Calibration Access and Data Handbook


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Procedure

This provides the shape of the XRT along the dispersion direction (integrated in cross-dispersion).

This function can have two modes of operations:

  1. Numerical integration of the parameterized PSF (see HIGH accuracy level described in section 3.2.1)
    This parameterization is numerically integrated along the cross-dispersion direction.
  2. Dedicated parameterization
    A dedicated set of parameters is provided that describes the PSF when integrated on the cross-dispersion direction. Gaussian and Lorentzian functions and a linear combination are allowed. The Gaussian parameterization has the form

    \begin{displaymath}G_i(\beta) = a_i\ e^{-\frac{1}{2} \left
( \frac{\mu_i + \beta-[C(\mu_i-\mu_0)]}{C\sigma_i} \right )^2} \ ,\end{displaymath}

    with the parameters $a_i$, $\mu_i$, and $\sigma_i$ provided by the CCF per Gaussian. Lorentzian parameterizations have the form

    \begin{displaymath}L_i(\beta) = \frac{1}{\pi}\frac{b_i\ C\ w_i/2}{
(\beta-[\nu_i+C(\nu_i-\mu_0)])^2 + \frac{C^2 w_i^2}{4}
} \ ,\end{displaymath}

    with the parameters $b_i$, $nu_i$, and $w_i$ provided by the CCF per Lorentzian.

    The functions are selected by a flag in the CCF. The parameters of the centers ($\mu$ and $\nu$) are given with respect to a common center to allow for an assymetric parameterizations. The common center $\mu_0$ is calculated by

    \begin{displaymath}a_i' = a_i\ \sigma_i\ \sqrt{2\pi} \end{displaymath}


    \begin{displaymath}\mu_{\rm G} = \frac{\sum a_i'\cdot \mu_i}{\sum a_i'}\end{displaymath}


    \begin{displaymath}\mu_{\rm L} = \frac{\sum b_i\cdot \nu_i}{\sum b_i}\end{displaymath}


    \begin{displaymath}\mu_0 = \frac{\mu_{\rm G}\sum a_i' + \mu_{\rm L}\sum b_i}
{\sum a_i' + \sum b_i} \ .\end{displaymath}

    The value of the input argument CENTER_CHAN is added to $\mu_0$, in order to provide the PSF correctly centered.

The width of the PSF of the telescope scales as a function of angle of incidence on the gratings and diffration angle of the RGA by the chromatic magnification function $C$. The chromatic magnification which is an imaging effect by the gratings that scales the width of the PSF in the dispersion direction as a function of angle of incidence and diffraction angle. It is defined as

\begin{displaymath}C = \frac{\sin\alpha}{\sin\beta_{\rm m}} \end{displaymath}

and is an input argument.

The function returns the probability distribution $P(j)$ which calculated from $p(\beta)$ by integration per channel $j$

\begin{displaymath}P(j) = \sum_{i=1}^{N_{\rm Gauss}}
\left ( a_i\ \sigma_i\ \sq...
... b_i
\int_{\beta_j}^{\beta_{j+1}} L_i(\beta)\ {\rm d}\beta
\ .\end{displaymath}

Note: The CAL inverts the parameterization that is in XRT_XPSF along the dispersion axis, because the RGA acts as a mirorr in this respect.


next up previous contents
Next: Calling Parameters Up: CAL_rgsgetXRTFigure Previous: CAL_rgsgetXRTFigure   Contents
Michael Smith 2011-09-20