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{{unsolved|physics|What causes the unexpected change in [[acceleration]] for [[gravitational slingshot|flybys]] of spacecraft?}}
The '''flyby anomaly''' is an unexpected energy increase during Earth [[gravity assist|flybys]] of spacecraft. This anomaly has been observed as shifts in the [[s band|S-band]] and [[x band|X-band]] [[Doppler effect|Doppler]] and ranging [[telemetry]].  Taken together, it causes a significant unaccounted velocity increase of up to 13&nbsp;mm/s during flybys.<ref name="Rosetta">{{Cite web|url=http://www.spaceref.com/news/viewpr.html?pid=29609|title=ESA's Rosetta spacecraft may help unravel cosmic mystery|date=November 12, 2009|publisher=European Space Agency |accessdate=13 March 2010}}</ref> Numerically larger discrepancies (400–1000 m) have been observed at least in one flyby ([[NEAR Shoemaker|NEAR]]) against [[Space Surveillance Network]] (SSN) radars.

==Observations==
[[gravitational slingshot|Gravitational assists]] are valuable techniques for [[Timeline of Solar System exploration|Solar System exploration]]. Because the success of these flyby maneuvers depends on the geometry of the [[trajectory]], the position and velocity of a spacecraft is continually tracked during its encounter with a planet by the [[Deep Space Network]] (DSN).

[[Image:AntreasianGuinn199803b.jpg|right|300px|thumb|Range residuals during the Earth flyby of NEAR]]
[[File:Mdis depart anot.ogv|thumb|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&nbsp;[[millihertz|mHz]] shift, which corresponds to a velocity increase of 3.92&nbsp;mm/s at [[Apsis|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 anomaly was detected after the second Earth flyby of the [[Galileo spacecraft]] in December 1992, where the measured velocity decrease matched that expected from atmospheric drag at the lower altitude of 303&nbsp;km.  However, the drag estimates had large error bars, and so an anomalous acceleration could not be ruled out.<ref>{{cite web |title=TRACKING GALILEO AT EARTH-2 PERIGEE USING THE TRACKING AND DATA RELAY SATELLITE SYSTEM |author=C, Edwards, J. Anderson, P, Beyer, S. Bhaskaran, J. Borders, S. DiNardo, W. Folkner, R. Haw, S. Nandi, F. Nicholson, C. 0ttenhoff, S. Stephens |url=http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.38.4256&rep=rep1&type=pdf |date=1993}}.  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.O mm/s [5], clearly consistent with the observed velocity change. In 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.</ref>

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

An analysis of the [[MESSENGER]] spacecraft (studying [[Mercury (planet)|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)|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.<ref name="ESA"/><ref>{{cite news
 | url=https://www.nav.ei.tum.de/fileadmin/w00bkq/layout/colloquium_biele_slides.pdf
 | title=Navigation of the interplanetary Rosetta and Philae spacecraft and the determination of the gravitational field of comets and asteroids - (DLR) @ TU München, 2012
 | author=J. Biele
 | date=2012
 | accessdate=2014-11-18 }}
</ref>

The 2013 flyby of Juno on the way to Jupiter yielded no anomalous acceleration.<ref name="JUNO">{{cite conference |author1=Thompson, Paul F.|author2=Matthew Abrahamson|author3=Shadan Ardalan|author4=John Bordi |title=Reconstruction of Earth flyby by the Juno spacecraft. |publisher=AAS, |conference=24th AAS/AIAA Space Flight Mechanics Meeting |location=Santa Fe, NM |date=2014 |pages=14–435 |url=http://ntrs.nasa.gov/search.jsp?R=20160008163}}</ref>

