Weak stability boundary: Difference between revisions

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" />

This boundary was defined for the first time by [[Edward Belbruno]] of [[Princeton University]] in 1987.<ref name=":1">{{Cite webconference |last=Belbruno |first=E. |date=May 1987 |title=Lunar Capture Orbits, A method of Constructing Earth-Moon Trajectories and the Lunar GAS Mission |urlchapter=https://static1.squarespace.com/static/5d1093ddd1691a0001033ebe/t/5d9db6d1cc827325b39149a5/1570617085139/Belbruno_Lunar_Capture_Orbits.pdfLunar |series=no. 87-1054 |publisher=Proceedings of AIAA/DGGLR/JSASS Inter. Elec. Propl. Conf. |access-date=2022-09-01 |archive-date=2022-08-01 |archive-url=https://web.archive.org/web/20220801122518/https://static1.squarespace.com/static/5d1093ddd1691a0001033ebe/t/5d9db6d1cc827325b39149a5/1570617085139/Belbruno_Lunar_Capture_Orbits.pdf |url-status=live }}</ref> He described a [[Low-energy transfer]] which would allow a spacecraft to changecapture orbits using very little fuel.<ref name=":1">{{Cite web |last=Belbruno |first=E |date=May 1987 |title=Lunar Capture Orbits, Aa method of Constructingconstructing Earth-Moonearth Trajectoriesmoon trajectories and the Lunarlunar GAS Missionmission |url=https://static1.squarespace.com/static/5d1093ddd1691a0001033ebe/t/5d9db6d1cc827325b39149a5/1570617085139/Belbruno_Lunar_Capture_Orbits.pdf |series=no. 87-1054 |publisher=Proceedings of the 19th AIAA/DGGLR/JSASS Inter.International Elec.Electric Propl.Propulsion Conf.Conference |access-date=20222023-09-0108 |archive-date=2022-08-01 |archive-url=https://web.archive.org/web/20220801122518/https://static1.squarespace.com/static/5d1093ddd1691a0001033ebe/t/5d9db6d1cc827325b39149a5/1570617085139/Belbruno_Lunar_Capture_Orbits.pdf |url-status=livedead |doi=10.2514/6.1987-1054 }}</ref><ref name="Rutgers">{{citeHe webdescribed |title=a [[Low-energy Fueltransfer]] Spacecraftwhich Trajectorieswould toallow thea Moonspacecraft |url=http://mpe.dimacs.rutgers.edu/2013/05/14/low-fuel-spacecraft-trajectories-to-the-moon/ |website=Mathematicschange oforbits Planetusing Earthvery |publisher=Rutgerslittle |access-date=1 September 2022 |archive-date=27 January 2021 |archive-url=https://webfuel.archive.org/web/20210127144414/http://mpe.dimacs.rutgers.edu/2013/05/14/low-fuel-spacecraft-trajectories-to-the-moon/ |url-status=live }}</ref> It was for motion about Moon (P2) with P1 = Earth. It is defined [[algorithmically]] by monitoring cycling motion of P about the Moon and finding the region where cycling motion transitions between stable and unstable after one cycle. ''Stable motion'' means P can completely cycle about the Moon for one cycle relative to a reference section, starting in weak capture. P needs to return to the reference section with negative [[Kepler's laws of planetary motion|Kepler]] energy. Otherwise, the motion is called ''unstable'', where P does not return to the reference section within one cycle or if it returns, it has non-negative Kepler energy.<ref name=":1" /><ref name=":0" /> The set of all transition points about the Moon comprises the weak stability boundary, W. The motion of P is sensitive or chaotic as it moves about the Moon within W. A mathematical proof that the motion within W is chaotic was given in 2004.<ref name=":0" /> This is accomplished by showing that the set 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 webmagazine |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 |websitemagazine=Discover}}</ref><ref>{{Cite journal |last=Belbruno |first=E. |date=May-JuneMay–June 1992 |title=“ThroughThrough the Fuzzy Boundary: A New Route to the Moon”,Moon |url=http://epizodyspace.ru/bibl/inostr-yazyki/the_planetary_report/1992/tpr-1992-v12n3.pdf |journal=Planetary Report |volume=7 |issue=3 |pages=8-108–10}}</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 "Gravity'san Rim:article Ridingin Chaos''Discover'' to the Moon"magazine, the WSB can 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 |last1=Garcia |first1=F. |last2=Gomez |first2=G. |date=2007 |title=A Note on the Weak Stability Boundary |url=https://static1.squarespace.com/static/5d1093ddd1691a0001033ebe/t/630f8ec5bb0d060a78f239e1/1661963974782/WSBGomez.pdf |journal=[[Celestial Mechanics and Dynamical Astronomy]] |volume=97 |pages=87–100 |doi=10.1007/s10569-006-9053-6 |s2cid=16767342 |access-date=2022-09-01 |archive-date=2022-09-01 |archive-url=https://web.archive.org/web/20220901232930/https://static1.squarespace.com/static/5d1093ddd1691a0001033ebe/t/630f8ec5bb0d060a78f239e1/1661963974782/WSBGomez.pdf |url-status=livedead }}</ref> It defines {{mvar|W}} relative to {{mvar|n}}-cycles, where {{mvar|n}} = 1,2,3,…..., yielding boundaries of order n. This gives a much more complex region consisting of the union of all the weak stability boundaries of order n. This definition was explored further in 2010.<ref>{{Cite journal |last1=Belbruno |first1=E. |last2=Gidea |first2=M. |last3=Topputo |first3=F. |date=2010 |title=Weak Stability Boundary and Invariant Manifolds |url=https://static1.squarespace.com/static/5d1093ddd1691a0001033ebe/t/630f8f532c0b0b596bb8b412/1661964121374/WSBJournalFinalBGT.pdf |journal=SIAM J.Journal on Applied Dynamical Systems |volume=9 |issue=3 |pages=1060–1089 |doi=10.1137/090780638 |access-date=2022-09-01 |archive-date=2022-09-01 |archive-url=https://web.archive.org/web/20220901232930/https://static1.squarespace.com/static/5d1093ddd1691a0001033ebe/t/630f8f532c0b0b596bb8b412/1661964121374/WSBJournalFinalBGT.pdf |url-status=livedead }}</ref> The results suggested that W consists, in part, of the hyperbolic network of invariant manifolds associated to the Lyapunov orbits about the L1, L2 [[Lagrange point|Lagrange points]]s near P2.  The explicit determination of the set {{mvar|W}} about P2 = Jupiter, where P1 is the Sun, is described in "Computation of Weak Stability Boundaries: Sun-Jupiter Case".<ref>{{Cite journal |last1=Topputo |first1=F. |last2=Belbruno |first2=E. |date=2009 |title=Computation of Weak Stability Boundaries: Sun-Jupiter Case |url=https://home.aero.polimi.it/topputo/data/uploads/papers/articles/article-2009-1.pdf |journal=[[Celestial Mechanics and Dynamical Astronomy]] |volume=105 |pages=3–17 |doi=10.1007/s10569-009-9222-5 |s2cid=121915109 |access-date=2022-09-01 |archive-date=2022-09-01 |archive-url=https://web.archive.org/web/20220901232933/https://home.aero.polimi.it/topputo/data/uploads/papers/articles/article-2009-1.pdf |url-status=live }}</ref> It turns out that a weak stability region can also be defined relative to the larger mass point, P1. A proof of the existence of the weak stability boundary about P1 was given in 2012,<ref name=":2">{{Cite journal |last1=Belbruno |first1=E. |last2=Gidea |first2=M. |last3=Topputo |first3=F. |date=2013 |title=Geometry of Weak Stability Boundaries |url=https://arxiv.org/abs/1204.1502 |journal=Qualitative Theory of Dynamical Systems |volume=12 |issue=3 |pages=53–55 |doi=10.1007/s12346-012-0069-x |arxiv=1204.1502 |s2cid=16086395 |via=ARXIV |access-date=2022-09-01 |archive-date=2022-01-28 |archive-url=https://web.archive.org/web/20220128060336/https://arxiv.org/abs/1204.1502 |url-status=live }}</ref>, but a different definition is used.  The chaos of the motion is analytically proven in "Geometry of Weak Stability Boundaries".<ref name=":2" /> The boundary is studied in "Applicability and Dynamical Characterization of the Associated Sets of the Algorithmic Weak Stability Boundary in the Lunar Sphere of Influence".<ref>{{Cite journal |last1=Sousa Silva |first1=P. A. |last2=Terra |first2=M. O. |date=2012 |title=Applicability and Dynamical Characterization of the Associated Sets of the Algorithmic Weak Stability Boundary in the Lunar Sphere of Influence |url=https://static1.squarespace.com/static/5d1093ddd1691a0001033ebe/t/630f8f9388d5cf482a519f1f/1661964183382/WSB+PaperSousaSilvaTerra2012.pdf |journal=[[Celestial Mechanics and Dynamical Astronomy]] |volume=113 |issue=2 |pages=141–168 |doi=10.1007/s10569-012-9409-z |bibcode=2012CeMDA.113..141S |s2cid=121436433 |access-date=2022-09-01 |archive-date=2022-09-01 |archive-url=https://web.archive.org/web/20220901232931/https://static1.squarespace.com/static/5d1093ddd1691a0001033ebe/t/630f8f9388d5cf482a519f1f/1661964183382/WSB+PaperSousaSilvaTerra2012.pdf |url-status=livedead }}</ref>

