PHYSICAL REVIEW D 69, 042001 ͑2004͒
Deflection of spacecraft trajectories as a new test of general relativity: Determining the parametrized post-Newtonian parameters  and ␥
James M. Longuski,1 Ephraim Fischbach,2,* Daniel J. Scheeres,3 Giacomo Giampieri,4 and Ryan S. Park3 1School of Aeronautics and Astronautics, Purdue University, West Lafayette, Indiana 47907-1282, USA 2Department of Physics, Purdue University, West Lafayette, Indiana 47907-1396, USA 3Department of Aerospace Engineering, The University of Michigan, Ann Arbor, Michigan 48109-2140, USA 4Space and Atmospheric Physics, Blackett Laboratory, Imperial College, London SW7 28W, England ͑Received 11 July 2003; published 20 February 2004͒ In a previous work, we proposed a new test of general relativity ͑GR͒ based on a general deflection formula which applies to all values of asymptotic speed Vϱ (0рVϱр1). The formula simplifies to Einstein’s light deflection result when Vϱϭ1. At low velocity, the general deflection equation reduces to the classical New- tonian contribution along with additional terms which contain the GR effect. A spacecraft, such as the proposed interstellar mission which involves a close pass of the Sun, can be used to exaggerate the GR effect so that it can be accurately measured. In this paper we provide a detailed derivation of the general deflection equation, expressed in terms of the parametrized post-Newtonian constants  and ␥. The resulting formula demonstrates that by measuring spacecraft trajectories we can determine  and ␥ independently. We show via a detailed covariance analysis that  and ␥ may be determined to a precision of ϳ4ϫ10Ϫ5 and ϳ8ϫ10Ϫ6, respectively, using foreseeable improvements in spacecraft tracking.
DOI: 10.1103/PhysRevD.69.042001 PACS number͑s͒: 04.80.Cc, 95.55.Pe
I. INTRODUCTION the limit when the satellite is ultrarelativistic. This is to be expected, since the hyperbolic ͑open͒ trajectory of a light ray In a recent paper ͓1͔ a new test of general relativity ͑GR͒ can be viewed as the limiting case of that for an ultrarelativ- was proposed based on the deflection of spacecraft trajecto- istic massive object. Finally, the third term in Eq. ͑2.29͒ is ries. One of the new ͑and unanticipated͒ features of this test proportional to the factor (2ϩ2␥Ϫ), which is also to be is that in principle it allows the parametrized-post-Newtonian expected, since this is the same factor that appears in the GR ͑PPN͒ parameters  and ␥ to be disentangled from each description of perihelion precession. Thus the two relativistic other, and hence to be determined separately in a single ex- terms in Eq. ͑2.29͒, which are proportional to ␥ and to (2 periment. In light of Ref. ͓1͔, the objectives of the present ϩ2␥Ϫ), respectively, can be understood as expressing the paper are twofold: ͑a͒ to supply the details of the formalism fact that in some sense a spacecraft in a hyperbolic orbit has underlying the analysis in Ref. ͓1͔ and ͑b͒ to explore quan- characteristics of both a light ray and of a massive object. titatively how precisely  and ␥ can be determined from a Finally we note that since the coefficients of ␥ and (2ϩ2␥ specific mission. As part of this discussion we address the Ϫ) in Eq. ͑2.29͒ have a different dependence on the space- question of how well we can determine not only some linear craft velocity, these can in principle be separately deter- combination of  and ␥, such as (2ϩ2␥Ϫ), but also  mined, thus yielding two independent equations from which and ␥ separately. As we shall see, the deflection of spacecraft  and ␥ can be inferred. trajectories as a test of GR is of interest not only because of The preceding discussion leads immediately to the ques- the theoretical possibility of discriminating  and ␥, but also tion of whether measuring the gravitational deflection of a because such an experiment appears to be feasible with tech- spacecraft to the requisite level of precision is technically nology that is either currently available or on the near hori- feasible. It was shown in Ref. ͓1͔ that with recent improve- zon. ments in spacecraft technology, particularly VLBI ͑very long Since the possibility of decoupling  and ␥ in a single baseline interferometry͒ tracking and drag-free systems, a experiment is one of the novel features of the proposed measurement of (2ϩ2␥Ϫ)toϳ10Ϫ3 would be techni- spacecraft mission, it is useful to explain in intuitive terms cally feasible in the foreseeable future. In our present work how this decoupling can come about. The main theoretical we provide a more detailed covariance analysis which shows result of our analysis is given by the general deflection equa- that  and ␥ may be measured to an accuracy of ϳ4 tion, which is Eq. ͑8͒ of Ref. ͓1͔ or Eqs. ͑2.27͒–͑2.29͒ in ϫ10Ϫ5 and ϳ8ϫ10Ϫ6 respectively using advanced K-band Sec. II of the present paper. We note from Eq. ͑2.29͒ that the radiometric tracking. deflection angle can be expressed as a sum of three contri- In Sec. III of the present paper we supply the details of butions, the first of which is purely Newtonian. The sum of our numerical analysis applied to a specific proposed mis- this Newtonian contribution and the second term ͑propor- sion, including a discussion of the contributions from the ␥ tional to ) yields the GR prediction for light deflection in quadrupole moment of the Sun, J2. Of particular interest is the question of how well  and ␥ can be determined sepa- rately with existing or available technology. Although disen- *Corresponding author. tangling  and ␥ in this ͑or any other͒ experiment will be