Summary of Earth-flyby spacecraft is provided in table below.<ref name="ESA">{{cite web|url=http://webservices.esa.int/blog/post/5/916 |title=Mystery remains: Rosetta fails to observe swingby anomaly |publisher=ESA |archiveurl=https://web.archive.org/web/20091223041138/http://webservices.esa.int/blog/post/5/916 |archivedate=2009-12-23 }}</ref><ref name="Anderson">{{citation|doi=10.1016/j.newast.2006.11.004|last=Anderson|first=John D.|author2=James K. Campbell |author3=Michael Martin Nieto |date=July 2007|title=The energy transfer process in planetary flybys|journal=New Astronomy|volume=12|issue=5|pages=383–397|arxiv=astro-ph/0608087|bibcode = 2007NewA...12..383A }}</ref>

{| class="wikitable" border="1"
|-
! Quantity
! [[Galileo (spacecraft)|Galileo I]]
! [[Galileo (spacecraft)|Galileo II]]
! [[NEAR Shoemaker|NEAR]]
! [[Cassini–Huygens|Cassini]]
! [[Rosetta (spacecraft)|Rosetta]]-I
! [[MESSENGER|Messenger]]
! [[Rosetta (spacecraft)|Rosetta]]-II
! [[Rosetta (spacecraft)|Rosetta]]-III
! [[Juno (spacecraft)|Juno]]
! [[Hayabusa 2]]
|-
| Date
| 1990-12-08
| 1992-12-12
| 1998-01-23
| 1999-08-18
| 2005-03-04
| 2005-08-02
| 2007-11-13
| 2009-11-13
| 2013-10-09
| 2015-12-03
|-
| 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
| —
| 12.739
| 19.03
| 10.517
| 10.389
| 12.49
| 13.34
| 
| 10.3
|-
| [[Impact parameter]], km
| 11261
|
| 12850
| 8973
| 22680.49
| 22319
|
|
|
|
|-
| Minimal [[altitude]], km
| 956
| 303
| 532
| 1172
| 1954
| 2336
| 5322
| 2483
| 561<ref>[http://www.universetoday.com/105451/nasas-juno-spacecraft-returns-1st-flyby-images-of-earth-while-sailing-on-to-jupiter/ NASA’S JUNO SPACECRAFT RETURNS 1ST FLYBY IMAGES OF EARTH WHILE SAILING ON TO JUPITER]</ref>
| 3090<ref>[http://global.jaxa.jp/press/2015/12/20151214_hayabusa2.html Hayabusa2 Earth Swing-by Result]</ref>
|-
| Spacecraft [[mass]], kg
| 2497.1
|
| 730.40
| 4612.1
| 2895.2
| 1085.6
| 2895
| 2895
| ~2720
| 590
|-
| [[Inclination#Orbits|Trajectory inclination to equator]], degrees
| 142.9
| 138.9
| 108.8
| 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<ref name="JUNO"/>
| ?
|-
| Speed increment at [[perigee]], mm/s
| 2.56±0.05
|
| 7.21±0.07
| −1.7±0.9
| 0.67±0.02
| 0.008±0.004
| ~0
| −0.004±0.044
| 
| ?
|-
| Gained energy, J/kg
| '''35.1±0.7'''
|
| '''92.2±0.9'''
|
| '''7.03±0.19'''
|
|
|
|
| ?
|}

==Future research==
Upcoming missions with Earth flybys include [[BepiColombo]] with its launch due in October 2018 and its Earth flyby due April 2020.

Some missions designed to study gravity, such as [[STEP (satellite)|STEP]], will make extremely accurate gravity measurement and may shed some light on the anomaly.<ref>{{cite journal |title=Probing the Flyby Anomaly with the future STE-QUEST mission |journal=Planetary and Space Science |volume=79-80 |pages=76–81 |arxiv=1210.7333 |doi=10.1016/j.pss.2013.02.005|year=2013 |last1=Páramos |first1=Jorge |last2=Hechenblaikner |first2=G. |bibcode=2013P&SS...79...76P }}</ref>

===Proposed equation===
An empirical equation for the anomalous flyby velocity change was proposed by J. D. Anderson et al.:

