We discuss the IAU resolutions B1.3, B1.4, B1.5, and B1.9 that were adopted during the 24th General Assembly in Manchester, 2000, and provides details on and explanations for these resolutions. It is explained why they present significant progress over the corresponding IAU 1991 resolutions and why they are necessary in the light of present accuracies in astrometry, celestial mechanics, and metrology. In fact, most of these resolutions are consistent with astronomical models and software already in use. The metric tensors and gravitational potentials of both the Barycentric Celestial Reference System and the Geocentric Celestial Reference System are defined and discussed. The necessity and relevance of the two celestial reference systems are explained. The transformations of coordinates and gravitational potentials are discussed. Potential coefficients parameterizing the post-Newtonian gravitational potentials are expounded. Simplified versions of the time transformations suitable for modern clock accuracies are elucidated. Various approximations used in the resolutions are explicated and justified. Some models (e.g., for higher spin moments) that serve the purpose of estimating orders of magnitude have actually never been published before.

Every three years the IAU Working Group on Cartographic Coordinates and Rotational Elements revises tables giving the directions of the poles of rotation and the prime meridians of the planets, satellites, minor planets, and comets. This report takes into account the IAU Working Group for Planetary System Nomenclature (WGPSN) and the IAU Committee on Small Body Nomenclature (CSBN) definition of dwarf planets, introduces improved values for the pole and rotation rate of Mercury, returns the rotation rate of Jupiter 123 102 B. A. Archinal et al.to a previous value, introduces improved values for the rotation of five satellites of Saturn, and adds the equatorial radius of the Sun for comparison. It also adds or updates size and shape information for the Earth, Mars' satellites Deimos and Phobos, the four Galilean satellites of Jupiter, and 22 satellites of Saturn. Pole, rotation, and size information has been added for the asteroids (21) Lutetia, (511) Davida, and (2867) Šteins. Pole and rotation information has been added for (2) Pallas and (21) Lutetia. Pole and rotation and mean radius information has been added for (1) Ceres. Pole information has been updated for (4) Vesta. The high precision realization for the pole and rotation rate of the Moon is updated. Alternative orientation models for Mars, Jupiter, and Saturn are noted. The Working Group also reaffirms that once an observable feature at a defined longitude is chosen, a longitude definition origin should not change except under unusual circumstances. It is also noted that alternative coordinate systems may exist for various (e.g. dynamical) purposes, but specific cartographic coordinate system information continues to be recommended for each body. The Working Group elaborates on its purpose, and also announces its plans to occasionally provide limited updates to its recommendations via its website, in order to address community needs for some updates more often than every 3 years. Brief recommendations are also made to the general planetary community regarding the need for controlled products, and improved or consensus rotation models for Mars, Jupiter, and Saturn.

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As a preparation step to compute Jacobian elliptic functions efficiently, we created a fast method to calculate the complete elliptic integral of the first and second kinds, K (m) and E(m), for the standard domain of the elliptic parameter, 0 < m < 1. For the case 0 < m < 0.9, the method utilizes 10 pairs of approximate polynomials of the order of 9-19 obtained by truncating Taylor series expansions of the integrals. Otherwise, the associate integrals, K (1 − m) and E(1 − m), are first computed by a pair of the approximate polynomials and then transformed to K (m) and E(m) by means of Jacobi's nome, q, and Legendre's identity relation. In average, the new method runs more-than-twice faster than the existing methods including Cody's Chebyshev polynomial approximation of Hastings type and Innes' formulation based on q-series expansions. Next, we invented a fast procedure to compute simultaneously three Jacobian elliptic functions, sn(u|m), cn(u|m), and dn(u|m), by repeated usage of the double argument formulae starting from the Maclaurin series expansions with respect to the elliptic argument, u, after its domain is reduced to the standard range, 0 ≤ u < K (m)/4, with the help of the new method to compute K (m). The new procedure is 25-70% faster than the methods based on the Gauss transformation such as Bulirsch's algorithm, sncndn, quoted in the Numerical Recipes even if the acceleration of computation of K (m) is not taken into account.