ASTR 1040 Recitation: Stellar Structure Ryan Orvedahl Department - - PowerPoint PPT Presentation

ASTR 1040 Recitation: Stellar Structure

Ryan Orvedahl

Department of Astrophysical and Planetary Sciences University of Colorado at Boulder Boulder, CO 80309 ryan.orvedahl@colorado.edu

February 12, 2014

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

Today’s Schedule

Comments on Homework How to Build a Stellar Structure Model

• R. Orvedahl (CU Boulder)

Stellar Structure Feb 12 3 / 16

How To Build A Star

What physics do you need to build a star?

• R. Orvedahl (CU Boulder)

Stellar Structure Feb 12 4 / 16

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

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

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

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

Gravity vs. Pressure

Hydrostatic Balance: dP dr = −GMr(r)ρ(r) r 2 How much mass? Mr(r) = r 4πr 2ρ(r)dr

• R. Orvedahl (CU Boulder)

Stellar Structure Feb 12 5 / 16

Gravity vs. Pressure

Q: What is the source of the pressure gradient outside the core in the equation for Hydrostatic Equilibrium?

• R. Orvedahl (CU Boulder)

Stellar Structure Feb 12 6 / 16

Gravity vs. Pressure

Q: What is the source of the pressure gradient outside the core in the equation for Hydrostatic Equilibrium? A: Energy transport mechanisms such as radiative diffusion or convection Radiation exerts a pressure Prad = aT 4/3

• R. Orvedahl (CU Boulder)

Stellar Structure Feb 12 6 / 16

Nuclear Reactions

Reaction rates, ri,j ri,j ≈ r0XiXjρα+1T β Energy released / kg / sec, ǫi,j ǫi,j = E0

ρ ri,j

(E0 = Energy / Rx) Combine the two equations ǫi,j = ǫ′

0XiXjραT β

• R. Orvedahl (CU Boulder)

Stellar Structure Feb 12 7 / 16

Nuclear Reactions

Luminosity ∝ Energy released dL = ǫdm where ǫ = ǫnuc + ǫgrav is the total energy released / kg / sec by all reactions and gravity dm = dMr = ρ(r)dV = 4πr 2ρ(r)dr dLr = ǫdMr = 4πr 2ρ(r)ǫdr ⇒ dLr dr = 4πr 2ρ(r)ǫ

• R. Orvedahl (CU Boulder)

Stellar Structure Feb 12 8 / 16

Nuclear Reactions

dLr dr = 4πr 2ρ(r)ǫ = 4πr 2ǫ0ρα+1T 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

Energy Transport

Remember radiation exerts a pressure Prad = aT 4/3 ⇒ dPrad

dr

= 4

3aT 3 dT dr

From Radiation Transport Theory

dPrad dr

= − ¯

κρ c Frad

Combine equations

dT dr = − 3 4ac ¯ κρ T 3Frad, where Frad = Lr 4πr2

Get T in terms of Lr dT dr = − 3 4ac ¯ κρ T 3 Lr 4πr 2

• R. Orvedahl (CU Boulder)

Stellar Structure Feb 12 10 / 16

What Equations Do We Have So Far?

1

dP dr = − GMr(r)ρ(r) r2

# Eqns = 1, # Variables = 3: P(r), Mr(r), ρ(r)

• R. Orvedahl (CU Boulder)

Stellar Structure Feb 12 11 / 16

What Equations Do We Have So Far?

1

dP dr = − GMr(r)ρ(r) r2

# Eqns = 1, # Variables = 3: P(r), Mr(r), ρ(r)

2

dMr(r) dr

= 4πr 2ρ(r)

Eqns = 2, Vars = 3: P(r), Mr(r), ρ(r)

• R. Orvedahl (CU Boulder)

Stellar Structure Feb 12 11 / 16

What Equations Do We Have So Far?

1

dP dr = − GMr(r)ρ(r) r2

# Eqns = 1, # Variables = 3: P(r), Mr(r), ρ(r)

2

dMr(r) dr

= 4πr 2ρ(r)

Eqns = 2, Vars = 3: P(r), Mr(r), ρ(r)

3

dLr dr = 4πr 2ρ(r)ǫ = 4πr 2ǫ0ρα+1T β

Eqns = 3, Vars = 5: P(r), Mr(r), ρ(r), Lr(r), T(r)

• R. Orvedahl (CU Boulder)

Stellar Structure Feb 12 11 / 16

What Equations Do We Have So Far?

1

dP dr = − GMr(r)ρ(r) r2

# Eqns = 1, # Variables = 3: P(r), Mr(r), ρ(r)

2

dMr(r) dr

= 4πr 2ρ(r)

Eqns = 2, Vars = 3: P(r), Mr(r), ρ(r)

3

dLr dr = 4πr 2ρ(r)ǫ = 4πr 2ǫ0ρα+1T β

Eqns = 3, Vars = 5: P(r), Mr(r), ρ(r), Lr(r), T(r)

4

dT dr = − 3 4ac ¯ κρ T 3 Lr 4πr2

Eqns = 4, Vars = 5: P(r), Mr(r), ρ(r), Lr(r), T(r)

• R. Orvedahl (CU Boulder)

Stellar Structure Feb 12 11 / 16

Equation of State

Gives the pressure in terms of density and temperature P = P(ρ, T) Ideal Gas: P = ρkT

¯ m ,

¯ m = mean atomic mass For the Sun: µ =

¯ m mH ≈ 1.6

Or Electron Degenerate Matter: P = (3π2)2/3

5 2 me

Z

A

• ρ

mH

5/3 Or ...

• R. Orvedahl (CU Boulder)

Stellar Structure Feb 12 12 / 16

Final Set of Equaions

1

dP dr = − GMr(r)ρ(r) r2

2

dMr(r) dr

= 4πr 2ρ(r)

3

dLr dr = 4πr 2ρ(r)ǫ = 4πr 2ǫ0ρα+1T β

4

dT dr = − 3 4ac ¯ κρ T 3 Lr 4πr2

5

P = ρkT

¯ m

Still cannot solve without Boundary Conditions ...

• R. Orvedahl (CU Boulder)

Stellar Structure Feb 12 13 / 16

Boundary Conditions

Mr → 0 as r → 0 Lr → 0 as r → 0 ρ → 0 as r → R∗ T → Teff as r → R∗ P → 0 as r → R∗ Now we can solve the system

• R. Orvedahl (CU Boulder)

Stellar Structure Feb 12 14 / 16

Numerically Integrate the System

• R. Orvedahl (CU Boulder)

Stellar Structure Feb 12 15 / 16

Numerically Integrate the System

• R. Orvedahl (CU Boulder)

Stellar Structure Feb 12 16 / 16