THE GRAVITATIONAL TWO-BODY PROBLEM

Transcription

1 THE GRAVITATIONAL TWO-BODY PROBLEM Danel J. Scheeres Department of Aerospace Engneerng Scences, The Unversty of Colorado, USA Keywords: Two-body problem, Kepleran soluton, dstrbuted bodes, rotatonal moton, translatonal moton. Contents. Introducton. Body and Mass Dstrbuton Specfcatons 3. Newtonan Gravtatonal Attracton 4. Equatons of Moton 5. Conservaton Prncples and Constrants 6. Constrants on Moton: Escape and Impact 7. Partcular Solutons 8. Soluton of the -Pont Mass or -Sphere Problem 9. Concluson Acknowledgements Glossary Bblography Bographcal Sketch Summary The gravtatonal two-body problem s defned and descrbed. The classcal Kepleran soluton for the moton of two pont masses s just one specalzed verson of ths problem, and n general the only one whch s completely ntegrable. Ths chapter wll provde a general defnton of the two-body problem makng no assumptons on the form of the mass dstrbutons that are mutually gravtatng. Then varous levels of approxmaton wll be ntroduced, descrbng the constrants and general results whch exst for ths problem. The general, or full, two-body problem actually couples rotatonal and translatonal moton n the general case, formng a non-ntegrable problem. Despte ths, there are strong constrants on the Hll and mpact stablty of ths problem. In addton, relatve equlbra and ther stablty can be dscussed n a general settng. The chapter culmnates n a dervaton of the Kepleran soluton for the dynamcs of two pont mass bodes orbtng each other.. Introducton The most basc problem n celestal mechancs s the gravtatonal two-body problem, specfcally the moton of two mutually attractng mass dstrbutons. The understandng of the smplest verson of ths problem occurred wth Coperncus placement of the sun at the center of the solar system and Kepler s subsequent frst descrpton of the ellptcal moton of the planets n the solar system. What Kepler dscovered based on observaton and deducton was a specal soluton to the general gravtatonal two-body problem, where the bodes n moton can be approxmated as spheres or pont masses.

2 Kepler s solutons provded the descrptve geometry of planetary moton. These results were not placed onto a frm mechancal bass untl Newton s celebrated development of the law of gravtaton, hs laws of moton, and the calculus. Taken together these advancements provded a complete development of Kepler s solutons based on physcal prncples. Beyond ths sgnfcant advance, Newton s work also enabled the complete gravtatonal two-body problem to be fully posed and analyzed. Detals of the dynamcs and solutons of the unapproxmated problem stll requred sgnfcant development and nsght, represented by many of the greats of celestal mechancs, physcs and mathematcs. Ths chapter addresses ths most fundamental problem n a general settng, placng an emphass on recent scholarshp and on developng a unfed vew of the two-body problem. Most strkng, t s noted that the full two-body problem s not solvable n most of ts statements, and can only be solved under some very restrctve approxmatons consstent wth Kepler s orgnal soluton. A complete dervaton of Kepler s soluton s gven at the close of the chapter, but frst we develop and focus on the more general statement of ths problem and detal the constrants, solutons and approxmatons that enable these problems to be understood. Ths general approach has been motvated by recent scholarshp on the general, or so-called full, two-body problem that accounts for the coupled rotatonal and translatonal moton of the two mass dstrbutons. Early work that focused on the couplng between translatonal and rotatonal moton traces back to Cassn s Laws on the moton of the moon, although n these cases the lbratonal rotaton s drven by the orbt, whle the nfluence of the rotaton on the orbt s generally neglected. Startng n the 950s, Duboshn studed the dynamcs of coupled rotatonal and translatonal moton and stated the general set of dfferental equatons and ther fundamental ntegrals of moton. In the 970s Knoshta developed perturbaton theores for these problems under the assumpton of relatvely weak couplng between the dfferent modes of moton. In the 990s research nto the generalzed verson of ths problem was explored by a number of researchers. Wang, Krshnaprasad, and Maddocks approached the problem from a geometrcal mechancs approach and explored the exstence and stablty of relatve equlbra, albet under the assumpton that one of the bodes was a sphere. Macejewsk consdered the fully general problem and explored the propertes of relatve equlbra as well as several dfferent ways to pose the problem. In the 000s a seres of artcles by Scheeres explored constrants on the solutons to the full two-body problem and appled ths problem to the dynamcs of bnary asterods and the evoluton of rubble ple asterods. Sgnfcant advances n the study of the averaged full two-body problem was made by Boué and Laskar, generalzng the classcal Cassn states and showng how they ft nto a larger set of ntegrable motons for the averaged problem. The approach taken n ths chapter s focused on sharp results that do not make strong assumptons on the moton, such as are found n averagng theores. Solutons whch yeld general and partcular solutons to the problem are gven, and constrants for the system whch act on the full soluton space are emphaszed. The goal of the current chapter s to develop a consstent and unfed approach to ths problem, relyng on classcal mechancs formulatons. Several theorems are ntroduced and developed to capture key results that hold for the general system and some specal subsystems. There are several results for the general full two-body problem that are not well known and are

