ASTR 1040 Recitation: Stellar Structure

Transcription

1 ASTR 1040 Recitation: Stellar Structure Ryan Orvedahl Department of Astrophysical and Planetary Sciences University of Colorado at Boulder Boulder, CO February 12, 2014

2 This Week MIDTERM: Thurs Feb 13 (regular class time, 9:30 am) Review Session: Wed Feb 12 (5:00-7:00 pm) Observing Session: Web Feb 12 (7:30 pm) R. Orvedahl (CU Boulder) Stellar Structure Feb 12 2 / 16

3 Today s Schedule Comments on Homework How to Build a Stellar Structure Model R. Orvedahl (CU Boulder) Stellar Structure Feb 12 3 / 16

4 How To Build A Star What physics do you need to build a star? R. Orvedahl (CU Boulder) Stellar Structure Feb 12 4 / 16

5 How To Build A Star What physics do you need to build a star? Gravity vs. Pressure R. Orvedahl (CU Boulder) Stellar Structure Feb 12 4 / 16

6 How To Build A Star What physics do you need to build a star? Gravity vs. Pressure Nuclear Reactions R. Orvedahl (CU Boulder) Stellar Structure Feb 12 4 / 16

7 How To Build A Star What physics do you need to build a star? Gravity vs. Pressure Nuclear Reactions Energy Transport R. Orvedahl (CU Boulder) Stellar Structure Feb 12 4 / 16

8 How To Build A Star What physics do you need to build a star? Gravity vs. Pressure Nuclear Reactions Energy Transport Equation of State R. Orvedahl (CU Boulder) Stellar Structure Feb 12 4 / 16

9 Gravity vs. Pressure Hyostatic Balance: How much mass? dp = GM r(r)ρ(r) r 2 M r (r) = r 0 4πr 2 ρ(r) R. Orvedahl (CU Boulder) Stellar Structure Feb 12 5 / 16

10 Gravity vs. Pressure Q: What is the source of the pressure gradient outside the core in the equation for Hyostatic Equilibrium? R. Orvedahl (CU Boulder) Stellar Structure Feb 12 6 / 16

11 Gravity vs. Pressure Q: What is the source of the pressure gradient outside the core in the equation for Hyostatic Equilibrium? A: Energy transport mechanisms such as radiative diffusion or convection Radiation exerts a pressure P rad = at 4 /3 R. Orvedahl (CU Boulder) Stellar Structure Feb 12 6 / 16

12 Nuclear Reactions Reaction rates, r i,j r i,j r 0 X i X j ρ α+1 T β Energy released / kg / sec, ɛ i,j ɛ i,j = E 0 ρ r i,j (E 0 = Energy / Rx) Combine the two equations ɛ i,j = ɛ 0X i X j ρ α T β R. Orvedahl (CU Boulder) Stellar Structure Feb 12 7 / 16

13 Nuclear Reactions Luminosity Energy released dl = ɛdm where ɛ = ɛ nuc + ɛ grav is the total energy released / kg / sec by all reactions and gravity dm = dm r = ρ(r)dv = 4πr 2 ρ(r) dl r = ɛdm r = 4πr 2 ρ(r)ɛ dl r = 4πr 2 ρ(r)ɛ R. Orvedahl (CU Boulder) Stellar Structure Feb 12 8 / 16

14 Nuclear Reactions dl r = 4πr 2 ρ(r)ɛ = 4πr 2 ɛ 0 ρ α+1 T β Reaction Name α β P-P Chain 1 4 CNO Cycle 1 15 Triple-α 2 40 R. Orvedahl (CU Boulder) Stellar Structure Feb 12 9 / 16

15 Energy Transport Remember radiation exerts a pressure P rad = at 4 /3 dp rad = 4 3 dt at 3 From Radiation Transport Theory dp rad = κρ c F rad Combine equations dt = 3 κρ F 4ac T 3 rad, where F rad = Lr 4πr 2 Get T in terms of L r dt = 3 κρ L r 4ac T 3 4πr 2 R. Orvedahl (CU Boulder) Stellar Structure Feb / 16

16 What Equations Do We Have So Far? 1 dp GMr (r)ρ(r) = r 2 # Eqns = 1, # Variables = 3: P(r), M r (r), ρ(r) R. Orvedahl (CU Boulder) Stellar Structure Feb / 16

17 What Equations Do We Have So Far? 1 dp GMr (r)ρ(r) = r 2 # Eqns = 1, # Variables = 3: P(r), M r (r), ρ(r) 2 dm r (r) = 4πr 2 ρ(r) Eqns = 2, Vars = 3: P(r), M r (r), ρ(r) R. Orvedahl (CU Boulder) Stellar Structure Feb / 16

18 What Equations Do We Have So Far? 1 dp GMr (r)ρ(r) = r 2 # Eqns = 1, # Variables = 3: P(r), M r (r), ρ(r) 2 dm r (r) = 4πr 2 ρ(r) Eqns = 2, Vars = 3: P(r), M r (r), ρ(r) 3 dl r = 4πr 2 ρ(r)ɛ = 4πr 2 ɛ 0 ρ α+1 T β Eqns = 3, Vars = 5: P(r), M r (r), ρ(r), L r (r), T (r) R. Orvedahl (CU Boulder) Stellar Structure Feb / 16

19 What Equations Do We Have So Far? 1 dp GMr (r)ρ(r) = r 2 # Eqns = 1, # Variables = 3: P(r), M r (r), ρ(r) 2 dm r (r) = 4πr 2 ρ(r) Eqns = 2, Vars = 3: P(r), M r (r), ρ(r) 3 dl r = 4πr 2 ρ(r)ɛ = 4πr 2 ɛ 0 ρ α+1 T β Eqns = 3, Vars = 5: P(r), M r (r), ρ(r), L r (r), T (r) 4 dt = 3 κρ 4ac T 3 L r 4πr 2 Eqns = 4, Vars = 5: P(r), M r (r), ρ(r), L r (r), T (r) R. Orvedahl (CU Boulder) Stellar Structure Feb / 16

20 Equation of State Gives the pressure in terms of density and temperature Ideal Gas: P = P(ρ, T ) P = ρkt m, m = mean atomic mass For the Sun: µ = m m H 1.6 Or Electron Degenerate Matter: Or... P = (3π2 ) 2/3 5 2 m e [ ( Z A ) ρ m H ] 5/3 R. Orvedahl (CU Boulder) Stellar Structure Feb / 16

21 Final Set of Equaions 1 dp GMr (r)ρ(r) = r 2 2 dm r (r) = 4πr 2 ρ(r) 3 dl r = 4πr 2 ρ(r)ɛ = 4πr 2 ɛ 0 ρ α+1 T β 4 dt = 3 κρ 4ac T 3 5 P = ρkt m L r 4πr 2 Still cannot solve without Boundary Conditions... R. Orvedahl (CU Boulder) Stellar Structure Feb / 16

22 Boundary Conditions M r 0 as r 0 L r 0 as r 0 ρ 0 as r R T T eff as r R P 0 as r R Now we can solve the system R. Orvedahl (CU Boulder) Stellar Structure Feb / 16

23 Numerically Integrate the System R. Orvedahl (CU Boulder) Stellar Structure Feb / 16

24 Numerically Integrate the System R. Orvedahl (CU Boulder) Stellar Structure Feb / 16