4.1.38. Vpro: velocity profile broadening model¶
This multiplicative model broadens an arbitrary additive component with an arbitrarily shaped Doppler profile, characterized by the half-width and a profile shape . The resulting spectrum is calculated from the original spectrum as
The function must correspond to a probability function, i.e. for all values of we have
In our implementation, we do not use but instead the cumulative probability density function , which is related to by
where obviously and . The reason for using the cumulative distribution is that this allows easier interpolation and conservation of photons in the numerical integrations.
If this component is used, you must have a file available which we call
vprof.dat (but any name is allowed). This is a simple ascii file,
with lines, and at each line two numbers: a value for
and the corresponding . The lines must be sorted
in ascending order in , and for to be a proper
probability distribution, it must be a non-decreasing function i.e. if
for all values of between 1 and
. Furthermore, we demand that and
Note that both and are dimensionless. The parameter serves as a scaling parameter for the total amount of broadening. Of course for a given profile there is freedom for the choice of both the -scale as well as the value of , as long as e.g. remains constant. In practice it is better to make a logical choice. For example, for a rectangular velocity broadening (equivalent to the vblo broadening model) one would choose with , , and , and then let do the scaling (this also allows you to have as a free parameter in spectral fits). If one would instead want to describe a Gaussian profile (for which of course also the vgau model exists, Vgau: gaussian velocity broadening model), one could for example approximate the profile by taking the -scale in units of the standard deviation; an example with a resolution of 0.1 standard deviation and a cut-off approximation at 5 standard deviations would be -5, -4.9, -4.8, , 4.8, 4.9, 5.0 with corresponding values for given by 0, 0.00000048, 0.00000079, , 0.99999921, 0.99999952, 1.
The parameters of the model are:
v: Velocity broadening parameter , in km/s. Default value: 1 km/s.
file: Ascii character string, containing the actual name of the vprof.dat file