[continues from 4.10.5]

## 4.11 The Point Spread Function

The Point Spread Function of an entire MECS unit ( Mirror Unit and Detector Unit) has been carefully calibrated on the ground [1,4] and verified in flight [2] . Its on-axis shape is remarkably well described by a radial distribution which applies also well within 6 (or may be 10) arcmin from the centre.
Some examples of monochromatic PSF taken on the ground are reported separately for on-axis and off-axis cases. The off-axis PSF shows a gradual deviation from a radial distribution, with an elongation of the core in the tangential direction, and the appearance of the effect of single reflections in the wings. Note that all scales are logarithmic.
See also the Cyg X-1 first light (off-axis, behind the strongback) and second light (on-axis) taken in flight during the commissioning phase.

The on-axis differential PSF can be modelled analytically as the sum of a Gaussian and a generalized Lorentzian component, whose coefficients depend on energy.

```     G(r) = cg exp [-0.5(r/sigma)2] = cgGs(r)
L(r) = cl [1+(r/rl)2]-m         = clLs(r)
```
An example of the differential PSF shape is shown here below (additional figures can be produced in a customized way).

Fig. 4.11-I : MECS differential PSF (analytical form) at 2 (solid) and 8 keV (dotted)

The normalization of the integral of the PSF over the entire plane to 1 allows to remove one free parameter, and use just R = cg/cl to reduce the expression to the following, taking advantage of the "simplified" forms of the Gaussian and Lorentzian Gs and Ls:

```    PSF(r) = N-1 {R Gs(r)               + Ls(r)       }    i.e.

PSF(r) = N-1 {R exp[-0.5(r/sigma)2] + [1+(r/rl)2]-m}    with

N      = 2*pi [ R sigma2 + rl2 /2(m-1)]
```
where the parameters R, rl, sigma and m depends on energy. Such dependency has been modelled empirically for each parameter using a function of the form :
`     parameter = d + eE + c exp[-(E-a)/b]`
where there is a different set of coefficients a,b,c,d,e for each parameter, and in addition the parameters with dimension of a length (rl and sigma) may be subject to a further scale factor, according to whether r is measured in mm or arcmin.
These coefficients are stored in files used for, and described with, the calculation of the integral PSF correction factor, while their energy dependency is shown here.

Fig. 4.11.II : Energy dependency of the MECS PSF parameters

Additional representations of (analytically-generated) differential PSF's can be obtained via the following form.

Select energy keV, representation and MECS unit