SOLAR SYSTEM STABILITY EXPLAINED UNDER THE N-BODY PROBLEM SOLUTION

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1 SOLAR SYSTEM STABILITY EXPLAINED UNDER THE N-BODY PROBLEM SOLUTION Jorg A Franco R E-ail: gorgafr@gailco Abstract: Th priction of th ovnt of a group of N gravitationally attracting bois aroun its cntr of ass CM, givn thir initial positions an vlocitis, was succssfully unrtakn in prvious paprs In orr to sktch thos lliptic an consir stabl otions, obtain quations of orbits wr put in function of thir si-ajor axis This prsntation siplifi th gotric consiration of initial conitions This work is th continuation of th prvious rprsntation of gravitational bois paths, but now applying th nst two-boy typ configurations as on of th possibl arrangnts allow by Nwton Equations unr th optics of th solution of th N-Boy Probl prsnt in prvious paprs onstrating an xplanation of th stability of th Solar Syst Kywors: Gravitational Two-Boy probl, Thr-Boy probl, N-Boy probl, Solar Syst INTRODUCTION Although Nwton Equations allow th analysis of any typ of path, in this work w consir only th stabl cass of lliptical orbits, as thos of th Plantary Solar Syst In th cas of th two-boy probl, if on of such asss is vry larg in coparison with th scon on, th iprssion is that th largr is fix an th othr ovs aroun this fix on as th only oving on It is known how to calculat th trajctory of a boy of ass that ovs aroun anothr of ass M consir fix (th Nwton s on-boy probl), pning on th knowlg of th raius r 0 an its sp v 0, at its closst approaching rgaring th as initial conitions For calculating th trajctory of a plant it os assu ovabl an Sun as th fix boy, both constituting an isolat syst (this is possibl by coinciing th origin at Sun) Naly, th influnc of othr bois of th solar syst is nglct Also it is assu that bois ar prfct sphrical solis Still, thortical rsults prict by this configuration ar practically intical to thos asur Upon th knowlg of th solution givn by Nwton in 687 of th on-boy probl an that coing fro that of th two-boy probl givn by Johan Brnoulli in 70, lt s ak th xrcis by oing first th on hunr tis largr than, rprsnting th unr th alray us Mathatica 7 softwar, an latr four hunr tis largr than, for stablishing iffrncs aong th For xapl, Sun ass is tis Earth ass Th solution of th Nwton s on-boy probl was cntral bcaus it constitut th founation for solving th two-boy probl which poss trining th otion of two bois of asss an that gravitationally intract btwn ach othr givn crtain initial conitions, siilar to thos prviously rfrr to th on-boy probl, but now rlativ to th cntr of ass of both bois, S (fix bcaus coorinat s origin is bing consir coinciing with this point) As it was asily onstrat in [], bas only upon Nwton s quations, that th two-boy probl of asss an oving aroun S can b ruc to that of th on-boy forulation in whr a

