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

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8. The adjusted model

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.

[FIGURE] Fig. 3. SAO-Hipparcos position difference field, averaged on a [FORMULA] grid

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

[EQUATION]

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

[EQUATION]

and the orthogonality relations:

[EQUATION]

Besides, [FORMULA] is a solenoidal and [FORMULA] an irrotational field:

[EQUATION]

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

[EQUATION]

whose complex Fourier coefficients are:

[EQUATION]

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:

[EQUATION]

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

One of the nice properties of the vector harmonics expansion is that global effects between two catalogues are neatly packed in the [FORMULA] harmonics. Indeed, let [FORMULA] 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 [FORMULA] coefficients in the form:

[EQUATION]

hereafter [FORMULA].

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

[EQUATION]

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

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© European Southern Observatory (ESO) 2000

Online publication: July 13, 2000
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