2.8

Question 1

Temperature Structure of the Sun (T(τ))

and its luminosity

effective temperature of the sun

Answer 1

Effective Temperature of the Sun (T_{eff, ⊙}):

5780 K

Blackbody with L_⊙ = 4 π R_⊙ σ T_eff⁴

• For τ=0 → T(τ=0) = (1 / 2)^1/4 T_eff = 0.84 T_eff = 4860 K
• For τ=2/3 → T(τ=2/3) = T_eff.
• For log τ = -1, 0, 0.5, 1
• T = 5030, 6110, 7520, 9941 K .

Question 2

Limb-Darkening Law for the Sun

Answer 2

• Normalized Intensity (I(0, μ) / I(0,1)): Normalized at center = 1
• At limb I(0, μ=0) / I(0,1) = 0.4. I(0, μ) follows a linear law
• I seen and measured as I(R / R_⊙), apparent norm. radial distance
• R / R_⊙ = sin θ
• The formula translates into:

I(R / R_⊙) / I(0) = 3/5 (sqrt(1-(R / R_⊙)²) + 2/3)

these just seem to be a collection of random things, idk

Question 3

Observed Solar Limb Darkening

Answer 3

• Agrees well with grey atmosphere for V-band
• Wavelength dependence:
  – Much stronger darkening in the UV
  – Weaker in the red
• Two parameter description for color effects: I(R / R_⊙) / I(0) = β / (1 + β) (sqrt(1-(R / R_⊙)²) + 1/β). (Eddington approximation for grey atmosphere: β = 3/2)
• Reasons for λ-dependence: decrease of temperature with height
• Short wavelength light from deep layers leads to strong limb darkening
• Long wavelength light from high up in the atmosphere leads to weak limb darkening

Question 4

Limb Darkening Parameter for Different Wavelengths

qualitative relationship

Answer 4

Wavelength (λ) and Limb Darkening Parameter (β): 380 nm (β = 10), 425 nm (β = 5.0), 480 nm (β = 3.0), 540 nm (β = 2.0), 680 nm (β = 1.2), 800 nm (β = 0.9).

truly, i dont think im learning this so i hope he doesnt ask, but would be quite random in my opinion, i think i will learn: lower for higher wavelengths

Question 5

Limb Darkening of Other Stars

Answer 5

can be measured as a by-product of planet transit curves