0556-2821/2004/69͑4͒/042001͑15͒/$22.5069 042001-1 ©2004 The American Physical Society LONGUSKI et al. PHYSICAL REVIEW D 69, 042001 ͑2004͒ difficult, the fact that it can be done at all serves to focus attention on strategies for maximizing the sensitivity to the individual parameters. In this connection it is worth noting that improvements in the classic tests of GR have been made possible by the introduction of new technologies, such as the Mo¨ssbauer effect and atomic clocks in the case of the gravi- tational redshift. Additionally, other related tests of GR, in- cluding lunar laser ranging ͓2–4͔ and tests of both the weak equivalence principle and the gravitational inverse-square law ͓5͔, have also benefited from the use of new improve- ments in technology. Recently ͓6͔, improvements in space- craft tracking techniques applied in the Cassini mission to Saturn have led to a new determination of ␥:(␥Ϫ1)ϭ(2.1 Ϫ Ϯ2.3)ϫ10 5. FIG. 1. Deflection of a spacecraft trajectory in a gravity field. Ϫ In Sec. IV we develop the theoretical formalism to show The spacecraft approaches with asymptotic velocity Vϱ , passes ͑ ͒ how  and ␥ can be disentangled in a single spacecraft de- through periapsis closest approach at distance r p , and leaves with ϩ flection experiment. This formalism characterizes the sensi- asymptotic velocity Vϱ . The spacecraft coordinates are given by tivity of the trajectory to  and ␥, which then leads to the the radial distance r from the center of the attracting body and the detailed covariance analysis presented in Sec. V. One out- angle with respect to the inertial direction jˆ. come of this analysis is the recognition that by using the full strength of the range and Doppler radiometric data, highly where A(r) and B(r) can be expanded in terms of the con- accurate VLBI measurements become less important. Our stants  and ␥ of the PPN metric, with cϭ1 ͓7͔: results and conclusions are presented in Sec. VI where we consider how standard and advanced tracking accuracies af- Gm A͑r͒ϭ1ϩ2␥ ϩ ͑2.3͒ fect the precision to which  and ␥ can be determined. r •••
II. DERIVATION OF THE GENERAL DEFLECTION Gm G2m2 ϭ Ϫ ϩ Ϫ␥ ϩ EQUATION B͑r͒ 1 2 2 ͑ ͒ . r r2 ••• To derive the general deflection equation we follow the ͑2.4͒ approach of Longuski et al. ͓1͔, but here we take the oppor- ͑ ͒ ͑ ͒ tunity to provide additional details. We begin by assuming In Eqs. 2.3 and 2.4 G is the Newtonian gravitational con- that a photon or a spacecraft ͑idealized as a massive particle͒ stant, m is the mass of the central body, and E and J are approaches a gravitating body from a very great distance constants given by Ϫ ͑starting with velocity Vϱ ) and is deflected by gravity. It 2 ϩ Eϭ1ϪVϱ , ͑2.5͒ recedes to a great distance with final velocity Vϱ ͑see Fig. ͒ ͑ 1 . Let (r) be the angle measured positively by the right- ϭ ͒Ϫ ϩ 2 1/2 ͑ ͒ J r p͓1/B͑r p 1 Vϱ͔ . 2.6 hand rule͒ from the inertial direction jˆ to the position vector ˆ →ϱ direction, er , as shown in Fig. 1. We then define (r ) Let us now examine the denominator term which appears ϵ ϭϪ ͑ ͒ ͑ ͒ ϱ , and also note that (r p) /2, where r p is the in Eq. 2.2 . Using Eq. 2.5 we have distance of closest approach as shown in the figure. From the symmetry between the approach asymptote and the departure 1 1 1 1 1 1 ͭ ͫ ϪEͬϪ ͮ ϭ ͫ Ϫ1ϩV2 ͬϪ . ϱ asymptote, we can express the total deflection due to gravity, J2 B͑r͒ r2 J2 B͑r͒ r2 ⌬ def ,as ͑2.7͒ ⌬ ϭ ͓ Ϫ͑ ͔͒Ϫ ͑ ͒ def 2 ϱ r p . 2.1 From Weinberg ͓7͔ the inverse of Eq. ͑2.4͒ is given by
We can now make use of the quadrature integral given by 1 2Gm 2G2m2 Weinberg ͓7͔, Ϸ1ϩ ϩ ͑2Ϫϩ␥͒. ͑2.8͒ B͑r͒ r r2 1/2͑ ͒ A r dr ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ϭϮ͵ , ͑2.2͒ Upon substituting Eqs. 2.3 , 2.7 and 2.8 into Eq. 2.2 , r2͕JϪ2͓BϪ1͑r͒ϪE͔ϪrϪ2͖1/2 we obtain, to order G2,
ϱ ͑rϪ1ϩ␥GmrϪ2͒dr ϱϪ͑r ͒ϭ ͵ . ͑2.9͒ p 2 Ϫ2 2 Ϫ2 2 2 Ϫ2 1/2 rp ͓VϱJ r ϩ2GmJ rϪ1ϩ2G m J ͑2Ϫϩ␥͔͒
042001-2 DEFLECTION OF SPACECRAFT TRAJECTORIES AS A . . . PHYSICAL REVIEW D 69, 042001 ͑2004͒
The integrals in Eq. ͑2.9͒ can be evaluated using the elemen- For the expression 1/ͱϪa in Eq. ͑2.10͒ we write tary results ͓8͔
ϩ ͒ Ϫϩ␥͒ 1/2 dy 1 by 2a 1 ͑Gm/r p ͑2 ͵ ϭ sinϪ1ͩ ͪ ͑for aϽ0͒, ϭͫ 1ϩ ͬ , ͑2.20͒ ͱ ͱϪ ͉ ͉ͱϪ ͱϪ ϩ 2 ͒ y X a y q a 1 r pVϱ/͑2Gm ͑2.10͒ Ӷ dy ͱX b dy and noting that Gm/r p 1 we have ͵ ϭϪ Ϫ ͵ ͑for aÞ0͒, y 2ͱX ay 2a yͱX ͑2.11͒ ͒ Ϫϩ␥͒ 1 ͓Gm/͑2r p ͔͑2 Ϸ1ϩ . ͑2.21͒ ͱϪ ϩ 2 ͒ where a 1 r pVϱ/͑2Gm Xϵaϩbyϩcy2, ͑2.12͒ Using Eqs. ͑2.16͒, ͑2.17͒, ͑2.19͒, and ͑2.21͒ we evaluate the qϵ4acϪb2. ͑2.13͒ integral of Eq. ͑2.10͒ for the upper and lower limits of ϱ and r , respectively: In our problem, Eq. ͑2.9͒, we have p
2 2 ϱ 2G m ϱ dy 1 byϩ2a aϭϪ1ϩ ͑2Ϫϩ␥͒, ͵ ϭ Ϫ1ͩ ͪͯ J2 ͱ ͱ sin ͱ rp y X Ϫa ͉y͉ Ϫq rp ϭ 2 ͒ Ϫϩ␥͒ b 2Gm/J , ͓Gm/͑2r p ͔͑2 Ϸͭ 1ϩ ͮ ϩ 2 ͒ 1 r pVϱ/͑2Gm 2 2 cϭVϱ/J , 1 ϫͭ sinϪ1ͫ ͬϩ ͮ . ͑2.22͒ ϩ 2 ͑ ͒ 2 4 G2m2 G2m2 1 r pVϱ/ Gm ϭϪ ͫ 2 ϩ Ϫ 2 ͑ Ϫϩ␥͒ͬ q Vϱ 2Vϱ 2 . J2 J2 J2 ͑2.14͒ By comparing Eqs. ͑2.9͒ and ͑2.11͒ we note that the factor Ϫ␥Gmb/(2a) will appear in the arcsine term ͓from Eq. 2 We evaluate the constants a, b, c, and q by expressing J in ͑2.10͔͒; it can be written as ͑ ͒ ͑ ͒ terms of r p and Vϱ . From Eqs. 2.6 and 2.8 we obtain
2 2 Ϫ␥ ␥ ͑ ͒ J ϭ2Gmr ͓1ϩ͑Gm/r ͒͑2Ϫϩ␥͒ϩr V /͑2Gm͔͒, Gmb Gm/ 2r p p p p ϱ ϭ . ͑2.23͒ ͑2.15͒ 2a ϩ 2 ͒ ͓1 r pVϱ/͑2Gm ͔
2 which gives J in terms of the physically measurable param- Collecting all the arcsine terms that result from Eq. ͑2.9͒,we ͑ ͒ ͑ ͒ eters r p and Vϱ . Substituting Eq. 2.15 into Eqs. 2.14 we obtain obtain
Ϫ Ϫ 2 ͑ ͒ ␥ ͒ ͒ Ϫϩ␥͒ 1 r pVϱ/ 2Gm Gm/͑2r p ͓Gm/͑2r p ͔͑2 aϭ , ͑2.16͒ ͫ 1ϩ ͬͭ 1ϩ ͮ ϩ͑ ͒͑ Ϫϩ␥͒ϩ 2 ͑ ͒ ϩ 2 ͒ ϩ 2 ͒ 1 Gm/r p 2 r pVϱ/ 2Gm 1 r pVϱ/͑2Gm 1 r pVϱ/͑2Gm Ϫ 1 1/r ϫͭ sin 1ͫ ͬϩ ͮ p bϭ , ͑2.17͒ 1ϩr V2 /͑Gm͒ 2 ϩ ͒ Ϫϩ␥͒ϩ 2 ͒ p ϱ 1 ͑Gm/r p ͑2 r pVϱ/͑2Gm ͒ Ϫϩ ␥͒ ͓Gm/͑2r p ͔͑2 2 ϭͭ 1ϩ ͮ 2 ͒ 1ϩr V2 /͑2Gm͒ Vϱ/͑2Gmrp p ϱ cϭ , ͑2.18͒ ϩ͑ ͒͑ Ϫϩ␥͒ϩ 2 ͑ ͒ 1 Gm/r p 2 r pVϱ/ 2Gm 1 ϫͭ sinϪ1ͫ ͬϩ ͮ . ͑2.24͒ ϩ 2 ͒ 2 1 r pVϱ/͑Gm Ϫ ͒2 ϩ 2 ͒ 2 ͑1/r p ͓1 r pVϱ/͑Gm ͔ qϭ . ͑2.19͒ ͓ ϩ͑ ͒͑ Ϫϩ␥͒ϩ 2 ͑ ͔͒2 1 Gm/r p 2 r pVϱ/ 2Gm The final term we must analyze from Eqs. ͑2.9͒ and ͑2.11͒ is
042001-3 LONGUSKI et al. PHYSICAL REVIEW D 69, 042001 ͑2004͒
ϱ Ϫ1 ͱX where we have used the identity ͓8͔ /2ϩsin (z) Ϫ␥Gm ͯ ϭcosϪ1(Ϫz). We recognize in Eqs. ͑2.27͒–͑2.29͒ the classi- ay rp cal nonrelativistic deflection of a spacecraft trajectory, ⌬ : Ϫ␥Gm NR ϭ ͱcϪ0 a 1 ⌬ ϵ2 sinϪ1ͩ ͪ . ͑2.30͒ NR 1ϩx ␥ ͒1/2 ͒ Ϫϩ␥͒ 1/2 Vϱ͑Gm/r p ͑2Gm/r p ͑2 ϭ ͫ 1ϩ ͬ The nonrelativistic deflection formula is well known to mis- ͓2ϩr V2 /͑Gm͔͒1/2 2ϩr V2 /͑Gm͒ p ϱ p ϱ sion designers ͓9͔ who use it to compute the effectiveness of ␥ ͒1/2 ͒ Ϫϩ␥͒ the gravity-assist technique, such as that used in the Voyager Vϱ͑Gm/r p ͑Gm/r p ͑2 Ϸ ͫ 1ϩ ͬ, missions to the outer planets. The ⌬ term is what re- ͓ ϩ 2 ͑ ͔͒1/2 ϩ 2 ͑ ͒ NR 2 r pVϱ/ Gm 2 r pVϱ/ Gm mains of Eq. ͑2.29͒ when the GR terms ͑i.e. the ⑀ terms͒ are ͑ ͒ ͑2.25͒ dropped. We can easily verify Eq. 2.30 by reprising our derivation of Eq. ͑2.29͒ with the simplifications where we note that the value of the function at r is zero, and p ϭ ͑ ͒ that the final expression is based on the approximation A 1, 2.31 Gm/r Ӷ1. p Gm Equations ͑2.24͒ and ͑2.25͒ provide the solution to the B͑r͒ϭ1Ϫ2 . ͑2.32͒ integral of Eq. ͑2.9͒: r
