Presented is an asymptotic series approximate solution of steady, axisymmetric natural convection within arbitrary-gap spherical annulus enclosures. The boundaries are of uniform and constant temperature, the inner shell being hotter than the outer. The fluid within the annulus is Boussinesq of arbitrary Prandtl number. Both the dimensionless momentum and energy equations involve the parameter R, the square root of the Grashof number based on outer radius. Temperature and streamfunction are represented as truncated power series of R and are even and odd functions of R respectively. We show that a natural representation for the temperature and streamfunction bases are separable series involving Legendre polynomials and associated Legendre functions of order 1 respectively. By invocation of orthogonality, an uncoupled system of linear ordinary differential equations is obtained for the radially dependent functions and are solved exactly in sequence by the symbolic manipulator MACSYMA. We present results for air (Pr = 0.7) for various values of R for an annular gap-width of 0.5 and compare the surface average Nusselt numbers with previous results.