  Most models of anomalous X-ray pulsars (AXPs) account for the observed X-ray
spectra and pulsations by means of radiation processes that occur on the
surfaces of neutron stars. For any such model, general relativistic deflection
of light severely suppresses the amplitude of the observed pulsations. We
calculate the expected pulsation amplitudes of AXPs according to various models
and compare the results with observations. We show that the high (<= 70%) pulse
amplitudes observed in some AXPs can be accounted for only if the surface
emission is localized (spot radius <40 degrees) and strongly beamed
(cos^n[theta'] with n>2, where theta' is the angle to the normal). These
constraints are incompatible with those cooling and magnetar models in which
the observed X-rays originate as thermal emission from the neutron-star
surface. Accretion models, on the other hand, are compatible with observations
for a wide range of parameters. Finally, definitive conclusions cannot be
reached on magnetospheric models, since their localization and beaming
properties are not well understood.
