and
  slide 7).
• kT 1 eV for 10,000 K, so for 10 million K it
  equals 1 keV
• classical probability e -(U/kT) essentially
  zero
• reaction only occurs through Q.M. tunneling.
• Tunneling probability exponential in ratio of
  barrier height U and energy of particle E
• U e-2/r
• r de Broglie wavelength h/p
• p mv
• E (1/2)mv2
• s exp(-2p2U/E) exp(-const.E -1/2)
• Product of Maxwellian distribution and tunneling
  gives Gamow Peak slide 8 and MathCAD

7
Coulomb barrier

• For discussion of coulomb barrier
• See http//hyperphysics.phy-astr.gsu.edu/hbase/nuc
  ene/coubar.html

8
tunneling, Gamow peak

• For discussion of Gamow tunneling peak
• See http//burro.astr.cwru.edu/Academics/Astr221/S
  tarPhys/coulomb.html