Hamiltonian mechanics

In Hamiltonian mechanics, a physical system is specified by a function, called Hamiltonian and denoted ${\displaystyle {\mathcal {H}}}$, of canonical coordinates in phase space. The canonical coordinates consist of the generalized coordinates ${\displaystyle x_{k}}$ in configuration space and their conjugate momenta ${\displaystyle p_{k}}$, for ${\displaystyle k=1,...N}$, for the N bodies in the system (${\displaystyle N=3}$ for the von Zeipel-Kozai–Lidov effect). The number of ${\displaystyle (x_{k},p_{k})}$ pairs required to describe a given system is the number of its degrees 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 a canonical transformation. The equations of motion for the system are obtained from the Hamiltonian through Hamilton'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

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 is enormously sensitive to any slight changes in the initial conditions, including even small uncertainties in determining the initial conditions, and rounding-errors in computer floating point arithmetic. The practical consequence is that, the three-body problem cannot be solved analytically for an indefinite amount of time, except in special cases.^[5]: 221 Instead, numerical methods are used for forecast-times limited by the available precision.^{[6]: 2, 10}

The Lidov–Kozai mechanism is a feature of hierarchical triple systems,^[7]: 86 that is systems in which one of the bodies, called the "perturber", is located far from the other two, which are said to comprise the inner binary. The perturber and the centre of mass of the inner binary comprise the outer binary.^[8]: §I Such systems are often studied by using the methods of perturbation theory to 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 term coupling the two orbits,^[8]

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

The coupling term is then expanded in the orders of parameter ${\displaystyle \alpha }$, defined as the ratio of the semi-major axes of the inner and the outer binary and hence small in a hierarchical system.^[8] Since the perturbative series converges rapidly, the qualitative behaviour of a hierarchical three-body system is determined by the initial terms in the expansion, referred to as the quadrupole (${\displaystyle \propto \alpha ^{2}}$), octupole (${\displaystyle \propto \alpha ^{3}}$) and hexadecapole (${\displaystyle \propto \alpha ^{4}}$) order terms,^[9]: 4–5

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

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.^[10]

Test particle limit

The simplest treatment of the von Zeipel-Lidov–Kozai mechanism assumes that one of the inner binary's components, the secondary, is a test particle – an idealized point-like object with negligible mass compared to the other two bodies, the primary and the distant perturber. These assumptions are valid, for instance, in the case of an artificial satellite in a low Earth orbit that is perturbed by the Moon, or a short-period comet that is perturbed by Jupiter.

The Keplerian orbital elements.

Under these approximations, the orbit-averaged equations of motion for the secondary have a conserved 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's eccentricity e and inclination i relative to the plane of the outer binary:

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

Conservation of L_z means that orbital eccentricity can be "traded for" inclination. Thus, near-circular, highly inclined orbits can become very eccentric. Since increasing eccentricity while keeping the semimajor axis constant reduces the distance between the objects at periapsis, this mechanism can cause comets (perturbed by Jupiter) to become sungrazing.

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

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

For values of L_z less than this critical value, there is a one-parameter family of orbital solutions having the same L_z but different amounts of variation in e or i. Remarkably, the degree of possible variation in i is independent of the masses involved, which only set the timescale of the oscillations.^[11]

Solar System

The von Zeipel-Lidov–Kozai mechanism causes the argument of pericenter (ω) to librate about either 90° or 270°, which is to say that its periapse occurs when the body is farthest from the equatorial plane. This effect is part of the reason that Pluto is dynamically protected from close encounters with Neptune.

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 the Hill 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.^[12]

The mechanism has been invoked in searches for Planet Nine, a hypothetical planet orbiting the Sun far beyond the orbit of Neptune.^[13]

A number of moons have been found to be in the Lidov–Kozai resonance with their planet, including Jupiter's Carpo and Euporie,^[14] Saturn's Kiviuq and Ijiraq,^[1]: 100 Uranus's Margaret,^[15] and Neptune's Sao and Neso.^[16]

Some sources identify the Soviet space probe Luna 3 as 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 the far side of the Moon. It burned in the Earth's atmosphere after completing eleven revolutions.^{[1]: 9–10} However, according to Gkolias et 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's oblateness.^[17]

Extrasolar planets

The von Zeipel-Lidov–Kozai mechanism, in combination with tidal friction, is able to produce Hot Jupiters, which are gas giant exoplanets orbiting their stars on tight orbits.^{[18][19][20][21]} The high eccentricity of the planet HD 80606 b in the HD 80606/80607 system is likely due to the Kozai mechanism.^[22]

Black holes

The mechanism is thought to affect the growth of central black holes in dense star clusters. It also drives the evolution of certain classes of binary black holes^[8] and may play a role in enabling black hole mergers.^[23]