  Comets can exhibit non-gravitational accelerations caused by recoil forces
due to anisotropic mass loss. So might active asteroids. We present an
astrometric investigation of 18 active asteroids in search of non-gravitational
acceleration. Statistically significant (signal-to-noise ratio (SNR) $> 3$)
detections are obtained in three objects: 313P/Gibbs, 324P/La Sagra and (3200)
Phaethon. The strongest and most convincing detection ($>$7$\sigma$ in each of
three orthogonal components of the acceleration), is for the $\sim$1 km
diameter nucleus of 324P/La Sagra. A 4.5$\sigma$ detection of the transverse
component of the acceleration of 313P/Gibbs (also $\sim$1 km in diameter) is
likely genuine too, as evidenced by the stability of the solution to the
rejection or inclusion of specific astrometric datasets. We also find a
3.4$\sigma$ radial-component detection for $\sim$5 km diameter (3200) Phaethon,
but this detection is more sensitive to the inclusion of specific datasets,
suggesting that it is likely spurious in origin. The other 15 active asteroids
in our sample all show non-gravitational accelerations consistent with zero. We
explore different physical mechanisms which may give rise to the observed
non-gravitational effects, and estimate mass-loss rates from the
non-gravitational accelerations. We present a revised momentum-transfer law
based on a physically realistic sublimation model for future work on
non-gravitational forces, but note that it has little effect on the derived
orbital elements.
