Simple Analytic Model of the Long-Term Evolution of Nanosatellite Constellations

The long-term evolution of a large constellation of nanosatellites is investigated using closed-form analytical methods. A set of partial differential equations is derived, the solution to which provides the evolution of the mean spatialnumberdensityoftheconstellationundertheactionofairdrag,on-orbitsatellitefailures,andthedeposition ofnewsatellitesinto theconstellation.Thesesolutionsgiveinsightintotheglobaldynamicsandlong-termevolution of large constellations of nanosatellites and display some interesting physical features. In particular, asymptotic solutions provided the steady-state distribution of nanosatellites and an estimate of the required rate of deposition of new nanosatellites to maintain the constellation. Nomenclature a = drag-induced acceleration B = ballistic coefe cient H = atmospheric scale height n = number density R = reference radius r = radial coordinate z = out-of-plane coordinate g = mean failure rate h = azimuthal coordinate l = gravitational parameter q = air density q 0 = base density at radius R x = orbital angular velocity