␥ ͒1/2 The result of these weak field approximations is that we ob- Vϱ͑Gm/r p ϱϪ͑r ͒Ϸ tain the Newtonian deflection. In this particular derivation p ϩ 2 ͒ 1/2 ͓2 r pVϱ/͑Gm ͔ the second term of the numerator in the integrand of Eq. ͑2.9͒, ␥GmrϪ2, vanishes so that only terms corresponding to ͒ Ϫϩ␥͒ ͑Gm/r p ͑2 Eq. ͑2.10͒ remain. An immediate consequence of the weak ϫͫ 1ϩ ͬ 2 field assumption is that the term (2Ϫϩ␥) which appears 2ϩr Vϱ/͑Gm͒ p in the constants a and q of Eq. ͑2.14͒ also vanishes, and ͒ Ϫϩ ␥͒ hence all terms containing  and ␥ are eliminated from Eq. ͑Gm/r p ͑2 2 ϩͭ 1ϩ ͮ ͑2.26͒. Since these are directly associated with ⑀ϵGm/r , ϩ 2 ͑ ͒ p 2 r pVϱ/ Gm we merely drop the ⑀ terms which appear explicitly in Eq. ͑ ͒ 2.30 to obtain the Newtonian deflection formula, Eq. Ϫ 1 ͑ ͒ ͑ ͒ ϫͭ sin 1ͫ ͬϩ ͮ 2.30 . Equation 2.30 gives the total turn angle of the vec- ϩ 2 ͒ 2 Ϫ 1 r pVϱ/͑Gm tor Vϱ ͑i.e., the angle between the approach velocity, Vϱ , ϩ Ӷ ͒ ͑ ͒ and the departure velocity, Vϱ ) based on Newton’s law of ͑for Gm/r p 1 . 2.26 gravity. If we substitute Vϱϭ1, or xϭ1/⑀, into Eq. ͑2.30͒, To obtain the general deflection equation we write we obtain the deflection of light predicted by Newtonian physics: ⌬ ϭ2͓ϱϪ͑r ͔͒Ϫ def p 1 ⌬ ͩ ͪ Х ⑀ ͑ ͒ x 1/2 ͑2ϩ2␥Ϫ͒ NR ⑀ 2 . 2.33 Х2␥⑀ͩ ͪ ϩ⑀ 2ϩx 2ϩx Similarly, setting Vϱϭ1 in Eq. ͑2.29͒ yields ͑2ϩ2␥Ϫ͒ 1 ϩ ͫ ϩ⑀ ͬ Ϫ1ͩ ͪ 1 4Gm 1ϩ␥ 2 1 ϩ sin ϩ , ⌬ ͩ ͪ ϭ ⑀͑ ϩ␥͒ϭ ͩ ͪ ͑ ͒ 2 x 1 x def ⑀ 2 1 , 2.34 r p 2 ͑2.27͒ where terms O(⑀2) and higher have been dropped. Equation where ͑2.34͒ yields Einstein’s formula for the deflection of light when ␥ is set to unity: twice the value given by Eq. ͑2.33͒. ⑀ϭ ϵ ϵ 2 ⑀ ͑ ͒ Gm/r p /r p , x Vϱ/ , 2.28 We note that Eq. ͑2.29͒ contains the same factor that ap- pears in the formula for the precession of perhelia ͓7͔, and where we have retained terms only to order ⑀.Itis convenient to rewrite our general deflection equation ͑2.27͒ 6Gm 2ϩ2␥Ϫ ⌬ ϭ ͩ ͪ , ͑2.35͒ in the final form prec L 3 1/2 Ϫ 1 x where ⌬ is the precession in radians per revolution, and ⌬ Х2 sin 1ͩ ͪ ϩ2␥⑀ͩ ͪ prec def 1ϩx 2ϩx L is the semilatus rectum of the elliptical orbit. What is re- markable about the factor (2ϩ2␥Ϫ) in the general deflec- ͑2ϩ2␥Ϫ͒ Ϫ1 ͑ ͒ ϩ2⑀ cosϪ1ͩ ͪ , ͑2.29͒ tion equation 2.29 is that the contribution from this term 2ϩx 1ϩx depends on the speed x. This means that an experiment based
042001-4 DEFLECTION OF SPACECRAFT TRAJECTORIES AS A . . . PHYSICAL REVIEW D 69, 042001 ͑2004͒
TABLE I. Representative values for spacecraft deflections.
Parameter Earth Jupiter Sun ͓ ͔ ϫ 6 r p km 6678 71 700 2.784 10 Vϱ ͓km/s͔ 9.000 5.455 37.92 Gm ͓km͔ 4.435ϫ10Ϫ6 1.410ϫ10Ϫ3 1.476 ⑀ 6.641ϫ10Ϫ10 1.966ϫ10Ϫ8 5.303ϫ10Ϫ7 x 1.357 1.684ϫ10Ϫ2 3.017ϫ10Ϫ2 ⌬ ͓ ͔ NR deg 50.21 159.1 152.2 ⌬ ͓ ͔ ϫ Ϫ9 ϫ Ϫ7 ϫ Ϫ6 E rad 3.229 10 1.767 10 4.673 10 a Ϫ8 Ϫ5 Ϫ2 ⌬ ͓km͔ 2.156ϫ10 1.267ϫ10 1.300ϫ10 rp aApproximate error tolerance on periapsis knowledge to obtain  and ␥ to 10Ϫ3.
many cases where the relativistic deflection of a spacecraft trajectory is greater than the deflection of light for the same FIG. 2. Plot of the function ⌬¯ in Eq. ͑2.37͒. As discussed in E periapsis distance, r , where periapsis is the point of closest the text, ⌬¯ gives the scaled contribution of GR to the deflection, p E approach. The question to be answered is whether an experi- plotted versus the scaled speed x. For an incident light ray ment can be devised to measure this effect. ⌬¯ ϭ E 1/2. ⌬ In Table I we estimate the total deflection angle def ͓from Eq. ͑2.27͔͒ and the deflection angle due to GR, ⌬ on the deflection equation can discriminate between the con- E ͓found by multiplying Eq. ͑2.36͒ by 2⑀(1ϩ␥)], for repre- tributions from  and ␥ by comparison to the precession of sentative hyperbolic spacecraft trajectories near the Earth, Mercury’s orbit. It is thus clear that at different speeds, an Jupiter, or the Sun. For these calculations we assume that experiment based on the general deflection equation can ϭ␥ϭ1. In anticipation of the more detailed analysis pre- separately determine the values of  and ␥. This is not true sented in Secs. IV and V below, we estimate the accuracy for other experiments, such as light deflection, radar time required to measure the relativistic deflection, ⌬ ,to delay, or planetary precession. E within 0.1% ͑the level of sensitivity necessary to determine We wish to obtain a formula that conveniently compares Ϫ  and ␥ to 10 3). For purposes of this estimate we use the the general relativistic effect on spacecraft deflection to light ͓ ͔͓ ͑ ͔͒ ϭ ⌬ approximation 1 see Eq. 3.10 ⌬r 0.1% Er p , deflection. One way to proceed is to define a quantity ⌬¯ p which sets a limit on the closest approach distance, r . obtained by subtracting the ͑often large͒ angle ⌬ in Eq. p NR ͑Other variables affect the sensitivity, but knowledge of r is ͑2.30͒ from the expression in Eq. ͑2.29͒, and to then normal- p the dominant error source.͒ Clearly the level of accuracy ize the result ͑i.e. divide͒ by the GR result 2⑀(1ϩ␥): required to perform the experiment with spacecraft deflec- tions at Earth or Jupiter is beyond present day technology, ⌬¯ ϵ͑⌬ Ϫ⌬ ͒/͓2⑀͑1ϩ␥͔͒ def NR because periapsis must be known to within 21.56 mor 1.267 cm, respectively. Since we evidently require a larger ϭ͓␥/͑1ϩ␥͔͓͒x/͑2ϩx͔͒1/2ϩ͑1ϩ␥͒Ϫ1 gravitational parameter, Gm, we turn our attention to the ϫ͓͑2ϩ2␥Ϫ͒/͑2ϩx͔͒cosϪ1͓Ϫ1/͑1ϩx͔͒. Sun—the largest gravitating body at our disposal. ͑2.36͒ III. PROPOSED EXPERIMENT For Einstein’s theory, ϭ␥ϭ1 and Eq. ͑2.36͒ becomes Mewaldt et al. ͓10͔ have proposed the Small Interstellar ⌬ 1/2 ͒ Ϫ E 1 x ͑3/2 1 Probe mission which would cross the solar wind termination ⌬¯ ϭ ϭ ͩ ͪ ϩ cosϪ1ͩ ͪ . E 2⑀͑1ϩ␥͒ 2 2ϩx ͑2ϩx͒ 1ϩx shock and heliopause and penetrate into nearby interstellar ͑ ͒ space. In order to accomplish its scientific objectives, the 2.37 Ϫ4 probe must attain VϱϷ1.3ϫ10 . To achieve this speed a ⌬¯ number of gravity assist scenarios are suggested ͓10͔, most The function E is plotted in Fig. 2. We note from Eq. ͑ ͒ ϭ ⌬¯ Х of which involve a final close flyby of the Sun at 4 solar radii 2.37 that when Vϱ 1, then E 0.5, as expected: the (4r᭪). At perihelion a maneuver is performed to change the ratio of the purely relativistic bending of light divided by the speed of the spacecraft by several km/s in order to send the total bending of light ͑including the Newtonian bending͒ is probe off on its hyperbolic trajectory. The Interstellar Probe 1/2. In contrast, for a parabolic trajectory Vϱϭ0 ͑i.e. x mission presents an ideal trajectory to observe the relativistic ϭ ⌬¯ ϭ ϭ ϭ 0) and E(0) 3 /4 2.36. When x 1, then Vϱ deflection, provided that the effects of non-gravitational ϭͱ Gm/r p which is the circular speed at a radius r p . Thus forces and the Newtonian deflection can be accounted for. the x variable is conveniently scaled in terms of ‘‘circular We will therefore use this mission as the basis for some ϭ ⌬¯ ϭ speeds’’ at r p . For x 1, E 1.34. We see that there are simple numerical estimates. In Secs. IV and V we present a
