aa r X i v : . [ phy s i c s . s p ace - ph ] S e p Curious numerical coincidence to the Pioneer anomaly
Liviu Iv˘anescu
Earth and Atmospheric Sciences Department, UQ `AM, Montreal, QC, Canada
ABSTRACT
One noticed a numerical coincidence between the Pioneer spacecrafts deceleration anomaly and ( γ − γ the Lorentz factor. The match is not only for distances larger than 20 AU, but even for the observed slopbetween 10 and 20 AU. Such numerical link may eventually lead to a scientific hypothesis for future theoreticalinvestigations. Keywords:
Pioneer anomaly, Lorentz factor, numerical coincidence, space.
1. INTRODUCTION
The Pioneer spacecrafts unmodeled deceleration (towards the Sun) of a P = (8 . ± . · − m/s , forheliocentric distances greater than 20 AU, was reported in 1998 and 2002, based on an initial dataset. Thisanomaly was confirmed recently using newly recovered and carefully verified data. The deceleration, observed forboth Pioneer spacecrafts, was confirmed by independent investigations. In this way, approximation algorithmsor errors in the navigation code have been ruled out as possible causes of the anomaly. Alternatively, severalphysical mechanisms came up and claimed being able to justify the target value of a P , by using standard or newphysics theories. For the time being, none seems to convince the scientific community.Another way to handle this issue is by reverse engineering, i.e. finding an expression which gives a numericalcoincidence to the anomaly and only then try to build a model. One such example is the fact that p G · m P /a P has the same order of magnitude as the Compton wavelength of a proton, with G the gravitational constant and m P the proton mass. Similarly, several authors explained why a P ≃ c · H could make sense, with H theHubble constant and c the speed of light in vacuum.Most of those models focus on a constant value, while it’s not certain that the anomaly is constant. Actually,the initial dataset suggests that the anomaly has different values at distances less that 20 AU, while having asmall gradient towards the larger distances. In addition, differences between the anomaly of Pioneer 10 and 11could be expected. The analysis of the newly recovered data may, hopefully, clarify those aspects.
2. NUMERICAL COINCIDENCE
One proposes here a reverse engineering challenge starting with the numerical coincidence that: a P ≃ k · ( γ − , (1)where k = 1 m/s , γ = 1 / p − β is the Lorentz factor, β = v/c , and v is the radial spacecraft velocity withrespect to the Sun ( ∼
12 Km/s at more than 20 AU). At a first look, it could make sense that a residual value, as a P , could be explained by an excess factor, as ( γ − a P .As a first check, for v = 12 Km/s, ( γ −
1) = 8 . · − , which is very well in the range of the observed a P values. Secondly, ( γ −
1) varies as a fonction of v , which changes with the heliocentric distance, and produces avery close match to the observed a P (figure 1). The trajectory data used here comes from the JPL HORIZONSon-line solar system data and ephemeris computation service. The time stamps corresponding to positions
Send correspondence to Liviu Iv˘anescu, [email protected]: D´epartement des sciences de la Terre et de l’Atmosph`ere, Universit´e du Qu´ebec `a Montr´eal (UQ `AM), Casepostale 8888, succursale Centre-Ville, Montreal, QC, H3C 3P8, Canada. −20−100102030 V e l o c i t y ( K m / s ) Pioneer 10 & 11 radial heliocentric velocity
P10P11 −1 Heliocentric distance (AU) A no m a l y ( m / s ) x − Pioneer 10 & 11 deceleration anomaly computed P10computed P11observed P10observed P11 V e l o c i t y ( K m / s ) Pioneer 10 & 11 radial heliocentric velocity
P10P11 A no m a l y ( m / s ) x − Pioneer 10 & 11 deceleration anomaly computed P10computed P11observed P10observed P11
Figure 1. Spacecrafts radial heliocentric velocity (upper graphs) and the corresponding computed anomaly using ( γ − with errorbars are presented. P10 and P11 stand forPioneer 10 and 11, respectively. are ranging from few minutes to 7 hours, in order to provide smooth trajectories, especially during Jupiter andSaturn flybys, at 5 and 9.4 AU, respectively.Analyzing the figure 1, one can observe that the expression (1) provides, for both spacecrafts, good approxi-mations to the observed a P values. Sometimes, ( γ −
1) is outside the errorbars, but the initial Pioneer datasetalso contained some bad values and therefore the errorbars may not be very accurate. A lack of accuracy issuggested as well by the fact that the observed values don’t follow a very smooth trend, as it should if theyfollow a certain model. One needs to emphasize that the strong anomaly slop, between 10 and 20 AU, suggestedby the Pioneer 11 observations, is pointed out by ( γ −
1) behavior too. Moreover, as the observed anomalyerrorbars represent the standard deviation over 10 days, this suggests important variation of the anomaly at thePioneer 11 Saturn flyby (9.4 AU). The values given by ( γ −
1) show such a behavior too.
