4 Stellar Structure To build a stellar model

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) • Energy Transport (radiative, convective, conductive transport) • Equation of State (relating P, T, and ρ ) • Opacity to radiative transport • Energy Generation (nuclear reaction parameters) 2

Pressure Radiation Energy Density

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

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

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

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,

Proton-Proton Chain

Nuclear Reaction Rates in Stars: Proton-Proton Chain X=Y=1/2, ρ = 100 g cm-3, T=15 x 106 Reaction PPI Energy Liberated (Me. V) non-neutrino 0. 16 0. 26 1. 02 5. 49 12. 86 Lifetime 8 x 109 years 1. 5 sec 2. 4 x 105 years 2 A+2 B+C⇒ 26. 2 Me. V in available energy plus 0. 5 Me. V lost to star via neutrinos PPIIA + B and primordial 4 He 1. 59 106 years 0. 06 0. 80 0. 4 years 17. 35 10 minutes 10 -16 seconds PPIII A+B+D 0. 13 10. 78 7. 2 66 years 1 second For T<10 x 106 K, the p-p chain terminates with (B). IF T> 10 x 106 K, (C) also occurs. If T>14 x 106 K, PPII and PPIII contribute, and in fact PPII will dominate over PPI in the production of energy.

CNO Cycle With a sufficient abundance of carbon and Tcore≥ 16 x 106 K, a catalytic reaction involving 12 C occurs. (Note: if T<13 x 106 K, the cycle time > age of the Sun (≥ 5 x 109 years)).

CNO Cycle solar-like core – T=15 x 106 K, ρ=100 g cm-3, X=0. 6 shell source of a post-main sequence 5 MSun star - T=30 x 106 K, ρ =26 g cm-3, X=0. 2 Reaction Energy non-neutrino 1. 94 1. 51 7. 55 7. 29 1. 76 4. 97 0. 71 1. 00 Time Solar-Like Shell Source 1. 5 x 106 yrs 270 yrs 14 minutes 3. 7 x 105 yrs 66 years 3 x 108 yrs 1500 yrs 3 minutes 104 yrs 0. 6 yrs Net result: If T>17 x 106 K, an alternate set of branches can occur 0. 3% of the time.

T 6 15 20 25 30 ν(PP) 3. 9 3. 5 3. 2 3. 0 T 6 20 30 40 50 60 70 80 90 100 ν(CNO) 18. 0 15. 6 14. 1 13. 1 12. 3 11. 6 11. 1 10. 6 10. 2

3 -α Process = “Helium-burning” When He is abundant, ρ ~ 104 – 105 g cm-3, and T> 108 K, the 3 -α He -burning reaction can occur (Salpeter 1952). The reaction releases 7. 27 Me. V per 12 C produced. This is often followed by

At T > 6 x 108 K, Carbon-burning occurs: At T > 1 -1. 5 x 109 K, Oxygen-burning occurs:

THE FUNDAMENTAL EQUATIONS OF STELLAR STRUCTURE