[continues from 4.11.2]

### 4.11.3 PSF correction factor

The factor f(E,r) which enters in the response matrix calculation is the fraction of counts falling in a circle of radius r mm, i.e. the integral of the radial PSF between 0 and r. More precisely one computes the Integral PSF as :
`     IPSF(q) = 2 pi * integral between 0 and q of PSF(r)rdr`
taking advantage of the fact that the Gaussian and Lorentzian components entering the definition of PSF(r) can be integrated analytically in r dr.

The resulting formula (where the fraction f is normalized to the detector radius rdet (FOV radius) of 15 mm) is of the form

```     f(E,r)  = IPSF(r)/IPSF(rdet)

IPSF(r) = IG(r) + IL(r)

IG(r)   = R sigma2 (1 - exp[ -0.5(r/sigma)2])

IL(r)   = rl {1 - 1 / [1 + (r/rl)2]m-1} / 2(m-1)
```

and depends on the same four parameters R, rl, sigma and m already described for the differential PSF(r). As already explained, each one of them in turn depends on energy via five coefficients

The relevant coefficients are kept in files m{1,2,3}_psf.coeff
while their energy dependency is shown elsewhere.

We report here some examples of the PSF correction factor (normalized integral PSF).

Fig. 4.11.3-I : MECS integral PSF (analytical form) at 2 (solid) and 8 keV (dotted)

while additional representations of (analytically-generated) integral PSF's can be obtained via the following form.

Select MECS unit then after having filled the following form
kind of representationparameterform
f(E,r) vs r at fixed energy E keV
f(E,r) vs E at fixed extraction radius mm
f(E,r) vs E at fixed extraction radius arcmin