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 law of universal gravitation.^{ [1] } The three-body problem is a special case of the n-body problem. Unlike two-body problems, no general closed-form solution exists,^{ [1] } as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.

- Mathematical description
- Restricted three-body problem
- Solutions
- General solution
- Special-case solutions
- Numerical approaches
- History
- Other problems involving three bodies
- n-body problem
- See also
- References
- Further reading
- External links

Historically, the first specific three-body problem to receive extended study was the one involving the Moon, Earth, and the Sun.^{ [2] } In an extended modern sense, a three-body problem is any problem in classical mechanics or quantum mechanics that models the motion of three particles.

In February 2024, several studies were reported which may suggest a solution.^{ [3] }^{ [4] }

The mathematical statement of the three-body problem can be given in terms of the Newtonian equations of motion for vector positions of three gravitationally interacting bodies with masses :

where is the gravitational constant.^{ [5] }^{ [6] } This is a set of nine second-order differential equations. The problem can also be stated equivalently in the Hamiltonian formalism, in which case it is described by a set of 18 first-order differential equations, one for each component of the positions and momenta :

where is the Hamiltonian:

In this case is simply the total energy of the system, gravitational plus kinetic.

This section needs additional citations for verification .(July 2023) |

In the *restricted three-body problem*, a body of negligible mass (the "planetoid") moves under the influence of two massive bodies. Having negligible mass, the force that the planetoid exerts on the two massive bodies may be neglected, and the system can be analysed and can therefore be described in terms of a two-body motion.^{ [5] }^{ [7] } With respect to a rotating reference frame, the two co-orbiting bodies are stationary, and the third can be stationary as well at the Lagrangian points, or move around them, for instance on a horseshoe orbit. It can be useful to consider the effective potential. Usually this two-body motion is taken to consist of circular orbits around the center of mass, and the planetoid is assumed to move in the plane defined by the circular orbits.

The restricted three-body problem is easier to analyze theoretically than the full problem. It is of practical interest as well since it accurately describes many real-world problems, the most important example being the Earth–Moon–Sun system. For these reasons, it has occupied an important role in the historical development of the three-body problem.

Mathematically, the problem is stated as follows.^{[ according to whom? ]} Let be the masses of the two massive bodies, with (planar) coordinates and , and let be the coordinates of the planetoid. For simplicity, choose units such that the distance between the two massive bodies, as well as the gravitational constant, are both equal to . Then, the motion of the planetoid is given by

where . In this form the equations of motion carry an explicit time dependence through the coordinates . However, this time dependence can be removed through a transformation to a rotating reference frame, which simplifies any subsequent analysis.

There is no general closed-form solution to the three-body problem,^{ [1] } meaning there is no general solution that can be expressed in terms of a finite number of standard mathematical operations. Moreover, the motion of three bodies is generally non-repeating, except in special cases.^{ [8] }

However, in 1912 the Finnish mathematician Karl Fritiof Sundman proved that there exists an analytic solution to the three-body problem in the form of a Puiseux series, specifically a power series in terms of powers of *t*^{1/3}.^{ [9] } This series converges for all real t, except for initial conditions corresponding to zero angular momentum. In practice, the latter restriction is insignificant since initial conditions with zero angular momentum are rare, having Lebesgue measure zero.

An important issue in proving this result is the fact that the radius of convergence for this series is determined by the distance to the nearest singularity. Therefore, it is necessary to study the possible singularities of the three-body problems. As is briefly discussed below, the only singularities in the three-body problem are binary collisions (collisions between two particles at an instant) and triple collisions (collisions between three particles at an instant).

Collisions, whether binary or triple (in fact, any number), are somewhat improbable, since it has been shown that they correspond to a set of initial conditions of measure zero. However, there is no criterion known to be put on the initial state in order to avoid collisions for the corresponding solution. So Sundman's strategy consisted of the following steps:

- Using an appropriate change of variables to continue analyzing the solution beyond the binary collision, in a process known as regularization.
- Proving that triple collisions only occur when the angular momentum
**L**vanishes. By restricting the initial data to**L**≠**0**, he removed all*real*singularities from the transformed equations for the three-body problem. - Showing that if
**L**≠**0**, then not only can there be no triple collision, but the system is strictly bounded away from a triple collision. This implies, by using Cauchy's existence theorem for differential equations, that there are no complex singularities in a strip (depending on the value of**L**) in the complex plane centered around the real axis (shades of Kovalevskaya). - Find a conformal transformation that maps this strip into the unit disc. For example, if
*s*=*t*^{1/3}(the new variable after the regularization) and if |ln*s*| ≤*β*,^{[ clarification needed ]}then this map is given by

This finishes the proof of Sundman's theorem.

