Given the population of each level, as computed by CLOUDY using the Sobolev approximation, the line profiles are calculated by using the formal solution of line profile synthesis. That is,
where p is the impact parameter and 
is defined by
In equation 6.2 
is the Source Function dependence with radius,
is the optical depth dependence with radius, z is the
distance along the line of sight ( 
) and 
is the
starting point of integration, which is 
for 
and
for 
.
The optical depth and source function for a given transition are calculated assuming a two-level atom. Following Mihalas mihalas:
where, 
and 
are the populations of the lower and upper levels
respectively (obtained from the run of CLOUDY), 
is the
frequency of the transition, 
and
are the Einstein transition probabilities, ds is the path
interval, h is the
Planck constant and 
is the local absorption profile, which is
assumed to be a
gaussian with width given by the local turbulent velocity, i.e.
\
where 
is the turbulent
width of the line [Johns & Basri 1995b].
The source function is given by
where all the symbols are the same as above and c is the speed of light. Re-writing 6.5 so that it is in units of the stellar continuum of temperature T results in
where k is the Boltzmann constant.
