1. 1 Tests of gravitational physics by ranging 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@boulder.nist.gov

2. 2 Outline 1.History of the present calculation 2.Characterizing the approach a.Analytical vs. numerical b.Worst-case systematics 3.The range observable 4.Choice of parameters a. orbital parameters b.solar system parameters c.cosmological parameters d.relativity parameters 5.Assumptions 6.Results

3. 3 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.

4. 4 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.

5. 5 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. The true worst-case uncertainties can be recovered by multiplying all quoted errors by 3. For random uncorrelated errors, the estimated uncertainties can be recovered by multiplying the quoted errors by where N is the number of observations.

6. 6 Parameters--Keplerian Orbital Elements Range is constructed from Keplerian orbital elements of Earth and Mercury:

7. 7 Unperturbed range observable 9 Orbital Parameters are selected:

8. 8 Additional parameters product of Newtonian gravitational constant and solar mass cosmological change in gravitational constant quadrupole moment of the sun Number of parameters so far: 9+3 = 12

9. 9 Relativity parameters preferred frame parameter, (deleted from consideration because it is now very well determined) Whitehead parameter: solar system -- milky way interaction measures nonlinear contribution of gravitational potential to Nordvedt parameter: effect of a third massive body on gravitational interaction of two bodies (violation of strong equivalence principle) measures spatial curvature produced by mass; of interest since some scalar-tensor theories predict values of order 10 -7 preferred frame parameter speed 377 m/s relative to CMWBR Total number of parameters: 9+3+6=18

10. 10 Quadrupole Moment of the Sun--J 2 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: FlatteningRotationHelioseismometry

11. 11 Orbital perturbations--solar J 2 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).

12. 12 Sample perturbations-  These perturbations are expressed with the help of the integrals

13. 13 Strong equivalence principle violation The nonlinear effect of the sun’s gravitational self-energy on two falling bodies (such as Jupiter and Mercury) is described by differential equations for the radial and tangential perturbations: q i is heliocentric position of planet i. The driving term can be expanded in power series in ratios such as (ratio of sun’s self energy to rest energy)

14. 14 Strong equivalence principle violation-cont’d 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:

15. 15 Strong equivalence principle violation-cont’d 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: Time in days Range perturbation (earth-moon) in meters It is known that the lunar range perturbation is about 8 m in amplitude if 

16. 16 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

17. 17 Correlations between  and J 2 However, the time signatures of various perturbations are instead highly correlated. Here is an example.

18. 18 Worst-case analysis The error in a parameter d i is correlated with the partial derivativefor 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 d i subject to the above constraint. could be bigger if the error in the residual Note there is a factor of N. Generally this error decreases but approaches a limit as the number of observations continues to increase.

19. 19 Error Correlations If the residuals are such that the m th parameter is most poorly determined, Then the error in the n th 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.

20. 20 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.)

21. 21 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 5 o of the sun’s center; Systematic range errors are subject to the constraint

22. 22 Perturbation Theory--outline of simulation 1.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) 6.Invert and calculate the worst-case uncertainties 7. Divide by 3 to get “modified worst-case uncertainties”

23. 23 Time in days Systematic error (m)

24. 24 Time in days Systematic error (m)

25. 25 Time in days Systematic error (m) 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.

26. 26 Results for 

27. 27 Calculation results-- uncertainty in solar quadrupole moment Present uncertainty level If General Relativity is correct If J 2 and  are included in the parameter list --isotropic case With all 18 parameters Uncertainty in solar quadrupole moment

28. 28 Present uncertainties in nonorbital parameters

29. 29 Results--assuming GR is correct with dG/dt Significant improvements are obtained after 1 year in all these parameters.

30. 30 Nonmetric theory results--15 &18 parameters **************************** * Significant improvement over present uncertainties