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Flyby anomaly

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Unsolved problem in physics:

What causes the unexpected change in acceleration for flybys of spacecraft?

The flyby anomaly is a discrepancy between current scientific models and the actual increase in speed (i.e. increase in kinetic energy) observed during a planetary flyby (usually of Earth) by a spacecraft. In multiple cases, spacecraft have been observed to gain greater speed than scientists had predicted, but thus far no convincing explanation has been found. This anomaly has been observed as shifts in the S-band and X-band Doppler and ranging telemetry. The largest discrepancy noticed during a flyby has been 13.46 mm/s.[1]


Gravitational assists are valuable techniques for Solar System exploration. Because the success of such flyby maneuvers depends on the exact geometry of the trajectory, the position and velocity of a spacecraft during its encounter with a planet is continually tracked with great precision by earth telemetry, e.g. via the Deep Space Network (DSN).

Range residuals during the Earth flyby of NEAR
During its flyby, MESSENGER did not observe any anomalies

The flyby anomaly was first noticed during a careful inspection of DSN Doppler data shortly after the Earth flyby of the Galileo spacecraft on 8 December 1990. While the Doppler residuals (observed minus computed data) were expected to remain flat, the analysis revealed an unexpected 66 mHz shift, which corresponds to a velocity increase of 3.92 mm/s at perigee. Investigations of this effect at the Jet Propulsion Laboratory (JPL), the Goddard Space Flight Center (GSFC) and the University of Texas have not yielded a satisfactory explanation.

No such anomaly was detected after the second Earth flyby of Galileo in December 1992, where the measured velocity decrease matched that expected from atmospheric drag at the lower altitude of 303 km. However, the drag estimates had large error bars, and so an anomalous acceleration could not be ruled out.[2]

On 23 January 1998 the Near Earth Asteroid Rendezvous (NEAR) spacecraft experienced an anomalous velocity increase of 13.46 mm/s after its Earth encounter. Cassini–Huygens gained around 0.11 mm/s in August 1999, and Rosetta gained 1.82 mm/s after its Earth flyby in March 2005.

An analysis of the MESSENGER spacecraft (studying Mercury) did not reveal any significant unexpected velocity increase. This may be because MESSENGER both approached and departed Earth symmetrically about the equator (see data and proposed equation below). This suggests that the anomaly may be related to Earth's rotation.

In November 2009, ESA's Rosetta spacecraft was tracked closely during flyby in order to precisely measure its velocity, in an effort to gather further data about the anomaly, but no significant anomaly was found.[3][4]

The 2013 flyby of Juno on the way to Jupiter yielded no anomalous acceleration.[5]

In 2018, a careful analysis of the trajectory of the presumed interstellar asteroid ʻOumuamua revealed a small excess velocity as it receded from the Sun. Initial speculation suggested that the anomaly was due to outgassing, though none had been detected.[6]

Summary of some Earth-flyby spacecraft is provided in table below.[3][7]

Galileo I Galileo II NEAR Cassini Rosetta-I MESSENGER Rosetta-II Rosetta-III Juno Hayabusa2 OSIRIS-REx[8] BepiColombo[9]
Date 1990-12-08 1992-12-08 1998-01-23 1999-08-18 2005-03-04 2005-08-02 2007-11-13 2009-11-13 2013-10-09 2015-12-03 2017-09-22 2020-04-10
Speed at infinity, km/s 8.949 8.877 6.851 16.01 3.863 4.056 4.7
Speed at perigee, km/s 13.738 8.877 12.739 19.03 10.517 10.389 12.49 13.34 14.93 10.3 8.5
Impact parameter, km 11261 12850 8973 22680.49 22319 19064
Minimal altitude, km 956 303 532 1172 1954 2336 5322 2483 561[10] 3090[11] 17237 12677
Spacecraft mass, kg 2497.1 2223.0 730.40 4612.1 2895.2 1085.6 2895 2895 ~2720 590 4000
Trajectory inclination to equator, degrees 142.9 138.9 108.0 25.4 144.9 133.1
Deflection angle, degrees 47.46 51.1 66.92 19.66 99.396 94.7 80
Speed increment at infinity, mm/s 3.92±0.08 −4.60±1.00 13.46±0.13 −2±1 1.82±0.05 0.02±0.01 ~0 ~0 0±0.8[5] ? ? ?
Speed increment at perigee, mm/s 2.560±0.050 −9.200±0.600 7.210±0.0700 −1.700±0.9000 0.670±0.0200 0.008±0.004 ~0.000±0.000 −0.004±0.044 ? ? ?
Gained energy, J/kg 35.1±0.7 92.2±0.9 7.03±0.19 ? ? ?

Anderson's empirical relation[edit]

An empirical equation for the anomalous flyby velocity change was proposed in 2008 by J. D. Anderson et al.:[12]

where ωE is the angular frequency of the Earth, RE is the Earth radius, and φi and φo are the inbound and outbound equatorial angles of the spacecraft. This formula was derived later by Jean Paul Mbelek from special relativity, leading to one of the possible explanations of the effect.[13] This does not, however, consider the SSN residuals – see "Possible explanations" below.

