4.1.44. 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.