2.6

Question 1

Approximation for 1-dimensional, or plane-parallel atmosphere

(radiative transfer for)

Answer 1

• Temperature T constant for horizontal plane
• structure varies only in vertical direction z

Question 2

Path d s in a plane-parallel atmosphere

Answer 2

ds = dz / cos(θ) = dz / μ

• μ = cos(θ)

Question 3

Radiative transfer equation for a plane-parallel atmosphere

Answer 3

μ ∂I_ν(z, μ) / ∂z = j_ν - α_ν I_ν

Question 4

Definition of optical depth τ

Answer 4

dτ_ν = -α_ν dz

• optical depth τ_ν(z) as function of vertical distance z
• τ_ν = 0 at top of atmosphere
• τ_ν increases with increasing depth or decreasing height

Question 5

Radiative transfer equation as a function of τ_ν(z)

Answer 5

μ ∂I_ν(τ_ν, ν) / ∂τ_ν = I_ν - S_ν

Question 6

Solution of radiative transfer equation

not what it is but how it is obtained

for a plane-parallel stellar atmosphere

Answer 6

Obtained by multiplying with e^{-τ_ν / μ} and integrating from a reference optical depth τ_ν,0

some additional context might be good if this is relevant

Question 7

Case I: Outward ray

radiative transfer as a function of optical depth

Answer 7

• 0 ≤ μ ≤ 1, cos(theta) positive
• ray starts very deep at τ_ν,0 = ∞

Question 8

Case II: Inward ray

Add Clarifier

Answer 8

0 ≥ μ ≥ -1, inward ray from surface τ_ν,0 = 0

Question 9

Important approximation for intensity I(τ_ν, μ)

Answer 9

If τ_ν » 1:

I_ν(τ_ν, μ) = B_ν(τ_ν) + μ dB_ν(τ_ν) / dτ_ν

Question 10

Anisotropic part of radiation field

Answer 10

- μ · dB_ν / dτ_ν,

• negative for inward direction
• positive for outward direction
• zero parallel to surface

so the first line is in fact the anis. part but the following seems to refer to mu = cos theta. im not sure if thats also called the anis. part. Also i am not sure what inward and outward are referring to here

Question 11

Radiation parameters U_𝛎, F_𝛎, P_𝛎 for plane-parallel atmosphere

Answer 11

• Energy density
• Flux
• Radiation pressure calculated using function A(cos θ)

not sure what the calculated using … is about

Question 12

Ratio between anisotropic and isotropic parts of the radiation field

Answer 12

(dB_ν / dτ_ν) / B_ν ≈ 3F_ν / cU_ν

Question 13

Integration over all wavelengths for anisotropic and isotropic terms

Answer 13

Anisotropic term / Isotropic term ≈ 3F / cU

Question 14

Energy density U for blackbody radiation

formula

Answer 14

U = aT⁴(τ),
where a = 4σ / c

Question 15

Total flux F emerging from surface with effective temperature T_eff

Answer 15

F = σT_eff⁴

Question 16

Anisotropy expressed in temperatures

Answer 16

Anisotropic term / Isotropic term ≈ 3/4 (T_eff / T(τ))⁴