Special relativity may account for the spacecraft flyby anomalies

Jean Paul Mbelek (1)

(1) Service d'Astrophysique, CEA-Saclay, Orme des Merisiers, 91191 Gif-sur-Yvette, France

September 10, 2008

Abstract

Recently, J. D. Anderson et al. [1] have proposed an empirical formula that accurately reproduces all the Earth flyby anomalies observed yet. Here, we show that special relativity (SR) transverse Doppler shift together with the addition of velocities may reproduce that formula.

PACS numbers: 03.30.+p, 95.10.Eg

Keywords: flyby anomaly, transverse Doppler shift, addition of velocities.

1 Introduction.

Galileo (in 1990 and 1992), NEAR (in 1998), Cassini (in 1999), Messenger (in 2005) and (in 2005) spacecraft showed an unexpected frequency change in the radio Doppler data during Earth flyby which are about ten times larger than the expected errors in measurement. The largest anomaly was recorded for NEAR (13 mm/s in excess) [1, 2]. During these Earth flyby, the incoming asymptotic velocity V∞i and outgoing asymptotic velocity V∞o of the spacecraft differed in magnitude by ∆V∞ not only by direction. The speeds V ∞i and V ∞o are respectively deduced by converting the measured two-ways Doppler shifts (δν/ν )i and (δν/ν )o into velocities from the usual nonrelativistic Doppler formula, thereby the so-called flyby anomaly ∆V∞ = V∞o - V ∞i. Recently, J. D. Anderson et al. [1] have proposed an empirical formula that accurately reproduces all the Earth flyby anomalies observed as yet. This formula involves respectively the declination δi and δo of the spacecraft's incoming and outgoing asymptotic velocities with respect to the geocentric reference frame and reads

∆V∞/V∞ = K (cos δi - cos δo), (1)

-6 -5 where K = 2 ωERE/c = 3.099 × 10 and, ωE = 7.292115 × 10 rad/s and RE = 6371 km denote respectively the angular velocity and the mean radius of the Earth. However, no underlying physical explanation for the anomaly has been put forward thus far to support the latter formula. Since the Earth's angular velocity and radius enter in the formula (1), the Lens-Thirring effect predicted by (GR) could be an explanation but this turns out to be negligibly small. Also, Lämmerzahl et al have ruled out many potential sources of the flyby anomalies [3]. In the following, we show that SR transverse Doppler shift together with the addition of velocities are sufficient to explain the flyby anomalies. Thus, GR does not need to be questioned and the flyby anomaly is merely due to an incomplete analysis using conventional physics.

2 Kinematics of the spacecraft flyby anomalies.

The photons of the communication signals between the ground based antennas and the spacecraft are subject to the transverse Doppler shift on account of the relative velocity, W, between the spinning Earth and the orbiting spacecraft. We argue that this well-known special relativistic effect has not been properly accounted for. Let ν0 and ν be respectively the frequencies of the emitted and received photons. The rotation of the Earth, at the orbit radius r of the spacecraft whose inclination of the orbital plane is I, implies the rotational velocity V E = ωE r cosI with respect to the geocentric reference frame. Hence, the one-way transverse Doppler shift reads

2 2 1/2 ν = ν0 (1 – W /c ) , (2) where W ≅ V∞ + VE, according to the addition of velocities in the low velocity approximation. Hence, the two-ways transverse Doppler shift for a given speed of the spacecraft with respect to the geocentric reference frame reads

2 2 1/2 2 2 2 2 2 δν/ν 0 = 2 [(1 – W /c ) – 1] ≅ - V∞ /c - VE /c - 2 V∞ . VE/c (3)

In the case where the declinations δi and δo of respectively the incoming (i) and outgoing (o) osculating asymptotic velocity vector differ with respect to Earth's equator, relation (3) above yields a overall Doppler shift in excess given by

2 ∆ν/ν 0 = ( δν/ν 0)o – ( δν/ν 0)i = 2 ( V∞ i - V∞ o ) . VE/c , (4) by setting V∞ o = V ∞ i = V∞, which means that there is no real flyby anomaly.

Now, if one forgets about the SR prediction of a transverse Doppler shift by assuming instead a nonrelativistic Doppler shift, namely ∆ν/ν 0 = ∆V∞ cos α/c, arranging relation (4) one obtains at the radius of closest approach r = R E + H,

∆V∞/V∞ = K (1 + H/R E) (cosI/cos α) (cos δi - cos δo). (5)

Clearly, relation (5) reduces to (1) for flybys at low altitude from the Earth geoid such that the estimate α ≅ ± I ± arccos(1 - H/R E + ε) holds, with | ε| « H/R E.

3 Conclusion.

Reconsidering the analysis of the Earth flyby anomalies observed up to now, and for which J. D. Anderson et al. have recently proposed an accurate empirical formula (1), we have found that SR transverse Doppler shift and the addition of velocities may provide the correct solution. Hence, we conclude that spacecraft flybys of heavenly bodies may be viewed as a new test of SR which has proven to be successful near the Earth. We call for a further follow- up of the spacecraft trajectory to test relation (5) well beyond the radius of closest approach.

References

[1] J. D. Anderson et al. , PRL 100, 091102 (2008)

[2] P.G. Antreasian and J.R. Guinn, Investigations into the unexpected delta-V increases during the Earth assists of and NEAR, AIAA 98-4287 (1998)

[3] C. Lämmerzahl, O. Preuss and H. Dittus, gr-qc/0604052