Possible explanations[edit]

There have been a number of proposed explanations of the flyby anomaly, including:

  • A postulated consequence of the assumption that the speed of light is isotropic in all frames, and invariant in the method used to measure the velocity of the space probes by means of the Doppler effect.[14] The inconsistent anomalous values measured: positive, null or negative are simply explained relaxing this assumption. During flyby maneuvers the velocity components of the probe in the direction of the observer Vo are derived from the relative displacement df of the radiofrequency f transmitted by the probe, multiplied by the local speed of the light c by the Doppler effect: Vo = (df / f) c. According to the Céspedes-Curé hypothesis,[15] the movement through variable gravitational energy density fields produces slight variations of the refractive index n of space and therefore of the speed of light c which leads to unaccounted corrections of the Doppler data that are based on an invariant c. This leads to incorrect estimates of the speed or energy change in the flyby maneuver on the Earth's frame of reference.
  • Unaccounted-for transverse Doppler effect—i.e. the redshift of light source with zero radial and non-zero tangential velocity.[13] However, this cannot explain the similar anomaly in the ranging data.
  • A dark-matter halo around Earth.[16]
  • The impact of general relativity, in its weak-field and linearized form yielding gravitomagnetic phenomena like frame-dragging, has been investigated as well:[17] it turns out to be unable to account for the flyby anomaly.
  • The classical time-retarded gravity explanation proposed by Joseph C. Hafele.[18]
    SSN range residuals for the NEAR flyby with range, delay
  • Range-proportional excess delay of the telemetry signal revealed by the United States Space Surveillance Network range data in the NEAR flyby.[19] This delay, accounting for the anomaly in both Doppler and range data, as well as the trailing Doppler oscillations, to within 10–20%, points to chirp modes in the reception due to the Doppler rate, predicting a positive anomaly only when the tracking by DSN is interrupted around perigee, and zero or negative anomaly if tracked continuously. No anomaly should occur in Doppler tracked by non-DSN stations.[20]
  • The action of a topological torsion current predicting flyby anomalies in retrograde direction, but null-effect when spacecraft approach the planet in prograde direction with respect to the planetary sense of rotation.[21]
  • The analysis of the Juno flyby looked at analysis errors that could potentially mimic the flyby anomaly. They found that a high-precision gravity field of at least 50×50 coefficients was needed for accurate flyby predictions. Use of a lower-precision gravity field (such as a model with 10×10 coefficients, sufficient for launch analysis), would yield a 4.5 mm/s velocity error.[5]

Related research[edit]

Some missions designed to study gravity, such as MICROSCOPE and STEP, are designed to make extremely accurate gravity measurements and may shed some light on the anomaly.[22] However, MICROSCOPE has completed its mission, finding nothing anomalous,[23] and STEP is yet to fly.

See also[edit]


