Simulations of the Three-Body Problem Finding Chaos in the Cosmos Craig Huber and Leah Zimmerman San Francisco State University ABSTRACT We studied the dynamical effects of three bodies interacting via Newton’s law of gravity This project attempted to map the dynamics of three-body gravitational interaction Our results clearly show the potential planets in our galaxy have of near chaotic behavior BACKGROUND/INTRODUCTION In the 2nd century C.E., Ptolemy, of the early Greeks, developed a geocentric scheme for the solar system A geocentric scheme calls for each of the heavenly bodies including the Sun, Moon, and other planets to orbit around a stationary common center, that being the Earth Ptolemy had to use the concept of epicycles in order to, as accurately as possible, describe the motion that he saw of the planets An epicycle is a circle whose center moves around the circumference of another circle The ancient Greeks could not discount the idea that planets must move in “perfect” circles Later, Nicolaus Copernicus proposed a heliocentric scheme in which all the planets orbit the sun Tycho Brahe was the last great astronomer to observe the motion of the planets without the use of a telescope and documented the paths of the planets with amazing precision Johannes Kepler interpreted these findings and showed that they were consistent with orbits in the shape of ellipses instead of circles He wrote three laws summarizing his observations: All planets move in elliptical orbits with the Sun at one focus A line joining any planet to the Sun sweeps out equal areas in equal times The square of the period of any planet about the Sun is proportional to the cube of the planet’s mean distance from the Sun Kepler’s laws, though accurate, are entirely empirical However, Sir Isaac Newton then engraved Kepler’s Laws of planetary motion in stone when he derived them from his new idea of universal gravitation In doing so, Newton unified the previously separate sciences of celestial mechanics and terrestrial mechanics Newton declared that any particle is attracted to any other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them Now, mathemeticians and physicists could predict the motion of orbiting planets But there was a problem Note that the N-body problem in n-dimensions has (n2+3n+2)/2 classical integrals of motion: n from the total momentum, n from the position of the center of mass, and n(n-1)/2 from the total angular momentum, and finally, one from the energy Because 2nN-1 integrals of motion were necessary to integrate the N-body problem, there are in general not enough classical integrals if N>2 Already the Newtonian three-body problem in n=2 contains all the complexity a dynamical system can have In this case, the three-body problem in two-dimensions has six classical integrals of motion while eleven integrals of motion are needed to integrate the problem The problem of three bodies in orbit around each other has haunted mathematicians and physicists alike since Newton derived Kepler’s laws and produced equations of motion for one planet orbiting the sun With ten coupled differential equations to solve and only eleven integrals of motion, the problem is not integrable However, with new mathematical tools, experimentalists can model approximate solutions to the three-body problem PSEUDOCODE Instead of attempting to solve the three-body problem analytically we wrote a program in Matlab to simulate the orbit of three masses under mutual gravitational potential attraction We assume that on a small timescale (i.e one-hundredth of a year for simulations of planets) the motion of a given mass is linear We also assume that the relative motion of the bodies is coplanar Each of three mass-elements is given initial conditions for mass, velocity and position in two Cartesian coordinates: mi, vxi, vyi, xi, yi The acceleration of each mass is derived from Newton’s gravitational force law in two dimensions, Gm m Gm1 m2 F 12 , r ( x1 x ) ( y1 y ) where G is Newton’s gravitational constant m3 G 6.726 10 11 kg s The acceleration of mass i due to masses (in parentheses) j and k is thus, Gm j j xi x j y i y j k Gmk xi xk yi y k with the total acceleration of mass i being the sum of ai(j) + ai(k) The total acceleration is then decomposed into its Cartesian coordinates xi xi a xi ai j k 1/ 2 1/ 2 xi x j y i y j xi x k y i y k a yi ai j yi x x y y 1/ 2 i j i j k x x i k yi yi yk 1/ The new position of each mass is calculated using the usual kinematic equations of motion First, the velocity in both the x- and y-directions are calculated from the accelerations above v xi v o a xi t v yi v o a yi t where vo is the initial velocity (or the velocity one iteration before) And the coordinates of mass i are thus, xi xo v xi t a xi t 2 y i y o v yi t a yi t 2 where xo and yo are the initial positions (or the positions one iteration prior) of mass i The new positions are calculated for a discrete amount of time, t, usually onehundredth of a year Depending on the system, the masses were allowed to iterate for anywhere between one and 24 years for a total of between 500 to 2,400 iterations The Earth, Jupiter, Sun system was allowed to iterate for a total of 24 years since the period of Jupiter is normally 12 years On the other hand, the Earth, Moon, Sun system was allowed to iterate for as little as one year Note that all of the above equations are valid for one