High precision methods for locating the celestial intermediate pole and origin

International audienceContext: .The precession-nutation transformation describes the changing directions on the celestial sphere of the Earth's pole and an adopted origin of right ascension. The coordinate system for the celestial sphere is the geocentric celestial reference system, and the two directions are the celestial intermediate pole (CIP) and the celestial intermediate origin (CIO), the latter having supplanted the equinox for this purpose following IAU resolutions in 2000. The celestial coordinate triad based on the CIP and CIO is called the celestial intermediate reference system; the prediction of topocentric directions additionally requires the Earth rotation angle (ERA), the counterpart of Greenwich sidereal time (GST) in the former equinox based system. Aims: .The purpose of this paper is to review the different ways of calculating the CIP and CIO directions to precisions of a few microarcseconds over a time span of several centuries, meeting the requirements of high-accuracy applications. Methods: .Various implementations are described, their theoretical bases compared and the relationships between the expressions for the relevant parameters are provided. Semi-analytical and numerical comparisons have been made, based on the P03 precession and the IAU 2000A nutation, with slight modifications to the latter to make it consistent with P03. Results: .We have identified which transformations between celestial and terrestrial coordinates involve a minimum number of variables and coefficients for given accuracy objectives. The various methods are consistent at the level of a few microarcseconds over several centuries, and equal accuracy is achievable using both the equinox/GST paradigm and the CIO/ERA paradigm. Given existing nutation models, the most concise expressions for locating the CIP are based on the Fukushima-Williams bias-precession-nutation angles. The CIO can be located to a few microarcseconds using the CIO locator s. The equation of the origins (EO) is sensitive to the precession-nutation, but can locate the CIO to a few microarcseconds as long as consistent models are used for EO and precession-nutation.