  The Yarkovsky--O'Keefe--Radzievskii--Paddack (YORP) effect governs the spin
evolution of small asteroids. The axial component of YORP, which alters the
rotation rate of the asteroid, is mostly independent of its thermal inertia,
while the obliquity component is very sensitive to the thermal model of the
asteroid.
  Here we develop a semi-analytic theory for the obliquity component of YORP.
We integrate an approximate thermal model over the surface of an asteroid, and
find an analytic expression for the obliquity component in terms of two YORP
coefficients.
  This approach allows us to investigate the overall evolution of asteroid
rotation state, and to generalize the results previously obtained in the case
of zero thermal inertia.
  The proposed theory also explains how a non-zero obliquity component of YORP
originates even for a symmetric asteroid due to its finite thermal inertia. In
many cases, this causes equatorial planes of asteroids to align with their
orbital planes.
  The studied non-trivial behavior of YORP as a function of thermal model
allows for a new kind of rotational equilibria, which can have important
evolutionary consequences for asteroids.
