On the Brumberg-Kopeikin and Damour-Soffel-Xu approaches
in the relativistic theory of reference systems

A.V. Krivov

Astronomical Institute, St. Petersburg University,
Stary Peterhof, 198904 St. Petersburg, Russia.
E-mail: krivov@aispbu.spb.su

There exist two recently developed powerful approaches in the general
relativistic theory of reference systems and time scales: that of
Brumberg and Kopeikin (BK) and Damour, Soffel and Xu (DSX).  This
paper aims at comparison of some basic elements of BK and DSX
methods, namely the definitions of lowest multipole moments of
extended bodies and coordinate transformations between global and
local reference systems.  Supposing that an extended body is the
Earth, and assuming that the global and local systems are BRS
(barycentric reference system of the Solar system) and GRS
(geocentric reference system) respectively, we have found the
following: (i) the difference between the Fock mass of the Earth
(used in BK approach) and its Blanchet-Damour mass (central to DSX
formalism) is negligible; (ii) Blanchet-Damour mass of the Earth can
be treated as practically conserved; (iii) BK and DSX definitions of
the geocenter practically coincide; (iv) the coordinate
transformation BRS-GRS in both approaches coincide at least to
O(c^{-4} r) terms (for time coordinate) and O(c^{-2}) accuracy (for
spatial coordinates), with r being the geocentric distance.  The
results above suggest that no perceptible ambiguities may arise from
using one or another approach when dealing with practical problems of
applied celestial mechanics, astrometry, and geodynamics.