:<math> \frac{dV}{V} = \frac{2 \omega_\text{E} R_\text{E} (\cos \varphi_\text{i} - \cos \varphi_\text{o})}{c}, </math>

where ''ω''<sub>E</sub> is the [[angular frequency]] of the Earth, ''R''<sub>E</sub> is the Earth radius, and ''φ''<sub>i</sub> and ''φ''<sub>o</sub> are the inbound and outbound equatorial angles of the spacecraft.<ref>{{Citation|author=Anderson|title=Anomalous Orbital-Energy Changes Observed during Spacecraft Flybys of Earth|journal=Phys. Rev. Lett.|volume=100|issue=9|pages=091102|date=7 March 2008|url=http://virgo.lal.in2p3.fr/NPAC/relativite_fichiers/anderson_2.pdf|postscript=.|doi=10.1103/physrevlett.100.091102 |pmid=18352689|bibcode = 2008PhRvL.100i1102A |display-authors=etal}}</ref> This formula was derived later by Jean Paul Mbelek from the special relativity, leading to one of the possible explanations of the effect.<ref name="mbelek"/> This does not, however, consider the SSN residuals – see "Possible explanations" below.

==Possible explanations==
There have been a number of proposed explanations of the flyby anomaly, including:
*Unaccounted [[transverse Doppler effect]]—i.e. the redshift of light source with zero radial and non-zero tangential velocity.<ref name="mbelek">{{cite arXiv |last=Mbelek |first=J. P. |eprint=0809.1888 |title=Special relativity may account for the spacecraft flyby anomalies |class=qr-qc |date=2009}}</ref> However, this cannot explain the similar anomaly in the ranging data.
*A [[dark-matter]] halo around Earth.<ref>{{citation|author=S.L.Adler|title=Can the flyby anomaly be attributed to Earth-bound dark matter?|date=2008|arxiv=0805.2895|bibcode=2009PhRvD..79b3505A|doi=10.1103/PhysRevD.79.023505|journal=Physical Review D|volume=79|issue=2 |pages=023505}}</ref>
*A modification of inertia resulting from a Hubble-scale [[Casimir effect]], related to the [[Unruh effect]] ([[MiHsC]]).<ref>{{citation|author=M. E. McCulloch|title=Modelling the flyby anomalies using a modification of inertia|journal=MNRAS Letters|volume=389 |issue=1|pages=L57–L60|doi=10.1111/j.1745-3933.2008.00523.x|date=2008 |bibcode=2008MNRAS.389L..57M|arxiv=0806.4159}}</ref><ref>https://www.technologyreview.com/s/601299/the-curious-link-between-the-fly-by-anomaly-and-the-impossible-emdrive-thruster/</ref>
*The impact of [[general relativity]], in its weak-field and linearized form yielding [[gravitoelectric]] and [[gravitomagnetic]] phenomena like [[frame-dragging]], has been investigated as well:<ref>{{citation|author=L. Iorio|title= The Effect of General Relativity on Hyperbolic Orbits and Its Application to the Flyby Anomaly|journal=Scholarly Research Exchange|volume=2009|doi=10.3814/2009/807695|id=807695|date=2009|pages=1–8 |bibcode=2009ScReE2009.7695I|arxiv=0811.3924}}</ref> it turns out to be unable to account for the flyby anomaly.
*The classical time-retarded gravity explanation proposed by [[Joseph C. Hafele]].<ref>http://www.ptep-online.com/index_files/2013/PP-33-01.PDF - Causal Version of Newtonian Theory by Time–Retardation of the Gravitational Field Explains the Flyby Anomalies</ref>
*Range-proportional excess delay of the telemetry signal revealed by the [[United States Space Surveillance Network]] range data in the NEAR flyby.<ref>{{cite conference |author1=Peter G. Antreasian |author2=Joseph R. Guinn |title=Investigations into the Unexpected Delta-V Increase During the Earth Gravity Assist of GALILEO and NEAR |url=https://trs-new.jpl.nasa.gov/bitstream/handle/2014/20322/98-1237.pdf?sequence=1&isAllowed=y |id=Article ID - AIAA 98-4287|conference=AIAA/AAS Astrodynamics Specialist Conf. and Exhibition|location=Boston, MA |publisher=AIAA|date=1998| accessdate=2017-05-06 }}</ref> 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.<ref>{{Citation |author1=V. Guruprasad|title=Observational evidence for travelling wave modes bearing distance proportional shifts |url=http://iopscience.iop.org/0295-5075/110/5/54001/article |doi=10.1209/0295-5075/110/54001 |journal=[[Europhysics Letters|EPL]] |volume=110 |issue=5|date=2015 |pages=54001  |arxiv = 1507.08222 |bibcode = 2015EL....11054001G }}</ref> 
[[Image:Epl17133f1 online.jpg|right|250px|thumb|SSN range residuals with range, delay]]

