Tests of gravitational physics by ranging to Mercury

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Tests of gravitational physics byranging to
Mercury
Neil Ashby, John Wahr Dept. of Physics,
University of Colorado at Boulder Peter
Bender Joint Institute for Laboratory
Astrophysics, Boulder
Affiliate, National Institute of Standards and
Technology, Boulder email ashby_at_boulder.nist.go
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Outline

• History of the present calculation
• Characterizing the approach
• Analytical vs. numerical
• Worst-case systematics
• The range observable
• Choice of parameters
• a. orbital parameters
• solar system parameters
• cosmological parameters
• relativity parameters
• Assumptions
• Results

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History and purpose

• Began 1974
• 1980-1982 NASA funding
• 1989-1995 various publications, conference
  talks/proceedings
• Most recent results published in Phys.
  Rev. D 75, 022001-022020 (2007)
• applied to BepiColombo
  mission to Mercury. 
• The purpose is to develop theory and associated
  computer code to
• support experiments to test alternative
  gravitational theories
• determine important solar system parameters
  (e.g. ).

in ranging experiments between the Earth and
- Mercury - Mars or Mars Mercury - A
close solar probe.
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Characterizing the approach-theoretical

• Orbital perturbations of the planets due to
  various relativity and/or
• cosmological effects are treated analytically to
  first order.
• The theoretical perturbation expressions are
  implemented in code for simulation of ranging
  missions of varying duration.

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Characterizing the approach-statistical
The approach is is a modified worst-case
approach. This means that errors are presumed to
be highly correlated. Specifically, systematic
ranging errors have time signatures that have
the worst possible effect on determination of
final uncertainties of a parameter of
interest. However, systematic errors cannot
maximize the uncertainties in all parameters
simultaneously, so we adopt a more reasonable
modified approach the worst-case
uncertainties are divided by 3.
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Parameters--Keplerian Orbital Elements
Range is constructed from Keplerian orbital
elements of Earth and Mercury
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Unperturbed range observable
9 Orbital Parameters are selected
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Additional parameters
product of Newtonian gravitational constant and
solar mass
Number of parameters so far 93 12
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Relativity parameters
Total number of parameters 93618
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Quadrupole Moment of the Sun--J2
Objective to develop better models of the solar
interior, explain -- energy generation, solar
evolution -- 11-year sunspot cycle --
neutrino flux --
Some information comes from observations of the
surface
Flattening
Rotation
Helioseismometry
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Orbital perturbations--solar J2
The effect of the solar quadrupole moment on
orbital elements was taken from the literature on
Lagrangian planetary perturbation theory, after
checking by numerical integration. This sample
is from Principles of Celestial Mechanics, by
P. M. Fitzpatrick (Academic Press, New York
(1971).
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Sample perturbations-b
These perturbations are expressed with the help
of the integrals

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Strong equivalence principle violation
The nonlinear effect of the suns gravitational
self-energy on two falling bodies (such as
Jupiter and Mercury) is described by differential
equations for the radial and tangential
perturbations
(ratio of suns self energy to rest energy)
qi is heliocentric position of planet i. The
driving term can be expanded in power series in
ratios such as
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Strong equivalence principle violation-contd
If the planets are coplanar, the equations for
radial and tangential (to the orbit)
perturbations can be expanded and expressed in
the form
where for planet i,
Particular solutions are
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Strong equivalence principle violation-contd
The second-order differential equations have
solutions that are superpositions of (a)
particular solutions (g) general solutions of
homogeneous equations --i.e., without driving
terms. Numerical solutions pick up contributions
including the general solutions unless the
boundary conditions are chosen properly.
Example for the earth-moon-sun system, the
solutions to the differential equations typically
look like this
It is known that the lunar range perturbation is
about 8 m in amplitude if h 1.
Range perturbation (earth-moon)
in meters
Time in days
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Covariance Analysis Worst Case Systematic Error
where is the range residual, the
difference between theoretically predicted range
with nominal values for the parameters, and the
measured range.
If errors in the range residuals were random and
uncorrelated, such that
Then it follows that the parameter error would be
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Correlations between b and J2
However, the time signatures of various
perturbations are instead highly correlated.
Here is an example.
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Worst-case analysis
The error in a parameter di
could be bigger if the error in the residual
is correlated with the partial derivative
for some n.
Suppose that over the entire data set we were
confident that the rms error in the residuals
could be limited or constrained by
Then we look for the maximum error in di subject
to the above constraint.
Note there is a factor of N. Generally this error
decreases but approaches a limit as the number of
observations continues to increase.
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Error Correlations
If the residuals are such that the mth parameter
is most poorly determined, Then the error in the
nth parameter is
So the inverse of the covariance matrix contains
a huge amount of information, For the BepiColombo
Mission, simulations have been carried out with
19, then 18 parameters.
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Assumptions
Launch January 1, 2012--Julian Date
2455928.0 Unperturbed Keplerian elements taken
from American Ephemeris Known Newtonian
perturbations assumed to be removed from
data Worst-Case uncertainties divided by 3 are
presented Mission duration is extended to 8
years in the calculation Nordtvedt parameter
can be treated as independent, or can be viewed
as dependent on other parameters, e.g.,
(Simulations have been done in this case but are
not presented here.)
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Further assumptions

• One normal range point per day is obtained
• No a priori knowledge of uncertainties of
  parameters is assumed
• Data is excluded if the line-of-sight passes
  within 5o of the suns center
• Systematic range errors are subject to the
  constraint

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Perturbation Theory--outline of simulation

• Use Relativistic equations of motion to obtain
  perturbing accelerations
• 2. Resolve perturbing accelerations into
  cartesian components radial (R),
• normal to radius in orbit plane (S), normal to
  orbit plane (T)
• 3. Integrate Lagrange Planetary Perturbation
  Equations to find the
• perturbed orbital elements (analytical,
  not numerical)
• 4. Calculate partial derivatives of the range
  with respect to each of the
• parameters of interest
• 5. Construct the covariance matrix each day (for
  up to 8 years)
• Invert and calculate the worst-case uncertainties
• 7. Divide by 3 to get modified worst-case
  uncertainties

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Systematic error (m)
Time in days
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Systematic error (m)
Time in days
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Comparisons of worst-case systematic errors
Since the worst-case systematic error cannot
simultaneously be worst for any two parameters,
the worst-case errors are divided by 3.
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Results for b
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Calculation results--uncertainty in solar
quadrupole moment
Present uncertainty level
If J2 and b are included in the parameter
list --isotropic case
Uncertainty in solar quadrupole moment
With all 18 parameters
If General Relativity is correct
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Present uncertainties in nonorbital parameters
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Results--assuming GR is correctwith dG/dt
Significant improvements are obtained after 1
year in all these parameters.
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Nonmetric theory results--15 18 parameters

Significant improvement over present
uncertainties