2.5

Question 1

Kirchhoff’s Law

Answer 1

• valid for optically thick objects in a cavity
• cavity in thermodynamic equilibrium
• the emission through a small hole equals black body radiation
  I_ν=B_ν(T)
• In thermodynamic equilibrium:
  S_ν=B_ν(T)
  S_ν=j_ν / α_ν
• leading to Kirchhoff’s law:
  j_ν=α_ν B_ν(T).

Question 2

Interpretation of Kirchhoff’s Law

Answer 2

• For atomic, molecular gases, and many solid particles: j_ν and α_ν have strong peaks
  → Optically thin emission has strong emission lines
  → absorbing gas has strong absorption lines
• In thermodynamic equilibrium, peaks in j_ν and α_ν must compensate each other

Question 3

Maxwellian Velocity Distribution

Answer 3

dn_v = 4π n (m/2πk_BT)^3/2 v² exp(-mv² / 2k_BT) dv
- Describes the distribution of particle velocities at temperature T.
- n_v = number of particles
- n = number of particles per unit volume
- v = velocity and m = particle mass

Question 4

Boltzmann Distribution Law

Answer 4

n_e/n₀ = exp(-E_e / k_B T)
- Describes the population of excited states (n_e) of an atom or molecule relative to the ground state (n₀)
- E = Energy of excited state above ground state

Question 5

Saha Equation

Answer 5

• Describes ionization state of the gas.
• x: fraction of atoms ionized:

x²/(1-x)=(2πm_e)^3/2/h³(k_BT)^5/2/P e^-χ/k_BT

• m_e: electron mass,
• P: pressure
• χ: ionization potential.

Question 6

Planck’s Law for Radiation Energy Density

Answer 6

U_ν dν = 8πh/c³ · ν³dν / (e^hν/k_BT-1).

• For hν > k_BT
  → Wien-part behaves like:

U_ν dν ≈ 8πh/c³ · ν³e^-hν/k_BT.

Question 7

Summary of High Energy Drop-off (1+4)

Answer 7

• All formulae have an exponential drop-off for high energies

∝e^-E/k_BT

where E is:

• kinetic energy mv² / 2
• excitation energy E_e
• ionization energy χ
• photon energy h_ν

Question 8

Local Thermodynamic Equilibrium (LTE)

Answer 8

LTE is valid for an isolated box filled with gas.

• All walls at T
• gas at T
• Planck radiation field U_ν(T)

LTE conditions for a volume in a star with diameter d for gas:

• T=const for gas (same velocity distribution in entire volume)
• d>Λ_p = mean free path of particles

for radiation:

• isotropic Planck radiation field in equilibrium with gas
• d>Λ_r = mean free path of photons between emission and absorption