Kozai mechanism

Incelestial mechanics,theKozai mechanismis a dynamical phenomenon affecting the orbit of abinary systemperturbed by a distant third body under certain conditions. The mechanism is also namedvon Zeipel-Kozai-Lidov,Lidov–Kozai,Kozai–Lidov,or some combination of Kozai, Lidov, and/or von Zeipel. It also termed an effect, oscillations, cycles, or resonance. This effect causes the orbit'sargument of pericentertooscillate about a constant value,which in turn leads to a periodic exchange between itseccentricityandinclination.The process occurs on timescales much longer than the orbital periods. It can drive an initially near-circular orbit to arbitrarily high eccentricity, andflipan initially moderately inclined orbit between aprograde and a retrograde motion.

The effect has been found to be an important factor shaping the orbits ofirregular satellitesof the planets,trans-Neptunian objects,extrasolar planets,andmultiple star systems.It hypothetically promotesblack hole mergers.It was described in 1961 byMikhail Lidovwhile analyzing the orbits of artificial and natural satellites of planets.In 1962,Yoshihide Kozaipublished this same result in application to the orbits ofasteroidsperturbed byJupiter.The citations of the papers by Kozai and Lidov have risen sharply in the 21st century. As of 2017,the mechanism is among the most studied astrophysical phenomena.It was pointed out in 2019 by Takashi Ito and Katsuhito Ohtsuka that the Swedish astronomerEdvard Hugo von Zeipelhad also studied this mechanism in 1909, and his name is sometimes now added.

Background

Hamiltonian mechanics

Main article:Hamiltonian mechanics

In Hamiltonian mechanics, a physical system is specified by a function, calledHamiltonianand denoted${\mathcal {H}}$,ofcanonical coordinatesinphase space.The canonical coordinates consist of thegeneralized coordinates$x_{k}$inconfiguration spaceand theirconjugate momenta$p_{k}$,for$k=1,...N$,for theNbodies in the system ($N=3$for the von Zeipel-Kozai–Lidov effect). The number of$(x_{k},p_{k})$pairs required to describe a given system is the number of itsdegrees of freedom.

The coordinate pairs are usually chosen in such a way as to simplify the calculations involved in solving a particular problem. One set of canonical coordinates can be changed to another by acanonical transformation.Theequations of motionfor the system are obtained from the Hamiltonian throughHamilton's canonical equations,which relate time derivatives of the coordinates to partial derivatives of the Hamiltonian with respect to the conjugate momenta.

The three-body problem

Main articles:Three-body problemandPerturbation theory

The dynamics of a system composed of three bodies system acting under their mutual gravitational attraction is complex. In general, the behaviour of a three-body system over long periods of time isenormously sensitive to any slight changes in the initial conditions,including even small uncertainties in determining the initial conditions, and rounding-errors in computerfloating pointarithmetic. The practical consequence is that, thethree-body problemcannot be solved analytically for an indefinite amount of time, except in special cases.Instead,numerical methodsare used for forecast-times limited by the available precision.

The Lidov–Kozai mechanism is a feature ofhierarchicaltriple systems,that is systems in which one of the bodies, called the "perturber", is located far from the other two, which are said to comprise theinner binary.The perturber and the centre of mass of the inner binary comprise theouter binary.Such systems are often studied by using the methods ofperturbation theoryto write the Hamiltonian of a hierarchical three-body system as a sum of two terms responsible for the isolated evolution of the inner and the outer binary, and a third termcouplingthe two orbits,

${\mathcal {H}}={\mathcal {H}}_{\rm {in}}+{\mathcal {H}}_{\rm {out}}+{\mathcal {H}}_{\rm {pert}}.$

The coupling term is then expanded in the orders of parameter$\ Alpha$,defined as the ratio of thesemi-major axesof the inner and the outer binary and hence small in a hierarchical system.Since the perturbative seriesconvergesrapidly, the qualitative behaviour of a hierarchical three-body system is determined by the initial terms in the expansion, referred to as thequadrupole ($\propto \ Alpha ^{2}$),octupole($\propto \ Alpha ^{3}$) andhexadecapole($\propto \ Alpha ^{4}$) order terms,

${\mathcal {H}}_{\rm {pert}}={\mathcal {H}}_{\rm {quad}}+{\mathcal {H}}_{\rm {oct}}+{\mathcal {H}}_{\rm {hex}}+O(\ Alpha ^{5}).$

For many systems, a satisfactory description is found already at the lowest, quadrupole order in the perturbative expansion. The octupole term becomes dominant in certain regimes and is responsible for a long-term variation in the amplitude of the Lidov–Kozai oscillations.

Secular approximation

The Lidov–Kozai mechanism is aseculareffect, that is, it occurs on timescales much longer compared to the orbital periods of the inner and the outer binary. In order to simplify the problem and make it more tractable computationally, the hierarchical three-body Hamiltonian can besecularised,that is, averaged over the rapidly varying mean anomalies of the two orbits. Through this process, the problem is reduced to that of two interacting massive wire loops.

Overview of the mechanism

Test particle limit

The simplest treatment of the von Zeipel-Lidov–Kozai mechanism assumes that one of the inner binary's components, thesecondary,is atest particle– an idealized point-like object with negligible mass compared to the other two bodies, theprimaryand the distant perturber. These assumptions are valid, for instance, in the case of an artificial satellite in alow Earth orbitthat is perturbed by theMoon,or ashort-period cometthat is perturbed byJupiter.

The Keplerianorbital elements.