2 ass, or ruc ass / ( + ), ovs aroun a ass = +, at a istanc r = r + r, with an attraction forc F = Gr By taking th cntr of ass position coinciing with th origin of th fra of rfrnc for asurnts, calculations ar siplifi an it allows consiring th CM fix Th orbit xprssions prsnt as raiuss quations wr foun to b proportional to that of th on-boy raius Also, it was fin a nw an vry iportant concpt in a siilar way to that of th ruc ass: th ruc raius SOLAR SYSTEM Whn forcs on bois ar not qual as in th cas of plantary Solar Syst is that thy ar not singl bois but a suprposition of two-boy otions constituting a nst of two-boy probls with iffrnt angular sps, which iplis that for xplaining such configuration it is ncssary to invstigat or to know th history of whn an how thy start bing part of th Solar syst, th volution of its propr gravitational instabilitis an othr uniaginabl circustancs happn until now aroun this procss For xapl, which was th first plant to bco attract by th Sun?, th scon?, or thy co out of a collision of galaxis, or? Howvr for instanc, a probabl approach to a nst of two-boy-typ configurations xplaining th cas of th solar plantary syst coul b th following on: Lt s suppos Sun an Mrcury asss oving at aroun thir sub-cntr of ass S, which by rucing this two-boy probl to th on-boy forulation has a si-ajor axis a Thn, by taking th su of asss of Sun an Mrcury,, as if thy wr locat at S oving at (NnC) aroun S, togthr with Vnus, which by rucing such two-boy probl for by at S an to th on-boy forulation has a si-ajor axis a Thn, taking th su of asss of Sun, Mrcury an Vnus,, as if thy wr locat at S an this ass, togthr with Earth, oving (NnC) at 4 aroun S 4, which by rucing th to th on-boy probl this has a si-ajor axis a 4 An so on until Nptun Finally, by taking th su of asss of Sun, Mrcury, Vnus, Earth, Mars, Jupitr, Saturrn, Uranus, an Nptun, , as if thy wr locat at S togthr with Pluto (tnth) oving both at (NnC) aroun th subcntr S , th two-boy probl constitut by locat at S , an Pluto s ass 0 can b ruc to th on-boy forulation probl having a si-ajor axis a In this way w can construct a probabl xplanation for th plantary Solar Syst s stability For xapl, in orr to unrstan stp by stp th way of obtaining what w ar trying to assrt, lt s stablish th quations of th cas of Sun an Mrcury, latr ths two with Vnus, which ar th sa as thos prsnt for any on of th thr last cass of th thr-boy, cas b), an latr with Earth thos of th four-boy probl, in ), s [] sction 5 Only w to chang th valus of th bois asss, by putting as th biggst ass Lt s rpat thos quations: a) For th two-boy probl: both asss an aroun thir CM at j r r Ur r r Ur, r r i i G G F F F h = r r + r = r + r v h cos Lt s rbr its gnral rprsntation through Mathatica 7:

3 Clar [,,,f,f,f,f,a,,p] =4 =6 = p=0 a=/(+) a=/(+) Manipulat[ParatricPlot[{{ a (-^)/(+ Cos[]) {Cos[],Sin[]}},{- a (-^)/(+ Cos[]) {Cos[],Sin[]}}},{,0,f},PlotStylThick,IagSiz{500,500},PlotRang{{-4,},{-,}},AspctRatioAutoatic,PlotRangClippingTru,ColorFunctionFunction[{x,y},ColorData["Rainbow "][y]],epilog{pointsiz[p ],Brown,Point[ a (-^)/(+ Cos[f]) {Cos[f],Sin[f]}],PointSiz[p ],,Point[- a (-^)/(+ Cos[f]) {Cos[f],Sin[f]}]}],{{,04},0,,Apparanc"Labl "},{{,RGBColor[,0,0]},ColorSttr,IagSizSall},{{f, Pi/5,"aniat orbit"},0,6 Pi,ControlTypTriggr,IagSizSall}] aniat orbit Not: In th aniation givn by Mathatica 7 angular sp is prsnt as constant Accoring to scon Kplr Law, what is constant is th prouct r This givs highst sp of asss at prig an th lowst on at apog (contrary to what is corrct) This warning is vali for th rst of graphics Now, its rprsntation for bing a vry big ass Not in th aniation that th biggr ass alost on t ov: Clar [,,,f,f,f,f,a,,p] =400 =5 =6 p=0 a=/(+) a=/(+) Manipulat[ParatricPlot[{{ a (-^)/(+ Cos[]) {Cos[],Sin[]}},{- a (-^)/(+ Cos[]) {Cos[],Sin[]}}},{,0,f},PlotStylThick,IagSiz{500,500},PlotRang{{-0,0},{-

4 0,0}},AspctRatioAutoatic,PlotRangClippingTru,ColorFunctionFunction[{x,y},ColorData["Rainbow "][y]],epilog{pointsiz[05 p ],Brown,Point[ a (-^)/(+ Cos[f]) {Cos[f],Sin[f]}],PointSiz[p ],,Point[- a (-^)/(+ Cos[f]) {Cos[f],Sin[f]}]}],{{,0},0,,Apparanc"Labl "},{{,RGBColor[,0,0]},ColorSttr,IagSizSall},{{f, Pi/5,"aniat orbit"},0,6 Pi,ControlTypTriggr,IagSizSall}] aniat orbit b) Thr-boy probl with nst configurations of th two-boy typ: two of th oving at an angular sp ij, aroun thir sub-cntr of ass, S ij, foring this sub-cntr with th thir ass a two-boy syst that ovs aroun thir CM at th sa angular vlocity (ij)k, not ncssarily coplanar (NnC) For asss an with a sub-cntr S, as if thy wr locat join at this sub-cntr, oving oppos with aroun CM: r ρ' ' Ur U r ρ' ' Ur U r r Ur ' h cos G F F h cos F h cos F r h cos G ' r Its gnral rprsntation, known fro [] Clar [,,,f,f,f,f,a,,p] = 4

