Radar detection range in higher spatial dimensions

Inspired by widely-studied physical models of the universe of dimension larger than three in modern physics, we analyze how radar detection range generalizes to higher spatial dimensions. The radar range equation in three spatial dimensions is a product of terms, each of which can be generalized to arbitrary spatial dimensions. A key step is to establish an equivalence relation based on gain that facilitates comparing antennas or targets in different numbers of spatial dimensions. Our expression for maximum hyper-dimensional detection range has a number of counter-intuitive properties. First, we find that maximum detection range is not monotonic with spatial dimensionality: detection range initially decreases as the number of spatial dimensions increase, but reaches a minimum and then increases asymptotically linearly for very large numbers of spatial dimensions. For low numbers of spatial dimensions, detection range depends, as expected, on radar transmit power, integration time, antenna gain, target radar cross section (RCS) and wavelength. In very high-dimensional spaces, however, detection range is simply proportional to the wavelength and to the dimension; it is asymptotically independent of the radar power, antenna gain and target RCS. Care must be taken in generalizing the radar range equation, because it assumes that the flux density from the radar is constant across the target and that the flux density reflected from the target is constant across the radar aperture. These assumptions may be violated at ranges where the target becomes spatially resolved by the radar antenna. Surprisingly, the electrical diameter of a fixed-gain antenna grows asymptotically linearly with dimension, regardless of its physical size, thus providing greater angular resolution in higher dimensions. This work has no application, that we are aware of, to space or remote sensing missions in our universe, however the results are unexpected and pedagogically interesting.