Bruno Bertotti Dipartimento di Fisica Nucleare e Teorica Università di Pavia, Italy Neil Ashby

1. Accurate light-time correction due to a gravitating massA mathematical follow-up to Cassini’s experiment Bruno Bertotti Dipartimento di Fisica Nucleare e Teorica Università di Pavia, Italy Neil Ashby DepartmentofPhysics Universityof Colorado, Boulder (USA) Paper in preparation

2. A carelessdismissal A test of General Relativity using radio links with the Cassini spacecraft B. Bertotti*, L. Iess†and P. Tortora‡ γ = 1 + (2.1 ± 2.3)10-5 A O Second-ordertermsneglected; but B Thisexpressionhas some second-ordercorrections, butnotall (no second-ordermetric)

3. Concern Whatis the orderofmagnitudeof the neglectedsecond-orderterms? Do they invalidate Cassini’s result?

4. Redress Cassini’s results are notaffected Secondorderdelayobtained in general Enhancementtamed

5. A mathematical follow-up to Cassini’s experimentisneeded In the DSN phasedifferences are measured, notfrequencychanges; the appropriate mathematicalobservableis the light-timebetweenanevent A and anevent B, greaterthan the geometricaldistancerABby the gravitationaldelayΔt. Standard formula: A O ODP doesnotuse the standard formula, but The correctionisofsecondorder, butitignoressecond-order metric and isinconsistent. Doesitaffect the result? B

6. Howlargeis the ODP correction ? In a closeconjunction Moyer’s “correction” isoforder Dangerous! Enhancementphenomenon

7. Asymptoticseries Selectionoftermscannotbebased on empiricalorderofmagnitudeestimates, but on automaticformalexpansions in powersofm/b0 , like: Asymptoticexpansions are definedbytheir coefficients and by the limitingproperty: asm → 0, at k-order the residual→ 0 asmk+1 They are notordinaryfunctions: e.g., The mainproblem: the orderofmagnitudeof the coefficientsΔs . IfrA /b0, rB/b0 = O(1), Δs = O(1). But in a closeconjunctionrA /b0, rB/b0 = O(R/b0) >> 1and second-ordertermsoforder

8. Orderofinfinitesimal m isnot a fixedquantity, but a parameterwhichtendstonought Asymptoticseriesprovideanautomatic and safe way to decide whichtermstokeep

9. Fermat’s Principle Isotropiccoordinates! Orbitalpropagation in solar system usesisotropiccoordinates, Lorentz-transformedtobarycentric system. Light propagates in spaceas in a refractive medium (Eikonalapproach) withrefractionindex , Ferrmat’s actionfunctional, has a minimum on the actualray. This minimum isequalto the light-timetB ¬ tA

10. The ray B m b Obtuse case (usual) (b0 =1) Acute case (notconsidered) A Actualray: b > b0. Wealsouseh = bN(b) (impact parameter) r A , r B , ΦABconstitute the experimentalsetup and are fixed.

11. The eikonal Solve for E (t,r) byseparationofvariables

12. The action LetS(h) beFermat’s functionalS(A,B) when the variabilityofitsargument (anylinebetween A and B) isrestrictedto the coordinate ofclosestapproachb (or h). S(h) iscalledreducedaction. Positive contributionsfromingoing and outgoingbranch. This minimum isequalto the light-time. To solve S’(h) =0,use the powerexpansion. Since m isinfinitesimal, in goingfrom h0 toh0 + mh1 the reducedactiondoesnotchange. Hence the first orderlight-timedoesnotdepend on h1; indeed, Similarly, in goingfromh0 + mh1 toh0 + mh1+ m2h2 the reducedactiondoesnotchange; the second-orderlight-timedoesnotdepend on h2. S(h) Light- time h “True” h

13. Secondorderlight-time(obtuse case) NotenhancedEnhanced

14. Enhancement Deflectionisdifferent: whereδs ~ 1 is pure number (δ1 = 2(1+ γ)) , because no dimensionalquantities are involved. Similarly, whenrA~ b 0 , rB~ b 0 , Δ s ~ 1; but 1 AU/Rsun = 200 >> 1 !!. WhenrA>> b 0 , rB>> b 0 Δ2 = O(R/ b 0) cannotbeexcluded; itcorrespondsto a change in γwhich can be 200 timesgreater, detectablewithσ γ~ 2 x 10-5, about Cassini’s value !! In anasymptoticexpansion, one can expectdimensionlesscoefficientstobeoforderunityonlyiftheirarguments are alsooforderunity A dangerous situation.

15. Enhancement Second-orderlight-time Find the limitof asb0/R →0 ODP formula recovered!

16. Light-timecorrectionsfor LISA Oneyearperiodicity; maskedbyaccelerationnoise

17. Conclusions Cassini’s results are notaffected Secondorderdelayobtained in general Enhancement in a closeconjunctionunderstood

18. Variational Lemma S(h) The changeofh withm isdescribedby butsinceS(h)is minimum at “true “ h, at orderm the light-timedoesnotdepend on h1.In fact, at first order Light- time h “True” h Forsimplicity, take h0 = b0 = 1. Similarly, at orderm2 the light-timedoesnotdepend on h2,, etc. Great simplification!