Content deleted Content added
→Applications: added content and sources |
→Applications: added content and citations |
||
Line 11:
== 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. 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’s ''[[Hiten (spacecraft)|Hiten]]'' spacecraft.<ref>{{Cite journal |last=Belbruno |first=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}}</ref> Other missions have used the same transfer type as ''[[Hiten (spacecraft)|Hiten]]'', that include ''[[GRAIL|Grail]], [[CAPSTONE|Capstone]], Denuri''. The WSB for Mars is studied in "Earth-Mars Transfers with Ballistic Capture"<ref>{{Cite journal |last=Topputo |first=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 |pages=329-346}}</ref> and ballistic capture transfers to Mars are computed. The [[BepiColombo|Bepi-Colombo]] mission of ESA achieved ballistic at the WSB of Mercury in 2022.
The WSB region can be used in the field of Astrophysics. It can be defined for stars within open star clusters. This is done in “Chaotic Exchange of Solid Material Between Planetary Systems: Implications for the Lithopanspermia Hypothesis”<ref>{{Cite journal |last=Belbruno |first=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 |pages=1-12}}</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|Lithopanspermia Hypothesis]].
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 resonance with Jupiter, which change resonance orbits by becoming weakly captured by Jupiter.<ref>{{Cite journal |last=Belbruno |first=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=Astron |volume=113 |pages=1433-44}}</ref> An example of such a comet is ''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 in quantum mechanics to the motion of an electron about the proton in a Hydrogen 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=015012}}</ref> This gives a classical model using chaotic dynamics with Newtonian gravity for the motion of an electron.
==References==
| |||