We compare two theoretical approaches to the data analysis of the Cassini
relativity experiment based on the Doppler tracking and the time delay
technique that were published correspondingly by Kopeikin et al in Phys. Lett.
A 367, 276 (2007) and by Bertotti et al in Class. Quant. Grav. 25, 045013
(2008). Bertotti et al believed that they found a discrepancy with our paper
and claimed that our analysis was erroneous. The present paper elucidates,
however, that the discrepancy is illusory and does not exist. The two
techniques give the same result making it evident that the numerical value of
the PPN parameter 'gamma' measured in the Cassini experiment is indeed affected
by the orbital motion of the Sun around the barycenter of the solar system.Comment: 6 pages, no figures. Accepted for publication to Physics Letters 

Anderson

Bertotti

Hardorp

Klioner

Kopeikin

Moyer

Sergei M. Kopeikin

Will

Physics Letters A

English

arXiv

We compare two theoretical approaches to the data analysis of the Cassini relativity experiment based on the Doppler tracking and the time delay technique that were published correspondingly by Kopeikin et al. [S.M. Kopeikin, A.G. Polnarev, G. Schäfer, I.Yu. Vlasov, Phys. Lett. A 367 (2007) 276] and by Bertotti et al. [B. Bertotti, N. Ashby, L. Iess, Class. Quantum Grav. 25 (2008) 045013]. Bertotti et al. believed that they found a discrepancy with our paper and claimed that our analysis was erroneous. The present Letter elucidates, however, that the discrepancy is illusory and does not exist. The two techniques give the same result making it evident that the numerical value of the PPN parameter γ measured in the Cassini experiment is indeed affected by the orbital motion of the Sun around the barycenter of the solar system

Kopeikin, Sergei M.