The corresponding series, however, converges very slowly. That is, obtaining a value of meaningful precision requires so many terms that this solution is of little practical use. Indeed, in 1930, David Beloriszky calculated that if Sundman's series were to be used for astronomical observations, then the computations would involve at least 10^{8000000} terms.^{ [10] }

In 1767, Leonhard Euler found three families of periodic solutions in which the three masses are collinear at each instant. See Euler's three-body problem.

In 1772, Lagrange found a family of solutions in which the three masses form an equilateral triangle at each instant. Together with Euler's collinear solutions, these solutions form the central configurations for the three-body problem. These solutions are valid for any mass ratios, and the masses move on Keplerian ellipses. These four families are the only known solutions for which there are explicit analytic formulae. In the special case of the circular restricted three-body problem, these solutions, viewed in a frame rotating with the primaries, become points referred to as L_{1}, L_{2}, L_{3}, L_{4}, and L_{5}, and called Lagrangian points, with L_{4} and L_{5} being symmetric instances of Lagrange's solution.

In work summarized in 1892–1899, Henri Poincaré established the existence of an infinite number of periodic solutions to the restricted three-body problem, together with techniques for continuing these solutions into the general three-body problem.

In 1893, Meissel stated what is now called the Pythagorean three-body problem: three masses in the ratio 3:4:5 are placed at rest at the vertices of a 3:4:5 right triangle. Burrau^{ [11] } further investigated this problem in 1913. In 1967 Victor Szebehely and C. Frederick Peters established eventual escape for this problem using numerical integration, while at the same time finding a nearby periodic solution.^{ [12] }

In the 1970s, Michel Hénon and Roger A. Broucke each found a set of solutions that form part of the same family of solutions: the Broucke–Henon–Hadjidemetriou family. In this family the three objects all have the same mass and can exhibit both retrograde and direct forms. In some of Broucke's solutions two of the bodies follow the same path.^{ [14] }

In 1993, a zero angular momentum solution with three equal masses moving around a figure-eight shape was discovered numerically by physicist Cris Moore at the Santa Fe Institute.^{ [15] } Its formal existence was later proved in 2000 by mathematicians Alain Chenciner and Richard Montgomery.^{ [16] }^{ [17] } The solution has been shown numerically to be stable for small perturbations of the mass and orbital parameters, which makes it possible that such orbits could be observed in the physical universe. However, it has been argued that this occurrence is unlikely since the domain of stability is small. For instance, the probability of a binary–binary scattering event^{[ clarification needed ]} resulting in a figure-8 orbit has been estimated to be a small fraction of a percent.^{ [18] }

In 2013, physicists Milovan Šuvakov and Veljko Dmitrašinović at the Institute of Physics in Belgrade discovered 13 new families of solutions for the equal-mass zero-angular-momentum three-body problem.^{ [8] }^{ [14] }

In 2015, physicist Ana Hudomal discovered 14 new families of solutions for the equal-mass zero-angular-momentum three-body problem.^{ [19] }

In 2017, researchers Xiaoming Li and Shijun Liao found 669 new periodic orbits of the equal-mass zero-angular-momentum three-body problem.^{ [20] } This was followed in 2018 by an additional 1223 new solutions for a zero-angular-momentum system of unequal masses.^{ [21] }

In 2018, Li and Liao reported 234 solutions to the unequal-mass "free-fall" three-body problem.^{ [22] } The free fall formulation of the three body problem starts with all three bodies at rest. Because of this, the masses in a free-fall configuration do not orbit in a closed "loop", but travel forward and backward along an open "track".

