  We develop a small distance expansion for the radiative heat transfer between
gently curved objects, in terms of the ratio of distance to radius of
curvature. A gradient expansion allows us to go beyond the lowest order
proximity transfer approximation. The range of validity of such expansion
depends on temperature as well as material properties. Generally, the expansion
converges faster for the derivative of the transfer than for the transfer
itself, which we use by introducing a near-field adjusted plot. For the case of
a sphere and a plate, the logarithmic correction to the leading term has a very
small prefactor for all materials investigated.