042001-5 LONGUSKI et al. PHYSICAL REVIEW D 69, 042001 ͑2004͒ more detailed analysis of such a mission and its ability to We next take the variation (␦) of the measurement, Eq. ͑3.4͒, disentangle the PPN parameters  and ␥. to compute how errors in measuring the eccentricity and the Let us first estimate how accurately the relevant param- unit vectors will contribute to errors in the measured value of eters must be known in order to discriminate between the ⌬: ϭ ϫ relativistic and Newtonian deflections. Using r p 4 6.960 5 6 ϫ10 kmϭ2.784ϫ10 km, and Gmϭ1.476 km, we find ⑀ 2 Ϫ1/2 Ϫ Ϫ ␦⌬ϭ͓e͑e Ϫ1͒ Ϫ͉rˆ ϫrˆϱ͉͔␦eϪe␦͉rˆ ϫrˆϱ͉. ϭ5.303ϫ10 7 and xϭ3.017ϫ10 2. Inserting these values p p ͑3.5͒ of ⑀ and x into Eq. ͑2.37͒, and multiplying by 4⑀ gives the ⌬ ϭ ϫ Ϫ6 total general relativistic deflection E 4.673 10 rad ͉ˆ ϫ ˆ ͉Хͱ 2Ϫ ͑ ͒ ϭ0.9639Љ. On the other hand, the nonrelativistic Newtonian Noting that rp rϱ e 1/e reduces Eq. 3.5 to ͑ ͒ ⌬ ϭ ϭ deflection is, from Eq. 2.30 , NR 2.656 rad 152.2°, which is very large compared to the relativistic deflection. ␦⌬ϭ Ϫ1͑ 2Ϫ ͒Ϫ1/2␦ Ϫ ͓͑ ˆϫ ˆ ͒ ␦ ˆ ϩ͑ ˆ ϫ ˆ ͒ ␦ ˆ ͔ e e 1 e e k rp • rϱ rϱ k • rp , Thus in order to observe the relativistic deflection we must ͑3.6͒ have very precise knowledge of the Newtonian contribution. ͑Of course in the case of the Interstellar Probe we will only observe the departure asymptote, namely half the deflections which represents the effect of variations in the angular posi- given by ⌬ and ⌬ .) We proceed to assess our ability tion of the probe at periapsis and at escape. Careful evalua- E NR tion of each term for a general flyby shows that the expres- to measure the relativistic effect, which will be proportional ͑ ͒ to the knowledge errors in the nonrelativistic effect. sion in square brackets in Eq. 3.6 can be expressed as We can view the rotation induced by general relativity on ϭ ␦͉ˆ ϫ ˆ ͉ϭ⌬ ϩ⌬ ͑ ͒ a hyperbolic trajectory as being a shift in the argument of e͓ ͔ e rp rϱ rϱ /rϱ r p /r p , 3.7 periapsis of the probe trajectory due to the gravitational in- ••• teraction, analogous to the advance in Mercury’s perihelion. where ⌬ denotes errors in distance measured normal to the Thus, in order to determine if this is a measurable effect, we radius vector. Since the eccentricity is, in turn, a function of must devise a series of ideal measurements to estimate the ϭ͓ shift in argument of periapsis between perihelion and escape. specific measurable quantities via the relation e 1 ϩ 2 ͔ At perihelion the argument of periapsis is related to the unit (r pVϱ/ ) , we have vector of the probe ͑assuming orbit plane coordinates͒ by the ␦ ϭ Ϫ ͒ ␦ ϩ ␦ Ϫ␦ ͑ ͒ equation e ͑e 1 ͓ r p /r p 2 Vϱ /Vϱ / ͔, 3.8
ˆ ϭ ͒ˆϩ ͒ˆ ͑ ͒ ␦ rp cos͑ i sin͑ j, 3.1 where r p denotes variations along the radius vector. If we combine the previous results and assume that the different ͑ measurements are uncorrelated, then the overall uncertainty where is the argument of periapsis arbitrarily set to zero in ⌬ is in Fig. 1͒ and ˆi and jˆ are unit vectors of our coordinate frame, with the third unit vector kˆϭˆiϫjˆ. When the probe is 2 Ϫ2 2 2 ⌬ϭe ͓͑eϪ1͒/͑eϩ1͔͓͒͑ /r ͒ ϩ4͑ /Vϱ͒ sufficiently far from the Sun on its escape trajectory, its as- rp p Vϱ 2 2 2 ymptote can similarly be specified by the unit vector ϩ͑ / ͔͒ϩ͓͑⌬ /r ͒ ϩ͑⌬ /rϱ͒ ͔, ͑3.9͒ rp p rϱ
ˆ ϭ ͑Јϩ ͒ˆϩ ͑Јϩ ͒ˆ ͑ ͒ rϱ cos ϱ i sin ϱ j, 3.2 where denotes the Gaussian standard deviation of the mea- sured quantity. where Ј is the new ͑shifted͒ argument of periapsis, and ϱ In general, the uncertainties in the first terms will be neg- ligible compared to the measured uncertainties ⌬ and is the limiting value of the true anomaly of the probe as it rp escapes from the Sun. In principle, each of these unit vectors ⌬rϱ. Additionally, at escape the probe unit vector direction can be measured, and the shift in argument of periapsis can can be measured extremely accurately using established be computed by comparing them. Specifically, VLBI techniques ͓11͔. This leaves the down-track measure- ment of the probe position at perihelion as the dominant ͉ˆ ϫ ˆ ͉ϭ Ϫ͒ ϩ Ϫ͒ error source, so that rp rϱ sin͑ Ј cos ϱ cos͑ Ј sin ϱ , ͑3.3͒ ⌬Ϸ⌬ /r . ͑3.10͒ rp p and we define ЈϪϵ⌬, which is the quantity we wish to measure. Noting that ⌬Ӷ1, cos ϱϭϪ1/e, and sin ϱ Current navigation practice would reduce ⌬r to the order ϭͱe2Ϫ1/e, where e is the eccentricity, we can solve for ⌬ p of 1–10 km ͓12͔. Taking ⌬ ϭ1 km for our numerical ex- in terms of measurable quantities: rp ample ͑where e is computed to be 1.03͒, we find that ⌬ ϭ3.6ϫ10Ϫ7 rad which, by comparison to half the deflection ⌬ϭͱ 2Ϫ Ϫ ͉ˆ ϫ ˆ ͉ ͑ ͒ ⌬ e 1 e rp rϱ . 3.4 angle E , represents an error of 16%. If this measurement
042001-6 DEFLECTION OF SPACECRAFT TRAJECTORIES AS A . . . PHYSICAL REVIEW D 69, 042001 ͑2004͒ uncertainty were reduced to the order of 10 m, then we would estimate that the contribution from the PPN param- eters  and ␥ could be found to 10Ϫ3. Measurement uncer- tainties of this order imply Earth-based measurement accu- racies on the order of 0.1 nrad. Based on operationally demonstrated measurements of the Deep Space Network’s VLBI system, its estimated accuracy at present is of order 5 nrad. Observations of natural radio sources made with the VLBI measurement technique have demonstrated accuracies of 0.8 nrad, and the fundamental limit on such measurements is of order 0.01 nrad ͓13͔. This precision is substantially better than the 0.1 nrad accuracy required to measure  and ␥ at the ϳ10Ϫ3 level. The feasibility of developing this tech- nology to the levels of accuracy needed for our proposed experiment and for near-Sun observations is considered in Ref. ͓14͔. We can presume that the increases in accuracy of VLBI will be accompanied by corresponding improvements in the infrastructure needed to support these measurements, FIG. 3. Geometry of the spacecraft trajectory, where the space- which would