  In planetary science, surface roughness is regarded to be a measure of
surface irregularity at small spatial scales, and causes the thermal-infrared
beaming effect (i.e. re-radiation of absorbed sunlight back towards to the
Sun). Typically, surface roughness exhibits a degeneracy with thermal inertia
when thermophysical models are fitted to disc-integrated thermal-infrared
observations of asteroids because of this effect. In this work, it is
demonstrated how surface roughness can be constrained for near-Earth asteroid
(433) Eros (i.e. the target of NASA's NEAR Shoemaker mission) when using the
Advanced Thermophysical Model with thermal-infrared observations taken during
an "almost pole-on" illumination and viewing geometry. It is found that the
surface roughness of (433) Eros is characterised by an RMS slope of 38 $\pm$
8{\deg} at the 0.5-cm spatial scale associated with its thermal-infrared
beaming effect. This is slightly greater than the RMS slope of 25 $\pm$ 5{\deg}
implied by the NEAR Shoemaker laser ranging results when extrapolated to this
spatial scale, and indicates that other surface shaping processes might
operate, in addition to collisions and gravity, at spatial scales under one
metre in order to make asteroid surfaces rougher. For other high obliquity
asteroids observed during "pole-on" illumination conditions, the
thermal-infrared beaming effect allows surface roughness to be constrained when
the sub-solar latitude is greater than 60{\deg}, and if the asteroids are
observed at phase angles of less than 40{\deg}. They will likely exhibit NEATM
beaming parameters that are lower than expected for a typical asteroid at all
phase angles up to 100{\deg}.
