2.3-4

Question 1

Specific Intensity in Empty Space

Answer 1

• Emission from a ‘point like source’

Assumptions:
- spherical symmetry
- emission into cone
- specific intensity I_x at r_x

→ geometric dilution of intensity: I(r) ∝ 1 / r².

Question 2

Emission of Resolved Source

Answer 2

Assumptions
- source is extended
- source is homogenous: no angular structure

→ I(r) = const.

• distance-independent surface brightness for resolved sourves

Question 3

Radiative Transfer with Matter

Answer 3

Radiation transfer equation:
dI_ν/ds = j_ν - α_ν I_ν
- Emission coefficient jν adds to radiation [Wm^-3 Hz^-1 sr^-1]
- absorption coefficient αν [m^-1] proportionate to Iν [Wm^-2 Hz^-1 sr^-1]

Question 4

Radiative Transfer - Only Absorption

Answer 4

• only absorption
• no emission
• j_ν = 0

→ dI / ds = - α_ν I_ν

I_ν(s) = I_ν(s₀) exp(- ∫ _s0^s α_ν(s’) ds’ )

Question 5

Radiative Transfer - Only Emission

Answer 5

• only emission
• no absorption

→ α_ν = 0

I_ν(s) = I₀ + ∫ _s0^s j_ν(s’) ds’

Question 6

General Solution of Radiative Transfer Equation

Answer 6

General solution:

I_ν(τ_ν) = I_ν(0) · exp(-τ_ν) + ∫ S_ν(τ_ν’) exp(-(τ_ν - τ_ν’)) dτ_ν

• For constant S_ν(τ_ν) = S_ν or emission and absorption properties of matter are constant:

I_ν(τ_ν) = I_ν(0) exp(-τ_ν) + S_ν(1 - exp(-τ_ν))

→ useful assumption for piecewise numerical integration for small dτ

Question 7

Two important solutions for the specific intensity I

Answer 7

• if τ » 1 → exp(-τ) « 1 = optically thick case:

I_ν = S_ν

• if τ « 1 → exp(-τ) ≈ 1 - τ_ν = optically thin case:

I_ν = I_ν(0) + S_ν · τ_ν = I_ν(0) + j_νds