X-ray Modulation Patterns

The bridge between radio and x-ray terminology is the modulation pattern. For an x-ray telescope, this is the two-dimensional field of probabilities that a photon coming from a certain direction will pass through the rotating collimator to the detector. (We neglect the effects of scattering and background here.) For a perfect collimator of pitch p, the cross section through the modulation pattern is a triangular waveform, which can be represented approximately by a sum of sinusoids:

$$M = a_0 + a_1 cos({\bf k} \cdot {\bf x} - \alpha) + \hbox{ higher order terms,} \qquad \eqno{(1)}$$

where $\bf{x}$ is the angular distance in the sky from the spin axis of the rotating subcollimator, and $\bf{k}$ is a vector whose direction is perpendicular to the slits, and whose magnitude is $2 \pi$divided by the pitch. The function M is always non negative, since it represents a probability.

During the rotation of a subcollimator, the x-rays emitted by a point source produce a modulation profile, which may be computed from equation 1 by choosing a source position $\bf{x}=(x_0,y_0)$, and letting the wavevector $\bf{k}$ rotate through a range of orientation angles $\phi$. The collimator phase $\alpha$, which represents the offset of the modulation pattern from the spin axis, will, in general, be a function of orientation angle, and will be known from the aspect system. The flux of x-rays will then be proportional to:

$$F = a_0 + a_1 cos(2\pi(x_0\ cos(\phi)+y_0\ sin(\phi))/p + \alpha) + \hbox{higher order terms,} \qquad \eqno{(2)}$$