==See also==
* [[List of unsolved problems in physics]]
* [[Modified Newtonian dynamics]]
* [[Pioneer anomaly]]

==References==
{{Reflist}}

==Literature==
* {{Citation |author1=J. D. Anderson |author2=J. G. Williams |title=Long-range tests of the equivalence principle |journal=[[Classical and Quantum Gravity|Class. Quantum Grav.]] |volume=18 |issue=13 |pages=2447–2456 |date=2001 |doi=10.1088/0264-9381/18/13/307 |bibcode = 2001CQGra..18.2447A }}.
* {{Citation |author1=C. Lämmerzahl |author2=O. Preuss |author3=H. Dittus |title=Is the physics within the Solar system really understood? |journal=Proceedings of the 359th WE-Heraeus Seminar on "Lasers, Clocks, and Drag-Free: Technologies for Future Exploration in Space and Tests of Gravity" |date=2006 |arxiv=gr-qc/0604052}}.
* {{Citation |author1=J. D. Anderson |author2=J. K. Campbell |author3=M. M. Nieto |title=The Energy Transfer Process in Planetary Flybys |arxiv=astro-ph/0608087 |date=2006 |bibcode = 2007NewA...12..383A |doi = 10.1016/j.newast.2006.11.004 |journal=New Astronomy |volume=12 |issue=5 |pages=383–397 }}.
*[http://www.space.com/scienceastronomy/080229-spacecraft-anomaly.html NASA Baffled by Unexplained Force Acting on Space Probes] (2008), at [[Space.com]].
* {{Citation |author1=J. D. Anderson |author2=J. K. Campbell |author3=J. E. Ekelund |author4=J. Ellis |author5=J. F. Jordan |title=Anomalous Orbital-Energy Changes Observed during Spacecraft Flybys of Earth |url=http://www2.phys.canterbury.ac.nz/editorial/Anderson2008.pdf |doi=10.1103/PhysRevLett.100.091102 |journal=[[Physical Review Letters|Phys. Rev. Lett.]] |volume=100 |issue=91102 |date=2008 |pages=091102 |bibcode=2008PhRvL.100i1102A |pmid=18352689}}.
* [http://www.economist.com/science/displaystory.cfm?story_id=10804075 Wanted: Einstein Jr] (2008), at [http://www.economist.com Economist.com].
* {{cite journal|author=K. Svozil |title=Microphysical analogues of flyby anomalies |journal=New Astronomy |volume=12 |issue=5 |pages=383–397 |arxiv=0804.2198 |date=2007 |bibcode=2007NewA...12..383A|doi=10.1016/j.newast.2006.11.004 }}

==External links==
* {{cite web |first= Claus |last= Lämmerzahl |publisher= University of Bremen |url= http://www.zarm.uni-bremen.de/fileadmin/images/laemmerzahlDatei/Pioneer_Utrecht.pdf |title= The Pioneer Anomaly or Do We Really Understand the Physics With the Solar System? |format= pdf; 6.25 MB, talk/slides |pages= 123 |date= 2008 |work= Center for Applied Space Technology and Microgravity }}
* {{cite web |first= Andreas |last= Aste |publisher= University of Basel  |url= http://phys-merger.physik.unibas.ch/~aste/flyby.pdf |title= Spacecraft Anomalies: An Update |format= pdf; 9.8 MB, talk/slides |pages= 29 |date= 2008 |work= Department of Physics }}

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