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.   No fuel is required for capture in this case.  This was numerically demonstrated in 1987.<ref name=":1" /> This is the first reference for ballistic capture for spacecraft and definition of the weak stability boundary.   The boundary was operationally demonstrated to exist in 1991 when it was used to find a ballistic capture transfer to the Moon for Japan’sJapan's ''[[Hiten (spacecraft)|Hiten]]'' spacecraft.<ref>{{Cite journal |last1=Belbruno |first1=E. |last2=Miller |first2=J. |date=1993 |title=Sun-Perturbed Earth-to-Moon Transfers with Ballistic Capture |url=https://static1.squarespace.com/static/5d1093ddd1691a0001033ebe/t/630e0687f121f13929cbad40/1661863561255/Paper-EB-JM-JGCD-1993.pdf |journal=[[Journal of Guidance, Control, and Dynamics]] |volume=9 |issue=4 |page=770 |doi=10.2514/3.21079 |bibcode=1993JGCD...16..770B |access-date=2022-09-01 |archive-date=2022-09-01 |archive-url=https://web.archive.org/web/20220901232931/https://static1.squarespace.com/static/5d1093ddd1691a0001033ebe/t/630e0687f121f13929cbad40/1661863561255/Paper-EB-JM-JGCD-1993.pdf |url-status=livedead }}</ref> Other missions have used the same transfer type as ''[[Hiten (spacecraft)|Hiten]]'', that includeincluding ''[[GRAIL|Grail]]'', ''[[CAPSTONE|Capstone]]'', Denuri''[[Danuri]]'', ''[[Hakuto-R Mission 1]]'' and ''[[Smart Lander for Investigating Moon|SLIM]]''. The WSB for Mars is studied in "Earth-Mars Transfers with Ballistic Capture"<ref>{{Cite journal |last1=Topputo |first1=F. |last2=Belbruno |first2=E. |date=2015 |title=Earth-Mars Transfers with Ballistic Capture |url=https://arxiv.org/abs/1410.8856 |journal=[[Celestial Mechanics and Dynamical Astronomy]] |volume=121 |issue=4 |pages=329–346 |doi=10.1007/s10569-015-9605-8 |arxiv=1410.8856 |bibcode=2015CeMDA.121..329T |s2cid=119259095 |access-date=2022-09-01 |archive-date=2022-08-25 |archive-url=https://web.archive.org/web/20220825004947/https://arxiv.org/abs/1410.8856 |url-status=live }}</ref> and ballistic capture transfers to Mars are computed.  The [[BepiColombo|Bepi-Colombo]] mission of ESA achievedwill achieve ballistic capture at the WSB of Mercury in 20222025.

