Physics 611
Astrophysics (Stellar Atmospheres)
Problem Assignment #6
Due: Monday, November 27, 2006
THE VOIGT PROFILE
(1) For a = 0.0025 plot on the same graph [log (αν/α0) versus u] three curves representing (a) the
Doppler core approximation for the Voigt profile    0 e  u , (b) the damping wings
2
approximation for the Voigt profile,   
profile,    0
a


e  y dy
 u2
, (c) the correct, numerically integrated Voigt
2


a 0
a 2  u  y 
2
. Let u range from 0 to 6. Let log (αν/α0) range from 0 to -8.
Your Doppler core approximation will extend beyond this range, but it is not important to plot it
outside this range since it is orders of magnitude less than the damping solution which totally
dominates the line wings.
(2) To four significant figures find the value of uI(a), the value of u at which the two approximations in
parts (1a) and (1b) above are equal.
(3) For a = 0.0025 plot on the same graph [log (αν/α0) versus u] two curves representing (a) the sum of
the Doppler core approximation for the Voigt profile and the damping wings approximation,

a 
, (b) the correct, numerically integrated Voigt profile,
 u2 
   0  e u 
2

   0
a


e  y dy
2


a 2  u  y 
2
.
(4) Now plot, as a function of u, the fractional difference in the true solution,
 true   0
a
e  y dy
 u2
a 




e

, i.e., plot
and
the
approximate
solution,
,

approx
0

 2
 u2 
a  u  y 



2
2
f(u) = (αν true - αν approx)/αν true = 1 - αν approx/αν true versus u. At what value of u (umax) is the fractional
difference the largest? What is the value of fmax = f(umax)? (In solving for umax avoid the vicinity of
the singularity in the approximate solution at u = 0.)