IntroductionVisualizing the ModelReducing the Model to Two SpaceQuantifying the ModelSolving the One-Body ProblemSolving for Kepler's First LawSolving for Kepler's Second LawConclusionIntroductionExpanding the ProblemReducing the Model to . . .Quantifying the Two . . .Solving the One-Body . . .Solving for Kepler’s . . .Solving for Kepler’s . . .ConclusionHome PageTitle PageJJ IIJ IPage 1 of 28Go BackFull ScreenCloseQuitA Presentation of the Two-Body ProblemNoah Brisby and Robin R. RumpleMay 10, 2002AbstractThis pap er will examine the motion of two particles in space according to their mutual gravita-tional attraction.1. IntroductionThe mechanics of bodies in space has fascinated man for many centuries. The vast universe has intriguedpeople worldwide to spend infinite numbers of hours to mathematically understand the motion of thecosmos. The specific time of “1525 to 1725” (Kaufmann 46) A.D. was a monumental time frame formathematical and observational development in understanding the mysteries of space. More specifically,two people in particular changed how people look at motion of planetary bodies. Those two people areJohannes Kepler and Isaac Newton.Kepler worked under Tycho Brahe for 22 months while making meticulous measurements of planetaryorbits. This data aided in Brahe’s theory of the Earth being “stationary, with the Sun and Moonrevolving around it, while all the other planets revolve around the Sun” (Kaufmann 48). Though the“‘Tychonic system”’ (Kaufmann 48) failed widespread acceptance, Kepler used Brahe’s data of planetarypositions after he died to formulate three laws of planetary motion. As told by Kaufmann, “Kepler’sfirst law [states that] every planet travels around the Sun along an elliptical orbit with the Sun at onefocus. According to his second law, the line joining the planet and the Sun sweeps out equal areas inequal intervals of time” (Kaufmann 43). Kepler’s third law states that, “the square of a planet’s siderealperiod is proportional to the cube of the length of its orbit’s semimajor axis” (Kaufmann 43).What is so unique about Kepler’s laws is that not only are they correct, but also the laws are basedon pure observational data.IntroductionExpanding the ProblemReducing the Model to . . .Quantifying the Two . . .Solving the One-Body . . .Solving for Kepler’s . . .Solving for Kepler’s . . .ConclusionHome PageTitle PageJJ IIJ IPage 2 of 28Go BackFull ScreenCloseQuitIsaac Newton took a different approach with describing the motion of bo dies both on Earth andin space. Newton formulated three laws of motion along with a law of physics that relates to theconservation of angular momentum. These are pure mathematical formulations that when juxtaposedwith Kepler’s observational laws, create a bond that is astronomical.Newton’s laws of motion coincided beautifully with Kepler’s three laws of planetary motion. Accord-ing to Kaufmann, “using his own three laws and Kepler’s three laws, Newton succeeded in formulatinga general statement describing the nature of the force called gravity that keeps planets in their or-bits” (50). This coalescence of these mathematical and observational laws may arguably be the mostimportant scientific discovery of all time.In 1687 Newton published Philosophae naturalis principia mathematica that presented mathematicsof motion, “forces, and gravitation” (Kaufmann 49). From this text the original Two-Body Problemcan be read.The purpose of the Two-Body Problem is to describe and predict the motion of two bodies in spaceaccording to their mutual gravitational attraction. Though the specific instance of this existing in realtime is very minimal, one cannot restrict the motion of one body purely to one other body withouttaking the entire system of the universe into consideration, the calculations presented by Newton havebeen of great support to present day astronomers and scientists. Furthermore, Newton’s Two-BodyProblem can be used to develop Kepler’s three laws of planetary motion.This paper will present Newton’s Two-Body Problem with the mathematics not exceeding first leveldifferential equations. Though the level of mathematical derivation of the problem can be related withvery high-end levels of knowledge, this timeless problem is only greater than b efore due to its manyinterpretable qualities.From the derivation of the Two-Body Problem into a simplified One-Body Problem, due to thefreedom of choosing coordinate systems, the first and second of Kepler’s laws of planetary motion willbe presented.2. Visualizing the ModelThe first step with developing the solution to the Two-Body Problem is to present an illustration of thegeometry.Figure 1 shows the basic concept of the Two-Body Problem. The picture begins with an isolatedsystem containing two masses m1and m2where vectors r1and r2extend from an arbitrary origin 0 toeach mass. The vector r extends from vector m2to m1. The dimension is three to mimic a realisticscenario.IntroductionExpanding the ProblemReducing the Model to . . .Quantifying the Two . . .Solving the One-Body . . .Solving for Kepler’s . . .Solving for Kepler’s . . .ConclusionHome PageTitle PageJJ IIJ IPage 3 of 28Go BackFull ScreenCloseQuityzxr1r2rm1(x1, y1, z1)m2(x2, y2, z2)OFigure 1: A view of the two bodies in three space .IntroductionExpanding the ProblemReducing the Model to . . .Quantifying the Two . . .Solving the One-Body . . .Solving for Kepler’s . . .Solving for Kepler’s . . .ConclusionHome PageTitle PageJJ IIJ IPage 4 of 28Go BackFull ScreenCloseQuitAt this point Newton’s laws of motion will be most helpful with formulating the orbit of the twomasses as seen from the origin 0.3. Reducing the Model to Two SpaceThe dimension of the system can actually be reduced to two space with aid of the cross product.Working with the Two-Body Problem in two space will simplify calculations immensely. The first stepwith developing a two dimensional model for the Two-Body Problem is to use Newton’s Second Law ofMotionF = ma.We can further define Newton’s Second Law to beF = F (r)ˆr,where F (r) is the force function of vector r. Remember, the m otion of the two bodies will solely dependon gravitational attraction. Thus, the only force that is acting on either body is gravity. The unitvector ˆr extends in the direction of r. With the substitution of m1a1and m2a2for F we have the twoequations below. The equations arem1a1= F (r)ˆrandm2a2= −F