4 Point spread function broadening due toattitude jitter

During an observation interval [t₁,t₂], 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 $T:\,\,=t_2-t_1$ 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 $t,\,t\in[t_1,t_2]$. 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 $w(x,s){\rm d}x$. Clearly,

$$\begin{split} w(x,s)\ge0,\qquad\int_{{\bf R}^2} w(x,s){\rm d}x=1. \end{split}$$ (31)

In presence of a jitter motion with density from (31) a PSF density $p(r;E,\epsilon)$ for a steadfast optical axis becomes transformed to

$$\begin{split} p_w(x,s):\,\,=\int_{{\bf R}^2}w(y,s)p(\vert x-y\vert;E,\vert y\vert){\rm d}y. \end{split}$$ (32)

Compared with the detector's field of view, the source motion s(t) takes place in a tiny neighbourhood of the nominal source position s. This combined with the fact that $p(r;E,\epsilon)$ varies slowly with respect to the off-axis angle $\epsilon$ allows the approximation $\vert y\vert\approx\vert s\vert$. Adoption of this approximation casts (31) into the shape

$$\begin{split} p_w(x,s):\,\,=\int_{{\bf R}^2}w(y,s)p(\vert x-y\vert;E,\vert s\vert){\rm d}y. \end{split}$$ (33)

Next, we assume two-dimensional normal distributions for p and w in (33). Remind the following.

Lemma. Let $X_k,\,k=1,2,$ be two independent, normally distributed random n-vectors with expectations ${\bf E}(X_k)=\,\,:\mu_k$ and $n\times n$ variance-covariance matrices ${\bf V}{\bf a}{\bf r}(X_k)=\,\,:V_k$. Then the sum $X:\,\,=X_1+X_2$ is again normally distributed with expectation $\mu:\,\,=\mu_1+\mu_2$and variance matrix $V:\,\,=V_1+V_2$. Thus, the sum X possesses the density

$$\begin{split} p_{X_1+X_2}(x):\,\,=&{1\over{(2\pi)^{n/2}\cdot... ...2)'(V_1+V_2)^{-1}(x-\mu_1-\mu_2)]},\ x\in{\bf R}^n. \end{split}$$ (34)

The prime in (34) means transposition.

We neglect the rest wobbling and confine us to plane circular symmetric normal densities $w(x,s):\,\,=w(\vert x-s\vert)$ with a standard deviation $\sigma_{{\rm Att}}>0$ for the attitude motion in

$$\begin{split} w(r):\,\,={1\over{2\pi}\sigma^2_{{\rm Att}}}{\... ...\sigma_{{\rm Att}}})^2},\qquad\sigma_{{\rm Att}}>0. \end{split}$$ (35)

A standard deviation of $\sigma_{{\rm Att}}=3$ 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 $\sigma$ in place of $\sigma_{{\rm Att}}$ and find the addition law for the considered standard deviation $\sigma$ in (35),

$$\begin{split} \sigma(\sigma_{{\rm Att}}):\,\,=\sqrt{\sigma^2+\sigma_{{\rm Att}}^2} \end{split}$$ (36)

due to the presence of a circular symmetric jitter distribution for the optical axis with standard deviation $\sigma_{{\rm Att}}>0$.

Finally, we apply the broadening rule (36) to the normal component in the PSPC having the standard deviation $\sigma(E,\epsilon)$ from (7) as well as to the two normal components in the HRI PSF with standard deviations $\sigma_1,\sigma_2(\epsilon)$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 $\sigma_{{\rm Att}}=3$ arcsec is negligible in the case of the WFC angular resolution.

Without further information, it cannot be decided whether a positive $\sigma_{{\rm Att}}$ 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 $\sigma _{{\rm Att}}=0$ 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 $\chi^2$ for 40-1=39 degrees of freedom amounts to $\chi_{{\rm red.},39}^2:\,\,=\chi^2/39=9.078$.

For a significance level of $90\%$ the critical value of the $\chi^2$-distribution with 39 degrees of freedom becomes $\chi^2_{{\rm crit.},39,0.9}=50.660$. This means a critical reduced $\chi^2$ value $\chi_{{\rm red.,crit.},39,0.9}^2=1.299<9.078$ which is grossly exceeded by the observed reduced $\chi^2_{{\rm red.},39}$. The rejection of the hypothesis of the equality of the HRI PSF distribution and the pulsar distribution underlying Fig. 23 is thus statistically justified.

Figure 22: The radial distribution histograms of the HRI observation of PSR J0437-4715 (black dots) and the HRI PSF distribution histogram (solid line) with $\sigma _{{\rm Att}}=0$ in the radial range [0,80] arcsec. 40 equal histogram classes were used in [0,120] arcsec. Centre is the pulsar position

An estimate for $\sigma_{{\rm Att}}$ was found to be $\hat\sigma_{{\rm Att}}=2.6$. Figure 23 shows the same observed histogram as in Fig. 22 along with the HRI PSF histogram now for $\sigma _{{\rm Att}}=2.6$ in place of the former $\sigma _{{\rm Att}}=0$. The observed reduced $\chi^2$ diminishes to $\chi^2_{{\rm red.},38}=0.922$ which is well below the critical value for 38 degrees of freedom $\chi^2_{{\rm red.,crit.},38}=1.303$. Now, the equality of radial pulsar and HRI PSF distributions cannot be rejected on the $90\%$ significance level.

Figure 23: The radial distribution histograms of the HRI observation of PSR J0437-4715 (black dots) and the HRI PSF distribution histogram (solid line) with $\sigma _{{\rm Att}}=2.6$ in the radial range [0,80] arcsec. 40 equal histogram classes were used in [0,120] arcsec. Centre is the pulsar position