Contents -- List of figures -- List of tables -- Preface -- New tools for determining the light travel time in static, spherically symmetric spacetimes beyond the order G2 -- 1 Introduction -- 2 Notations and conventions -- 3 Generalities -- 4 Specific assumptions on the metric and the light rays -- 4.1 Post-Minkowskian expansion of the metric -- 4.2 Time transfer function for a quasi-Minkowskian light ray -- 5 Fundamental properties of functions T(n) -- 5.1 Recurrence relation satisfied by functions T(n) -- 5.2 Analyticity of the functions T(n) -- 6 First procedure: determination of the T(n)'s from the recurrence relation for n = 1, 2, 3 -- 7 Second procedure: determination of the T(n)'s from the geodesic equations -- 7.1 Null geodesic equations -- 7.2 Post-Minkowskian expansion of the impact parameter -- 7.3 Implementation of the method -- 8 Simplification of the second procedure -- 8.1 Use of a constraint equation -- 8.2 Explicit calculation of T(1), T(2), and T(3) -- 9 Direction of light propagation up to order G3 -- 10 Light ray emitted at infinity -- 11 Enhanced terms in T(1), T(2), and T(3) -- 12 Application to some Solar System experiments -- 13 Concluding remarks -- References -- Testing relativistic gravity with radio pulsars -- 1 Introduction -- 1.1 Radio pulsars and pulsar timing -- 1.2 Binary pulsar motion in gravity theories -- 1.3 Gravitational spin effects in binary pulsars -- 1.4 Phenomenological approach to relativistic effects in binary pulsar observations -- 2 Gravitational wave damping -- 2.1 The Hulse-Taylor pulsar -- 2.2 The Double Pulsar - The best test for Einstein's quadrupole formula, and more -- 2.3 PSR J1738+0333 - The best test for scalar-tensor gravity -- 2.4 PSR J0348+0432 - A massive pulsar in a relativistic orbit.

2.5 Implications for gravitational wave astronomy -- 3 Geodetic precession -- 3.1 PSR B1534+12 -- 3.2 The Double Pulsar -- 4 The strong equivalence principle -- 4.1 The Damour-Schäfer test -- 4.2 Direct tests -- 5 Local Lorentz invariance of gravity -- 5.1 Constraints on a1 from binary pulsars -- 5.2 Constraints on a2 from binary and solitary pulsars -- 5.3 Constraints on a3 from binary pulsars -- 6 Local position invariance of gravity -- 7 A varying gravitational constant -- 8 Summary and outlook -- References -- Lunar laser ranging and relativity -- 1 Introduction -- 2 Model -- 2.1 Overview -- 2.2 Ephemerides -- 3 Analysis -- 3.1 Software package LUNAR -- 3.2 Newtonian parameters -- 4 Results for relativistic parameters -- 4.1 Gravitational constant -- 4.2 Equivalence principle -- 4.3 Yukawa term -- 4.4 Geodetic precession -- 4.5 Metric parameter β -- 4.6 Preferred-frame parameters α1, α2 -- 5 Summary and outlook -- References -- Dragging of inertial frames, fundamental physics, and satellite laser ranging -- 1 Introduction -- 2 Dragging of inertial frames -- 3 Tests of string theory and the LAGEOS and LARES space experiments -- 4 LAGEOS and Gravity Probe B: two independent space experiments measuring frame dragging -- 5 The LARES mission -- 5.1 The LARES satellite -- 5.2 The LARES satellite and geodesic motion -- 5.3 Preliminary orbital analyses -- 5.4 Error analyses -- 5.5 Monte Carlo simulations -- 6 Conclusions -- References -- Elliptic functions and elliptic integrals for celestial mechanics and dynamical astronomy -- 1 Introduction -- 2 Notations -- 2.1 Glossary -- 2.2 First input argument: ϕ, u and x -- 2.3 Second input argument: k, u and x -- 2.4 Sign of n -- 2.5 Third input argument: n and α -- 2.6 Ordering of arguments -- 2.7 Omission of parameters -- 3 Elliptic functions.

3.1 General elliptic function -- 3.2 Weierstrass elliptic function -- 3.3 Jacobian elliptic functions -- 3.4 Jacobi's amplitude function -- 3.5 Differential equations of Jacobian elliptic functions -- 3.6 Addition theorem of Jacobian elliptic functions -- 3.7 Jacobi's form of incomplete elliptic integrals -- 3.8 Addition theorem of incomplete elliptic integrals -- 3.9 Jacobi's original form of incomplete elliptic integral of the third kind -- 4 Elliptic integrals -- 4.1 General elliptic integral -- 4.2 Legendre's form of incomplete elliptic integrals -- 4.3 Associate incomplete elliptic integrals -- 4.4 Complete elliptic integrals -- 4.5 Generalized elliptic integrals -- 4.6 Symmetric elliptic integrals -- 5 Numerical computation of elliptic functions and elliptic integrals -- 5.1 Overview -- 5.2 Transformation method -- 5.3 Example of transformation method -- 5.4 Simultaneous computation of Jacobian elliptic functions -- 5.5 Better computation of Jacobian elliptic functions -- 5.6 Computation of Jacobi's form of incomplete elliptic integrals -- 5.7 Computation of Legendre's form of incomplete elliptic integral of the first kind -- 5.8 Computation of other incomplete elliptic integrals -- 5.9 Computation of complete elliptic integrals other than II(n

4 Planetary theories with the aid of the expansions of elliptic functions -- 4.1 Notations -- 4.2 Expansion of the right-hand members of the equations: A change of the time variable -- 4.3 Application to planetary problems -- 5 Reference frames, time scales, and units for planetary ephemerides -- 5.1 Victor Brumberg's contributions -- 5.2 Planetary ephemerides -- 5.3 Conclusions -- References -- Atomic time, clocks, and clock comparisons in relativistic spacetime: a review -- 1 Introduction -- 2 Atomic time and atomic clocks -- 2.1 Atomic time scales -- 2.2 Atomic clocks -- 3 Relativistic framework for time and frequency comparisons -- 3.1 Introduction -- 3.2 Simultaneity and synchronization -- 3.3 Relativistic definitions of spacetime coordinate systems -- 3.4 Time scales in the barycentric and geocentric systems -- 3.5 Relativistic theory for time transformations in the Solar System (BCRS) -- 4 Relativistic treatment for time and frequency comparisons in the vicinity of the Earth (GCRS) -- 4.1 One-way time transfer -- 4.2 Two-way time transfer using artificial satellites -- 4.3 Frequency comparisons -- 5 Time and frequency transfer techniques -- 5.1 Established time and frequency transfer techniques: GNSS and two-way time transfer -- 5.2 Some novel two-way techniques -- 6 Clocks in relativistic geodesy -- 6.1 Review of chronometric geodesy -- 6.2 The chronometric geoid -- 7 Conclusions and prospects -- References -- Index.