likely be developed in concert with improved craft position and velocity vectors are rជ and vជ . The Earth-to- VLBI technology. spacecraft position vector is ជ . Subscripts ‘‘o’’ and ‘‘t’’ refer to An important perturbation not directly addressed here is initial and later epochs. The location of the Earth with respect to the ⌽ due to the J2 gravity coefficient of the Sun. The shift induced spacecraft at perihelion is indicated by the phase angle, ESC . in periapsis due to this will be on the order of 1% of the shift Measurements of  and ␥ are computed assuming the optimal lo- ⌽ induced by general relativity. Thus, it will introduce addi- cation of the Earth ( ESC). tional errors in the determination of  and ␥. In order to discriminate for this effect an inclined orbit can be used ͓12͔. be able to separately determine  and ␥ to a precision of Alternately, tracking the spacecraft near perihelion may al- ϳ4ϫ10Ϫ5 and ϳ8ϫ10Ϫ6, respectively. low the signature of the J2 perturbation to be recognized and discriminated without having to resort to an inclined orbit. One approach to disentangle the parameters  and ␥ in IV. ESTIMATING GENERAL RELATIVITY PARAMETERS Eq. ͑2.29͒ would be to rely on experiments which determine FROM RADIOMETRIC TRACKING OF HELIOCENTRIC ␥ separately ͓6,15͔. However, as we show in the subsequent TRAJECTORIES  ␥ sections, and can actually be disentangled in a single A. General mission measuring the deflection of a spacecraft. In order to extract the general relativistic contribution to In this section we analyze in greater detail the precision to  the spacecraft’s trajectory from a mission such as the Small which we can determine the general relativistic parameters ␥ Interstellar Probe it will be necessary to deal with perturbing and by measuring spacecraft trajectories. Specifically, we  ␥ non-gravitational forces. For a typical spacecraft these forces will focus on the question of disentangling and by an arise from radiation pressure, solar wind, interplanetary dust, appropriate set of measurements. In Sec. III an estimate of atmospheric drag, magnetic fields, propellant leakage, and the necessary measurement accuracy of these parameters was spacecraft radiation ͓4,16͔. There are two ways to address made under a number of simplifying assumptions and ap- these perturbations: They can be measured directly by plac- proximations. In this section we relax some of these approxi- ing sufficiently sensitive accelerometers on-board the space- mations and perform a more detailed and rigorous analysis of craft, and then treating these accelerometers as data which the problem. Figure 3 illustrates the geometry of the space- allow the non-gravitational forces to be directly estimated. craft trajectory. The spacecraft position and velocity vectors ជ ជ ជ However, it is likely that another technique will be necessary at perihelion are ro and vo , and o is the Earth-to-spacecraft to sidestep these perturbations, which is to employ a drag- position vector at this time. The subscript ‘‘t’’ refers to these free spacecraft using small thrusters to null out the non- vectors at a later time. The phase angle at the initial time, ⌽ gravitational forces. Fortunately the necessary drag-free ESC , is the Earth-Sun-Spacecraft angle. Figure 3 shows the ⌽ ϭ ͑ technology is already under development, since it is a preq- Earth’s location for ESC 0, /2, , and 3 /2. Since radio- uisite for the ongoing Gravity Probe B mission ͓17͔, as well metric measurements are highly sensitive to the relative ge- as for the proposed STEP ͓18͔ and Galileo Galilei ͓19͔ mis- ometry of the spacecraft with respect to the Sun, we analyze  ␥ ⌽ sions. estimates of and as a function of ESC .) Although the In the next section we develop the theoretical formalism analysis which follows is semi-analytical, it includes realistic needed to demonstrate how  and ␥ can be disentangled ͑at models of the trajectory dynamics and measurement noise. least in principle͒ in a single spacecraft deflection mission We provide detailed estimates of how accurately the trajec- through an appropriate set of measurements. Using this for- tory must be measured to set new limits on the parameters  malism we show that a satellite deflection experiment may and ␥.
042001-7 LONGUSKI et al. PHYSICAL REVIEW D 69, 042001 ͑2004͒
B. Transient effect of  and ␥ In Eqs. ͑4.3͒ i is the orbit inclination, is the argument of ⍀ From Ref. ͓2͔͑Sec. 7͒ the perturbing relativistic accelera- periapsis, is the longitude of the ascending node, and n ϭͱ ͉ ͉3 tion, to first PN order, can be written as ͑taking Gϭcϭ1 and m/ a is the normalized mean motion. noting that the spacecraft mass is much smaller than the solar After substitution of the perturbing relativistic accelera- ͑ ͒ mass͒ tion components from Eq. 4.2 into the hyperbolic Lagrange equations, the changes in the orbital elements from periapsis m mrជ passage to some value of the true anomaly can be approxi- ជ 2ជ ជ ជ ជ ␦aϭ ͫ 2͑␥ϩ͒ Ϫ␥v rϩ2͑␥ϩ1͒͑r v͒vͬ, mated by keeping the elements on the right-hand side con- 3 r • r stant and allowing the true anomaly to vary. We thus find ͑4.1͒ ជ ជ 2em where r represents the spacecraft position vector and v rep- ⌬aϭ ͓͑2ϩ2ϩ3␥ϩ2e2ϩ␥e2͒⌬ cos f resents the spacecraft velocity vector normalized by the ͑e2Ϫ1͒2 speed of light. We note that the relativistic perturbation is Ϫ͑ ϩϩ ␥͒ ⌬ 2 ͔ ͑ ͒ present only in the orbital plane. The acceleration compo- 2 2 e sin f , 4.4a nents decomposed into the radial (R), transverse (S), and out-of-plane ͑W͒ directions are m ⌬eϭϪ ͓͑␥ϩ2ϩ4e2ϩ3␥e2͒⌬ cos f a͑e2Ϫ1͒ m2͑1ϩe cos f ͒2 2 Rϭ ͓͑1Ϫe ͒␥ϩ2͑1ϩe cos f ͒ Ϫ͑ ϩϩ ␥͒ ⌬ 2 ͔ ͑ ͒ a3͑e2Ϫ1͒3 2 2 e sin f , 4.4b ϩ ͑␥ϩ ͒ 2 2 ͔ ͑ ͒ 2 1 e sin f , 4.2a m ⌬ϭ ͫ͑2Ϫϩ2␥͒⌬ f ͑ 2Ϫ ͒ m2͑1ϩe cos f ͒2 a e 1 ϭ ͓ ͑␥ϩ ͒͑ ϩ ͒ ͔ S 2 1 1 e cos f e sin f , 2 a3͑e2Ϫ1͒3 2ϩ͑1Ϫe ͒␥ Ϫͩ ͪ ⌬ sin f ͑4.2b͒ e Wϭ0. ͑4.2c͒ Ϫ͑2ϩϩ2␥͒⌬͑sin f cos f ͒ͬ, ͑4.4c͒ In Eqs. ͑4.2͒, a denotes the semi-major axis, e is the eccen- tricity of the orbit, and f is the true anomaly ͑to be distin- 3m guished from the mean anomaly M). The hyperbolic ⌬Mϭ ͓␥ϩϩ͑2ϩ␥͒e2 Lagrange planetary equations ͓20,21͔, with proper changes a͑e2Ϫ1͒2 ͑i.e., e2Ͼ1, a→Ϫa) can be represented as ϩ͑2ϩ3␥ϩ2ϩ͑2ϩ␥͒e2͒eϩ͑2ϩ2␥ϩ͒e2͔n⌬t da 2a3/2 ϭϪ ͓Re sin f ϩS͑1ϩe cos f ͔͒, m dt ͱ ͑ 2Ϫ ͒ Ϫ ͫ͑2ϩ2␥ϩ͒⌬͑sin f cos f ͒ m e 1 ͱ 2Ϫ ͑4.3a͒ a e 1 ␥ϩ2ϩ͑4ϩ3␥͒e2 m de a͑e2Ϫ1͒ S p r ϩ ⌬ sin f ͬϪ͑2ϩ␥͒ ⌬F, ϭͱ ͫR sin f ϩ ͩ ϩ ͪͬ, e a dt m e r a ͑4.3b͒ ͑4.4d͒
