Vpro: velocity profile broadening model¶
This multiplicative model broadens an arbitrary additive component with an arbitrarily shaped Doppler profile, characterized by the half-width \(v\) and a profile shape \(f(x)\). The resulting spectrum \(S_c(E)\) is calculated from the original spectrum \(S(E)\) as
The function \(f(x)\) must correspond to a probability function, i.e. for all values of \(x\) we have
and furthermore
In our implementation, we do not use \(f(x)\) but instead the cumulative probability density function \(F(x)\), which is related to \(f(x)\) by
where obviously \(F(-\infty)=0\) and \(F(\infty)=1\). 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
here vprof.dat (but any name is allowed). This is a simple ascii file,
with \(n\) lines, and at each line two numbers: a value for
\(x\) and the corresponding \(F(x)\). The lines must be sorted
in ascending order in \(x\), and for \(F(x)\) to be a proper
probability distribution, it must be a non-decreasing function i.e. if
\(F(x_{i})\le F(x_{i+1})\) for all values of \(i\) between 1 and
\(n-1\). Furthermore, we demand that \(F(x_1)\equiv 0\) and
\(F(x_n)\equiv 1\).
Note that both \(x\) and \(F(x)\) are dimensionless. The parameter \(v\) 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 \(x\)-scale as well as the value of \(v\), as long as e.g. \(x_n v\) 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 \(n=2\) with \(x_1=-1\), \(x_2=1\), \(F(x_1)=0\) and \(F(x_2)=1\), and then let \(v\) do the scaling (this also allows you to have \(v\) 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 \(x\)-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 \(x=\)-5, -4.9, -4.8, \(\ldots\), 4.8, 4.9, 5.0 with corresponding values for \(F\) given by \(F=\) 0, 0.00000048, 0.00000079, \(\ldots\), 0.99999921, 0.99999952, 1.
The parameters of the model are:
v : Velocity broadening parameter \(v\), in km/s. Default
value: 1 km/s.file : Ascii character string, containing the actual name of the
vprof.dat file