Global Energetics

Total stellar energy

W = gravitational potential energy Ω + kinetic energy due to bulk motions (turbulence, pulsation) + internal energy from microscopic processes U Gravitational Potential Energy

Gravitational potential energy

Ω : energy required to assemble the star by collecting material from the outside universe. = negative of energy required to disperse the star.

∞ Gmdm Gm dΩ = − dr′ = − dm r ∫ 2 r r′ r

M Gm dm Ω = − ∫ dm 0 r

which depends on the density profile ρ(r) Gravitational Potential Energy 4 ⎫ 3 1 m = ρ0 πr ⎪ 3 3 e.g., for ρ r = ρ 3 ⎪ m ⎛ r ⎞ ⎛ m ⎞ ( ) 0 ⎬ → = ⎜ ⎟ → r = R⎜ ⎟ 4 3 ⎪ M ⎝ R⎠ ⎝ M ⎠ M = ρ0 π R 3 ⎭⎪

M Gm GM 1 3 M 3 GM 2 Ω = − ∫ dm = − ∫ m2 3 dm = − 0 r R 0 5 R

GM 2 3 In general, Ω = − q and q > since ρ ↓ with r R 5

2 r−1 q −2 ρ  → = ρ  r → q = 1 3 Internal and Kinetic Energy

Internal energy

U : Kinetic energy due to motions of particles (thermal, not bulk) + energy in atomic and molecular bonds.

U = − E dm Where E : local specific internal energy (erg/g) ∫ m m M

Kinetic energy

K : Total kinetic energy (thermal + bulk) The Virial Theorem

  Consider all particles in a star at positions r with momenta p i i

  Q = ∑ pi ⋅ ri i   pi r dQ i evaluate dt

dQ m ⋅ v ⋅ r 2 what are the units of ? (energy units: = m ⋅ v ) dt t The Virial Theorem   Q = ∑ pi ⋅ ri i

dQ d   = m ⋅ r ⋅ r dt dt ∑ i i i 2 ∑m ⋅ ri i d d ⎛ 1 2 ⎞ = ∑ ⎜ m ⋅ ri ⎟ dt i dt ⎝ 2 ⎠ is just the total moment of inertia of the system. 1 d 2  m r 2 = ∑ 2 ( ⋅ i ) 2 i dt dQ 1 d 2 I = dt 2 dt 2 The Virial Theorem   Q = ∑ pi ⋅ ri i   dQ dpi   dri = ∑ ⋅ ri + ∑ pi ⋅ dt i dt i dt           A B

  dri   B: ∑ pi ⋅ = ∑m ⋅ vi ⋅vi i dt i

1  2 = 2∑ m ⋅ vi = 2K (2 x total kinetic energy) i 2 The Virial Theorem  dpi  d       A: ⋅ r = m ⋅ r ⋅ r = m ⋅ r ⋅ r = F ⋅ r ∑ dt i ∑ dt ( i ) ∑ i i ∑ i i i i i i   F is the sum of all gravitational forces on particle i i = ∑Fij j ≠i   r r    mi m j ( j − i ) mi m j    rj − ri Fij = G 2 = − G 3 ri − rj r r r r ( ) i ij ij ij        rj   Fi ⋅ ri = Fij ⋅ ri = ⋅⋅⋅ + F ⋅ r + F ⋅ r + ⋅⋅⋅ ∑ ∑∑ ij i ji j i i j ≠i         F = −F = ⋅⋅⋅ + F ⋅ r − r + ⋅⋅⋅ = F ⋅ r − r ( ij ji ) ij ( i j ) ∑∑ ij ( i j ) i j >i

mi m j   2 mi m j gravitational = −G 3 ⋅ ri − rj = −G ∑∑ r ( ) ∑∑ r = Ω potential i j >i ij i j >i ij energy The Virial Theorem

1 d 2 I = 2K + Ω 2 dt 2

d 2 I In a globally static system: = 0 2K = −Ω dt 2

1 For a collisionless system: W = K + Ω W = Ω = −K 2 bound!

What about a collisional system? How does the 2K term relate to the internal energy of a gas? The Virial Theorem

Kinetic theory: pressure in a gas is caused by the sum of collisions of particles and momentum transfer

Particle i colliding with wall in x-direction

Δpi = 2 px,i = 2mivx,i

Time between collisions of same particle with wall 2l l Δt = vx,i Force due to particle i Δp m v2 F = i = i x,i i Δt l Total force on wall 2 mivx,i 1 2 F = ∑ = ∑mivx,i i l l i The Virial Theorem

Kinetic theory: pressure in a gas is caused by the sum of collisions of particles and momentum transfer

Total force on all 6 walls

2 2 2 2 2 2 F = ∑mi (vx,i + vy,i + vz,i ) = ∑mivi l i l i Total force on 1 wall

l 1 2 2 F = ∑mivi = K 3l i 3l

Pressure F 2K 2K P = = = 2K = 3P ⋅V A 3l ⋅ A 3V The Virial Theorem

In reality, pressure is changing over volume 2K = 3 PdV (can think of our box as a volume element dV) ∫ V

In a star’s spherical shells 3P dm = ρdV 2K = ∫ dm Depends on the equation of state M ρ

For a relation btw P and Em of P = (γ − 1)ρEm ( γ -law equation of state)

2K = 3 γ − 1 E dm = 3 γ − 1 E dm = 3 γ − 1 U ∫ ( ) m ( ) ∫ m ( ) M M 5 K is only equal to U if γ = i.e., for an ideal monoatomic gas 3 The Virial Theorem

