Saturn's moon Enceladus emits a plume of water vapour and micrometre-sized ice particles from a series of warm fissures located near its south pole. This geological activity could be powered or controlled by variations in the tidal stresses experienced by Enceladus as it moves around its slightly eccentric orbit. The specific mechanisms by which these varying stresses are converted into heat, however, are still being debated. Furthermore, it has proved difficult to find a clear correlation between the predicted tidal forces and measured temporal variations in the plume's gas content or the particle flux from individual sources. Here we report that the plume's horizontally integrated brightness is several times greater when Enceladus is near the point in its eccentric orbit where it is furthest from Saturn (apocentre) than it is when near the point of closest approach to the planet (pericentre). More material therefore seems to be escaping from beneath Enceladus' surface at times when geophysical models predict its fissures should be under tension and therefore may be wider open.