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.

We compare three methods for computing invariant Lyapunov exponents (LEs) in general relativity. They involve the geodesic deviation vector technique (M1), the two-nearby-orbits method with projection operations and with coordinate time as an independent variable (M2), and the two-nearby-orbits method without projection operations and with proper time as an independent variable (M3). An analysis indicates that M1 and M3 do not need any projection operation. In general, the values of LEs from the three methods are almost the same. As an advantage, M3 is simpler to use than M2. In addition, we propose to construct the invariant fast Lyapunov indictor (FLI) with two-nearby-trajectories and give its algorithm in order to quickly distinguish chaos from order. Taking a static axisymmetric spacetime as a physical model, we apply the invariant FLIs to explore the global dynamics of phase space of the system where regions of chaos and order are clearly identified.PACS numbers: 95.10. Fh, 95.30.Sf

Lyapunov exponents (LEs) are key indicators of chaos in dynamical systems. In general relativity the classical definition of LE meets difficulty because it is not coordinate invariant and spacetime coordinates lose their physical meaning as in Newtonian dynamics. We propose a new definition of relativistic LE and give its algorithm in any coordinate system, which represents the observed changing law of the space separation between two neighboring particles (an "observer" and a "neighbor"), and is truly coordinate invariant in a curved spacetime. : 95.10.Fh, 95.30.Sf Chaos is a popular phenomenon in dynamical systems. One of its main features is the exponential sensitivity on small variations of initial conditions. The exhibition of chaos in the motion of Pluto makes it particularly attractive for scientists to investigate the dynamical behavior of the solar system[1]. Key words: chaotic dynamics, relativity and gravitation PACSIn Newtonian mechanics, Lyapunov exponents (LEs), as a key index for mea-

ASTROD I is the first step of ASTROD (Astrodynamical Space Test of Relativity using Optical Devices). This mission concept has one spacecraft carrying a payload of a telescope, five lasers, and a clock together with ground stations (ODSN: * Corresponding author. Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing, 210008, China. Optical Deep Space Network) to test the optical scheme of interferometric and pulse ranging and yet give important scientific results. These scientific results include a better measurement of the relativistic parameters, a better sensitivity in using optical Doppler tracking method for detecting gravitational waves, and measurement of many solar system parameters more precisely. The weight of this spacecraft is estimated to be about 300-350 kg with a payload of about 100-120 kg. The spacecraft is to be launched with initial period about 290 days and to pass by Venus twice to receive gravity-assistance for achieving shorter periods. For a launch on August 4, 2010, after two encounters with Venus, the orbital period can be shortened to 165 days. After about 370 days from launch, the spacecraft will arrive at the other side of the Sun for the determination of relativistic parameters.

J. W. Moffat and V. T. Toth submitted recently a comment [1] on our latest paper "Modified scalar-tensor-vector gravity theory and the constraint on its parameters" [2]. We reply to each of their comments and justify our work and conclusions. Especially, their general STVG (MOG) theory has to be modified to fit the modern precision experiments.