University of Missouri: MOspace

Physics Letters A 373 (2009) 2605–2607Contents lists available at ScienceDirectPhysics Letters Awww.elsevier.com/locate/plaOn the two approaches to the data analysis of the Cassini interplanetaryrelativity experimentSergei M. KopeikinDepartment of Physics & Astronomy, University of Missouri, Columbia, MO 65211, USAa r t i c l e i n f o a b s t r a c tArticle history:Received 23 April 2009Received in revised form 5 May 2009Accepted 12 May 2009Available online 28 May 2009Communicated by V.M. AgranovichPACS:04.20.-q04.80.CcKeywords:GravitationRelativityReference framesCassini spacecraftWe compare two theoretical approaches to the data analysis of the Cassini relativity experiment basedon the Doppler tracking and the time delay technique that were published correspondingly by Kopeikinet al. [S.M. Kopeikin, A.G. Polnarev, G. Schäfer, I.Yu. Vlasov, Phys. Lett. A 367 (2007) 276] and by Bertottiet al. [B. Bertotti, N. Ashby, L. Iess, Class. Quantum Grav. 25 (2008) 045013]. Bertotti et al. believedthat they found a discrepancy with our paper and claimed that our analysis was erroneous. The presentLetter elucidates, however, that the discrepancy is illusory and does not exist. The two techniques givethe same result making it evident that the numerical value of the PPN parameter γ measured in theCassini experiment is indeed affected by the orbital motion of the Sun around the barycenter of the solarsystem.© 2009 Elsevier B.V. All rights reserved.In 2002 a measurement of the effect of solar gravity upon thephase of coherent microwave beams passing near the Sun hasbeen carried out with the Cassini mission, allowing a very accu-rate measurement of the PPN parameter γ [1,2]. The data havebeen analyzed with NASA’s Orbit Determination Program (ODP).Relativistic ranging time delay, incorporated to the NASA ODP code,was originally calculated by Moyer [3] under assumption that thegravitating body that deﬂects light, does not move. Regarding theSun, it means that the ODP code derives the ranging delay in theheliocentric frame. Let us introduce the heliocentric coordinatesXα = (X0, Xi) = (cT , X), and use notation xα = (x0, xi) = (ct, x) forthe barycentric coordinates of the solar system, which origin is atthe center of mass of the solar system. The Sun moves with respectto the barycentric frame with velocity v = dx/dt amounting to∼ 15 m/s due to the cumulative gravitational attraction of Jupiter,Saturn, and other planets [4]. We have discovered [5] that thoughthis velocity looks small, it affects the measured value of the PPNparameter γ and cannot be neglected in the data analysis of suchhigh-precision relativity experiment as Cassini [1,2].A legitimate question arises whether the ODP code accounts forthe solar motion or not. We analyzed this question in [5] by doingcalculations of the Doppler shift caused by the gravitational ﬁeldof the moving Sun. We came to the conclusion that the ODP codeE-mail address: kopeikins@missouri.edu.0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2009.05.058does take into account the motion of the Sun and that this mo-tion affects the Cassini ranging data. The original papers on Cassiniexperiment [1,2] did not analyze the impact of the solar motionon the results of the experiment. Hence, Bertotti et al. [6] havedecided to repeat our analysis [5] by making direct use of a differ-ent approach based on the light-time equation that is the Shapiroranging time delay. Unfortunately, insuﬃciently elaborated com-parison of the two different mathematical techniques did not allowBertotti et al. [6] to reproduce our results. The goal of the presentLetter is to show that the result of Bertotti et al. paper [6] exactlycoincides with that obtained earlier in our paper [5].The ranging time delay in the heliocentric coordinates with theSun located at the origin of this frame is well known [7]. Aftermaking use of the heliocentric coordinates it readsT2 − T1 = 1c|X2 − X1| + T , (1)T = (1+ γ )GMc3ln[R2 + R1 + R12R2 + R1 − R12], (2)where γ is a parameter of the PPN formalism [7], X2 and X1 arethe heliocentric coordinates of observer on the Earth and emit-ter (Cassini spacecraft) respectively, distance of the emitter fromthe Sun is R2 = |X2|, distance of the observer from the Sun isR1 = X1, and R12 = |X2 − X1| is the null-cone heliocentric dis-tance between the emitter and observer. This equation coincidesexactly (after reconciling our and Moyer’s notations for distances)2606 S.M. Kopeikin / Physics Letters A 373 (2009) 2605–2607with the ODP time-delay equation (8-38) given in Section 8 of theODP manual on page 8-19 [3].Moyer [3] had transformed the argument of the logarithm inthe heliocentric ranging delay (2) to the barycentric frame by mak-ing use of substitutionsX2 ⇒ r2 = x2 − x(t2), X1 ⇒ r2 = x1 − x(t1), (3)where x2 = x(t2), x1 = x(t1) are the barycentric coordinates of theobserver and the emitter taken at time of observation, t2, andemission, t1, respectively. The ODP manual [3] does not provideany evidence that these substitutions in the ranging time delay (2)are consistent with relativity and do not violate the Lorentz sym-metry of the Cassini experiment. Nonetheless, Eqs. (2), (3) are le-gitimate transformations from the heliocentric to the barycentricframe in the sense that they take into account velocity of the Sunin the argument of the ranging time delay in the linearized approx-imation. This is because the solar barycentric coordinate x(t1) atthe time of emission t1 is not the same as the solar coordinatex(t2) at the time of observation t2x(t2) = x(t1) + v(t1)(t2 − t1) + O(|t2 − t1|2), (4)due to the non-zero velocity v(t1) of the Sun.However, spatial transformations (3) are not suﬃcient in or-der to get all velocity-dependent terms of the ﬁrst order in theranging time delay. The reason is that the Newtonian part of theranging delay (1) contains time in its left side, which must betransformed from one frame to another with taking into accountthe post-Newtonian correction: T ⇒ t + vx/c2. This was done byBertotti et al. [6] who obtained that the ranging time delay in theheliocentric and barycentric frames must be related by the