In 2023, Ivan Hristov, Radoslava Hristova, Dmitrašinović and Kiyotaka Tanikawa published a search for "periodic free-fall orbits" three-body problem, limited to the equal-mass case, in which they found 12,409 distinct solutions.^{ [23] }

Using a computer, the problem may be solved to arbitrarily high precision using numerical integration although high precision requires a large amount of CPU time. There have been attempts of creating computer programs that numerically solve the three-body problem (and by extension, the n-body problem) involving both electromagnetic and gravitational interactions, and incorporating modern theories of physics such as special relativity.^{ [24] } In addition, using the theory of random walks, an approximate probability of different outcomes may be computed.^{ [25] }^{ [26] }

The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his * Philosophiæ Naturalis Principia Mathematica,* when Newton was trying to figure out if any long term stability is possible, especially the system of our Earth, the Moon, and the Sun. He was guided under the major Renaissance astronomers Nicolaus Copernicus, Tycho Brahe and Johannes Kepler to the beginning of the gravitational three-body problem.^{ [27] } In Proposition 66 of Book 1 of the *Principia*, and its 22 Corollaries, Newton took the first steps in the definition and study of the problem of the movements of three massive bodies subject to their mutually perturbing gravitational attractions. In Propositions 25 to 35 of Book 3, Newton also took the first steps in applying his results of Proposition 66 to the lunar theory, the motion of the Moon under the gravitational influence of Earth and the Sun.^{ [28] } Later, this problem was also applied to other planets' interactions with the Earth and the Sun.^{ [27] }

The physical problem was first addressed by Amerigo Vespucci and subsequently by Galileo Galilei, as well as Simon Stevin, but they did not realize what they contributed. Though Galileo determined that the speed of fall of all bodies changes uniformly and in the same way, he did not apply it to planetary motions.^{ [27] } Whereas in 1499, Vespucci used knowledge of the position of the Moon to determine his position in Brazil.^{ [29] } It became of technical importance in the 1720s, as an accurate solution would be applicable to navigation, specifically for the determination of longitude at sea, solved in practice by John Harrison's invention of the marine chronometer. However the accuracy of the lunar theory was low, due to the perturbing effect of the Sun and planets on the motion of the Moon around Earth.

Jean le Rond d'Alembert and Alexis Clairaut, who developed a longstanding rivalry, both attempted to analyze the problem in some degree of generality; they submitted their competing first analyses to the Académie Royale des Sciences in 1747.^{ [30] } It was in connection with their research, in Paris during the 1740s, that the name "three-body problem" (French : *Problème des trois Corps*) began to be commonly used. An account published in 1761 by Jean le Rond d'Alembert indicates that the name was first used in 1747.^{ [31] }

From the end of the 19th century to early 20th century, the approach to solve the three-body problem with the usage of short-range attractive two-body forces was developed by scientists, which offered P.F. Bedaque, H.-W. Hammer and U. van Kolck an idea to renormalize the short-range three-body problem, providing scientists a rare example of a renormalization group limit cycle at the beginning of the 21st century.^{ [32] } George William Hill worked on the restricted problem in the late 19th century with an application of motion of Venus and Mercury.^{ [33] }

At the beginning of the 20th century, Karl Sundman approached the problem mathematically and systematically by providing a function theoretical proof to the problem valid for all values of time. It was the first time scientists theoretically solved three-body problem. However, because there was not enough qualitative solution of this system, and it was too slow for scientists to practically apply it, this solution still left some issues unresolved.^{ [34] } In the 1970s, implication to three-body from two-body forces had been discovered by V. Efimov, which was named as Efimov Effect.^{ [35] }

In 2017, Shijun Liao and Xiaoming Li applied a new strategy of numerical simulation for chaotic systems called the clean numerical simulation (CNS), with the use of a national supercomputer, to successfully gain 695 families of periodic solutions of the three-body system with equal mass.^{ [36] }

In 2019, Breen et al. announced a fast neural network solver for the three-body problem, trained using a numerical integrator.^{ [37] }

In September 2023, several possible solutions have been found to the problem according to reports.^{ [38] }^{ [39] }

The term "three-body problem" is sometimes used in the more general sense to refer to any physical problem involving the interaction of three bodies.

A quantum-mechanical analogue of the gravitational three-body problem in classical mechanics is the helium atom, in which a helium nucleus and two electrons interact according to the inverse-square Coulomb interaction. Like the gravitational three-body problem, the helium atom cannot be solved exactly.^{ [40] }

In both classical and quantum mechanics, however, there exist nontrivial interaction laws besides the inverse-square force that do lead to exact analytic three-body solutions. One such model consists of a combination of harmonic attraction and a repulsive inverse-cube force.^{ [41] } This model is considered nontrivial since it is associated with a set of nonlinear differential equations containing singularities (compared with, e.g., harmonic interactions alone, which lead to an easily solved system of linear differential equations). In these two respects it is analogous to (insoluble) models having Coulomb interactions, and as a result has been suggested as a tool for intuitively understanding physical systems like the helium atom.^{ [41] }^{ [42] }

