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In [[celestial mechanics]], the '''Kozai mechanism''', or the '''Lidov–Kozai mechanism''', refers to the orbit of a satellite that is perturbed by another body orbiting farther out. Due to the perturbation, the orbit of the satellite experiences [[libration]] (oscillation about a constant value) of its [[argument of pericenter]]. As the orbit librates, there is a | |||
periodic exchange between its [[inclination]] and its [[Orbital eccentricity|eccentricity]]. | |||
The effect was described in 1961 by the Soviet specialist in space dynamics [[Michael Lidov]] (Russian: [[:ru:Михаил Львович Лидов]]) while analyzing the orbits of artificial and natural satellites of planets,<ref> {{cite journal |last= Lidov|first= M. L. |year=1962 |title= The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies|journal= Planetary and Space Science|volume=9 |issue= |pages=719–759|doi= 10.1016/0032-0633(62)90129-0|url= |accessdate=4 April 2013|bibcode = 1962P&SS....9..719L }}</ref> | |||
and in 1962 by the Japanese astronomer [[Yoshihide Kozai]] while analyzing the orbits of the [[asteroid]]s.<ref>{{cite journal |last=Kozai |first=Y.|year=1962 |title=Secular perturbations of asteroids with high inclination and eccentricity |journal=The Astronomical Journal |volume=67 |issue= |page=591 |publisher= |doi= 10.1086/108790|url=http://adsabs.harvard.edu/cgi-bin/bib_query?1962AJ.....67..591K |accessdate=4 April 2013|bibcode = 1962AJ.....67..591K }} | |||
</ref> Since then this effect has been found to be an important factor shaping the orbits of [[irregular satellite]]s of the planets, [[trans-Neptunian object]]s, and a few [[extrasolar planets]] and [[multiple star system]]s. | |||
== Kozai mechanism == | |||
In the hierarchical, restricted [[three-body problem]], it is assumed that the satellite has negligible mass compared with the other two bodies (the "primary" and the "perturber"), and that the distance between the primary and perturber is much greater than the distance from the primary to the satellite. These assumptions would be 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. | |||
Under these approximations, the orbit-averaged equations of motion for the satellite have a [[integral of motion|conserved quantity]]: the component of the satellite's orbital angular momentum parallel to the angular momentum of the primary/perturber angular momentum. This conserved quantity can be expressed in terms of the satellite's [[Orbital eccentricity|eccentricity]] ''e'' and [[inclination]] ''i'' relative to the plane of the outer binary: | |||
:<math> L_z = \sqrt{(1-e^2)} \cos i .</math> | |||
Conservation of ''L''<sub>z</sub> 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 comet|sungrazing]]. | |||
Kozai oscillations will be present if ''L''<sub>z</sub> is lower than a certain value. At the critical value of ''L''<sub>z</sub>, a "fixed-point" orbit appears, with constant inclination given by | |||
:<math>\arccos\left(\sqrt\frac{3}{5}\right) \approx 39.2^{o}</math> | |||
For values of ''L''<sub>z</sub> less than this critical value, there is a one-parameter family of orbital solutions having the same ''L''<sub>z</sub> 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.<ref name=DEGN/> | |||
==Consequences== | |||
The Kozai mechanism causes the [[argument of pericenter]] 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 Kozai mechanism places restrictions on the orbits possible within a system, for example | |||
*for a regular moon: 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]] | |||
==Timescale== | |||
The basic timescale associated with Kozai oscillations is<ref name="DEGN">{{cite book|last=Merritt|first=David|authorlink=David Merritt|title=Dynamics and Evolution of Galactic Nuclei|year=2013|publisher=Princeton University Press|location=Princeton, NJ|page=575|url=http://openlibrary.org/works/OL16802359W/Dynamics_and_Evolution_of_Galactic_Nuclei}} | |||
</ref> | |||
:<math> | |||
T_\mathrm{Kozai} = 2\pi\frac{\sqrt{GM}}{Gm_2}\frac{a_2^3}{a^{3/2}}\left(1-e_2^2\right)^{3/2} = \frac{M}{m_2}\frac{P_2^2}{P}\left(1-e_2^2\right)^{3/2} | |||
</math> | |||
where ''a'' indicates semimajor axis, ''P'' is orbital period, ''e'' is eccentricity and ''m'' is mass; variables with subscript "2" refer to the outer (perturber) orbit and variables lacking subscripts refer to the inner (satellite) orbit; ''M'' is the mass of the primary. | |||
The period of oscillation of all three variables (''e'', ''i'', ω) is the same, but depends on how "far" the orbit is from the fixed-point orbit, becoming very long for the [[separatrix (dynamical systems)|separatrix]] orbit that separates librating (Kozai) orbits from oscillating orbits. | |||
==See also== | |||
* [[Jacobi integral]] | |||
* [[Tisserand's relation]] | |||
==References== | |||
{{reflist}} | |||
== External links == | |||
*[http://www.orbitsimulator.com/gravity/articles/kozai.html Kozai mechanism visualization] | |||
[[Category:Orbital perturbations]] | |||
Revision as of 13:27, 21 January 2014
In celestial mechanics, the Kozai mechanism, or the Lidov–Kozai mechanism, refers to the orbit of a satellite that is perturbed by another body orbiting farther out. Due to the perturbation, the orbit of the satellite experiences libration (oscillation about a constant value) of its argument of pericenter. As the orbit librates, there is a periodic exchange between its inclination and its eccentricity.
The effect was described in 1961 by the Soviet specialist in space dynamics Michael Lidov (Russian: ru:Михаил Львович Лидов) while analyzing the orbits of artificial and natural satellites of planets,[1] and in 1962 by the Japanese astronomer Yoshihide Kozai while analyzing the orbits of the asteroids.[2] Since then this effect has been found to be an important factor shaping the orbits of irregular satellites of the planets, trans-Neptunian objects, and a few extrasolar planets and multiple star systems.
Kozai mechanism
In the hierarchical, restricted three-body problem, it is assumed that the satellite has negligible mass compared with the other two bodies (the "primary" and the "perturber"), and that the distance between the primary and perturber is much greater than the distance from the primary to the satellite. These assumptions would be 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.
Under these approximations, the orbit-averaged equations of motion for the satellite have a conserved quantity: the component of the satellite's orbital angular momentum parallel to the angular momentum of the primary/perturber angular momentum. This conserved quantity can be expressed in terms of the satellite's eccentricity e and inclination i relative to the plane of the outer binary:
Conservation of Lz 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.
Kozai oscillations will be present if Lz is lower than a certain value. At the critical value of Lz, a "fixed-point" orbit appears, with constant inclination given by
For values of Lz less than this critical value, there is a one-parameter family of orbital solutions having the same Lz 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.[3]
Consequences
The Kozai mechanism causes the argument of pericenter 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 Kozai mechanism places restrictions on the orbits possible within a system, for example
- for a regular moon: 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
Timescale
The basic timescale associated with Kozai oscillations is[3]
where a indicates semimajor axis, P is orbital period, e is eccentricity and m is mass; variables with subscript "2" refer to the outer (perturber) orbit and variables lacking subscripts refer to the inner (satellite) orbit; M is the mass of the primary. The period of oscillation of all three variables (e, i, ω) is the same, but depends on how "far" the orbit is from the fixed-point orbit, becoming very long for the separatrix orbit that separates librating (Kozai) orbits from oscillating orbits.
See also
References
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Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
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