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4 - Stellar Structure

4 - Stellar Structure. 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)

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4 - Stellar Structure

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  1. 4 - Stellar Structure

  2. 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 ρ) • Opacity to radiative transport • Energy Generation (nuclear reaction parameters) 2

  3. Pressure Radiation Pressure Radiation Energy Density

  4. Equation of State & Gas Pressure “Ideal” or “Perfect” Gas Law Non-degenerate: For a fully ionized gas (deep interiors):

  5. Gas Degeneracy Non-relativistic Degenerate: Relativistic Degenerate: (both are in cgs units)

  6. Equation of Mass Continuity Note: the gravitational potential energy of the star will then be:

  7. 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,

  8. Proton-Proton Chain

  9. Nuclear Reaction Rates in Stars: Proton-Proton Chain X=Y=1/2, ρ = 100 g cm-3, T=15x106 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 2A+2B+C⇒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 A+B+D 0.13 66 years 10.78 7.2 1 second For T<10x106K, the p-p chain terminates with (B). IF T> 10x106K, (C) also occurs. If T>14x106K, PPII and PPIII contribute, and in fact PPII will dominate over PPI in the production of energy.

  10. CNO Cycle With a sufficient abundance of carbon and Tcore≥16x106 K, a catalytic reaction involving 12C occurs. (Note: if T<13x106 K, the cycle time > age of the Sun (≥5x109 years)).

  11. CNO Cycle solar-like core – T=15x106K, ρ=100 g cm-3, X=0.6 shell source of a post-main sequence 5MSun star - T=30x106K, ρ =26 g cm-3, X=0.2 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 T>17x106K, an alternate set of branches can occur 0.3% of the time.

  12. T6 ν(PP) 15 3.9 20 3.5 25 3.2 30 3.0 T6 ν(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

  13. 3-α Process = “Helium-burning” When He is abundant, ρ ~ 104 – 105 g cm-3, and T> 108K, the 3-α He-burning reaction can occur (Salpeter 1952). The reaction releases 7.27 MeV per 12C produced. This is often followed by

  14. At T > 6x108K, Carbon-burning occurs: At T > 1-1.5x109K, Oxygen-burning occurs:

  15. THE FUNDAMENTAL EQUATIONS OF STELLAR STRUCTURE

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