Table Of ContentAstrophysics and Space Science Library 441
Ivan I. Shevchenko
The Lidov-Kozai
Effect –
Applications in
Exoplanet Research
and Dynamical
Astronomy
The Lidov-Kozai Effect –
Applications in Exoplanet Research
and Dynamical Astronomy
Astrophysics and Space Science Library
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W.B.BURTON,NationalRadioAstronomyObservatory,Charlottesville,
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A.HECK,StrasbourgAstronomicalObservatory,France
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V.M.KASPI,McGillUniversity,Montreal,Canada
J.M.E.KUIJPERS,UniversityofNijmegen,TheNetherlands
H.VANDERLAAN,UniversityofUtrecht,TheNetherlands
P.G.MURDIN,InstituteofAstronomy,Cambridge,UK
B.V.SOMOV,AstronomicalInstitute,MoscowStateUniversity,Russia
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Moreinformationaboutthisseriesathttp://www.springer.com/series/5664
Ivan I. Shevchenko
The Lidov-Kozai Effect –
Applications in Exoplanet
Research and Dynamical
Astronomy
123
IvanI.Shevchenko
PulkovoObservatoryoftheRussian
AcademyofSciences
St.Petersburg,Russia
ISSN0067-0057 ISSN2214-7985 (electronic)
AstrophysicsandSpaceScienceLibrary
ISBN978-3-319-43520-6 ISBN978-3-319-43522-0 (eBook)
DOI10.1007/978-3-319-43522-0
LibraryofCongressControlNumber:2016951301
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Preface
Uptonow,itnearlymakesmelaughandcryofrapturewhenI
recallthisresult.
VladimirBeletsky,SixDozens(Memoirs)
The Lidov-Kozai effect (LKE), as any other basic astrophysical phenomena, has
several “faces”, or manifestations. The most familiar face, granted in Wikipedia,
isthephenomenonofcoupledperiodicvariations(whichcanbeverylarge)ofthe
inclination and eccentricity of an orbiting body, which (the variations) may take
placeinthepresenceofaninclined-enoughperturber.
Aface,lessfamiliartothegeneralaudience,butphysicallythegenericone,isthe
so-called phenomenonof !-libration; i.e., libration of the argument of pericenter
of an orbiting body,when an inclined-enoughperturberis present. In fact, the !-
librationisatthecoreoftheLKE:itisjustalarge-amplitude!-librationthatentails
thementionedcoupledvariationsininclinationandeccentricity.
Anotherconcept,frequentlypresentinthemodernastrophysicalliterature,isthe
LKEasavaluabletooltoexplainvariousmergerevents.Indeed,ifitis“necessary
to merge”anygravitatingbinary(saya binaryconsistingoftwo ordinarystars, or
black holes, or asteroids, or a star and a “hotJupiter,” or a planetand an artificial
satellite,etc.)intoasingleobject,theLKEiscommonlythefirstoneinthequeue
of possible explaining mechanisms that come to mind of an astrophysicist. This
is just because it is rapid and no dissipation is needed—only the presence of a
perturber.This third face revealed itself already in the pioneeringworks of Lidov
(whooutlinedaconditionforamergerofasatellitewiththeEarth,inparticularthe
Moon with the Earth), and since then it has been becoming only more and more
attractiveforresearchers.
To date, the LKE,in all ofthese faces, hasbeenverifiedto be importantin the
dynamics of a lot of kinds of astrophysical objects. Historically the applications
started with satellites and asteroids; now they comprise comets, trans-Neptunian
objects, exoplanets, multiple stars, and many others. Perhaps, of most interest for
theastronomicalandastrophysicalcommunityatpresentisitsrelevanceformany
exoplanetarysystems.
RecentyearswitnessedmajoradvancementsinthetheoryofLKE.Inparticular,
anotherfaceoftheLKEemerged:theso-calledflip(orbitturnover)phenomenon.
ItwouldbenoexaggerationtosaythatatpresenttheLidov-Kozaieffectbecomes
oneof themoststudied astrophysicaleffects;this ismanifested,in particular,in a
v
vi Preface
sharpriseofcitationsofthepioneerarticlesofLidovandKozaiduringrecentyears
(seeFig.1).