3 somewhat surprsng and at odds wth the classcal Kepler soluton. Frst s that the full two-body problem s n general non-ntegrable and can exhbt chaotc moton (although we wll not establsh these results here). Ths occurs ether due to the couplng of rotatonal moton of the two bodes wth ther relatve translatonal moton or due to the non-sphercal mass dstrbutons of ether one of the bodes. Second, for the general evoluton of the two-sphere problem s that a system wth a fxed value of angular momentum can have multple crcular orbts, some of whch can be unstable. Only when the system s lmted to two pont-mass dstrbutons does t become a fully ntegrable problem. In the followng we work through a number of these dfferent results and only arrve at the ntegrable verson of the problem n the last secton.. Body and Mass Dstrbuton Specfcatons The core assumpton we make n ths chapter s that the mass dstrbutons of the two bodes are rgd, meanng that we do not account for any deformaton n ther shape or mass dstrbuton. Fgure provdes a graphcal defnton of the problem, wth the followng secton provdng a mathematcal descrpton. Fgure. The full gravtatonal -Body Problem and ts degrees of freedom.. Mass and Center of Mass Assume that there exst two rgd bodes wth dstrbuted masses, characterzed by havng fnte denstes and well defned lmts. Ther dfferental masses are defned as dm ρ dv () where s the varyng densty of the th body,,, and ρ s the poston of the mass element referenced to some frame. The densty s zero when ρ s taken outsde of the body and s fnte wthn the body, denoted as. Wth ths defnton the total mass of each body equals

4 M dm () and the locaton of each of the body s center of mass s computed as r dm M ρ (3) The jont mass dstrbuton of the system s defned va the dfferental mass element dm dm dm (4) and the jont bodes as,. Ths allows the total mass of the system to be defned as MM dm (5) The barycenter of the system s then found as R dm M M ρ (6) As wll be dscussed later, we can take.. Relatve Orentatons R 0 n general. The orentaton of the bodes s defned by rotaton dyadcs that transform a vector n the body-fxed frame nto nertal space, denoted as A and A. Then the complete specfcaton of each body n an nertal frame s r, r, A, A and the relatve poston and orentaton of the bodes wth respect to each other s ra, where the relatve poston and atttude of body relatve to body s defned as r r r (7) T A A A (8) As each of these terms represent 3 degrees of freedom, to specfy the relatve poston and orentaton of two rgd bodes requres a total of 6 degrees of freedom. These are called the nternal or relatve degrees of freedom. To orent these nternal degrees of freedom wth respect to an nertal frame requres an addtonal 3 degrees of freedom, represented by the rotaton dyadc A. Fgure shows a graphcal representaton of ths full system. We note that each of the rgd bodes has an angular velocty that defnes the nstantaneous rate of rotaton between the body-fxed frame and an nertal frame.

5 .3. Moments of Inerta An addtonal mass dstrbuton quantty of nterest s the nerta dyadcs of each body, defned by I U ρρ dm (9) where U s the dentty dyadc and the product of two vectors, e.g. ρρ, s a dyad. The coeffcents of ths dyadc can be computed n closed form for specal cases such as constant densty spheres, ellpsods, and polyhedra. The nerta dyadc of the entre system can also be specfed as I U ρρ dm (0) and smplfes nto the ndvdual nerta dyadcs and the nerta dyadc of the two masses f the system s computed relatve to the barycenter MM I U rr ˆˆ I I () r M M where the hat desgnates a vector as a unt vector. Of nterest later s the moment of nerta relatve to a fxed drecton, Ĥ, computed as I Hˆ IH ˆ () H Note that I H s a functon of the relatve poston of the two masses and ther orentaton, all relatve to the unt vector Ĥ. Also of nterest s the polar moment of nerta, computed as one-half of the trace of the total nerta IP trace I (3) MM trace r M M I I (4) Note that the polar moment of nerta for a -body system s only a functon of the separaton between the bodes, or I r, and s ndependent of the orentaton of the P two mass dstrbutons. An mportant nequalty can be proven between the polar moment of nerta and the moment of nerta relatve to a fxed drecton

