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 ( \lesssim 70 % ) pulse amplitudes observed in some AXPs can be accounted for only if the surface emission is localized ( spot radius < 40 ^ { \circ } ) and strongly beamed ( \sim \cos ^ { n } \theta ^ { \prime } with n \gtrsim 2 , where \theta ^ { \prime } 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 can not be reached on magnetospheric models , since their localization and beaming properties are not well understood .