CLASSICAL MECHANICS PHYS 7221
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PHYS 7221 The ThreeBody Problem Special Lecture Wednesday October 11 2006 Juhan Frank LSU 1 The ThreeBody Problem in Astronomy The classical Newtonian threebody gravitational problem occurs in Nature exclusively in an as tronomical context and was the subject of many investigations by the best minds of the 18th and 19th centuries Interest in this problem has undergone a revival in recent decades when it was real ized that the evolution and ultimate fate of star clusters and the nuclei of active galaxies depends crucially on the interactions between stellar and black hole binaries and single stars The general threebody problem remains unsolved today but important advances and insights have been enabled by the advent of modern computational hardware and metho s The longterm stability of the orbits of the Earth and the Moon was one of the early concerns when the age of the Earth was not wellknowni Newton solved the twobody problem for the orbit of the Moon around the Earth and considered the effects of the Sun on this motion This is perhaps the earliest appearance of the threebody problemi The rst and simplest periodic exact solution to the threebody problem is the motion on collinear ellipses found by Euler 1767 Also Euler 1772 studied the motion of the Moon assuming that the Earth and the Sun orbited each other on circular orbits and that the Moon was masslessi This approach is now known as the restricted threebody problemi At about the same time Lagrange 1772 discovered the equilateral triangle solution described in Goldstein 2002 and Hestenes 1999 The collinear and equilateral triangle solutions are the only explicit solutions known for arbitrary masses and a handful of solutions for special cases are also known Montgomery 2001 i The basis for the modern theory of the restricted threebody problem was developed by Jacobi 1836 Delaunay 1860 and Hill 1878 The classical period ends with the powerful methods of surfaces of section phase space and deterministic chaos developed by Poincare who was awarded in 1889 the prize established by Sweden7s King Oscar 11 for the rst person to solve the n body problemi Although strictly speaking Poincare did not solve the general 3body problem let alone the n body problem his insights in uenced much of the work that followed 1 will review some of the known exact solutions valid for special cases sketch out a few aspects of the restricted threebody problem and conclude by discussing some numerical results and astrophysical applications 2 The General ThreeBody Problem Just as in the twobody problem it is most convenient to work in the centerofmass CM system with xi denoting the position of mass mi The Newtonian equations of motion in this system are of the form 1 where i j k stand for 1 2 3 and the two ordered permutations of these indexesi These three second order vector differential equations are equivalent to 18 rst order scalar differential equations The CM condition and its rst derivative Gkakg lxrxkl Xi Xj 7G 7 X1 m 2mm lt2 Z 3 are 6 constraints that reduce the order of the system to 12 In the absence of external forces and torques the energy and angular momentum are conserved quantities or integrals of the motion These further reduce the order of the system to 12 7 4 8 As in the twobody problem one could eliminate the time and reduce the order by one and using an analog procedure to xing the line of nodes reduce it again to 6 Even if the motion was restricted to a plane xed in space the order is reduced to 4 which is still unsolvable in general Figure 1 Position vectors in the CM system and relative position vectors for the threebody problem Hestenes 1987 In 1973 Broucke amp Lass realized that the equations of motion could be written in a more symmetrical form by using the relative position vectors Si xj 7 xk labeled in such a way that the Si is the side opposite to the vertex of the triangle containing the mass mi see Fig l and that 5152530 4 In terms of these relative position vectors the equations of motion 1 adopt the symmetrical form s si 7GMmZG 5 i where M m1 7212 m3 is the total mass and the vector G is given by 3 1 G i m m 6 H Note that the rst term on the rihis of 5 is identical to what one gets in the standard treatment of the twobody Kepler problem which admits conic sections as orbital solutions It is the second term that is responsible for the dif culty in this problem since it couples the equations for the Si 3 Euler s Solution If all particles are collinear all the vectors Si xi and G are proportional to one another Without loss of generality let7s suppose that m2 lies in between the other two massesi Then 53 points from ml to ma s1 points in the sarne direction and sense as s3 from m to me and s2 points back from mg to 7711 Therefore we can write 51 Ass 52 71 Msa 7 where A is a positive scalar Expressing everything in the equations of motion in terrns of 53 and lambda one obtains after some algebra see Hestenes 1987 for details a fth degree polynomial in A with one single positive real root which is a function of the three masses and 53 obeys a twoebody equation of the form 7712 m3lt172 GMSS 8 m2 ma1 A s 39 537 Thus the particles rnove along confocal ellipses