di r where F is the hyperbolic eccentric anomaly ͓note that ⌬t in ϭ W cos͑ϩ f ͒, ͑4.3c͒ ͑ ͒ dt h Eq. 4.4d is the actual time multiplied by the speed of light c]. d⍀ rW sin͑ϩ f ͒ Figures 4 and 5 show the change in the orbital elements ϭ , ͑4.3d͒ due to the relativistic effect. The initial epoch is at the peri- dt h sin i ϭ ϭ apsis with r p 4r᭪ and Vϱ 39 km/s, where r p and r᭪ are the perihelion distance and radius of the Sun, respectively. A d 1 a͑e2Ϫ1͒ 2ϩe cos f ϭ ͱ ͫϪR cos f ϩS sin f ͬ conclusion that we draw by studying these perturbations is dt e m 1ϩe cos f that most of the changes in orbital elements due to GR occur very early in the trajectory ͑usually within a few days of d⍀ ͒ Ϫ cos i, ͑4.3e͒ perihelion . dt Of most interest are the partial derivatives of the orbital elements with respect to the relativistic constants,  and ␥, 2 2 dM 1 2r ͑e Ϫ1͒ ͑e Ϫ1͒ r based on Eqs. ͑4.4͒. These indicate the sensitivity of the ϭnϪ ͫ Ϫ cos f ͬRϩ ͫ ͬS. dt na a e nae p trajectory to the PPN parameters, and give us an indication ͑4.3f͒ of the information content related to  and ␥ in the trajec-
042001-8 DEFLECTION OF SPACECRAFT TRAJECTORIES AS A . . . PHYSICAL REVIEW D 69, 042001 ͑2004͒ e mץ ϭ ͓͑1ϩ3e2͒͑1Ϫcos f ͒ϩ2e sin2 f ͔, ͑4.6b͒ a͑e2Ϫ1͒ ␥ץ
m ͑e2Ϫ1͒ץ ϭ ͫ2 f Ϫ2 sin f cos f ϩ sin f ͬ, a͑e2Ϫ1͒ e ␥ץ ͑4.6c͒
M 3mץ ϭ ͓1ϩ3eϩ3e2ϩe3͔M a͑e2Ϫ1͒2 ␥ץ
m ͑1ϩ3e2͒ m Ϫ ͫ2 sin f cos f ϩ sin f ͬϪ F. aͱe2Ϫ1 e a ͑ ͒ FIG. 4. Change in the semi-major axis ⌬a and eccentricity ⌬e 4.6d due to the relativistic effect. These figures, as well as Fig. 5͑a͒, indicate that most of the GR effects are observable within a few Similarly, the partial derivatives of the orbital elements with days of perihelion. respect to  are tory. If we let E be the set of orbital elements, a change in E ϭ a 2emץ due to relativistic effects can be represented as E Eo ϩ⌬E, where E denotes the initial orbital elements. Taking ϭϪ ͓2͑1Ϫcos f ͒ϩe sin2 f ͔, ͑4.7a͒ 2 2 ץ o partials with respect to the GR parameters yields ͑e Ϫ1͒ ⌬ץ ͒ ⌬ϩ ץ ץ E ͑Eo E E ϭ ϭ . ͑4.5͒ e mץ ͒,␥͑ץ ͒,␥͑ץ ͒,␥͑ץ ϭ ͓2͑1Ϫcos f ͒ϩe sin2 f ͔, ͑4.7b͒ a͑e2Ϫ1͒ ץ We will use these partials later to form the state transforma- tion from the GR parameters to the data measurements. The partial derivatives of the orbital elements with respect to ␥ m 2 sin fץ are given by ϭ ͫϪ f Ϫ Ϫsin f cos f ͬ, ͑4.7c͒ a͑e2Ϫ1͒ e ץ a 2emץ ϭϪ ͓͑3ϩe2͒͑1Ϫcos f ͒ϩ2e sin2 f ͔, ͑4.6a͒ e2Ϫ1͒2͑ ␥ץ M 3mץ ϭ ͓1ϩ2eϩe2͔M a͑e2Ϫ1͒2 ץ
m 2 Ϫ ͫsin f cos f ϩ sin f ͬ. ͑4.7d͒ aͱe2Ϫ1 e
Figures 6–8 show how the partial derivatives of the orbital elements with respect to the GR parameters change as the spacecraft travels on the hyperbolic trajectory, where the ini- tial conditions are the same as assumed above. It is crucial to note that the partials of with respect to  and ␥ are quite different from each other—not only their signs, but also their ratios. These partials essentially represent the amount of in- formation contained in our measurements of  and ␥, and thus their ratios represent the correlation between  and ␥. The slow convergence of the ratio of the partials with respect to in Fig. 8, as compared to the other ratios, shows that there is a possibility of obtaining separate estimates of FIG. 5. Change in the argument of perihelion ⌬, and mean the PPN parameters by tracking the spacecraft close to anomaly ⌬M, due to the relativistic effect. See also caption to Fig. perihelion, which is a novel feature of the GR test we are 4. proposing.
042001-9 LONGUSKI et al. PHYSICAL REVIEW D 69, 042001 ͑2004͒
FIG. 6. Change in the partial derivatives of semi-major axis and FIG. 8. Ratios of orbital-element partial derivatives with respect eccentricity with respect to the GR parameters. The variable y de- to the GR parameters. The time variation for ͑lower left plot͒ notes either  or ␥ as appropriate. demonstrates the potential to separately determine  and ␥.
with range measurements, VLBI can determine the V. COVARIANCE ANALYSIS AND LEAST SQUARES 3-dimensional position of the spacecraft. The final data type APPROXIMATION we consider is Doppler measurements, Z˙ , which measure the frequency shift ͑Doppler effect͒ in the transmitted sig- A. Measurement data types nals. The frequency shift directly gives the range rate and, Having established the sensitivity of the trajectory to  due to the Hamilton-Melbourne effect ͓23͔, provides angular and ␥, we can consider estimates of how well  and ␥ can information on the trajectory. All of these measurement data be determined by measuring the trajectory ͑Fig. 3͒. For our types are analyzed using a variety of phase angles between analysis, three different measurement data types are consid- the Earth and the spacecraft trajectory—i.e., the initial Earth- ⌽ ered. The first is a two-way radar range measurement (Z), Sun-spacecraft angle, ESC , as shown in Fig. 3. which measures the distance between the spacecraft and the tracking station based on the travel time of the uplink and B. State to be estimated downlink signals. The second data type we consider is VLBI measurements (Zm,n), which measure the longitudinal and At epoch, the spacecraft is located at perihelion of its latitudinal angles of the spacecraft trajectory in the plane of heliocentric trajectory with the orbital elements the sky at the location of the tracking station ͓22͔. Combined ϭ ϫ 7 ͑ ͒ ao 8.725 10 km ,
ϭ eo 1.0319,
ϭ ϭ⍀ ϭ ϭ ͑ ͒ io o o M o 0. 5.1
These elements, in addition to the PPN parameters  and ␥, ϭ ជ ជ ␥T define the epoch state of our system: Y o ͓ro vo ͔ ϭ ␥T ͓xo y o zo uo o wo ͔ . The trajectories of the space- craft and of the Earth are assumed to be coplanar. ͑We are ignoring the issue of disentangling the effect of J2 in this analysis.͒ The spacecraft escapes the Sun with Vϱ ϭ 39 km/s, which corresponds to a periapsis velocity of V p ϭ311 km/s. The hypothetical trajectory from Ref. ͓1͔ will most likely fly into perihelion as an elliptic orbit and then boost to a hyperbolic escape trajectory. Hence the initial state, Eqs. ͑5.1͒, can be considered as the condition at epoch. Table II presents the conservative initial uncertainties that are as- FIG. 7. Change in the partial derivatives of the argument of sumed at epoch for the initial covariance matrix ͑assuming perihelion and mean anomaly with respect to the GR parameters. an accurate on-board measurement of the ⌬V maneuver ap- The variable y denotes  or ␥ as appropriate. plied at perihelion͒.