3. CONCLUSIONS
The reported Pioneer spacecrafts deceleration anomaly is computed from the observed trajectory. Here it wasidentified a curious numerical link between the observed deceleration and the relativistic term ( γ − γ −
1) factor.
REFERENCES [1] J. D. Anderson, P. A. Laing, E. L. Lau, A. S. Liu, M. M. Nieto, and S. G. Turyshev, “Indication, fromPioneer 10/11, Galileo, and Ulysses Data, of an Apparent Anomalous, Weak, Long-RangeAcceleration,”
Physical Review Letters (Oct., 1998) 2858–2861, arXiv:gr-qc/9808081 .22] J. D. Anderson, P. A. Laing, E. L. Lau, A. S. Liu, M. M. Nieto, and S. G. Turyshev, “Study of theanomalous acceleration of Pioneer 10 and 11,” Phys. Rev. D (Apr., 2002) 082004, 55 pages, arXiv:gr-qc/0104064 .[3] S. G. Turyshev and V. T. Toth, “The Pioneer Anomaly in the Light of New Data,” Space Science Reviews (June, 2009) 19 pages, arXiv:0906.0399 .[4] C. B. Markwardt, “Independent Confirmation of the Pioneer 10 Anomalous Acceleration,”
ArXiv GeneralRelativity and Quantum Cosmology e-prints (Aug., 2002) 29 pages, arXiv:gr-qc/0208046 .[5] Ø. Olsen, “The constancy of the Pioneer anomalous acceleration,”
A&A (Feb., 2007) 393–397.[6] A. Levy, B. Christophe, P. B´erio, G. M´etris, J.-M. Courty, and S. Reynaud, “Pioneer 10 Doppler dataanalysis: Disentangling periodic and secular anomalies,”
Advances in Space Research (May, 2009)1538–1544, arXiv:0809.2682 .[7] V. T. Toth, “Independent Analysis of the Orbits of Pioneer 10 and 11,” International Journal of ModernPhysics D (2009) 717–741, arXiv:0901.3466 .[8] S. G. Turyshev, J. D. Anderson, O. Bertolami, B. Dachwald, H. Dittus, U. Johann, D. Izzo,C. L¨ammerzahl, M. M. Nieto, A. Rathke, S. Reynaud, and W. Sebolt, “Pioneer anomaly investigation atISSI.” Web site, 2009. .[9] J. M¨akel¨a, “Pioneer Effect: An Interesting Numerical Coincidence,” ArXiv General Relativity andQuantum Cosmology e-prints (Oct., 2007) 15 pages, arXiv:0710.5460 .[10] L. M. Tomilchik, “Hubble law, Accelerating Universe and Pioneer Anomaly as effects of the space-timeconformal geometry,”
ArXiv General Relativity and Quantum Cosmology e-prints (June, 2008) 15 pages, arXiv:0806.0241 .[11] JPL, “Horizons on-line solar system data and ephemeris computation service.” Web site, 2009. http://ssd.jpl.nasa.gov/?horizonshttp://ssd.jpl.nasa.gov/?horizons