The WSB region can be used in the field of [[Astrophysics]]. It can be defined for stars within open [[star cluster]]s. This is done in “Chaotic"Chaotic Exchange of Solid Material Between Planetary Systems: Implications for the [[Lithopanspermia]] Hypothesis”Hypothesis"<ref>{{Cite journal |last1=Belbruno |first1=E. |last2=Moro-Martin |first2=A. |last3=Malhotra |first3=R. |last4=Savransky |first4=D. |date=2012 |title=Chaotic Exchange of Solid Material Between Planetary Systems: Implications for the Lithopanspermia Hypothesis |url=https://arxiv.org/abs/1205.1059 |journal=Astrobiology |volume=12 |issue=8 |pages=754–774 |doi=10.1089/ast.2012.0825 |pmid=22897115 |pmc=3440031 |arxiv=1205.1059 |access-date=2022-09-01 |archive-date=2022-05-03 |archive-url=https://web.archive.org/web/20220503162119/https://arxiv.org/abs/1205.1059 |url-status=live }}</ref> to analyze the capture of solid material that may have arrived on the Earth early in the age of the Solar System to study the validity of the [[Panspermia|Lithopanspermialithopanspermia Hypothesishypothesis]].

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 resonance orbits about the Sun in [[orbital resonance]] with Jupiter, which change resonance orbits by becoming weakly captured by Jupiter.<ref>{{Cite journal |last1=Belbruno |first1=E. |last2=Marsden |first2=B. |date=1997 |title=Resonance Hopping in Comets |url=https://ui.adsabs.harvard.edu/link_gateway/1997AJ....113.1433B/ADS_PDF |journal=AstronThe Astronomical Journal |volume=113 |pages=1433–44 |doi=10.1086/118359 |bibcode=1997AJ....113.1433B |access-date=2022-09-01 |archive-date=2022-09-01 |archive-url=https://web.archive.org/web/20220901232933/https://articles.adsabs.harvard.edu/pdf/1997AJ....113.1433B |url-status=live }}</ref>  An example of such a comet is ''[[39P/Oterma]].''

This property of change of resonance of orbits about P1 when P is weakly captured by the WSB of P2 has an interesting application to the field of Quantumquantum Mechanicsmechanics to the motion of an electron about the proton in a Hydrogenhydrogen atom. The transition motion of an electron about the proton between different energy states described by the [[Schrödinger equation|Schrodinger equation]] is shown to be equivalent to the change of resonance of P about P1 via weak capture by P2 for a family of transitioning resonance orbits.<ref>{{Cite journal |last=Belbruno |first=E. |date=2020 |title=Relation Between Solutions of the Schrodinger Equation with Transitioning Resonance Solutions of the Gravitational Three-Body Problem |url=https://iopscience.iop.org/article/10.1088/2399-6528/ab693f |journal=Journal of Physics Communications |volume=4 |issue=15012 |page=015012 |doi=10.1088/2399-6528/ab693f |arxiv=1905.06705 |bibcode=2020JPhCo...4a5012B |s2cid=211076278 |access-date=2022-09-01 |archive-date=2020-02-16 |archive-url=https://web.archive.org/web/20200216213938/https://iopscience.iop.org/article/10.1088/2399-6528/ab693f |url-status=live }}</ref>  This gives a classical model using chaotic dynamics with Newtonian gravity for the motion of an electron.

* 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