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The basic idea is to minimize the difference between the true source coordinates and the ones computed by the eboxdetect as a function of the three angles. More explicitly, the following steps are carried out:
  1. $n=0$

  2. Set $(\phi_0, \theta_0, \psi_0) = (0, 0, 0)$, i.e. assume perfect alignment between the star tracker/spacecraft and instrument boresight frame in Fig. 1.

  3. run the task attcalc on the event list $EL_{n-1}$ producing a new event list $EL_n$ with sky coordinates corresponding to the tuple $(\phi_n, \theta_n, \psi_n)$.

  4. accumulate a sky image $IMG_n$ from event list $EL_n$ with task evselect

  5. run eboxdetect on $IMG_n$ which gives a list of source positions in absolute sky coordinates $(\alpha_i, \delta_i)_n$, $i=1,\ldots,N$

  6. identify the reference sources in the new source list and compute

      $\displaystyle \epsilon_n = \sum_{i=1}^N (\alpha_i^{\mbox{ref}}-\alpha_i)^2
+ (\delta_i^{\mbox{ref}}-\delta_i)^2
$ (1)

  7. if $\epsilon_n$ is small enough the sought set of angles is found:
      $\displaystyle (\phi_n, \theta_n, \psi_n)
$ (2)
    else endif
Thus, the procedure is a minimization of the sum of squares of the absolute deviations of the reference source positions from the true values in the 3-dimensional parameter space of the boresight misalignment angles. The actual minimization scheme employed is the Nelder-Mead Simplex method[3] which uses only function evaluations. It has been found that the Simplex scheme is much better suited than conventional conjugate gradient methods which often fail to efficiently find the direction to the global minimum. The dependency of $\epsilon$ on the third angle $\psi$, i.e., the calibration around the optical axis, has been found to be fairly weak. For the sake of efficiency, the search for the global minimum has therefore been split into two stages:
  1. Minimization of $\epsilon=\epsilon(\phi, \theta, \psi=\mbox{const})$ with a fairly relaxed stopping criterion.
  2. Starting from the local minimum found in step 1, full minimization of $\epsilon=\epsilon(\phi, \theta, \psi)$ with a more stringent stopping criterion.

XMM-Newton SOC -- 2021-11-30