simpleequationt = (1+ γ − k · β)2GMc3ln[r2 + r1 + r12r2 + r1 − r12], (5)where r12 = |r2 − r1|, r1 = |r1|, r2 = |r2|, β = v/c, and k is aunit vector along the light ray from the emitter to the observer.Formula (5) was derived previously in our work [8] (see also [9]for the case γ = 1).Bertotti et al. [6, Section 4] noticed that our paper [5] neglectedthe post-Newtonian correction in the transformation of the timecoordinate: T ⇒ t + vx/c2. Hence, [6] believed that [5] missedthe velocity-dependent term in front of the logarithmic function inEq. (5). However, our paper [5] dealt with the gravitational Dopplershift of the Cassini radio frequency, that is with the time deriva-tive of the original heliocentric equation (1). Transformation of thegravitational Doppler shift does not require to transform time inorder to get all linear velocity-dependent corrections, because theheliocentric equation for the Doppler shift is already proportionalto velocities of observer and emitter. Hence, the only transforma-tion, which remains to complete is the transformation of the ve-locities, which can be done after making use of transformations (3)of spatial coordinates differentiated with respect to time (see [5,Eqs. (12)–(15)]). Our equation for the gravitational Doppler shift,zgr , written down in the barycentric coordinates is [5,8]zgr = −(r1r12β2 +r2r12β1)· αB−(γ¯r1r12β2 + γ¯r2r12β1 − β)· αB , (6)where β1 = (1/c)dx1/dt – velocity of the emitter, β2 = (1/c)dx2/dt– velocity of the observer, γ¯ = γ − 1, and the dot between twovectors denotes a scalar dot product. VectorαB = α R2 d, (7)dwhere R – radius of the Sun, d – radius-vector of the impactparameter of the radio signal with respect to the Sun, d = |d|, andα = 8.5×10−6 rad is the solar gravitational deﬂection of light onits limb.The ﬁrst term in the right side of Eq. (6) was obtained byBertotti and Giampieri [10] under condition that the Sun does notmove. The second term in the right side of Eq. (6) would lead todeviation from general relativity in case of γ¯ = 0. The solar veloc-ity β also enters this term, has the same magnitude as the termswith γ¯ and, hence, correlates with the measurement of γ¯ . ThisαBβ  3 × 10−13 correction comes from the time derivative ofthe argument of the logarithmic function in Eq. (5) as shown in [5].Differentiation of the solar-velocity term in front of the logarithmin Eq. (5) would give corrections of the next order of magnitude tothe right side of the Doppler shift equation (6). We did not forgetthat term but neglected it due to its smallness. We conclude thatthe remark Bertotti et al. [6] on that we missed the velocity termin front of the logarithm in Eq. (5) is irrelevant for our Dopplershift formula calculations [5].The paper by Bertotti et al. [6] also claims that the velocity-dependent terms appear in the time delay (5) only in front of thelogarithmic function. This claim is rather naive and was the rea-son for misinterpretation of the impact of the solar motion onthe Cassini data present in paper [6]. It only seems like that theargument of the logarithm in Eq. (5) does not contain velocity-dependent terms explicitly. One should keep in mind that the dis-tance r12 is not taken on a single hypersurface of constant timebut connects two different positions of the Sun, which are not thesame in the barycentric coordinates because the Sun moves. Thepost-Newtonian expansion of distance r12 yieldsr12 = r − r · β + O(β2), (8)where r = |r| and r = x2 − x1 = kr is a null-cone vector connect-ing the observer to the emitter along the radio wave path fromthe Cassini to the observer. The post-Newtonian expansion of theranging delay (5) ist = (1+ γ − k · β)2GMc3ln[r2 + r1 + r − r · βr2 + r1 − r + r · β]+ O(2GMβ2c3), (9)which explicitly reveals the presence of the velocity-dependentterms in the argument of the ranging time delay. Eq. (9) has beenderived in our paper [5], and its logarithmic part is just a partialderivative of the heliocentric time delay with respect to the PPNparameter γ as shown in [5, Eq. (25)].Bertotti et al. [6] claimed that expression (9) for the rangingtime delay is not used in the ODP code and cannot be appliedfor theoretical analysis of the Cassini experiment as we did in [5].Therefore, Bertotti et al. [6] have concluded that our numerical es-timates of the gravitational shift of frequency caused by motionof the Sun with respect to the barycenter of the solar system asgiven in [5], are incorrect. These statements of Bertotti et al. [6]cannot be accepted by any rationally thinking researcher as theauthors of [6] have trivially overlooked that expression (9) has ex-actly the same logarithmic function as in Eq. (5) with the argumentexpressed in terms of the null-cone distance r and velocity of theSun, v , which are related to distance r12 via self-consistent math-ematical transformation (8). The equivalence of Eqs. (5) and (9)means that our numerical estimates and theoretical conclusionsgiven in paper [5] with regard to the impact of the solar motionon the Cassini measurement of γ parameter, are ﬁrmly conﬁrmedby the independent study of Bertotti et al. [6].S.M. Kopeikin / Physics Letters A 373 (2009) 2605–2607 2607References[1] B. Bertotti, L. Iess, P. Tortora, Nature 425 (2003) 374.[2] J.D. Anderson, G. Lau, E.L. Giampieri, in: Proceedings of the 22nd Texas Sympo-sium on Relativistic Astrophysics, Stanford, 2004, eConf C041213, 0305, avail-able on http://www.slac.stanford.edu/econf/C041213/papers/0305.PDF.[3] T.D. Moyer, Formulation for Observed and Computed Values of DeepSpace Network Data Types for Navigation, John Wiley & Sons, Hoboken,2003.[4] J. Hardorp, Bull. Am. Astron. Soc. 17 (1985) 592.[5] S.M. Kopeikin, A.G. Polnarev, G. Schäfer, I.Yu. Vlasov, Phys. Lett. A 367 (2007)276.[6] B. Bertotti, N. Ashby, L. Iess, Class. Quantum Grav. 25 (2008) 045013.[7] C.M. Will, Theory and Experiment in Gravitational Physics, Cambridge Univer-sity Press, Cambridge, 1993.[8] S.M. Kopeikin, G. Schäfer, Phys. Rev. D 60 (1999) 124002.[9] S.A. Klioner, Astron. Astrophys. 404 (2003) 783.[10] B. Bertotti, G. Giampieri, Class. Quantum Grav. 9 (1992) 777.

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