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3 Pages

EddingtonLineProfileLec21

Course: A 453, Fall 2009
School: Rochester
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Word Count: 383

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realistic More line profiles, using Eddington Approximation for T structure Te := 5800 K 4 model the solar spectrum T ( ) := Te 3 2 + 4 3 Eddington Approximation for T structure 1.5 .10 4 T (0) = 4877 K T 2 = 5800 K 3 T 0 i ( ) 1 .10 4 T (1) = 6133 K 2 Gaussian line profile 5000 0 0 i 5 10 0 line ( , 0 , 0) := 0 e line 0 is the ratio of line to continuum optical depth...

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realistic More line profiles, using Eddington Approximation for T structure Te := 5800 K 4 model the solar spectrum T ( ) := Te 3 2 + 4 3 Eddington Approximation for T structure 1.5 .10 4 T (0) = 4877 K T 2 = 5800 K 3 T 0 i ( ) 1 .10 4 T (1) = 6133 K 2 Gaussian line profile 5000 0 0 i 5 10 0 line ( , 0 , 0) := 0 e line 0 is the ratio of line to continuum optical depth is the optical depth in the line down to the continuum optical depth = 1 level. It is proportional to the absorber's column density above the continuum optical depth = 1 level I ( ) = B T = ( ( ) 1) 1 1 Intensity = source function at optical depth = 1 in general, we see to total optical depth, line+ continuum = 1 continuum optical depth when total optical depth = 1 = c + c line = c , 0 , 0 := ( 1 + line ( , 0 , 0) I , 0 , 0 := B , T c , 0 , 0 ( ) ( ( ( ))) Simplified radiative transfer, we "see" to optical depth = 1. 0 := i := 500 nm line near peak of Planck function i := 0 0 .. 100 5 + 14 i 10 4 .10 ( ) I( i , 0 , 2) 2 .1014 I( i , 0 , 20) I i , 0 , .2 3 .10 14 1 .10 14 0 499.7 499.8 499.9 500 500.1 500.2 500.3 i nm 0 := i := 5000 nm IR line in Rayleigh-Jeans tail of Planck function i := 0 0 .. 100 5 + 11 i 10 8 .10 ( ) I( i , 0 , 2) 4 .1011 I( i , 0 , 20) I i , 0 , .2 6 .10 11 2 .10 11 0 4999.7 4999.8 4999.9 5000 5000.1 5000.2 5000.3 i nm 0 := i := 200 nm UV line in Wien tail i := 0 0 .. 100 5 + 13 i 10 3 .10 ( ) I( i , 0 , 2) I( , i 0 , 20) . I i , 0 , .2 2 10 13 1 .10 13 0 199.7 199.8 199.9 200 i nm 200.1 200.2 ----------------------------------------------------------------------------------------------------------------------------The above simulations correspond to theta = 0, i.e. lines from the center of the disk of the sun. How would these lines change for a distant star, where the intensity I is integrated over all theta's to give the flux F? Hint, consider an absorption line from the limb of the sun, i.e. theta = 90 degrees. Hint, what will the flux be at the bottom of an optically thick line be? Hint, what will the flux in the continuum be? We "see" to optical depth 1 at the center of the disk, but only 2/3 when averaged over the disk. Compare 2/3 to 1 at different wavelengths B (500 nm , T (.667)) B ( 500 nm , T ( 1) ) = 0.762 line near peak of Planck fct. B (5000 nm , T (.667)) B ( 5000 nm , T ( 1) ) = = 0.932 IR, Rayleig...

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