  1. ^ "ESA's Rosetta spacecraft may help unravel cosmic mystery". European Space Agency. November 12, 2009. Retrieved 13 March 2010.[permanent dead link]
  2. ^ Edwards, C.; Anderson, J; Beyer, P; Bhaskaran, S.; Borders, J.; DiNardo, S.; Folkner, W.; Haw, R.; Nandi, S.; Nicholson, F.; 0ttenhoff, C.; Stephens, S. (1993-08-16). Tracking Galileo at Earth-2 Perigee Using The Tracking and Data Relay Satellite System (PDF) (Report). CiteSeerX hdl:2014/34792. Archived (PDF) from the original on 2022-04-18.{{cite report}}: CS1 maint: numeric names: authors list (link). The two [measurement] methods yielded similar fits to the data. Within an uncertainty of eight percent, both methods yielded a decrease in velocity along track of −5.9±0.2 mm/s. A priori predictions for the drag-induced velocity change, based on the Jacchia–Roberts model, were −6.2±4.0 mm/s [5], clearly consistent with the observed velocity change. By contrast, DSN data from the December 1990 Earth flyby, at altitude 956 km, indicated an unexplained increase in along-track velocity of 4 mm/s, after accounting for the much smaller drag effects. Given the uncertainty in drag models, we cannot conclusively rule out the possibility that a similar increase occurred at Earth 2. For example, an unmodeled increase of 4 mm/s and a drag decrease of −10 mm/s would be compatible with our results and our a priori atmospheric model. Significantly larger anomalous velocity increases, however, would appear inconsistent with the drag model.
  3. ^ a b "Mystery remains: Rosetta fails to observe swingby anomaly". ESA. Archived from the original on 2009-12-23.
  4. ^ J. Biele (2012). "Navigation of the interplanetary Rosetta and Philae spacecraft and the determination of the gravitational field of comets and asteroids - (DLR) @ TU München, 2012" (PDF). Archived from the original (PDF) on 2014-11-29. Retrieved 2014-11-18.
  5. ^ a b c Thompson, Paul F.; Matthew Abrahamson; Shadan Ardalan; John Bordi (2014). Reconstruction of Earth flyby by the Juno spacecraft. 24th AAS/AIAA Space Flight Mechanics Meeting. Santa Fe, NM: AAS. pp. 14–435.
  6. ^ Is the Interstellar Asteroid Really a Comet?
  7. ^ Anderson, John D.; James K. Campbell; Michael Martin Nieto (July 2007), "The energy transfer process in planetary flybys", New Astronomy, 12 (5): 383–397, arXiv:astro-ph/0608087, Bibcode:2007NewA...12..383A, doi:10.1016/j.newast.2006.11.004, S2CID 15913052
  8. ^ Stephen Clark (September 22, 2017). "OSIRIS-REx asteroid mission receives gravitational boost from planet Earth". Spaceflight Now.
  11. ^ Hayabusa2 Earth Swing-by Result
  12. ^ Anderson; et al. (7 March 2008). "Anomalous Orbital-Energy Changes Observed during Spacecraft Flybys of Earth" (PDF). Physical Review Letters. 100 (9): 091102. Bibcode:2008PhRvL.100i1102A. doi:10.1103/physrevlett.100.091102. PMID 18352689. Archived from the original (PDF) on 4 June 2016. Retrieved 15 February 2011.
  13. ^ a b Mbelek, J. P. (2009). "Special relativity may account for the spacecraft flyby anomalies". arXiv:0809.1888 [qr-qc].
  14. ^ Greaves, Eduardo D.; Bracho, Carlos; Mikoss, Imre (2020). "A Solution to the Flyby Anomaly Riddle". Progress in Physics. 16 (1): 49.
  15. ^ Cespedes-Cure, Jorge (2002). Einstein on Trial or Metaphysical Principles of Natural Philosophy (1st ed.). Venezuela: et al. Organization. ISBN 0-9713873-0-3.
  16. ^ Adler, S. L. (2009), "Can the flyby anomaly be attributed to Earth-bound dark matter?", Physical Review D, 79 (2): 023505, arXiv:0805.2895, Bibcode:2009PhRvD..79b3505A, doi:10.1103/PhysRevD.79.023505, S2CID 13152802
  17. ^ Iorio, L. (2009). "The Effect of General Relativity on Hyperbolic Orbits and Its Application to the Flyby Anomaly". Scholarly Research Exchange. 2009: 7695. arXiv:0811.3924. Bibcode:2009ScReE2009.7695I. doi:10.3814/2009/807695.
  18. ^ Hafele, Joseph C. (April 2013). "Causal Version of Newtonian Theory by Time-Retardation of the Gravitational Field Explains the Flyby Anomalies" (PDF). Progress in Physics. 2 (2): 3. Bibcode:2013PrPh....9b...3H. ISSN 1555-5534.
  19. ^ Antreasian, Peter G.; Guinn, Joseph R. (1998-08-10). Investigations into the Unexpected Delta-V Increase During the Earth Gravity Assist of GALILEO and NEAR (PDF). AIAA/AAS Astrodynamics Specialist Conference and Exhibition. Boston, Massachusetts: AIAA. CiteSeerX hdl:2014/20322. AIAA 98-4287. Archived (PDF) from the original on 2022-01-19. Retrieved 2017-05-06.
  20. ^ Guruprasad, V. (2015). "Observational evidence for travelling wave modes bearing distance proportional shifts". EPL. 110 (5): 54001. arXiv:1507.08222. Bibcode:2015EL....11054001G. doi:10.1209/0295-5075/110/54001. S2CID 42285652.
  21. ^ Pinheiro, Mario J. (2016). "Some effects of topological torsion currents on spacecraft dynamics and the flyby anomaly". Monthly Notices of the Royal Astronomical Society. 461 (4): 3948–3953. arXiv:1606.00691. Bibcode:2016MNRAS.461.3948P. doi:10.1093/mnras/stw1581.
  22. ^ Páramos, Jorge; Hechenblaikner, G. (2013). "Probing the Flyby Anomaly with the future STE-QUEST mission". Planetary and Space Science. 79–80: 76–81. arXiv:1210.7333. Bibcode:2013P&SS...79...76P. doi:10.1016/j.pss.2013.02.005. ISSN 0032-0633. S2CID 119287334.
  23. ^ Touboul, P.; Métris, G.; Rodrigues, M.; Bergé, J.; Robert, A.; Baghi, Q.; et al. (2022). "MICROSCOPE Mission: Final Results of the Test of the Equivalence Principle". Physical Review Letters. 129 (12): 121102. arXiv:2209.15487. Bibcode:2022PhRvL.129l1102T. doi:10.1103/PhysRevLett.129.121102. PMID 36179190. S2CID 252468544.


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