5 = =0 a=0 =6 p=0 f=/(+) f=/(+) f=(+)/(++) f=/(++) Manipulat[ParatricPlot[{{ f (-^)/(+ Cos[ t]) Cos[ t]+a f (-^)/(+ Cos[t]) Cos[t], f (- ^)/(+ Cos[ t]) Sin[ t]+a f (-^)/(+ Cos[t]) Sin[t]},{- f (-^)/(+ Cos[ t]) Cos[ t]+a f (-^)/(+ Cos[t]) Cos[t],- f (-^)/(+ Cos[ t]) Sin[ t]+a f (-^)/(+ Cos[t]) Sin[t]}, {a f (-^)/(+ Cos[t]) Cos[t],a f (-^)/(+ Cos[t]) Sin[t]}, {-a f (-^)/(+ Cos[t]) Cos[t],-a f (-^)/(+ Cos[t]) Sin[t]}},{t,0,tf},PlotStyl- >Thick,IagSiz{500,500},PlotRang{{-,0},{- 0,0}},AspctRatioAutoatic,PlotRangClippingTru,ColorFunction- >Function[{x,y},ColorData["Rainbow"][y]],Epilog{PointSiz[8 p /f],brown,point[{ f (-^)/(+ Cos[ tf]) Cos[ tf]+a f (-^)/(+ Cos[tf]) Cos[tf], f (-^)/(+ Cos[ tf]) Sin[ tf]+a f (-^)/(+ Cos[tf]) Sin[tf]}],PointSiz[5p /f],,point[{- f (-^)/(+ Cos[ tf]) Cos[ tf]+a f (-^)/(+ Cos[tf]) Cos[tf],- f (-^)/(+ Cos[ tf]) Sin[ tf]+a f (-^)/(+ Cos[tf]) Sin[tf]}],PointSiz[ 6 p /f],black,point[a f (-^)/(+ Cos[tf]) {0,0}],PointSiz[ p /f],black,point[a f (-^)/(+ Cos[tf]) {Cos[tf],Sin[tf]}],PointSiz[8 p /f],r,point[-a f (-^)/(+ Cos[tf]) {Cos[tf],Sin[tf]}],PointSiz[ 6 p /f],black,point[a f (-^)/(+ Cos[tf]) {0,0}]}],{{,},0,4,Apparanc"Labl "},{{a,0},0,0,apparanc"labl "},{{,06},0,,Apparanc"Labl "},{{,0},0,,Apparanc"Labl "},{{,RGBColor[0,,0]},ColorSttr,IagSizSall},{{tf,Pi/,"aniat orbit"},0,pi,controltyptriggr,iagsizsall}] a aniat orbit Big valus of (brown) ak lliptical paths of othr asss s to b oving as aroun : Clar [,,,f,f,f,f,a,,p] =00 5