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TABLE II. Assumed initial uncertainties for various physical D. Implementation of the analysis quantities. We can exhibit the partial derivatives of the initial orbital elements and the GR parameters with respect to the initial Quantity xx , yy , zz uu , , ww  , ␥␥ ij ϫ (km2) (km2/s2) for iϭ j position, velocity, and the GR parameters as an 8 8 matrix, ” represented as follows: Uncertainty 1 10Ϫ6 10 ͒ E͑tץ ͪ o ͩ ץ W 0 ϫ X͑t ͒ 6 2ץ o ϭͫ o 6ϫ6 ͬ . ͑5.6͒ ץ C. Computation of the information and covariance matrix Y o 0 ϫ I ϫ To compute the state uncertainty at epoch, given a number 2 6 2 2 ץ ץ ץ ץ -of measurements, requires the computation of the informa Here, E(to)/ X(to) is the inverse of X(to)/ E(to) and tion matrix ⌳, which is given by can be obtained from the analytic relations Ϫ1 ͒ ץ ͒ ץ ͒ ץ T ͒ ץ N 1 Zi͑Y o Zi͑Y o E͑to X͑to ⌳ϭ͚ͫ ͩ ͪ ͩ ͪ ͬ. ͑5.2͒ ͩ ͪ ϭͩ ͪ ͒ E͑tץ ͒ X͑tץ Yץ Yץ 2 i i o o o o o o ជ͑ ͒ Tץ ជ ͑ ͒ T ץ v to r to Here Zi are the measurements, i are the noise factors in the ϭ ͑ ͒ͫͩ ϩ ͪ ͩ Ϫ ͪ ͬ , ͒ ͑ ץ ͒ ͑ ץ P E,E measurements, Y o is the epoch state, and N is the number of E to E to measurements taken. Given the information matrix, the co- ͑5.7͒ variance matrix P of the initial state and the GR parameters is then where P is an antisymmetric 6ϫ6 matrix made up of the Poisson brackets Pϭ⌳Ϫ1, ͑5.3͒ T T Eץ Eץ Eץ Eץ ͑ ͒ϭͩ i ͪͩ j ͪ Ϫͩ j ͪͩ i ͪ ͑ ͒ where P ϭ for i, j͕x ,y ,z ,u , ,w ,␥,͖. The P Ei ,E j . 5.8 vជץ rជץ vជץ rជץ ij ij o o o o o o standard deviation in our measured ␥ or  parameter will ϭͱ ϭͱ then be ␥ 77 and  88 and their correlation will Although the computation of P is quite complicated, there be 78 / ␥  . exist only five independent nonzero terms ͓20͔: The initial uncertainties in Eq. ͑5.3͒ are provided by Table II. It follows from Eq. ͑5.2͒ that to compute the information 2 P͑a,M ͒ϭϪP͑M,a͒ϭϩ , ͑5.9a͒ matrix we must analyze the sensitivity of a measurement na with respect to the initial conditions and the GR parameters: ͱe2Ϫ1 W P͑e,͒ϭϪP͑,e͒ϭ , ͑5.9b͒ץ Eץ Xץ Zץ Zץ ϭ o ͑ ͒ na2e 5.4 , ͪ ץͩͪ ץͩͪ ץͩͪ ץ ͩ ץ Y o X E Wo Y o e2Ϫ1 where P͑e,M ͒ϭϪP͑M,e͒ϭ , ͑5.9c͒ na2e ϭ ⍀ ␥  T ͑ ͒ Wo ͓ao eo io o o M o o o͔ , 5.5a 1 ͑ ⍀͒ϭϪ ͑⍀ ͒ϭϩ ͑ ͒ Eϭ͓aei ⍀ M͔T, ͑5.5b͒ P i, P ,i , 5.9d na2ͱe2Ϫ1 sin i ជ ˜ ˜ r xPϩyQ 1 Xϭͫ ͬϭͫ ͬ ϭ͓xyzu w͔T. ͑5.5c͒ ͑ ͒ϭϪ ͑ ͒ϭ ជ P i, P ,i . v ˜x˙ Pϩ˜y˙ Q na2ͱe2Ϫ1 tan i ͑5.9e͒ For the current analysis these partials are computed analyti- cally, and the initial values of the variables are denoted by It is important to note that these partials can be calculated based on the initial conditions, and are constants throughout the subscript ‘‘o.’’ ͑Figure 3 depicts ជ , rជ , and vជ .) o o o the numerical computation. The Gaussian vectors P and Q are functions of the orbital We next consider the partial derivatives of the orbital el- elements i,, and ⍀ and define the geometry of the orbit in ements with respect to the initial orbital elements and the GR ͑ ͓ ͔ ͒ ˜ ˜ space Ref. 20 , Sec. 2.7 . The scalars x and y are functions parameters. These can be considered as the orbital-element of the orbital elements a, e, and M which define the coordi- transition matrix plus the partial derivatives of the orbital nates inside the orbital plane. The Gaussian vectors P and Q elements with respect to the GR parameters given in Sec. IV. are constant and can be computed based on the initial epoch, For an unperturbed hyperbolic Keplerian orbit, the orbital whereas ˜x and ˜y must be computed at each instant in time. elements are constants except for the mean anomaly M(t), Finally, Z denotes the data measurement. which can be represented as
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͒Ϫ ͒ϭ Ϫ ͒ ͑ ͒ ϫ M͑t M͑to n͑t to . 5.10 gives rise to a 6 6 identity matrix with one non-vanishing The partial derivatives of the orbital elements with respect off-diagonal element: to the initial orbital elements and the GR parameters can be M͑t͒ 3nץ written as ϭϪ ͑ Ϫ ͒ ͑ ͒ t to . 5.12 ͒ ץ a͑to 2a E͑t͒ץ E͑t͒ץ Eץ ϭͫͩ ͪ ͩ ͪ ͬ. ͑5.11͒ The partial derivatives of the orbital elements with respect to ͒,␥͑ץ ͒ E͑tץ Wץ o o 6ϫ6 6ϫ2 the GR parameters are given in Eqs. ͑4.6͒ and ͑4.7͒. E) of theץ/Xץ) Next we consider the partial derivatives The mean motion is a function of the semi-major axis, a, and state vector with respect to the orbital elements, which can ץ ץ hence the orbital-element transition matrix, E(t)/ E(to), be represented by a 6ϫ6 matrix:
y˜ץ x˜ץ Qץ Pץ Qץ Pץ Qץ Pץ y˜ץ x˜ץ y˜ץ x˜ץ Pϩ Q Pϩ Q ˜x ϩ˜y ˜x ϩ˜y ˜x ϩ˜y Pϩ Q Mץ Mץ ⍀ץ ⍀ץ ץ ץ iץ iץ eץ eץ aץ aץ Xץ ϭ ͑ ͒ 5.13 . ˙ ˙ ץ ץ ץ ץ ץ ץ ˙ ˙ ˙ ˙ ͬ y˜ץ x˜ץ y ˙ P ˙ Q ˙ P ˙ Q ˙ P ˙ Q˜ץ x˜ץ y˜ץ x˜ץ ͫ Eץ ˜ ϩ˜ ˜ ϩ˜ ˜ ϩ˜ ⍀ Pϩ Qץ⍀ yץ xץ yץx ץ y ץ Pϩ Q Pϩ Q x Mץ Mץ e i iץ eץ aץ aץ 6ϫ6
For a given initial condition, we can compute each of these partials based on two-body relations ͓20,24͔. With proper changes ͑i.e. a→Ϫa), the equations for two-body hyperbolic motions yield the relations
ͱma ͱma rϭ a͑e cosh FϪ1͒, ˜xϭa͑eϪcosh F͒, ˜x˙ ϭ sinh F, ˜yϭaͱe2Ϫ1 sinh F, ˜y˙ ϭ ͱe2Ϫ1 cosh F, ͑5.14͒ r r where we solve for the hyperbolic eccentric anomaly ͑F͒ using the modified Kepler’s equation ͓21͔
m ͱ ͑tϪ͒ϭe sinh͑F͒ϪF, ͑5.15͒ ͉a͉3 where is the time of perihelion passage. We now take the partials of the coordinates ˜x and ˜y and their time derivatives, ˜x˙ and ˜y˙ , with respect to the orbital elements a, e, and M, which gives
˜ 2 ˜x y ˜x˙ ͩ aϩ ͪ a r͑e2Ϫ1͒ n x,˜y͒T˜͑ץ ϭ , ͑5.16͒ T ˜˜ ͬ ˙a,e,M ͒ ͫ ˜y xy ˜y͑ץ Ϫͩ ͪ a r͑e2Ϫ1͒ n
2 ˜ ˜ 2 3 ˜x˙ a x e y a Ϫ Ϫ˜x˙ ͩ ͪ ͩ 2ͩ ͪ ϩ ͩ ͪ ͪ Ϫnͩ ͪ ˜x 2a r a e2Ϫ1 a r x˙ ,˜y˙ ͒T˜͑ץ ϭ . ͑5.17͒ T 2 ˜ 2 ˜ 2 3 ͬ a,e,M ͒ ͫ ˜y˙ n a x y a͑ץ Ϫ Ϫ ͩ ͪ ͩ Ϫ ͪ Ϫnͩ ͪ ˜y 2a ͱe2Ϫ1 r r a͑e2Ϫ1͒ r
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The partials of P and Q with respect to i, , and ⍀ are provided in Ref. ͓20͔͑Sec. 7͒. We note that these partials are constants with respect to time, and can be evaluated based on the spacecraft’s initial epoch. X of the dataץ/Zץ Next we evaluate the partial derivatives measurements with respect to the state vector. The first data type we consider are range measurements
ϭ͉ជϪជ Ϫជ ͉ϭ͉ជ͉ ͑ ͒ Z r rE rTS , 5.18 the distance between the spacecraft and the tracking station ͑ ͒ ជ ͑ TS . The vector rE represents the position of the Earth with respect to the Sun͒ whose orbit is assumed to be circular with 1 year period and radius of 1 AU. The partial derivative of Z with respect to X is given by
T FIG. 9. Occulation effect due to the Sun. No radiometric mea- Z ˆ ͑t͒ץ ϭϭͫ ͬ ͑ ͒ surements can be made when the spacecraft passes in front of ͑or 5.19 , ץ X 0 ϫ behind͒ the Sun. See text for further details. 3 1 6ϫ1 which is widely used for interplanetary missions. These mea- where ˆ is the unit position vector of the spacecraft as mea- surements determine the shift in frequency due to the Dop- sured from the Earth ͑Fig. 3͒. We consider several assumed pler effect, and contain both range and angular information. values for the precision of the range measurement, , in the Ϫ4 Ϫ2 The partial derivative of Z• results in interval 10 kmрр10 km. ˆ Tץ ,An additional measurement data type considered is VLBI ˙ ជ • ץ -which yields accurate angular measurements of the space Z ជ r • ͬ , ͑5.24͒ץ craft relative to a radio source. We represent this measure- ϭͫ Xץ ment as a set of angles, ˆ 6ϫ1 ϭ T ͑ ͒ Z(m,n) ͓ Zm Zn͔ , 5.20 where ˆ 1ץ where Z and Z are the longitudinal and the latitudinal ϭ Ϫˆ ˆ T ͑ ͒ m n ͓I3 ͔, 5.25 rជ ץ -angular measurements, respectively. Taking partials with re spect to X yields ϫ • and I3 is the unit 3 3 matrix. The accuracies, , assumed ˆ T Ϫ6 Ϫ7 mo for the Doppler measurement are 10 ,10, and 0 ϫ Ϫ8 . 1 3 10 km/s for integration over a 60-s period ץ Z(m,n) ϭ , ͑5.21͒ ͬ X ͫ ˆ Tץ no 0 ϫ E. Solar occultation effects 1 3 2ϫ6 When the spacecraft passes in front of ͑or behind͒ the Sun ͑Fig. 9͒, radiometric measurements cannot be obtained. where we define Since the trajectory originates close to the Sun, solar inter- ference of the measurements can be an important effect in ˆ ϭˆ l o , the early stage of the experiment. Let us define
