Calculation of shape factors between parallel ring sectors sharing a common centerline

Thermal control calculations for spin-stabilized spacecraft often require determination of the shape factor for diffuse radiant-energy exchange between parallel ring sectors sharing a common centerline. The present paper utilizes the contour integral method to derive a closed-form expression for the shape factor from a differential area on one ring sector to the other (finite-size) ring sector. This expression is then integrated numerically to determine the desired shape factors between two finite-size ring sectors. Typical results are presented for both configurations. In particular, the shape factor is a straight line when the difference of angular displacements which define location of surface 1 with respect to x-axis is equal to 360 deg.