particular mass The indices relating to which mass the variable refers need just be cycles through each of the three bodies to get equations for the other masses The sun was given coordinates at the origin since its motion due to gravitational effects of the earth and Jupiter is negligible after only several years In fact, all calculations are done in the center of mass frame for ease of viewing RESULTS We first attempted to generate plots of existing stable orbits of planets within our own solar system Using figures from various astronomical data tables we compiled our program choosing various real masses to orbit each other Name Sun Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Halley (comet) Semimajor axis (au) -0.3871 0.7233 1.0000 1.5237 5.2028 9.5388 19.191 30.061 39.529 18 Period (yr) -0.2408 0.6152 1.0000 1.8809 ~12 29.456 84.07 164.81 248.53 76 Mass (earth mass) 333,480 0.0553 0.8150 1.0000 0.1074 317.89 ?? 14.56 17.15 0.002 ~10-10 Taking the Earth, Moon, and Sun as a starting point and then trying the Earth, Venus, and the Sun, we found that our program accurately displays three bodies orbiting each other under mutual gravitational attraction Fig (left) Earth (green) and the Moon (red) orbiting the Sun (blue) (right) Earth (green) and Venus (red) orbiting the Sun (blue) After one year, and iterations every one-hundredth of a year, plots show the Earth-MoonSun and the Earth-Venus-Sun systems in stable orbits, as they should be Changing the input parameters slightly, we were able to generate plots of the Earth-Jupiter-Sun system This system showed a greater tendency toward chaos, with only sometimes slight deviations from “normal” initial conditions leading to non-stable orbits Fig.2 The Earth (green), Jupiter (red), and Sun (blue) system with a period of twelve years By moving Jupiter’s initial position much closer to the Sun and Earth made obvious perturbations in the Earth’s orbit about the Sun In fact, with Jupiter nearly as close to the Sun as the Earth is (1.1 astronomical units instead of 5.208 AU), showed evident signs of chaos, especially sensitive dependence on initial conditions Fig.3 (left) Jupiter with an initial distance of 1.1 AU from the Sun (right) Jupiter has an initial distance of 1.105 AU from the Sun This is a change of less than 0.5% Clearly, Jupiter shows sensitive dependence on initial conditions As Jupiter’s initial distance from the Sun changes by less than half a percent (from 1.1 AU to 1.105 AU, its position twelve years later is vastly different Additionally, if Jupiter were much more massive, its perturbative effects on the Earth’s orbit become more prominent Fig (left) Jupiter’s mass is 100 times its normal size The Earth shows a stable, periodic, toroidal orbit (right) Jupiter’s mass is now 1,000 times its real size In this case, the Earth’s orbit is clearly unstable In the first instance, with Jupiter’s mass 100 times its normal size (31,700 Earth masses instead of 317 Earth masses), the Earth’s orbit takes on a toroidal shape The system is stable after twelve years And we see the Sun beginning to rotate about the center of mass of the system However if Jupiter’s mass is taken to be 1,000 times its real size, the Earth’s orbit becomes erratic The Sun (333,480 Earth masses) and Jupiter are now clearly rotating around their mutual center of mass, since the mass of the Earth is nearly negligible in this case Now, the Sun and Jupiter act more like a binary star system instead of a planet orbiting a sun This is because Jupiter’s mass is now comparable to that of the Sun Indeed, if the Earth’s initial position is skewed slightly, its orbit displays sensitive dependence on initial conditions Fig Finding chaos in scenarios not necessarily derived found in nature CONCLUSION By running simulations instead of attempting to solve the three-body problem analytically, we found definite occasions of chaotic behavior in real examples For instance the Earth, Jupiter, Sun system showed sensitive dependence on initial conditions when we varied Jupiter’s mass and initial location Movies and phase plots of our results can be found at the URL http://www.physics.sfsu.edu/~lzimmer/math490.html, as well as our powerpoint presentation REFERENCES Diacu, Florin and Holmes, Philip Celestial Encounters: The Origins of Chaos and Stability Princeton University Press, Princeton, New Jersey 1996 Halliday, David, Kenneth Krane, and Robert Resnick Physics vol I, 4th Ed John Wiley & Sons, Inc., New York 1992 Leimanis, E and Minorsky, N Dynamics and Nonlinear Mechanics John Wiley & Sons, Inc., New York 1958 Marion, Jerry and Stephen Thornton Classical Dynamics of Particles and Systems, 3rd Ed Harcourt Brace & Company, Fort Worth 1995 Peterson, Ivars Newton’s Clock: Chaos in the Solar System W H Freeman and Company, New York 1993 ... dependence on initial conditions Fig Finding chaos in scenarios not necessarily derived found in nature CONCLUSION By running simulations instead of attempting to solve the three-body problem analytically,... without the use of a telescope and documented the paths of the planets with amazing precision Johannes Kepler interpreted these findings and showed that they were consistent with orbits in the shape... has six classical integrals of motion while eleven integrals of motion are needed to integrate the problem The problem of three bodies in orbit around each other has haunted mathematicians and