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Satellite services are fundamental to the global economy, and their design reflects a tradeoff between coverage and cost. Here, we report the discovery of two alternative 4-satellite constellations with 24- and 48-hour periods, both of which attain nearly continuous global coverage. The 4-satellite constellations harness energy from nonlinear orbital perturbation forces (e.g., Earth's geopotential, gravitational effects of the sun and moon, and solar radiation pressure) to reduce their propellant and maintenance costs. Our findings demonstrate that small sacrifices in global coverage at user-specified longitudes allow operationally viable constellations with significantly reduced mass-to-orbit costs and increased design life. The 24-hour period constellation reduces the overall required vehicle mass budget for propellant by approximately 60% compared to a geostationary Earth orbit constellation with similar coverage over typical satellite lifetimes. Mass savings of this magnitude permit the use of less expensive launch vehicles, installation of additional instruments, and substantially improved mission life.