Fig.1 Theper-yearnumberofcitationsofthepioneerarticlesofLidovandKozai,asafunction
oftime,uptoyear2015inclusive(ThecitationdataoftheNASAADSdatabasehavebeenused
toconstructthediagram)
Thetopics,coveredinthisbook,include:historicalbackgroundforappearance
of works by Lidov and Kozai in the beginning of the 1960s (Luna-3, etc.);
modern secular theories and integrable approximationsin celestial mechanics; an
overview of classical works on the Lidov-Kozai effect; modern advancements
and generalizations in the Lidov-Kozai theory, in particular in the framework of
noncircularandnonrestrictedproblems;theLidov-Kozaimechanismexplainingthe
observedorbitalconfigurationsofirregularsatellites,explainingcomets-sungrazers,
andexplainingtheobserveddynamicalpatternsintheasteroidandKuiperbelts;the
role of the Lidov-Kozai mechanism (in particular the Lidov-Kozai migration) in
sculptingexoplanetarysystems;applicationsinstellardynamics,suchasscenarios
for formation of close binary stars; and explanations for highly eccentric stellar
orbitsintheGalacticcenter.
Theinitialaimofthisbookwastoprovidethemostfullcoverageoftheeffect’s
theoryandapplications.However,duetoalargeamountofthepublishedandnewly
appearing advancements, it is practically impossible to fulfill this aim in detail.
Therefore,apartfromtheprovidedbibliography,forfurtherreadingonthesubject,
IwouldrecommendthematerialontheLidov-Kozaieffectinasteroidaldynamics,
presented in Sects.8.2 and 11.2.2 of the book Modern Celestial Mechanics by
A. Morbidelli (2002), and the material on the LKE in exoplanetary dynamics
presented in Sects.7 and 8 of the review “The Long-Term Dynamical Evolution
ofPlanetarySystems”byM.B.Daviesetal.(2014).
Preface vii
Thepresentedbookisself-contained:onlybasicknowledgeinmathematicsand
mechanicsisrequiredforunderstandingthematerial,ifreadfromthebeginning.I
hopethatthebookcanbehelpfulforaresearcherworkinginastrophysics,celestial
mechanics, stellar dynamics, theoretical mechanics, and space mission design, at
any level (researcher, graduate student, undergraduate student), depending on the
interestsofthereader.
I am mostgratefulto Vladislav Sidorenko,KonstantinKholshevnikov,Mikhail
Vashkovyak, and Alessandro Morbidelli for valuable remarks and comments. I
am especially thankful to Vladislav Sidorenko for providing rare bibliographic
materialsonthesubject.
SaintPetersburg,Russia IvanI.Shevchenko
2016
Contents
1 DynamicalEssenceandHistoricalBackground.......................... 1
1.1 TheKeplerianOrbitalElements........................................ 2
1.2 The“omega-libration”.................................................. 5
1.3 TheBreakthroughPremises ............................................ 6
1.4 FromLuna-3toModernSpaceMissions.............................. 9
2 AveragingandNormalizationinCelestialMechanics ................... 13
2.1 TheHamiltonianFormalism............................................ 14
2.2 TheTwo-BodyProbleminaHamiltonianForm ...................... 15
2.3 AnN-BodyProbleminaHamiltonianForm.......................... 16
2.4 TheDelaunayVariables................................................. 17
2.5 Near-IntegrableSystems................................................ 19
2.6 ThevonZeipelMethod................................................. 20
2.7 TheHori–DepritMethod ............................................... 23
3 ClassicalResults.............................................................. 27
3.1 ASingle-AveragedR3BP............................................... 28
3.2 TheDouble-AveragedR3BP ........................................... 31
3.2.1 TheLidov-KozaiHamiltonian ................................. 31
3.2.2 EquationsandConstantsofMotion............................ 34
3.2.3 ClassificationofOrbits......................................... 36
3.2.4 TheLidov-KozaiDiagrams .................................... 39
3.2.5 TheSolutionintheJacobiEllipticFunctions ................. 41
3.3 LKE-PreventingPhenomena ........................................... 46
3.3.1 PerturbationsbyAdditionalOrbitingBodies.................. 47
3.3.2 Primary’sOblateness........................................... 48
3.3.3 Tides............................................................. 50
3.3.4 GeneralRelativity .............................................. 52
3.3.5 TheOrbitalPrecessioninTotal ................................ 53
3.4 CriticalRadii............................................................ 54
ix