Within the point vortex model, the motion of vortices in a two-dimensional ideal fluid is described by equations of motion that contain only first-order time derivatives. I.e. in contrast to Newtonian mechanics, it is the *velocity* and not the acceleration that is determined by their relative positions. As a consequence, the three-vortex problem is still integrable,^{ [43] } while at least four vortices are required to obtain chaotic behavior.^{ [44] } One can draw parallels between the motion of a passive tracer particle in the velocity field of three vortices and the restricted three-body problem of Newtonian mechanics.^{ [45] }

The gravitational three-body problem has also been studied using general relativity. Physically, a relativistic treatment becomes necessary in systems with very strong gravitational fields, such as near the event horizon of a black hole. However, the relativistic problem is considerably more difficult than in Newtonian mechanics, and sophisticated numerical techniques are required. Even the full two-body problem (i.e. for arbitrary ratio of masses) does not have a rigorous analytic solution in general relativity.^{ [46] }

The three-body problem is a special case of the n-body problem, which describes how n objects move under one of the physical forces, such as gravity. These problems have a global analytical solution in the form of a convergent power series, as was proven by Karl F. Sundman for *n* = 3 and by Qiudong Wang for *n* > 3 (see n-body problem for details). However, the Sundman and Wang series converge so slowly that they are useless for practical purposes;^{ [47] } therefore, it is currently necessary to approximate solutions by numerical analysis in the form of numerical integration or, for some cases, classical trigonometric series approximations (see n-body simulation). Atomic systems, e.g. atoms, ions, and molecules, can be treated in terms of the quantum n-body problem. Among classical physical systems, the n-body problem usually refers to a galaxy or to a cluster of galaxies; planetary systems, such as stars, planets, and their satellites, can also be treated as n-body systems. Some applications are conveniently treated by perturbation theory, in which the system is considered as a two-body problem plus additional forces causing deviations from a hypothetical unperturbed two-body trajectory.

In physics, **angular momentum** is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity – the total angular momentum of a closed system remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Bicycles and motorcycles, flying discs, rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes form spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system, but it does not uniquely determine it.

A **centripetal force** is a force that makes a body follow a curved path. The direction of the centripetal force is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". In the theory of Newtonian mechanics, gravity provides the centripetal force causing astronomical orbits.

In astronomy, **Kepler's laws of planetary motion**, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbits and epicycles with elliptical trajectories, and explaining how planetary velocities vary. The three laws state that:

- The orbit of a planet is an ellipse with the Sun at one of the two foci.
- A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
- The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit.

In celestial mechanics, an **orbit** is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.

In physics, **equations of motion** are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system. The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics.

In Newtonian physics, **free fall** is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on it.

In physics, the **center of mass** of a distribution of mass in space is the unique point at any given time where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may be applied to cause a linear acceleration without an angular acceleration. Calculations in mechanics are often simplified when formulated with respect to the center of mass. It is a hypothetical point where the entire mass of an object may be assumed to be concentrated to visualise its motion. In other words, the center of mass is the particle equivalent of a given object for application of Newton's laws of motion.

**Newton's law of universal gravitation** says that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Separated objects attract and are attracted as if all their mass were concentrated at their centers. The publication of the law has become known as the "first great unification", as it marked the unification of the previously described phenomena of gravity on Earth with known astronomical behaviors.

In classical mechanics, the **two-body problem** is to predict the motion of two massive objects which are abstractly viewed as point particles. The problem assumes that the two objects interact only with one another; the only force affecting each object arises from the other one, and all other objects are ignored.

**Quantum chaos** is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship between quantum mechanics and classical chaos?" The correspondence principle states that classical mechanics is the classical limit of quantum mechanics, specifically in the limit as the ratio of Planck's constant to the action of the system tends to zero. If this is true, then there must be quantum mechanisms underlying classical chaos. If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, how can exponential sensitivity to initial conditions arise in classical chaos, which must be the correspondence principle limit of quantum mechanics?

In classical mechanics, the **gravitational potential** is a scalar field associating with each point in space the work per unit mass that would be needed to move an object to that point from a fixed reference point. It is analogous to the electric potential with mass playing the role of charge. The reference point, where the potential is zero, is by convention infinitely far away from any mass, resulting in a negative potential at any finite distance.

In classical mechanics, the **Laplace–Runge–Lenz (LRL) vector** is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit; equivalently, the LRL vector is said to be *conserved*. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.

