Theory of a Polyhedral Heliotrope on an Artificial Satellite

Abstract

AbstractThe heliotrope, invented by GAUSS, is a geodetic instrument for transmitting signals between distant stations by reflection of solar rays from a small, flat mirror. Many such mirrors on an artificial satellite, forming a polyhedral heliotrope, would automatically send useful signals and greatly increase the satellite visibility.It will be shown that the apparent stellar magnitude of a heliotrope signal is given by m = − 36.85 − 2.5 log10 k \documentclass{article}\pagestyle{empty}\begin{document}$ \left({\frac{D}{\varrho}} \right)^2 \sin \frac{\Theta}{2} $\end{document}, where k is the reflectivity factor, D the effective diameter of the mirror face, ϱ the distance from the observer, and Θ the elongation of the satellite from the sun. Thus, the mean stellar magnitude at opposition from a 1 cm face is about +0.m9 when 500 km away.The theory of the duration and statistical frequency of such flashes from a quasi‐regular polyhedron has been worked out in quantitative form. It turns out that the mean approximate signal rate per second for a polyhedron of radius 25 cm would be \documentclass{article}\pagestyle{empty}\begin{document}$\bar n = \frac{{2{\rm I}\,\omega}}{{D^2 }} $\end{document}, where ω is the number of satellite rotations per second. Thus, there would be 21 flashes per second from a satellite with 1 cm faces rotating once per second, each flash lasting less than 0.s001, which would correspond to about 2′′ or 3′′ of motion on the photographic trail of the satellite.Some such arrangement of the reflecting surface of the satellite is necessary for any study of its rotational motion. Also, it would obviously facilitate all observations during the expected 99% of the satellite lifetime after its radio power has been exhausted.