On the antenna pattern of an orbiting interferometer

The response of an interferometer changing its orientation with respect to a fixed reference frame is analysed in terms of the beam-pattern factors and the polarization-averaged antenna power pattern. Given the motion of the antenna, the latter quantity describes the directionality of the antenna as a function of time. An interesting case is represented by the class of motions where the plane of the detector is constrained to move on the surface of a cone of constant aperture; at the same time, the two arms are rotating around a vertical axis. This picture describes, in particular, the motion of LISA, a proposed space-based laser interferometer, as well as of other planned missions. The overall sky coverage, and that of the Galactic plane in particular, is provided as a function of the aperture of the cone. Similarly, one can consider the case of an Earth-based interferometer. Using the same formalism, one can derive a simple expression for the antenna pattern, averaged over the time of arrival of the signal, as a function of the position and orientation on the Earth's surface. In particular, there turn out to be particular values for the terrestrial latitude and the inclination angle with respect to the local parallel which render the time-averaged antenna response perfectly isotropic. In the frequency domain, the general result is that the motion of the detector introduces in the instrumental response to a long-duration continuous signal a few harmonics of the orbital frequency, whose magnitude depends on the position of the source in the sky. In particular, we describe the response of LISA to sinusoidal waves coming from a few known binary systems in our Galaxy.