In astrodynamics or celestial mechanics, an **elliptic orbit** or **elliptical orbit** is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1. In a wider sense, it is a Kepler orbit with negative energy. This includes the **radial elliptic orbit**, with eccentricity equal to 1.

In astronomy, **perturbation** is the complex motion of a massive body subjected to forces other than the gravitational attraction of a single other massive body. The other forces can include a third body, resistance, as from an atmosphere, and the off-center attraction of an oblate or otherwise misshapen body.

In physics and astronomy, **Euler's three-body problem** is to solve for the motion of a particle that is acted upon by the gravitational field of two other point masses that are fixed in space. This problem is exactly solvable, and yields an approximate solution for particles moving in the gravitational fields of prolate and oblate spheroids. This problem is named after Leonhard Euler, who discussed it in memoirs published in 1760. Important extensions and analyses were contributed subsequently by Lagrange, Liouville, Laplace, Jacobi, Darboux, Le Verrier, Velde, Hamilton, Poincaré, Birkhoff and E. T. Whittaker, among others.

The **Schrödinger–Newton equation**, sometimes referred to as the **Newton–Schrödinger** or **Schrödinger–Poisson equation**, is a nonlinear modification of the Schrödinger equation with a Newtonian gravitational potential, where the gravitational potential emerges from the treatment of the wave function as a mass density, including a term that represents interaction of a particle with its own gravitational field. The inclusion of a self-interaction term represents a fundamental alteration of quantum mechanics. It can be written either as a single integro-differential equation or as a coupled system of a Schrödinger and a Poisson equation. In the latter case it is also referred to in the plural form.

The **two-body problem in general relativity** is the determination of the motion and gravitational field of two bodies as described by the field equations of general relativity. Solving the Kepler problem is essential to calculate the bending of light by gravity and the motion of a planet orbiting its sun. Solutions are also used to describe the motion of binary stars around each other, and estimate their gradual loss of energy through gravitational radiation.

In celestial mechanics, a **Kepler orbit** is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, solar radiation pressure, a non-spherical central body, and so on. It is thus said to be a solution of a special case of the two-body problem, known as the Kepler problem. As a theory in classical mechanics, it also does not take into account the effects of general relativity. Keplerian orbits can be parametrized into six orbital elements in various ways.

In physics, the **n-body problem** is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally. Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets, and visible stars. In the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem. The n-body problem in general relativity is considerably more difficult to solve due to additional factors like time and space distortions.

In classical mechanics, the **central-force problem** is to determine the motion of a particle in a single central potential field. A central force is a force that points from the particle directly towards a fixed point in space, the center, and whose magnitude only depends on the distance of the object to the center. In a few important cases, the problem can be solved analytically, i.e., in terms of well-studied functions such as trigonometric functions.

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*G*has been set to 1, and the initial conditions are**r**_{1}(0) = -**r**_{3}(0) = (-0.97000436, 0.24308753);**r**_{2}(0) = (0,0);**v**_{1}(0) =**v**_{3}(0) = (0.4662036850, 0.4323657300);**v**_{2}(0) = (-0.93240737, -0.86473146). The values are obtained from Chenciner & Montgomery (2000). - 1 2 Šuvakov, M.; Dmitrašinović, V. "Three-body Gallery" . Retrieved 12 August 2015.
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*Master of Science Thesis at the Faculty of Physics, Belgrade University*. Retrieved 5 February 2019. - ↑ Li, Xiaoming; Liao, Shijun (December 2017). "More than six hundreds new families of Newtonian periodic planar collisionless three-body orbits".
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*New Astronomy*.**70**: 22–26. arXiv: 1805.07980 . Bibcode:2019NewA...70...22L. doi:10.1016/j.newast.2019.01.003. S2CID 89615142. - ↑ Hristov, Ivan; Hristova, Radoslava; Dmitrašinović, Veljko; Tanikawa, Kiyotaka (2023). "Three-body periodic collisionless equal-mass free-fall orbits revisited". arXiv: 2308.16159 [physics.class-ph].
- ↑ "3body simulator".
*3body simulator*. Retrieved 2022-11-17. - ↑ Technion (6 October 2021). "A Centuries-Old Physics Mystery? Solved".
*SciTechDaily*. SciTech . Retrieved 12 October 2021. - ↑ Ginat, Yonadav Barry; Perets, Hagai B. (23 July 2021). "Analytical, Statistical Approximate Solution of Dissipative and Nondissipative Binary-Single Stellar Encounters".
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