Problem

Investigation of line formation and the curve of growth for a simple Lorentz profile of the form

where 7 is the full-width half-maximum of the line profile and w is the frequency. As described in Section 4.1 (pp. 150-152), the Lorentz profile resulting from van der Waals interactions in the high-pressure M and L dwarf atmospheres often dominates the line broadening. To simulate line absorption, we will conduct a Monte-Carlo simulation where the wavelength at which the absorption occurs is chosen from a Lorentz profile probability distribution.

(a) Show that the Lorentz profile can be written in the form f (x) = C/[1 + (x - x0)2/a2]

where C is a constant, a is related to 7 and x is a dimensionless wavelength variable. x is often measured in units of the Doppler width of the line, Sw^,

Sw/w = £/c ^ SWD = £0Wq/c where is the Doppler velocity = sqrt (2kT/m) = 12.85 (T/104 A)05km/sec, w0 is the line center frequency and A is the atomic weight of the absorbing atom.

(b) A random number chosen on a uniform distribution may be converted to one chosen from a specific probability distribution (see e.g., discussion in Numerical Recipes [P1], Section 7.2). To do so, we must evaluate the inverse indefinite integral of the Lorentz profile. Show that the form f (x) given in part (a) integrates to an arctangent, so that the inverse indefinite integral we seek is given by:

where U is a random number drawn uniformly between 0 and 1.

(c) Write a program to choose randomly from a Lorentz distribution for an arbitrary number of times, and use the program to plot the line profile assuming that each time a random wavelength is chosen, an absorption occurs which removes one flux unit. Use an initial continuum of 1,000 flux units. Start with a = 1. Produce plots choosing between a few hundred and a few million times. Note that the flux cannot go below zero. To make the plotting feasible, only consider wavelengths between —100 to 100 wavelength units of the center. (Hint: in IDL, use routines 'randomu' and 'histogram'.)

(d) Now calculate the curve of growth, a plot of log (equivalent width) against log (number of absorptions). Recall from Figure 4.4, that there is a linear part, a flat part and a square root part on the curve of growth. Can you identify all of these parts on your graph? If not, why not? Also, what approximation are we making that eventually will make the profile inaccurate? (Hint: the equivalent width is an integral of the line under the continuum.) You don't need an elaborate integration method. Simple trapezoidal integration assigning the flux to its value at the computed points is adequate (in IDL, use routine 'total').

(e) What happens to the curve of growth for values of a = 0.1 and a = 10 (show plots)? Why? (Hint: what does a correspond to?).