ជ ͑Ϫជ ͒ mˆ ϭˆl ϫnˆ , Ϫ rE o o o ϭcos 1ͫ • ͬ, ͑5.26͒ r ˆϪ ˆ ˆ ͒ˆ z ͑z l o l o nˆ ϭ • , ͑5.22͒ where represents the spacecraft-Earth-Sun angle. Based on o ͉ˆϪ͑ˆ ˆ ͒ˆ ͉ z z•l o l o the geometry of the Earth and Sun, and assuming that the ជ Earth is in circular orbit about the Sun, the angle between rE where zˆϭ͓001͔T. In Eq. ͑5.21͒ is the range from Earth and the tangent vector from center of the Earth to the outer to the spacecraft as defined earlier. The precisions assumed radius of the Sun, , can be computed and its value is ap- for the angular measurements are 5, 1, and 0.1 nrad. proximately 0.267°. We assume that no Doppler or VLBI The final data measurement type we analyze is Doppler, measurements are taken if рϩ0.5°, corresponding to ap- proximately 3r᭪ , and that no range measurements are taken d ជ ជ ជ ˙ if рϩ5°, corresponding to approximately 20r᭪ . The ef- Z˙ ϭ ͉rϪr Ϫr ͉ϭˆ ជ , ͑5.23͒ dt E TS • fects of measurement geometry and solar occulations on the
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FIG. 10. Dependence of measurement accuracies and correla- FIG. 11. Uncertainties in  and ␥ determinations as functions of  ␥ ⌽ tions of and on the phase angle, ESC , of the Earth. time for standard and advanced tracking accuracies. Standard accu- Ϫ3 Ϫ7 racy (X band͒ assumes ϭ10 km, ˙ ϭ10 km/s, and Є accuracies and correlations of  and ␥ depend on the phase ϭ1 nrad for range, range rate, and cross track uncertainties, respec- ͒ ϭ Ϫ4 angle of the Earth, ⌽ , as shown in Fig. 10. tively. Advanced accuracy (K band assumes 10 km, ˙ ESC Ϫ8 ϭ10 km/s, and Єϭ0.1 nrad. In the legend ‘‘current’’ refers to standard technology, and ‘‘future’’ refers to advanced technology. VI. RESULTS determinations made using fewer data ͑and hence with re- For our analysis, the spacecraft is assumed to be initially ͒ at periapsis of the heliocentric hyperbolic trajectory, with duced precision may be less correlated. This is a subtle issue ϭ ϭ and should be investigated in more detail to determine how r p 4r᭪ and Vϱ 39 km/s. All of the data measurements  ␥ considered are analyzed with different initial phase angles the measurement of may be optimized with respect to . ⌽  ␥ Our results are summarized in Table III. All of the values ( ESC) in order to study the sensitivity of and estimates ⌽ ͑ ͒ of  and ␥ shown in Table III are the final ones taken at to ESC Fig. 10 . The measurements are assumed to be taken every 15 min over a 10-day time span. It is important the end of the time span, without the solar occultation effect, to note that uncertainties in  and ␥ vary over an order of thus indicating how accurately the GR parameters can be magnitude, which indicates that the relative geometry of the determined. Two obvious ways to increase the accuracy of spacecraft with respect to the tracking station is a critical these parameters are to either take more measurements or to factor in this test. The results of this analytical approach are improve the noise factors. It is important to note, however, consistent with numerical simulations that were carried out. that our analysis neglects a number of possible systematic The results shown in Figs. 11 and 12 are based on track- error sources that may be present in the measurements. ing the spacecraft under the same data schedule as defined above, except that we disregard the solar occultation effect ͑which provides a lower bound on the uncertainty in our  and ␥ estimates͒. Figure 11 shows the uncertainties in  and ␥ when the range, VLBI, and Doppler measurements are combined. The plots of ␥ and  are shown for ‘‘standard accuracy’’ and ‘‘advanced accuracy’’ which we characterize as follows. Standard technology can provide noise factors of Ϫ3 Ϫ7 ϭ10 km, ˙ ϭ10 km/s, and Єϭ1 nrad using X-band radiometric tracking. These noise factors are directly related to how much information can be obtained from the spacecraft trajectory. Advanced accuracy noise factors are one order of magnitude more sensitive than the standard ac- curacy case, and are consistent with the K-band radiometric tracking system. Figure 12 shows the correlations between ␥ and  ͑i.e., 78 / ␥ ). We observe that the GR parameters become more correlated as the spacecraft moves away from periap- FIG. 12. Correlation of  and ␥ for range, VLBI, and Doppler Ϫ3 sis. Thus, as time increases, the most accurate determinations measurements, with standard accuracies ϭ10 km, ˙ Ϫ7 of the PPN parameters may be correlated with each other, but ϭ10 km/s, and Єϭ1 nrad.
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TABLE III. Uncertainties  and ␥ in the determination of the PPN parameters  and ␥, respectively, for various assumed uncertainties in range, VLBI, and Doppler measurements, as described in the text. The time span for the measurements is 30 days.
Accuracy status ͑km͒ • ͑km/s͒ Є ͑nrad͒  ␥
Standard (X band͒ 10Ϫ3 10Ϫ7 1 3.7ϫ10Ϫ4 7.8ϫ10Ϫ5 Advanced (K band͒ 10Ϫ4 10Ϫ8 0.1 3.7ϫ10Ϫ5 7.8ϫ10Ϫ6
Based on these results, we find that the original discussion pressure, and systematic measurement errors in an analysis in Ref. ͓1͔ is conservative, and that tracking technology cur- of this experiment. rently being implemented may already allow this experiment Note added. After this work was completed, we learned of to determine the PPN parameters  and ␥ to an accuracy of similar research by Vulkov ͓25͔. ϳ4ϫ10Ϫ4 and ϳ8ϫ10Ϫ5, respectively ͑as shown in Table III͒. By performing a detailed covariance analysis of the pro- ACKNOWLEDGMENTS posed experiment, the full strength of the range and Doppler We are indebted to Jennifer Coy, Damon Landau, Anas- radiometric data can be accounted for, weakening the re- tassios Petropoulos, Nancy M. Schnepp, and Sharon Wise for quirement for highly accurate VLBI measurements. There their assistance in preparing this paper. The work of E.F. was are, however, additional issues that must still be addressed. supported in part by the U.S. Department of Energy under The current results serve as an impetus to continue this in- Contract No. DE-AC02-76ER01428. D.J.S. acknowledges vestigation, as we have determined that the spacecraft trajec- support from the IND Technology Program by a grant from tory clearly has sufficient information content to allow for the Jet Propulsion Laboratory, California Institute of Tech- this unique measurement to be performed. Work in progress nology, which is under contract with the National Aeronau- will include the effects of solar oblateness, solar radiation tics and Space Administration.
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