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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: [email protected] 1 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 2 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. GM , J 2 ). in ranging experiments between the Earth and - Mercury - Mars or Mars & Mercury - A close solar probe. 3 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. 4 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 3 N where N is the number of observations. 5 Parameters--Keplerian Orbital Elements Range is constructed from Keplerian orbital elements of Earth and Mercury: a1 , a2 semimajor axes; e1 , e2 eccentricities; 1 , 2 longitude of perihelion; 0,i 2 inclination; --, 2 longitude of ascending node; L10 , L20 initial longitude; 6 Unperturbed range observable 2 r12 r22 2r1r2 cos 12 ; cos 12 cos f1 (1 L10 ) ( 2 L20 ) ( L20 L10 ) cos[ f 2 ( 2 L20 ) ( 2 L20 )] cos I 2 sin f1 (1 L10 ) ( 2 L20 ) ( L20 L10 ) sin[ f 2 ( 2 L20 ) ( 2 L20 )] a(1 e2 ) r 1 e cos f 9 Orbital Parameters are selected: a1 , a2 , e1 , e2 ; 1 L10 , 2 L20 , 2 L20 , I 2 ; L20 L10 7 Additional parameters GM J2 G G product of Newtonian gravitational constant and solar mass quadrupole moment of the sun cosmological change in gravitational constant Number of parameters so far: 9+3 = 12 8 Relativity parameters 1 2 3 measures nonlinear contribution of gravitational potential to g 00 measures spatial curvature produced by mass; of interest since some scalar-tensor theories predict values of order 10-7 preferred frame parameter preferred frame parameter speed 377 m/s relative to CMWBR preferred frame parameter, (deleted from consideration because it is now very well determined) Whitehead parameter: solar system -- milky way interaction Nordvedt parameter: effect of a third massive body on gravitational interaction of two bodies (violation of strong equivalence principle) Total number of parameters: 9+3+6=18 9 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 10 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). 11 Sample perturbations- These perturbations are expressed with the help of the integrals Sij df sin f cos f f i j f0 a 2me 2 S10 2eS11 / 4 m 2 e a / 2ea GM c2 F F F0 1 e2 I 0 2 ~ m S00 2 S01 / e (sin f cos f / a M 3m 1 e cos f 0 n (t ) / a 4 2 m 2 S01 / e (sin f cos f / ( a ) . 12 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: G( M M i ) d 2q i GM j qi dt 2 qi3 M c 2 qij 3 qij qi q j , 6 (ratio of sun’s self energy 3.52 10 to rest energy) M c2 qi is heliocentric position of planet i. The driving term can be expanded in power series in ratios such as aearth amercury , . a jupiter a jupiter 13 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 2 rr 2i rt 3i rr An cos(nSi ); n 1 rt 2i rr Bn sin(nSi ), n 1 where for planet i, i2 G( M M i ) , ai3 Si (i j )t it . Particular solutions are: ni An 2i Bn rr , 2 2 2 n 1 ni ( n i i ) 2ni i An (n 2i 2 3i 2 ) Bn rt . 2 2 2 ni (n i i ) n 1 14 Strong equivalence principle violation-cont’d Range perturbation (earth-moon) in meters 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 1. Time in days 15 Covariance Analysis & Worst Case Systematic Error Ckj N 18 i 1 (ti ) (ti ) d k d j dl C 18 j 1 1 (correlation matrix) (ti ) lj d (ti ) i 1 j N (correction to parameter) where (ti ) 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 (ti ) (t j ) ij 2 , Then it follows that the parameter error would be dl (C 1 )ll . 16 Correlations between and J2 However, the time signatures of various perturbations are instead highly correlated. Here is an example. 17 Worst-case analysis The error in a parameter di could be bigger if the error in the residual i / d n , 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: 1 N (ti ) 2 . 2 i Then we look for the maximum error in di subject to the above constraint. dl N (C 1 )ll . Note there is a factor of N. Generally this error decreases but approaches a limit as the number of observations continues to increase. 18 Error Correlations If the residuals are such that the mth parameter is most poorly determined, Then the error in the nth parameter is: N 1 dn ( C )nl 1 (C )ll 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. 19 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., 2 4 3 1 2 3 (Simulations have been done in this case but are not presented here.) 20 Further assumptions • One normal range point per day is obtained; • No a priori knowledge of uncertainties of parameters is assumed; o • Data is excluded if the line-of-sight passes within 5 of the sun’s center; • Systematic range errors are subject to the constraint 1 N N 2 2 2 ( t ) (4.5cm) i i 1 21 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” 22 Systematic error (m) Time in days 23 Systematic error (m) G /G Time in days 24 Systematic error (m) Comparisons of worst-case systematic errors G /G Time in days Since the “worst-case” systematic error cannot simultaneously be worst for any two parameters, the worst-case errors are divided by 3. 25 Results for 5 105 26 Uncertainty in solar quadrupole moment Calculation results-uncertainty in solar quadrupole moment Present uncertainty level If J2 and are included in the parameter list --isotropic case With all 18 parameters If General Relativity is correct 27 Present uncertainties in nonorbital parameters 28 Results--assuming GR is correct with dG/dt Significant improvements are obtained after 1 year in all these parameters. 29 Nonmetric theory results--15 &18 parameters * * * * * * * * * * * * * * * Significant improvement over present uncertainties 30