Astron. Astrophys. 359, 1195-1200 (2000)

The differences in the data set (position and proper motion) generate two vector fields on the sphere. Fig. 3 shows the field of position differences, averaged on a suitable grid.

 Fig. 3. SAO-Hipparcos position difference field, averaged on a grid

The analysis of vector fields on a sphere is better done using vector spherical harmonics (Mignard & Morando 1990). Let be the spherical angular coordinates, the usual spherical harmonics and the unit vectors in the and directions on the sphere. Then, two families of vector functions are defined through the equations:

called toroidal and spheroidal vector harmonics. They form together a complete basis for vector functions on the sphere. They obey the symmetry properties:

and the orthogonality relations:

Besides, is a solenoidal and an irrotational field:

Any square summable field on the sphere can be expanded in a Fourier series:

whose complex Fourier coefficients are:

Our model assumes a finite Fourier expansion of real spherical vector fields. The reality conditions, together with the conjugation symmetry (5 y 6) imply that our model series has the form:

where the superscripts denote the real and imaginary part of the function. The number of coefficients in the expansion is .

One of the nice properties of the vector harmonics expansion is that global effects between two catalogues are neatly packed in the harmonics. Indeed, let be an infinitesimal rotation of the reference system. Then, it can be shown that the Cartesian components of the rotation vector are related to the coefficients in the form:

hereafter .

A global glide between the reference systems is defined through and irrotational vector , whose components are related to the s coefficients:

hereafter . This is important, since the main differences between both reference frames are global transformations.

© European Southern Observatory (ESO) 2000

Online publication: July 13, 2000