1. Rome MORE meeting Feb 16-17, 2009 1 1 Strong Equivalence Principle Violation in Coplanar 3-Body Systems Neil Ashby University of Colorado Boulder, CO USA Email: ashby@boulder.nist.gov

2. Rome MORE meeting Feb 16-17, 2009 2 2 Purpose of talk The purpose is to discuss some stumbling blocks that Peter Bender and I encountered while studying SEP violation effects on the orbits of Earth and Mercury, arising from orbital perturbations due to Jupiter. SEP violation affects the range between Earth and Mercury, so has an influence on the MORE experiment. We first attempted to integrate the orbital equations directly but ran into numerical difficulties; these will be described and demonstated. We then adopted an analytical approach for idealized orbits.

3. Rome MORE meeting Feb 16-17, 2009 3 3 Effect of SEP violation The ratio of gravitational self-energy of the sun to its rest energy is described by the parameter SEP violation modifies Newtonian gravity, so that the earth-sun attraction is modified by the mass of Jupiter.

4. Rome MORE meeting Feb 16-17, 2009 4 4 Some orbital elements of Jupiter, Earth, and Mercury JupiterEarthMercury Eccentricity, e0.048390.01670.2056 Inclination, I1.31 deg07.005 deg The inclination and eccentricity of Jupiter are small, so in a first approximation we treat the Jupiter-Earth-Sun system as coplanar and the orbits as nominally circular. Such an approximation applied to Mercury, but Mercury’s orbit is “stiffer” So the effect of eccentricity or inclination is smaller still.

5. Rome MORE meeting Feb 16-17, 2009 5 5 A simpler system –the Earth-Moon Sun system In the Earth-Moon Sun system, SEP violation causes modification of the Newtonian equations of motion, arising from the self-energy of the earth. The ratio of gravitational self-energy to rest mass energy of earth is The system is still coplanar. The analogy is: Earth----Sun (main attracting body) Moon---Earth or Mercury Sun-----Jupiter The system is analagous to the system of real interest, but the solution is simpler; SEP violation in the earth-moon system is non-controversial and solutions to the orbit perturbation problem can be found in the literature; The earth-moon system is subject to the same numerical integration difficulties that were encountered in the study of Jupiter-Earth-Mercury-Sun systems, and the solution is similar. So the Earth-Moon-Sun system is described here as a preliminary to describing the real systems of interest.

6. Rome MORE meeting Feb 16-17, 2009 6 6 Equations of motion of moon in geocentric coordinates Notation: radius vector, earth to moon: radius vector, sun to earth: Attraction towards earth: SEP violating acceleration: Tidal acceleration: (This is neglected)

7. Rome MORE meeting Feb 16-17, 2009 7 7 Equations of motion Equation of motion: Separating the equations into radial and tangential parts: synodic frequency

8. Rome MORE meeting Feb 16-17, 2009 8 8 Equations of motion— radial and tangential perturbations The solution involves 4 constants: the initial values of The general solution is a sum of a particular solution of the inhomogeneous equations (synodic frequency) and a general solution of the homogeneous equations (frequency of unperturbed moon motion)

9. Rome MORE meeting Feb 16-17, 2009 9 9 General Solution to equations of motion In order to prevent the solution of the homogeneous equations from contributing, the initial conditions must be chosen as follows:

10. Rome MORE meeting Feb 16-17, 2009 10Rome MORE meeting Feb 16-17, 2009 10 How much accuracy is need in initial conditions? Earth-moon range is nominally Tangential velocity is nominally SEP perturbations in range and tangential velocity have amplitudes To avoid significant contributions from unwanted solutions of the homogeneous equations, the initial conditions have to be specified to a few parts in