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{{Short description|Physics algorithm}}
{{Technical|date=December 2022}}
'''Weak stability boundary''' or a [[Low-energy transfer]] is a concept introduced by [[Edward Belbruno]] in 1987. The concept explained how a spacecraft could change orbits using very little fuel. ▼
▲'''Weak stability boundary'''
Weak stability boundary is defined for the [[three-body problem]]. This problem considers the motion of a particle P of negligible mass moving with respect to two larger bodies, P1, P2, modeled as point masses, where these bodies move in circular or elliptical orbits with respect to each other, and P2 is smaller than P1.<ref name=":0"/>
The force between the three bodies is the classical Newtonian [[gravitational force]]. For example, P1 is the Earth, P2 is the Moon and P is a spacecraft; or P1 is the Sun, P2 is [[Jupiter]] and P is a comet, etc. This model is called the [[Restricted three body problem|restricted three-body problem]].<ref name=":0">{{Cite book |last=Belbruno |first=Edward |url=https://press.princeton.edu/books/hardcover/9780691094809/capture-dynamics-and-chaotic-motions-in-celestial-mechanics |title=Capture Dynamics and Chaotic Motions in Celestial Mechanics |publisher=Princeton University Press |year=2004 |isbn=9780691094809 |access-date=2022-09-01 |archive-date=2019-06-01 |archive-url=https://web.archive.org/web/20190601162732/https://press.princeton.edu/titles/7687.html |url-status=live }}</ref> The weak stability boundary defines a region about P2 where P is temporarily captured. This region is in position-velocity space. Capture means that the Kepler energy between P and P2 is negative. This is also called ''weak capture.''<ref name=":0" /> ==Background==
This boundary was defined for the first time by [[Edward Belbruno]] of [[Princeton University]] in 1987
The set of all transition points about the Moon comprises the weak stability boundary, {{mvar|W}}. The motion of {{mvar|P}} is sensitive or chaotic as it moves about the Moon within {{mvar|W}}. A mathematical proof that the motion within {{mvar|W}} is chaotic was given in 2004.<ref name=":0" /> This is accomplished by showing that the set {{mvar|W}} about an arbitrary body P2 in the restricted three-body problem contains a hyperbolic invariant set of fractional dimension consisting of the infinitely many intersections [[Hyperbolic manifold]]s.<ref name=":0" />
The weak stability boundary was originally referred to as the ''fuzzy boundary''.<ref name=":3">{{Cite magazine |last=Frank |first=Adam |date=September 1, 1994 |title=Gravity's Rim: Riding Chaos to the Moon |url=https://www.discovermagazine.com/the-sciences/gravitys-rim |magazine=Discover}}</ref><ref>{{Cite journal |last=Belbruno |first=E. |date=May–June 1992 |title=Through the Fuzzy Boundary: A New Route to the Moon |url=http://epizodyspace.ru/bibl/inostr-yazyki/the_planetary_report/1992/tpr-1992-v12n3.pdf |journal=Planetary Report |volume=7 |issue=3 |pages=8–10}}</ref> This term was used since the transition between capture and escape defined in the algorithm is not well defined and limited by the numerical accuracy. This defines a "fuzzy" location for the transition points. It is also due the inherent chaos in the motion of P near the transition points. It can be thought of as a fuzzy chaos region. As is described in an article in ''Discover'' magazine, the WSB can be roughly viewed as the fuzzy edge of a region, referred to as a [[gravity well]], about a body (the Moon), where its force of gravity becomes small enough to be dominated by force of gravity of another body (the Earth) and the motion there is chaotic.<ref name=":3" />
A much more general [[algorithm]] defining {{mvar|W}} was given in 2007.<ref>{{Cite journal |
== Applications ==
There are a number of important applications for the weak stability boundary (WSB). Since the WSB defines a region of temporary capture, it can be used, for example, to find transfer trajectories from the Earth to the Moon that arrive at the Moon within the WSB region in weak capture, which is called [[ballistic capture]] for a spacecraft.
The WSB region can be used in the field of [[Astrophysics]]. It can be defined for stars within open [[star
Numerical explorations of trajectories for P starting in the WSB region about P2 show that after the particle P escapes P2 at the end of weak capture, it moves about the primary body, P1, in a near resonant orbit, in resonance with P2 about P1. This property was used to study comets that move in
This property of change of resonance of orbits about P1 when P is weakly captured by the WSB of P2 has an interesting application
==References==
{{reflist}}
== Further
* Belbruno, E.; Green, J (2022). [https://static1.squarespace.com/static/5d1093ddd1691a0001033ebe/t/630f8fcbeb141b29d29d8222/1661964239009/DarkMatter-MNRAS-Final.pdf “When Leaving the Solar System: Dark Matter Makes a Difference”], ''Monthly Notices of the Royal Astronomical Society'', V510, 5154.
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* Osserman, J (April 2005) [https://www.msri.org/people/staff/osserman/papers/fea-osserman.pdf “Mathematics of the Heavens”], ''Notices of the American Mathematical Society'', V52, No. 4
* Ross, Shane (April 2008) [https://www.ams.org/notices/200804/200804FullIssue.pdf Book Review of ''Fly me to the Moon''], ''Notices of American Mathematical Society'', Volume 55, No. 4, 478-430
* Casselman, R (April 2008). [https://www.ams.org/notices/200804/200804FullIssue.pdf “Chaos in the Weak Stability Boundary”], Cover of ''Notices of American Mathematical Society'', p549
* Mathematics of Planet Earth [https://web.archive.org/web/20210127144414/http://mpe.dimacs.rutgers.edu/2013/05/14/low-fuel-spacecraft-trajectories-to-the-moon/ "Low Fuel Spacecraft Trajectories to the Moon"]
[[Category:Physics theorems]]
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