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Weak stability boundary

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Weak stability boundary or a Low-energy transfer 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.[1]

Description

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.[1] 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.[1]

Background

This boundary was defined for the first time in 1987.[2] 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. Stablemotion 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 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.[2][1] 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.[1] 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 manifolds.[citation needed]

A much more general algorithm defining W was given in 2007.[3] It defines W relative to n-cycles, where 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.[4] 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 points near P2.  The explicit determination of the set W about P2 = Jupiter, where P1 is the Sun, is described in "Computation of Weak Stability Boundaries: Sun-Jupiter Case".[5] 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[6], but a different definition is used.  The chaos of the motion is analytically proven in "Geometry of Weak Stability Boundaries".[6] 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".[7]

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.[2] 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.[8] Other missions have used the same transfer type as Hiten, that include Grail, Capstone, Denuri. The WSB for Mars is studied in "Earth-Mars Transfers with Ballistic Capture"[9] and ballistic capture transfers to Mars are computed.  The 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”[10] 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 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.[11]  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 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.[12]  This gives a classical model using chaotic dynamics with Newtonian gravity for the motion of an electron.

References

  1. ^ a b c d e Belbruno, Edward (2004). Capture Dynamics and Chaotic Motions in Celestial Mechanics. Princeton University Press. ISBN 9780691094809.
  2. ^ a b c Belbruno, E (May 1987). "Lunar Capture Orbits, A method of Constructing Earth-Moon Trajectories and the Lunar GAS Mission" (PDF). no. 87-1054. Proceedings of AIAA/DGGLR/JSASS Inter. Elec. Propl. Conf.
  3. ^ Garcia, F; Gomez, G (2007). "A Note on the Weak Stability Boundary" (PDF). Celestial Mechanics and Dynamical Astronomy. 97: 87–100.
  4. ^ Belbruno, E; Gidea, M; Topputo, F (2010). "Weak Stability Boundary and Invariant Manifolds" (PDF). SIAM J. Applied Dynamical Systems. 9 (3): 1060–1089.
  5. ^ Topputo, F; Belbruno, E (2009). "Computation of Weak Stability Boundaries: Sun-Jupiter Case" (PDF). Celestial Mechanics and Dynamical Astronomy. 105: 3–17.
  6. ^ a b Belbruno, E; Gidea, M; Topputo, F (2013). "Geometry of Weak Stability Boundaries". Qualitative Theory of Dynamical Systems. 12 (3): 53–55 – via ARXIV.
  7. ^ Sousa Silva, P.A.; Terra, M.O. (2012). "Applicability and Dynamical Characterization of the Associated Sets of the Algorithmic Weak Stability Boundary in the Lunar Sphere of Influence" (PDF). Celestial Mechanics and Dynamical Astronomy. 113: 141–168.
  8. ^ Belbruno, E; Miller, J (1993). "Sun-Perturbed Earth-to-Moon Transfers with Ballistic Capture" (PDF). Journal of Guidance, Control, and Dynamics. 9 (4).
  9. ^ Topputo, F; Belbruno, E (2015). "Earth-Mars Transfers with Ballistic Capture". Celestial Mechanics and Dynamical Astronomy. 121: 329–346.
  10. ^ Belbruno, E; Moro-Martin, A; Malhotra, R; Savransky, D (2012). "Chaotic Exchange of Solid Material Between Planetary Systems: Implications for the Lithopanspermia Hypothesis". Astrobiology. 12: 1–12.
  11. ^ Belbruno, E; Marsden, B (1997). "Resonance Hopping in Comets". Astron. 113: 1433–44.
  12. ^ Belbruno, E (2020). "Relation Between Solutions of the Schrodinger Equation with Transitioning Resonance Solutions of the Gravitational Three-Body Problem". Journal of Physics Communications. 4 (015012).

Further Reading

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