Strong Equivalence Principle Violation in Coplanar 3Body Systems

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Strong Equivalence Principle Violation in
Coplanar 3-Body Systems
Neil Ashby University of Colorado Boulder, CO
USA Email ashby_at_boulder.nist.gov
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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.
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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.
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Some orbital elements of Jupiter, Earth, and
Mercury
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 Mercurys
orbit is stiffer So the effect of eccentricity
or inclination is smaller still.
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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.
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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
(This is neglected)
Tidal acceleration
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Equations of motion
Separating the equations into radial and
tangential parts
Equation of motion
synodic frequency
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Equations of motionradial 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)
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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
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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