4 - Stellar Structure

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4 - Stellar Structure
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• To build a stellar model, we need to develop a
  series of equations
• Mass Conservation (distribution of mass with
  radius)
• Hydrostatic Equilibrium (balance of internal
  pressure gradient with gravity)
• Thermal Equilibrium (luminous energy versus
  radius)
• Energy Transport (radiative, convective,
  conductive transport)
• Equation of State (relating P, T, and ?)
• Opacity to radiative transport
• Energy Generation (nuclear reaction parameters)

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Pressure
Radiation Pressure
Radiation Energy Density
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Equation of State Gas Pressure
Ideal or Perfect Gas Law
Non-degenerate
For a fully ionized gas (deep interiors)
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Gas Degeneracy
Non-relativistic Degenerate
Relativistic Degenerate
(both are in cgs units)
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Equation of Mass Continuity
Note the gravitational potential energy of the
star will then be
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Hydrostatic Equilibrium
The force downward due to gravity on a mass
element of volume Adr is
The net force outward on the same mass element
due to the pressure gradient is
For equilibrium,
So,
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Proton-Proton Chain
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Nuclear Reaction Rates in Stars Proton-Proton
Chain XY1/2, ? 100 g cm-3, T15x106
Energy Liberated (MeV) Reaction
non-neutrino neutrino Lifetime PPI 0.16
0.26 8x109 years 1.02 5.49 1.5
sec 12.86 2.4x105 years 2A2BC?26.2 MeV
in available energy plus 0.5 MeV lost to star via
neutrinos PPII A B and primordial
4He 1.59 106 years 0.06 0.80 0.4
years 17.35 10 minutes 10-16
seconds PPIII ABD 0.13 66 years
10.78 7.2 1 second For Tlt10x106K,
the p-p chain terminates with (B). IF Tgt 10x106K,
(C) also occurs. If Tgt14x106K, PPII and PPIII
contribute, and in fact PPII will dominate over
PPI in the production of energy.
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CNO Cycle With a sufficient abundance of carbon
and Tcore16x106 K, a catalytic reaction
involving 12C occurs. (Note if Tlt13x106 K, the
cycle time gt age of the Sun (5x109 years)).
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CNO Cycle solar-like core T15x106K, ?100 g
cm-3, X0.6 shell source of a post-main sequence
5MSun star - T30x106K, ? 26 g cm-3,
X0.2 Energy Time Reaction
non-neutrino neutrino Solar-Like
Shell Source 1.94 1.5x106 yrs 270
yrs 1.51 0.71 14 minutes 7.55 3.7x105
yrs 66 years 7.29 3x108 yrs
1500 yrs 1.76 1.00 3 minutes 4.97 104
yrs 0.6 yrs Net result If Tgt17x106K, an
alternate set of branches can occur 0.3 of the
time.
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T6 ?(CNO) 20 18.0 30 15.6 40 14.1 50 13.1 60 12.3
70 11.6 80 11.1 90 10.6 100 10.2
T6 ?(PP) 15 3.9 20 3.5 25 3.2 30 3.0
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3-a Process Helium-burning When He is
abundant, ? 104 105 g cm-3, and Tgt 108K, the
3-a He-burning reaction can occur (Salpeter
1952). The reaction releases 7.27 MeV per 12C
produced. This is often followed by
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At T gt 6x108K, Carbon-burning occurs
At T gt 1-1.5x109K, Oxygen-burning occurs
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THE FUNDAMENTAL EQUATIONS OF STELLAR STRUCTURE
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