A Numerical Scheme to Integrate the Rotational Motion of a
Quasi-Rigid body and Its Application to the Earth Rotation
TOSHIO FUKUSHIMA
National Astronomical Observatory
2-21-1, Ohsawa, Mitaka, Tokyo 181, JAPAN
(Internet) toshio@spacetime.mtk.nao.ac.jp
A new numerical scheme to integrate the orientation of a quasi-
rigid body is presented. The basic variables are (L, L_X, L_Y,
L_A, L_B, f); the magnitude and the X-, Y-, A-, and B-components
of the spin angular momentum vector L, and the longitude of A-
axis measured from X-axis along the great circle perpendicular
to L. The orientation matrix is generated by the five succes-
sive rotations in the sense of 1-2-3-2-1 as
( e_A, e_B, e_C)
= R_1(sigma) R_2(-phi) R_3(-f) R_2(beta) R_1(-xi)
where sigma = tan^(-1) (L_Y/sqrtL^2 - (L_X^2 + L_Y^2)), phi =
sin^(-1) (L_X/L), beta= sin^(-1) (L_A/L), and xi= tan^(-1)
(L_B/sqrtL^2 - (L_A^2 + L_B^2)). Not all these basic variables
but the corrections to their nominal linear motions for some com-
ponents instead such as (Delta L, Delta L_X, Delta L_Y, L_A, L_B,
Delta f) are actually integrated in the inertial coordinate sys-
tem whose Z-axis is chosen to be close to L at the initial epoch
of integration. These small variables directly correspond to the
variation of LOD, the nutations in obliquity and in declination,
the polar motion, and the variation of UT1 in terms of the Earth
rotation. Numerical simulations showed that the new scheme in-
tegrates the Earth orientation matrix 6-7 digits more precisely
than the ordinary Eulerian approach does while the required com-
putational time does not change significantly.