Under these approximations, the orbit-averaged equations of motion for the secondary have aconserved quantity:the component of the secondary's orbital angular momentum parallel to the angular momentum of the primary / perturber orbit. This conserved quantity can be expressed in terms of the secondary'seccentricityeandinclinationirelative to the plane of the outer binary:

$L_{\mathrm {z} }={\sqrt {1-e^{2}\;}}\,\cos i=\mathrm {constant}$

Conservation ofL_zmeans that orbital eccentricity can be "traded for" inclination. Thus, near-circular, highly inclined orbits can become very eccentric. Since increasing eccentricity while keeping thesemimajor axisconstant reduces the distance between the objects atperiapsis,this mechanism can cause comets (perturbed byJupiter) to becomesungrazing.

Lidov–Kozai oscillations will be present ifL_zis lower than a certain value. At the critical value ofL_z,a "fixed-point" orbit appears, with constant inclination given by

$i_{\mathrm {crit} }=\arccos \left({\sqrt {{\frac {3}{5}}\,}}\,\right)\approx 39.2^{\mathsf {o}}$

For values ofL_zless than this critical value, there is a one-parameter family of orbital solutions having the sameL_zbut different amounts of variation ineori.Remarkably, the degree of possible variation iniis independent of the masses involved, which only set the timescale of the oscillations.

Timescale

The basic timescale associated with Kozai oscillations is

$T_{\mathrm {Kozai} }=2\pi \,{\frac {\,{\sqrt {G\,M\;}}\,}{G\,m_{2}}}\,{\frac {\,a_{2}^{3}\,}{a^{3/2}}}\left(1-e_{2}^{2}\right)^{3/2}={\frac {\,P_{2}^{2}\,}{P}}\,\left(1-e_{2}^{2}\right)^{3/2}$

whereaindicates the semimajor axis,Pis orbital period,eis eccentricity andmis mass; variables with subscript "2" refer to the outer (perturber) orbit and variables lacking subscripts refer to the inner orbit;Mis the mass of the primary. For example, withMoon's period of 27.3 days, eccentricity 0.055 and theGlobal Positioning Systemsatellites period of half a (sidereal) day, the Kozai timescale is a little over 4 years; forgeostationary orbitsit is twice shorter.

The period of oscillation of all three variables (e,i,ω– the last being theargument of periapsis) is the same, but depends on how "far" the orbit is from the fixed-point orbit, becoming very long for theseparatrixorbit that separates librating orbits from oscillating orbits.

Astrophysical implications

Solar System

The von Zeipel-Lidov–Kozai mechanism causes theargument of pericenter(ω) tolibrateabout either 90° or 270°, which is to say that itsperiapseoccurs when the body is farthest from the equatorial plane. This effect is part of the reason thatPlutois dynamically protected from close encounters withNeptune.

The Lidov–Kozai mechanism places restrictions on the orbits possible within a system. For example:

For a regular satellite

If the orbit of a planet's moon is highly inclined to the planet's orbit, the eccentricity of the moon's orbit will increase until, at closest approach, the moon is destroyed by tidal forces.

For irregular satellites

The growing eccentricity will result in a collision with a regular moon, the planet, or alternatively, the growing apocenter may push the satellite outside theHill sphere.Recently, the Hill-stability radius has been found as a function of satellite inclination, also explains the non-uniform distribution of irregular satellite inclinations.

The mechanism has been invoked in searches forPlanet Nine,a hypothetical planet orbiting the Sun far beyond the orbit of Neptune.

A number of moons have been found to be in the Lidov–Kozai resonance with their planet, including Jupiter'sCarpoandEuporie,Saturn'sKiviuqandIjiraq,Uranus'sMargaret,and Neptune'sSaoandNeso.

Some sources identify the Soviet space probeLuna 3as the first example of an artificial satellite undergoing Lidov–Kozai oscillations. Launched in 1959 into a highly inclined, eccentric, geocentric orbit, it was the first mission to photograph thefar side of the Moon.It burned in the Earth's atmosphere after completing eleven revolutions.However, according to Gkoliaset al..(2016) a different mechanism must have driven the decay of the probe's orbit since the Lidov–Kozai oscillations would have been thwarted by effects of the Earth'soblateness.

Extrasolar planets

The von Zeipel-Lidov–Kozai mechanism, in combination withtidal friction,is able to produceHot Jupiters,which aregas giantexoplanets orbiting their stars on tight orbits.The high eccentricity of the planetHD 80606 bin theHD 80606/80607system is likely due to the Kozai mechanism.

Black holes

The mechanism is thought to affect the growth of centralblack holesin densestar clusters.It also drives the evolution of certain classes ofbinary black holesand may play a role in enablingblack hole mergers.

History and development

The effect was first described in 1909 by the Swedish astronomerHugo von Zeipelin his work on the motion of periodic comets inAstronomische Nachrichten.In 1961, the Soviet space scientistMikhail Lidovdiscovered the effect while analyzing the orbits of artificial and natural satellites of planets. Originally published in Russian, the result was translated into English in 1962.

Lidov first presented his work on artificial satellite orbits at theConference on General and Applied Problems of Theoretical Astronomyheld in Moscow on 20–25 November 1961.His paper was first published in a Russian-language journal in 1961.The Japanese astronomerYoshihide Kozaiwas among the 1961 conference participants.Kozai published the same result in a widely read English-language journal in 1962, using the result to analyze orbits ofasteroidsperturbed byJupiter.Since Lidov was the first to publish, many authors use the term Lidov–Kozai mechanism. Others, however, name it as the Kozai–Lidov or just the Kozai mechanism.