6 =5 =0 a=0 =6 p=0 w=4 f=/(+) f=/(+) f=(+)/(++) f=/(++) Manipulat[ParatricPlot[{{ f (-^)/(+ Cos[w t]) Cos[w t]+a f (-^)/(+ Cos[t]) Cos[t], f (- ^)/(+ Cos[w t]) Sin[w t]+a f (-^)/(+ Cos[t]) Sin[t]},{- f (-^)/(+ Cos[w t]) Cos[w t]+a f (-^)/(+ Cos[t]) Cos[t],- f (-^)/(+ Cos[w t]) Sin[w t]+a f (-^)/(+ Cos[t]) Sin[t]}, {a f (-^)/(+ Cos[t]) Cos[t],a f (-^)/(+ Cos[t]) Sin[t]}, {-a f (-^)/(+ Cos[t]) Cos[t],-a f (-^)/(+ Cos[t]) Sin[t]}},{t,0,tf},PlotStyl- >Thick,IagSiz{500,500},PlotRang{{-0,0},{- 0,0}},AspctRatioAutoatic,PlotRangClippingTru,ColorFunction- >Function[{x,y},ColorData["Rainbow"][y]],Epilog{PointSiz[ p /f],brown,point[{ f (-^)/(+ Cos[w tf]) Cos[w tf]+a f (-^)/(+ Cos[tf]) Cos[tf], f (-^)/(+ Cos[w tf]) Sin[w tf]+a f (-^)/(+ Cos[tf]) Sin[tf]}],PointSiz[9p /f],,point[{- f (-^)/(+ Cos[w tf]) Cos[w tf]+a f (-^)/(+ Cos[tf]) Cos[tf],- f (-^)/(+ Cos[4 tf]) Sin[4 tf]+a f (-^)/(+ Cos[tf]) Sin[tf]}],PointSiz[ 05 p /f],black,point[a f (-^)/(+ Cos[tf]) {0,0}],PointSiz[ 5 p],black,point[a f (-^)/(+ Cos[tf]) {Cos[tf],Sin[tf]}],PointSiz[9 p /f],r,point[-a f (-^)/(+ Cos[tf]) {Cos[tf],Sin[tf]}],PointSiz[ 5 p],black,point[a f (-^)/(+ Cos[tf]) {0,0}]}],{{,0},0,40,Apparanc"Labl "},{{a,7},0,50,apparanc"labl "},{{,00},0,,Apparanc"Labl "},{{,00},0,,Apparanc"Labl "},{{,RGBColor[0,,0]},ColorSttr,IagSizSall},{{tf,Pi/,"aniat orbit"},0, Pi/,ControlTypTriggr,IagSizSall}] a aniat orbit c) Four-boy probl with nst two-boy typ A pair at an sub-sub-cntr S with a thir boy at, (NnC) aroun th sub-cntr S, (BP, NnC), an this last sub-cntr with th 6

7 fourth boy aroun th CM at an angular sp (BP, NnC): r ρ' r ρ' ρ' ρ', ' ', U U ' ' U U U U,, r4 ' ρ',, Ur 4 U r 4 r4 Ur 4 r r4, 4, ' r ',r4 4 ',, ', r4 ' h, cos h r 4 cos,, ' h cos h cos F F 4 h cos G F F ' G, 4 h,,, cos, F F 4 G ' r, 4 r Th gnral rprsntation of four asss nst in configurations of two-boy typ follows []: Clar[,,,f,f,f,f,a,,p] =4 =6 =0 = = 4=7 4=40 a=5 b=0 = p=0 f=/(+) f=/(+) f=(+)/(++) f=/(++) f4=(++)/(+++4) f=4/(+++4) Manipulat[ParatricPlot[{{ f (-^)/(+ Cos[ t]) Cos[ t]+a f (-^)/(+ Cos[7 t]) Cos[7 t]+b f (-^)/(+ Cos[t]) Cos[t], f (-^)/(+ Cos[ t]) Sin[ t]+a f (-^)/(+ Cos[7 t]) Sin[7 t]+b f (- ^)/(+ Cos[t]) Sin[t]},{- f (-^)/(+ Cos[ t]) Cos[ t]+a f (-^)/(+ Cos[7 t]) Cos[7 t]+b f (- ^)/(+ Cos[t]) Cos[t],- f (-^)/(+ Cos[ t]) Sin[ t]+a f (-^)/(+ Cos[7 t]) Sin[7 t]+b f (- ^)/(+ Cos[t]) Sin[t]},{a f (-^)/(+ Cos[7 t]) Cos[7 t]+b f (-^)/(+ Cos[t]) Cos[t],a f (-^)/(+ Cos[7 t]) Sin[7 t]+b f (-^)/(+ Cos[t]) Sin[t]},{-a f (-^)/(+ Cos[7 t]) Cos[7 t]+b f (-^)/(+ Cos[t]) 7

