sol4-f7

# sol4-f7 - Astro 405/505 fall semester 2007 Homework 4th set...

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Astro 405/505, fall semester 2007 Homework, 4th set, solutions. Problem 7: Radiation spectra under LTE The supernova remnant subtends a solid angle element d Ω = 1 . 2 · 10 - 6 sr. The intensity therefore is I ν = F ν d Ω = 1 . 3 · 10 - 13 erg / cm 2 / s / Hz / sr which we need to compare with the Planckian, because that is the solution of the radiation transport equation for LTE and large optical depth. B ν ( T ) = 2 h ν 3 c 2 1 exp ± kT ² - 1 B 100 ( T ) = (1 . 5 · 10 - 23 cgs) 1 exp ± kT ² - 1 I ν = B 100 ( T ) requires ¿ kT or, more precisely, kT = (8 . 7 · 10 9 ) T 4 · 10 7 K The radiation maximum occurs at about 3 kT or about 2 · 10 18 Hz. The emission is not necessarily optically thick, in which case the radiating material can only be hotter than calculated here. Problem 8: Stellar emission This is what we did in session 7, where we derive the intensity as a function of aspect angle (Eq.7.4) I ν ( μ, τ z = 0) = 1 μ Z 0 0 z S ν exp - τ 0 z μ ! Here μ = cos θ = ~e k · ~e n which is unity for normal incidence (we look at the center of the sun) and approaching zero for grazing incidence (we look at the edge of the sun). In addition our assumption of LTE implies S ν = B ν ( T ) . a) If the temperature T is constant, then the source function can be pulled out of the integral

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sol4-f7 - Astro 405/505 fall semester 2007 Homework 4th set...

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