of the sarne eccentricity ie sirnilar ellipses and the sarne orbital period around the c rnrnon center of mass always lined up and separated by distances obeying eq 7 is describes one family of solutions The other two families can be found by putting one of the other particles in the middle On a sober note these collinear solutions are not realized in nature because they are unstable to small perturbations Figure 2 Euler s collinear solution for masses in the ratio 7711 mg mg 1 2 3 Hestenes 1987 4 Lagrange s Solution This case is realized when G 0 and the equations for the sz decouple The three decoupled equations have the twoebody forrn whose solutions are ellipses for bound cases The condition for 0 is that 51 52 53 in other words the particles sit at the vertexes of an equilateral triangle at all tirnes even as this triangle changes size and rotates see Fig 3 Each particle follows an ellipse of the sarne eccentricity but oriented at diderent angles with the cornrnon center of rnass at two Montgornery 2001 describes several other solutions that exist when all the particles have the sarne rnass eg a gureeeight solution for 3 particles which is stable and even more cornplicated solutions with up to eleven l particles Although these exact solutions are fun they are of very little practical irnportance since they require very special initial conditions to be realized 5 The Pythagorean Problem Burrau 1913 considered a well de ned but arbitrarily selected initial con guration of three bodies of rnasses 3 4 and 5 placed at the corners of a Pythagorean triangle facing sides of length proportional Figure 3 Lagrange s equilateral triangle solution for masses in the ratio 7711 m2 m3 l 2 3 Hestenes 1987 to each mass The initial con guration is shown on Fig 4 The masses are at rest initially and begin to move due their mutual attraction After a very complex interaction the two heavier masses bind in a stable binary while the light object escapes and all particles recede without limit from the common center of mass Fig 5 shows the timedevelopment of the orbital paths of the three Pi1i 3 m3 P262 1 P2 1 1 quot12 4 m 5 Figure 4 lnitial con guration for Burrau s pythagorean problem shown in the OM reference frame The masses are in the ratios m1 m2 m3 3 4 5 and are released from rest Valtonen dc Karttunen 2006 particles This behavior turns out to be quite common when three particles of roughly comparable masses are allowed to interact gravitationally with randomly selected initial conditions Modern computer experiments have explored the outcomes of hundreds ofthousands olinitial con gurations and have allowed the development of a statistical understanding of the interactions between three particles of a binary with a third object and of binaries with binaries 6 The Restricted ThreeBody Problem Roche s Potential We illustrate the standard techruoues involved in the restricted threebody problem by considering the motion of gas or alternatively nonrlnteractlng particles in and around an interacting binar with a circular orbit adapted from Frank King dc Raine 2002 Noneinteracting particles feel the Figure 5 Trajectories for Burrau s pythagorean problem shown in the CM reference frame The last two panels are identical since both the binary and the light particle have left the frame Valtonen amp Karttunen 2006 external gravity of the stars but do not interact with each other The motion of gas is subject to pressure gradient forces in addition to gravity of the stars The mass of the particles or uid elements is almost always negligible compared to the masses of the binary components and thus induces negligible accelerations on the stars Under these conditions the binary components orbit the common center of mass on circular orbits of constant radius and will be at rest in a suitably chosen corotating framei Any gas ow between the two stars is governed by the Euler equation dv 8v 1 CTEvVv7V 7VP 9 where 39i39 is the gravitational potential of the stars For noninteracting particles we drop the pressure gradient and advective terms and integrate directly v following the particles motloni It is convenient to write this in a frame of reference rotating with the binary system with angular velocity w relative to an inertial frame since in the rotating frame the two stars are xed This introduces extra terms in the Euler equation to take account of centrifugal and Coriolis forcesi With the assumptions made for the Roche problem the Euler equation takes the form 6v 1 5 vVv 7V R72wAvi 7VP 10 p with the angular velocity of the binary w given in terms of a unit vector e normal to the orbital plane by G M 12 w T e a The term 72w V is the Coriolis force per unit mass 7V R includes the effects of both gravitation and centrifugal force R is known as the Roche potential Fig 6 and is given by 11 where r1 r2 are the position vectors of the centres of the two stars We gain considerable insight ng the equipotential surfaces of R and in particular their sections ber that some of the the accreting gas are not represented by Ri into accretion problems by plotti in 39 A forces in particular the Coriolis 3939mns wuwtn quotquotm 39 nu quot1 lllo t MW39lquot ma W Figure 6 A surface representing the Roche potential for a binary system with mass ratio 