6 IH I (5) P whch holds for any fxed drecton Ĥ..4. Body Shapes and Geometry In the followng we wll assume that both bodes are convex. Ths s not an essental assumpton but makes t smpler to dscuss stuatons when the two bodes can come nto contact. If they are both convex, then at every relatve confguraton of the system d r,a ˆ defned as the there s a well defned mnmum dstance between the bodes radus at whch the two bodes touch for ther relatve confguraton. Ths dstance changes smoothly wth ˆr and A and s constant for all rotatons of A about the unt vector ˆr. If non-convex bodes are assumed then there s the potental for multple dstances between the dstrbutons at a gven relatve confguraton and dscontnutes n the mnmum dstance as a functon of ˆr and A. The maxmum of these mnmum dstances can be specfed as D max d, ra ra. ˆ, ˆ Ths quantty s of fundamental nterest for any two body system as beyond ths dstance the two bodes can never mpact wth each other. Ths lmt can be defned ndependent of whether the bodes are convex or not. 3. Newtonan Gravtatonal Attracton Havng defned the two bodes and ther mass dstrbuton, we next consder the relatve forces that these dstrbutons apply to each other due to Newton s law of gravtatonal attracton. 3.. Relatve Forces Newton s fundamental law of gravtatonal attracton states that two dfferental mass elements wll experence an attracton between them proportonal to the product of the masses, nversely proportonal to the square of the dstance between them and drected along ther relatve poston. Gven our defnton of the relatve poston vector r as gong from body to body the dfferental force that a mass element n body places on a mass element n body s dm dm df r A ρ A ρ 3 r A ρ A ρ (6) where we recall that r r r and ρ s the poston of a mass element of body relatve to ts center of mass. The dfferental force of a mass element n body on a mass element n body s smply df n accordance wth Newton s law of acton and reacton. To compute the total force that body exerts on body, and vce-versa, these dfferentals are ntegrated over both mass dstrbutons:

7 F df (7) and F F. The dfferental force can also be derved from a scalar potental functon, defned as the dfferental potental gravtatonal energy between the bodes dm dm (8) r A ρ A ρ du then df df du r ρ du r ρ (9) (0) Note that du du r ρ r n general. The gravtatonal potental energy between the bodes s then defned by ntegratng ths dfferental potental over both bodes U r, r, A, A du () dm ρ dm ρ r r A ρ A ρ () where the ntegraton varables ρ are expressed n ther respectve body-fxed frames and all terms n the vector magntude are specfed n nertal space. The forces of these bodes on each other are then computed as F U r (3) F U r (4) 3.. Relatve Moments For bodes wth fnte mass dstrbutons t s also necessary to compute the mutual moments of the force (or moments) that are exerted on each other. The moment that

8 body exerts on body s found by ntegratng the dfferental moment over both bodes: dm A ρ df (5) M A ρ df (6) The moment can also be related to the mutual potental, although the detals are more nvolved. Ths s found by takng the partal of the mutual potental wth respect to the nfntesmal rotatons about each axs of body, whch we represent as M U θ. In practce, once the relatve atttude of body s defned explctly usng A the partals wth respect to the angles can be computed from ths relatonshp. The moment actng on body, M, s smlarly computed as U θ. It must be noted that the mutual moments are not equal and opposte, but that ther sum equals the negatve moment of the total gravtatonal force between the bodes M M r r F 0 (7) meanng that we do not need to solve for one of the moments f the force s known TO ACCESS ALL THE 45 PAGES OF THIS CHAPTER, Vst: Bblography A.J. Macejewsk. Reducton, relatve equlbra and potental n the two rgd bodes problem. Celestal Mechancs and Dynamcal Astronomy, 63(): 8, 995. [A comprehensve summary and dscusson of the Full -Body problem] D.J. Scheeres. Stablty n the full two-body problem. Celestal Mechancs and Dynamcal Astronomy, 83():55 69, 00. [Ths paper presents rgorous results on the stablty of moton n the Full -Body problem] D.J. Scheeres. Relatve Equlbra for General Gravty Felds n the Sphere-Restrcted Full -Body Problem. Celestal Mechancs and Dynamcal Astronomy, 94:37 349, March 006. [Ths paper presents an algorthm for computng general relatve equlbra for the sphere-restrcted Full -Body problem] D.J. Scheeres. Stablty of the planar full -body problem. Celestal Mechancs and Dynamcal Astronomy, 04():03 8, 009. [Ths paper presents detaled examples of the Full -Body problem when both bodes are non-sphercal] D.J. Scheeres. Orbtal Moton n Strongly Perturbed Envronments : Applcatons to Asterod, Comet and Planetary Satellte Orbters. Sprnger-Praxs, London (UK), 0. [Ths book contans much nformaton

