Abstract The longitude-dependent part of the geopotential can give rise to significant changes in inclination for a close satellite when its mean motion is commensurable with the Earth's rotation. For a decaying satellite passing through resonance, the total change in inclination depends on the value of a resonant variable at exact commensurability, which is an essentially random quantity. Many different gravity coefficients may contribute significantly, with relative amplitudes which are highly dependent on inclination. The equations for general β α resonance also reveal a basic distinction between (βt- α) even and odd. When the drag significantly exceeds the resonance forces, an approximate solution can be found in terms of Fresnel integrals. This shows that the inclination is almost equally likely to increase or decrease, and that the total change is proportional to (drag) − 1 2 , i.e. to the time taken to pass through resonance. The effect offers a way of deriving gravity coefficients of medium order (e.g. m=15) from the observed magnitude and shape of the variation in inclination. The magnitudes of even higher order gravity coefficients obtained from some resonance with α=2 (e.g. 29 2 ) or even with α=3 (e.g. 44 3 ) might yield information on the depth of the sources of the high order gravity field. The effect is also of special interest in deriving upper-atmosphere mean winds from the changes in inclination of decaying satellite orbits since the satellite may pass through a strong resonance.