Virial theorem: 3(γ − 1)U + Ω = 0

Total energy must be < 0 for star to Total W U energy = Ω + be stable. Otherwise, it has enough Ω energy to disperse itself. = Ω − 4 3(γ − 1) → γ > 3 ⎛ 1 ⎞ = Ω⎜1− ⎟ ⎛ 5⎞ ⎝ 3(γ − 1)⎠ • Ideal gas ⎜ γ = ⎟ : safely bound ⎝ 3⎠ 3γ − 4 ⎛ 4 ⎞ = Ω • Radiation γ = : unstable 3(γ − 1) ⎝⎜ 3⎠⎟

As radiation pressure becomes more important, star becomes less bound. The Virial Theorem

3γ − 4 Virial theorem: 3(γ − 1)U + Ω = 0 W = Ω 3(γ − 1)

γ = 5 3

Ω

γ = 4 3

As a star radiates, it loses energy (without other energy sources).

⎧Ω < 0 W < 0 → ⎨ star contracts and heats up U > 0 ⎩ The Virial Theorem: Applications

Internal temperature 3(γ − 1)U + Ω = 0

5 Ω • For an ideal monoatomic gas: γ = → U = − 3 2 3 3 • Also, the energy density is: E = nkT → U = nkTV 2 2 ⎧ n : number density of particles ⎫ ⎪ ⎪ ρN A ⎨ µ : mean molecular weight ⎬ n = µ ⎪ ’ ⎪ ⎩N A : Avogadro s number ⎭

3 ρN 3 N 2 µ U U = A kTV = A kMT → T = 2 µ 2 µ 3 N Ak M Ω q GM 2 • Virial theorem: U = − = 2 2 R The Virial Theorem: Applications

Internal temperature

−1 q µG M ⎛ q ⎞ ⎛ M ⎞ ⎛ R ⎞ T = T = 5 × 106 K 3 N k R ⎝⎜ 3 5⎠⎟ ⎜ M ⎟ ⎜ R ⎟ A ⎝  ⎠ ⎝  ⎠ (actual central temperature of sun is 15 million K)

• Internal T >> surface T è strong T gradient

• Most of the sun is highly ionized

• T is sufficiently high for nuclear fusion (~1 million K)

• Fusion happens due to gravitational energy Timescales

Dynamical timescale 1 d 2 I 2 = 2K + Ω d 2 I GM 2 2 dt • Virial theorem: ≈ dt 2 R

2 1 2 I GM ⎛ R ⋅ I ⎞ ⎫ 3 1 2 ≈ → tdyn ≈ ⎪ ⎛ R ⎞ −1 2 2 ⎝⎜ GM 2 ⎠⎟ → t ≈ ≈ Gρ tdyn R ⎬ dyn ⎜ ⎟ ( ) ⎪ ⎝ GM ⎠ I ≈ MR2 ⎭

2 GM • Escape velocity: vesc  R R R −1 2 • Timescale for collapse: tdyn   tdyn  (Gρ) v 1 2 esc ⎛ GM ⎞ ⎝⎜ ⎠⎟ R For sun, tdyn~ 1 hour Timescales

Kelvin-Helmholtz timescale Timescale for star to radiate away its thermal/gravitational energy.

• Star contracts slowly maintaining hydrostatic equilibrium GM 2 ΔR < 0 Ω = −q ⎫ R ⎪ GM 2 ⎬ → ΔΩ = +q ΔR dΩ GM 2 R2 ΔΩ < 0 = q ⎪ dR R2 ⎭

5 Ω • Virial theorem: γ = → W = → ΔW < 0 3 2 • Energy is lost from the system: radiation ΔΩ ΔW = half goes to increasing U and half goes to luminosity 2 Timescales

Kelvin-Helmholtz timescale Timescale for star to radiate away its thermal/gravitational energy.

• Suppose contraction is solely responsible for maintaining luminosity

dW d ⎛ Ω⎞ 1 dΩ dR q GM 2 dR L = − = − = − = − dt dt ⎝⎜ 2 ⎠⎟ 2 dR dt 2 R2 dt

dR R q GM 2 ≈ − tKH = dt tKH 2 LR

2 −1 −1 ⎛ M ⎞ ⎛ L ⎞ ⎛ R ⎞ t = 2 × 107 yr KH ⎜ M ⎟ ⎜ L ⎟ ⎜ R ⎟ ⎝  ⎠ ⎝  ⎠ ⎝  ⎠

• Too short a timescale to have happened. Timescales

Nuclear timescale Timescale for star to burn its H fuel and leave the Main Sequence.

2 total available energy ε fcore Mc t = tnuc = nuc luminosity L ε = efficiency of nuclear burning (41H è 4He) ~ 0.007

mass fraction of core (star leaves main sequence after core burns) ~ 0.1 fcore =

For sun, tnuc~ 10 billion years 3.5 L ⎛ M ⎞ For stars of solar mass or greater: ≈ L ⎜ M ⎟  ⎝  ⎠ −2.5 ⎛ M ⎞ t ≈ 1010 yr nuc ⎜ M ⎟ ⎝  ⎠ Timescales

tdyn ~ 1 hour

tKH ~ 20 million years

tnuc ~ 10 billion years

t  t  t dyn KH nuc

It is valid to assume that stars on the main sequence are in dynamical and thermal equilibrium.