4 MgMl 025 the same as in Fig 7 The larger pit is around the more massive star The downward curvature near the edges is due to the centrifugal7 term a test particle attempting to corotate with the binary at these distances experiences a net outward forcei The shape of the equipotentials is governed entirely by the mass ratio 4 while the overall scale is given by the binary separation a Figure 7 is drawn for the case 4 025 but its qualitative features apply for any mass ratio Matter orbiting at large distances 7 gtgt a from the system sees it as a point mass concentrated at the centre of mass Thus the equipotentials at large distances are just those of a point mass viewed in a rotating frame Similarly there are circular equipotential sections around the centres of each of the two stars r1 r2 the motion of matter here is dominated by the gravitational pull of the nearer stari Hence the potential R has two deep valleys or pits centred on r1 rgi The most interesting and important feature of Fig 7 is the gureof eight area the heavy line which shows how these two valleys are connected In three dimensions this critical Figure 7 Sections in the orbital plane of the Roche equipotentials R constant for a binary system with mass ratio 4 MgMl 025i Shown are the center of mass CM and Lagrange points L17L5i The equipotentials are labeled 1 in order of increasing Ri Thus the saddle point L1 the inner Lagrange point forms a pass7 between the two Roche lobes7 the two parts of the gureof eight equipotential 3 The Roche lobes are roughly surfaces of revolution about the line of centers Mlngi L4 and L5 the Trojan asteroid7 points are local maxima of R but Coriolis forces stabilize synchronous orbits of test bodies at these points surface7 has a dumbbell shape the part surrounding each star is known as its Roche lobe The lobes join at the inner Lagrange point L1 which is a saddle point of R to continue the analogy L1 is like a high mountain pass between two valleys This means that material inside one of the lobes in the vicinity of L1 nds it much easier to pass through L1 into the other lobe than to escape the critical surface altogether The two offaxis Lagrange points L4 and L5 are local maxima of the effective potential and are unstable at rst sight However Coriolis forces act to con ne the orbits of particles released in the vicinity of these points A more careful treatment shows that orbits around L4 and L5 are stable provided that M1 7 M2M gt 2327 2 Hestenes 1987 or q lt 004 It also turns out that the collinear saddle points L1 L2 and L3 are capable of supporting stable quasiperiodic orbits known as halo orbits In fact several of the best known past and future NASA missions were or are planned to go on libration orbits around L1 SOHO LlSA and L2 WMAP NGST 7 Annotated References Diacu F amp Holmes P 1996 Celestial Encounters The Origins of Chaos and Stability Princeton University Press A Very readable and nonmathematical historical account of the development of the n body problem nonlinear dynamics chaos and stability including the more recent developments due to Arnold Kolmogorov Liapunov and Moser Frank 1 King AR amp Raine DJ 2002 Accretion Power in Astrophysics Cambridge University Press An introductory graduatelevel book on accretion in binaries and active galactic nuclei The restricted threebody problem forms the basis for the study of accretion ows in interacting binary stars Goldstein H Poole CP amp Safko JL 2002 Classical Mechanics 3rd ed AddisonWesley Good introductory section 312 describing some exact solutions and some aspects of the restricted threebody problem but with one major error and other minor problems Hestenes D 1987 and 1999 New Foundations for Classical Mechanics 1st and 2nd ed D Reidel LSU s library has the corrected 1st edition The overall avor of this book is fairly mathematical and should be considered more as an introduction to geometric algebra and spinors However section 5 of chapter 6 derives correctly the results described in Goldstein and can be read pro tably without familiarity with the rest of the machinery Montgomery R 2001 A New Solution to the ThreeBody Problem Not Am Math Soc 48 471481 Contains some curiosities URLs to animations and the references to the classical work of Euler in Latin and Lagrange in French Szebehely VG 1967 Theory of Orbits Academic Press The most comprehensive treatment of the restricted threebody problem in the literature Contains vastly more than you ever want to know unless you plan to become an expert in calculating space probe orbits or do research in certain areas of stellar dynamics Szebehely VG amp Mark H 1998 Adventures in Celestial Mechanics 2nd ed Wiley A very readable introductory text on orbital dynamics with historical digressions and an account of modern results on chaos strange attractors and orbital stability Valtonen M amp Karttunen H 2006 The ThreeBody Problem Cambridge University Press This book was just published and based on what I ve been able to glean from the web it looks destined to become a must read for theoretical astrophysicists working on widely diverse elds like solar system dynamics globular clusters and active galactic nuclei Mauri Valtonen is the best known theorist still pursuing the slingshot mechanism for quasars