4 Point spread function broadening due toattitude jitter

During an observation interval [*t*_{1},*t*_{2}], the instrument's optical axis jitters
around a nominal direction due to the behaviour of the attitude control loop. An
additional deliberate wobbling motion is usually superimposed in pointed observation mode and
corrected for.
In the standard ROSAT data analysis, a software correction for the rest wobbling
motion and non-perfect attitude data is applied leaving a smaller uncorrected rest wobbling.
With or without a rest
wobbling the jitter motion remains. The movement of the optical axis manifests itself
in the detector plane as a translation of any image point. Consider in the sequel
a true source point *s* along with its related jitter trajectory *s*(*t*).

When viewing the composite motion, jitter plus possible wobbling, as an instrument property we are confronted with the problem to determine the PSF under this composite motion. The jitter motion is statistically independent from the infall of photons and from the detector dynamics.

First, we assume a sufficient long observation span
to the effect
that an area element in the detector plane around the hypothetical source position
*s* is traversed often enough by the source trajectory *s*(*t*) evolving in time
.
Under this assumption it is reasonable to assume a probability
distribution density *w*(*x*,*s*) for the composite source motion. The probability
to find the source in a neighbourhood with area *dx* around a point *x* is thus
.
Clearly,

In presence of a jitter motion with density from (31) a PSF density for a steadfast optical axis becomes transformed to

Compared with the detector's field of view, the source motion

Next, we assume two-dimensional normal distributions for

**Lemma**. Let
be two independent, normally distributed random *n*-vectors with
expectations
and
variance-covariance matrices
.
Then the
sum
is again normally distributed with expectation
and variance matrix
.
Thus, the sum *X* possesses the density

The prime in (34) means transposition.

We neglect the rest wobbling and confine us to plane circular symmetric normal
densities
with a standard deviation
for the
attitude motion in

A standard deviation of arcsec is to be expected for the observation under analysis. A statistical investigation on the variability of this value from observation to observation is not yet carried out.

We apply the above lemma to a circular symmetric normal density
as the one in (35) but with standard deviation
in
place of
and find the addition law for the
considered standard deviation
in (35),

due to the presence of a circular symmetric jitter distribution for the optical axis with standard deviation .

Finally, we apply the broadening rule (36) to the normal component in the PSPC having the standard deviation from (7) as well as to the two normal components in the HRI PSF with standard deviations from (11).

The weights of the remaining PSF components are so small that no further (more difficult) correction efforts are made. This approximation is deemed to be good for the HRI and acceptable for the PSPC. No jitter broadening correction was foreseen for the WFC because a jitter in the order of arcsec is negligible in the case of the WFC angular resolution.

Without further information, it cannot be decided whether a positive is due to an extended celestial source or due the behavior of the observing instrument.

The following Figs. 22 and 23 demonstrate the effect of a PSF correction for the jitter motion and provide also a way to obtain an estimate for the jitter amplitude.

A ROSAT HRI pointed observation of the millisecond pulsar PSR J0437-4715 was performed in 1994 in the usual wobbling mode. A question on source extension could not be decided on the base of this observation (see Becker & Trümper 1998). Therefore, a re-observation without wobbling on the same object was carried out in 1997. Here, only the jitter motion of the optical axis broadens the HRI PSF. Figure 22 shows the histogram of the observed ring-integrated surface brightness distribution (black dots) with error bars according to Poisson distributions in an 80 arcsec circle centered at the pulsar position. Totally 40 histogram classes of equal width were used to cover the larger concentric disk of radius 120 arcsec.

The observed histogram was compared with the histogram of the HRI PSF taken with arcsec (solid line) on the same radial range [0,120] arcsec. The residual plot suggests already that the observed histogram and the HRI PSF histogram are not in good agreement. The reduced for 40-1=39 degrees of freedom amounts to .

For a significance level of the critical value of the -distribution with 39 degrees of freedom becomes . This means a critical reduced value which is grossly exceeded by the observed reduced . The rejection of the hypothesis of the equality of the HRI PSF distribution and the pulsar distribution underlying Fig. 23 is thus statistically justified.

An estimate for was found to be . Figure 23 shows the same observed histogram as in Fig. 22 along with the HRI PSF histogram now for in place of the former . The observed reduced diminishes to which is well below the critical value for 38 degrees of freedom . Now, the equality of radial pulsar and HRI PSF distributions cannot be rejected on the significance level.

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