Weak stability boundary (WSB), including

Low-energy transfer A low-energy transfer, or low-energy trajectory, is a route in space that allows spacecraft to change orbits using significantly less fuel than traditional transfers. These routes work in the Earth–Moon system and also in other systems, such as ...

, is a concept introduced by

Edward Belbruno Edward Belbruno (born August 2, 1951, as a U.S. Citizen in Heidelberg, Germany) is an artist, mathematician and scientist whose interests are in celestial mechanics, dynamical systems, dynamical astronomy, and aerospace engineering. His artistic ...

in 1987. The concept explained how a spacecraft could change orbits using very little fuel. Weak stability boundary is defined for the

three-body problem In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...

. 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. The force between the three bodies is the classical Newtonian

gravitational force In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...

. For example, P1 is the Earth, P2 is the Moon and P is a spacecraft; or P1 is the Sun, P2 is

Jupiter Jupiter is the fifth planet from the Sun and the List of Solar System objects by size, largest in the Solar System. It is a gas giant with a mass more than two and a half times that of all the other planets in the Solar System combined, but ...

and P is a comet, etc. This model is called the

restricted three-body problem In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...

. 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.''

Background

This boundary was defined for the first time by

Princeton University Princeton University is a private university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...

in 1987. He described a

which would allow a spacecraft to change orbits using very little fuel. 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 Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws o ...

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. 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. 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 In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, res ...

s. The weak stability boundary was originally referred to as the ''fuzzy boundary.'' '' ''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's Rim: Riding Chaos to the Moon" the WSB can can be roughly viewed as the fuzzy edge of a region, referred to as a

gravity well The Hill sphere of an astronomical body is the region in which it dominates the attraction of satellites. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere. That moon would, in turn, have a Hill sp ...

, 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. A much more general

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...

defining W was given in 2007. 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. 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 In celestial mechanics, the Lagrange points (; also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the influence of two massive orbiting bodies. Mathematically, this involves the solution of th ...

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". 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, but a different definition is used. The chaos of the motion is analytically proven in "Geometry of Weak Stability Boundaries". 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".

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 Ballistic capture is a low energy method for a spacecraft to achieve an orbit around a distant planet or moon with no fuel required to go into orbit. In the ideal case, the transfer is ballistic (requiring zero Delta-v) after launch. In the tradit ...

for a spacecraft. No fuel is required for capture in this case. This was numerically demonstrated in 1987. 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 Hiten may refer to: * Hiten (name), Indian given name * Hiten (spacecraft), Japanese lunar probe *Tennin , which may include , , and the specifically female version, the , are a divine kind of spiritual beings found in Japanese Buddhism, the ...

'' spacecraft. Other missions have used the same transfer type as ''

'', that include ''

Grail The Gravity Recovery and Interior Laboratory (GRAIL) was an American lunar science mission in NASA's Discovery Program which used high-quality gravitational field mapping of the Moon to determine its interior structure. The two small spacecraf ...

Capstone CAPSTONE (Cislunar Autonomous Positioning System Technology Operations and Navigation Experiment) is a lunar orbiter that will test and verify the calculated orbital stability planned for the Lunar Gateway space station. The spacecraft is a 12- ...

Danuri The Korea Pathfinder Lunar Orbiter (KPLO), officially Danuri, is South Korea's first lunar orbiter. The orbiter, its science payload and ground control infrastructure are technology demonstrators. The orbiter will also be tasked with surveying ...

''. The WSB for Mars is studied in "Earth-Mars Transfers with Ballistic Capture" and ballistic capture transfers to Mars are computed. The Bepi-Colombo mission of ESA achieved ballistic capture at the WSB of Mercury in 2022. The WSB region can be used in the field of

Astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the h ...

. It can be defined for stars within open

star cluster Star clusters are large groups of stars. Two main types of star clusters can be distinguished: globular clusters are tight groups of ten thousand to millions of old stars which are gravitationally bound, while open clusters are more loosely clust ...

s. This is done in “Chaotic Exchange of Solid Material Between Planetary Systems: Implications for the

Lithopanspermia Panspermia () is the hypothesis, first proposed in the 5th century BCE by the Greek philosopher Anaxagoras, that life exists throughout the Universe, distributed by space dust, meteoroids, asteroids, comets, and planetoids, as well as by spacecr ...

Hypothesis” 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 orbits about the Sun in

orbital resonance In celestial mechanics, orbital resonance occurs when orbiting bodies exert regular, periodic gravitational influence on each other, usually because their orbital periods are related by a ratio of small integers. Most commonly, this relationsh ...

with Jupiter, which change resonance orbits by becoming weakly captured by Jupiter. An example of such a comet is ''

39P/Oterma 39P/Oterma is a currently inactive List of periodic comets, periodic comet with an orbital period of nearly 20 years that stays outside the orbit of Jupiter. The comet nucleus, nucleus has a diameter around 4–5 km. It was last observed in Augu ...

.'' 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 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. This gives a classical model using chaotic dynamics with Newtonian gravity for the motion of an electron.

References

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Background

Applications

References

Further reading