9 on the computaton of gravtatonal potentals for the modelng of ther dynamcal evoluton] D.J. Scheeres. Mnmum energy confguratons n the n-body problem and the celestal mechancs of granular systems. Celestal Mechancs and Dynamcal Astronomy, 3: 9030, 0. [Treats fnte densty mnmum energy confguratons n the N -body problem] G. Boué and J. Laskar. Spn axs evoluton of two nteractng bodes. Icarus, 0(): , 009. [Ths paper presents recent results on averaged dynamcs n the Full -Body problem] G.N. Duboshn. The dfferental equatons of translatonal-rotatonal moton of mutally attractng rgd bodes. Sovet Astronomy, :39, 958. [A classcal dscusson of the couplng of rotatonal and translatonal moton] H. Knoshta. Frst-order perturbatons of the two fnte body problem. Publcatons of the Astronomcal Socety of Japan, 4:43, 97. [A classcal treatment of the Full -Body problem] H. Pollard. Celestal mechancs. The Carus Mathematcal Monographs, Provdence: Mathematcal Assocaton of Amerca, 976. [A classc text on celestal mechancs] L.S. Wang, P.S. Krshnaprasad, and JH Maddocks. Hamltonan dynamcs of a rgd body n a central gravtatonal feld. Celestal Mechancs and Dynamcal Astronomy, 50(4): , 990. [Ths paper presents a geometrc mechancs approach to the Full -Body problem] P. Pravec, D. Vokrouhlckỳ, D. Polshook, DJ Scheeres, AW Harrs, A. Galád, O. Vaduvescu, F. Pozo, A. Barr, P. Longa, et al. Formaton of asterod pars by rotatonal fsson. Nature, 466(730): , 00. [Ths paper provdes evdence of the fsson of asterods nto rgd bodes whch subsequently escape from each other] S. A. Jacobson and D. J. Scheeres. Dynamcs of rotatonally fssoned asterods: Source of observed small asterod systems. Icarus, 4:6 78, July 0. [Ths paper presents numercal experments regardng the dsrupton of Full -Body problems wth postve energy] S. Smale. Topology and mechancs.. Inventones mathematcae, ():45 64, 970. [Develops the theory behnd the amended potental] V.I. Arnold, V.V. Kozlov, and A.I. Neshtadt. Mathematcal aspects of classcal and celestal mechancs. Sprnger, 006. [Ths book provdes a clear explanaton of the applcaton of the amended potental to the N -body problem] V. Gubout and D.J. Scheeres. Stablty of Surface Moton on a Rotatng Ellpsod. Celestal Mechancs and Dynamcal Astronomy, 87:63 90, November 003. [Ths paper dscusses the stablty of surface restng ponts on a rotatng ellpsod] Bographcal Sketch Danel Jay Scheeres (born n 963 n Royal Oak, Mchgan, USA) has a Bachelor s of Scence n Letters and Engneerng from Calvn College (985), a Bachelor s and Master s of Scence n Aerospace Engneerng from The Unversty of Mchgan (987 and 988, respectvely), and a PhD. n Aerospace Engneerng from The Unversty of Mchgan (99) where he studed wth Nguyen Xuan Vnh. Snce 008 he has been the A. Rchard Seebass Endowed Char Professor n the Department of Aerospace Engneerng Scences at The Unversty of Colorado Boulder. Pror to ths he held academc postons at the Unversty of Mchgan and Iowa State Unversty. Pror to that he was a Senor Member of the Techncal Staff at the Jet Propulson Laboratory / Calforna Insttute of Technology. He s past char of the Amercan Astronomcal Socety s Dvson on Dynamcal Astronomy and the vce-presdent of the Celestal Mechancs Insttute. Hs research nterests nclude celestal mechancs of dstrbuted bodes, the mechancs of comets and asterods, the navgaton and dynamcs of spacecraft, and the long-term evoluton of orbt debrs.