Transformation of Tesseral Harmonics under Rotation

Abstract

Summary As the transformation formulae relating tesseral harmonics referred to two different sets of rectangular right-handed axes are important in the theory of satellites (see, for example, Kaula 1961, G. E. Cook 1963), it may be useful to give an elementary and self-contained account of a method for their derivation. The method depends essentially on the relation of the transformation to the unimodular unitary (2 x 2) group; this can be seen explicitly from the definition given by Jeffreys and Jeffreys (1962), and a full discussion of the rotation group such as that given by Wigner (1959) is not required. The difficulties that arise come from the variety of conventions used, firstly in the definition of Euler’s angles and secondly in the choice of the standard tesseral harmonics. I have tried to use a notation that will not lead to confusion in astronomical applications. This accounts for the rather unusual nomenclature for Euler’s angles. A consequence of importance in the theory of artificial satellites is briefly discussed.