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Calibration Access and Data Handbook

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
   
   $$G_i(\beta) = a_i\ e^{-\frac{1}{2} \left ( \frac{\mu_i + \beta-[C(\mu_i-\mu_0)]}{C\sigma_i} \right )^2} \ ,$$
   
   
   with the parameters $a_i$, $\mu_i$, and $\sigma_i$ provided by the CCF per Gaussian. Lorentzian parameterizations have the form
   
   $$L_i(\beta) = \frac{1}{\pi}\frac{b_i\ C\ w_i/2}{ (\beta-[\nu_i+C(\nu_i-\mu_0)])^2 + \frac{C^2 w_i^2}{4} } \ ,$$
   
   
   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
   

   $$a_i' = a_i\ \sigma_i\ \sqrt{2\pi}$$
   
   
   
   
   $$\mu_{\rm G} = \frac{\sum a_i'\cdot \mu_i}{\sum a_i'}$$
   
   
   
   
   $$\mu_{\rm L} = \frac{\sum b_i\cdot \nu_i}{\sum b_i}$$
   
   
   
   
   $$\mu_0 = \frac{\mu_{\rm G}\sum a_i' + \mu_{\rm L}\sum b_i} {\sum a_i' + \sum b_i} \ .$$
   
   

   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

$$C = \frac{\sin\alpha}{\sin\beta_{\rm m}}$$

and is an input argument.

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

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

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.