sol3 - dr ( * ) with T φ ' -˙ M r j =-˙ M r r 2 Ω to...

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Astro 405/505, fall semester 2004 Homework, 3rd set, solutions Problem 1: optically thick line emission Use L ν = 4 π Z 2 π 0 Z 0 dr I ν ( ~e k ) r 2 | ~e k · ~ ( ~e k · ~ V ) | | V ( r ) | Θ( ν - ν 0 + ν r ) Θ( ν 0 + ν r - ν ) with ν r = ν 0 c | V ( r ) | and I ν = B ν ( T ) optically thick conditions!! Orient ~e k along z-axis. Then for the velocity profile in question | ~e k · ~ ( ~e k · ~ V ) | = | ∂z ± z t ² | = 1 t and for the power spectrum L ν = 8 π 2 B ν ( T 0 ) Z r max 0 dr r Θ ³ r - c t ν 0 | ν 0 - ν | ´ L ν = 4 π 2 B ν ( T 0 ) r 2 max 1 - µ ν - ν 0 ν 0 V max c ! 2 for | ν - ν 0 | ≤ ν 0 V max c That is the same profile as in the optically thin case. Problem 2: accretion disks a) Steady-state! Therefore ˙ M = M star / (10 9 yrs) ˙ M ± = ˙ M NS ' 6 · 10 16 g sec - 1 ˙ M BH ' 6 · 10 24 g sec - 1 b) Keplerian disks! Therefore Ω 2 r 3 = G M star . The inner radius and the angular frequency there are r ± = 7 · 10 11 cm r NS = 10 7 cm r BH = 3 · 10 14 cm at r min Ω ± = 2 · 10 - 5 sec - 1 Ω NS = 370 sec - 1 Ω BH = 2 . 5 · 10 - 5 sec - 1
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Use our heating-cooling balance argument to find the temperature 2 b SB T 4 eff = 1 2 π r T φ d Ω
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Unformatted text preview: dr ( * ) with T φ ' -˙ M r j =-˙ M r r 2 Ω to obtain at r min T ± ' 500 K T NS ' 2 · 10 6 K T BH ' 5 · 10 4 K For the total power either integrate the differential power spectrum L ν over all frequencies or directly use the Stefan-Boltzmann law multiplied with the surface area A L ' A b SB T 4 eff = Z ∞ r min dr 2 π r b SB T 4 eff ≈ π r 2 min b SB T 4 eff where the last step corresponds to a very rough approximation. Better is to work out the r-dependence of T 4 eff via Eq.* and to solve the integral over radius. The very rough estimate would yield L ± = 4 · 10 30 erg / sec L NS = 3 · 10 35 erg / sec L BH = 10 44 erg / sec...
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This note was uploaded on 02/29/2012 for the course PHYS 227 taught by Professor Rabe during the Fall '08 term at Rutgers.

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sol3 - dr ( * ) with T φ ' -˙ M r j =-˙ M r r 2 Ω to...

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