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 v and a profile shape f(x). The resulting spectrum S_c(E) is calculated from the original spectrum S(E) as

S_c(E) = \int f \bigl( \frac{E-E_0}{E_0} \frac{v}{c}  \bigr)
S(E_0) {\mathrm d}E_0.

The function f(x) must correspond to a probability function, i.e. for all values of x we have

f(x)\ge 0

and furthermore

\int_{-\infty}^{\infty} f(x) {\mathrm d}x = 1.

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

F(x)\equiv \int_{-\infty}^{x} f(y){\mathrm d}y,

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