8 Cos[t],-a f (-^)/(+ Cos[7 t]) Sin[7 t]+b f (-^)/(+ Cos[t]) Sin[t]},-b f4 (-^)/(+ Cos[t]) {Cos[t],Sin[t]},b f (-^)/(+ Cos[t]) {Cos[t],Sin[t]}},{t,0,tf},PlotStylThick,IagSiz{500,500},PlotRang{{-5,0},{- 0,0}},AspctRatioAutoatic,PlotRangClippingTru,ColorFunctionFunction[{x,y},ColorData["Rainbow"][y]],Epi log{pointsiz[ p ],Brown,Point[{ f (-^)/(+ Cos[ tf]) Cos[ tf]+a f (-^)/(+ Cos[7 tf]) Cos[7 tf]+b f (-^)/(+ Cos[tf]) Cos[tf], f (-^)/(+ Cos[ tf]) Sin[ tf]+a f (-^)/(+ Cos[7 tf]) Sin[7 tf]+b f (-^)/(+ Cos[tf]) Sin[tf]}],PointSiz[p ],,Point[{- f (-^)/(+ Cos[ tf]) Cos[ tf]+a f (-^)/(+ Cos[7 tf]) Cos[7 tf]+b f (-^)/(+ Cos[tf]) Cos[tf],- f (-^)/(+ Cos[ tf]) Sin[ tf]+a f (-^)/(+ Cos[7 tf]) Sin[7 tf]+b f (-^)/(+ Cos[tf]) Sin[tf]}],PointSiz[ p /f],black,point[{a f (- ^)/(+ Cos[7 tf]) Cos[7 tf]+b f (-^)/(+ Cos[tf]) Cos[tf],a f (-^)/(+ Cos[7 tf]) Sin[7 tf]+b f (- ^)/(+ Cos[tf]) Sin[tf]}],PointSiz[ p ],R,Point[{-a f (-^)/(+ Cos[7 tf]) Cos[7 tf]+b f (-^)/(+ Cos[tf]) Cos[tf],-a f (-^)/(+ Cos[7 tf]) Sin[7 tf]+b f (-^)/(+ Cos[tf]) Sin[tf]}],PointSiz[ p 4],Blu,Point[{-b f4 (-^)/(+ Cos[tf]) {Cos[tf],Sin[tf]}}],PointSiz[ p /f],black,point[{b f (-^)/(+ Cos[tf]) {Cos[tf],Sin[tf]}}],PointSiz[ p /f],black,point[{-b f (-^)/(+ Cos[tf]) {0,0}}]}],{{,},0,4,Apparanc"Labl "},{{a,},0,0,apparanc"lfbl "},{{b,7},0,0,apparanc"lfbl "},{{,0},0,,Apparanc"Labl "},{{,0},0,,Apparanc"Labl "},{{,RGBColor[0,,0]},ColorSttr,IagSizSall},{{tf,Pi/,"aniat orbit"},0, Pi/,ControlTypTriggr,IagSizSall}] a b aniat orbit As with thr bois, by oing (brown) th biggst ass with a ass qual to 50 units it can b obsrv that th paths of th scon grn ass an thos of th thir r on ar not wll fin in th sns that squntial paths of th sa ass o not follow prvious trajctoris S also that th sub-cntr S (black point that ovs insi th brown ass) has a vry coplx otion (a ix of spiral otions) Also, it can b not in aniat o that th otion of th brown ass is yt or coplx than that of S Th ia of putting th biggst ass with a not so high valu was prcisly to prciv or clarly th rsulting iffrnt ovnts of ach ass an 8

9 th ix ffct of th intractiv gravitational forcs prsnt aong such asss As always, black points intify sub-cntrs of ass an th CM of th syst Its rprsntation follows: Clar[,,,f,f,f,f,a,,p] =50 = =+ =4 =++ 4=5 4=+++4 a=5 b=000 = p=04 w=7 w=4 s= Pi/ s=- Pi/ f=/(+) f=/(+) f=(+)/(++) f=/(++) f4=(++)/(+++4) f=4/(+++4) Manipulat[ParatricPlot[{{ f (-^)/(+ Cos[w t+s]) Cos[w t+s]+a f (-^)/(+ Cos[w t+s]) Cos[w t+s]+b f (-^)/(+ Cos[t]) Cos[t], f (-^)/(+ Cos[w t+s]) Sin[w t+s]+a f (-^)/(+ Cos[w t+s]) Sin[w t+s]+b f (-^)/(+ Cos[t]) Sin[t]},{- f (-^)/(+ Cos[w t+s]) Cos[w t+s]+a f (-^)/(+ Cos[w t+s]) Cos[w t+s]+b f (-^)/(+ Cos[t]) Cos[t],- f (-^)/(+ Cos[w t+s]) Sin[w t+s]+a f (-^)/(+ Cos[w t+s]) Sin[w t+s]+b f (-^)/(+ Cos[t]) Sin[t]},{a f (- ^)/(+ Cos[w t+s]) Cos[w t+s]+b f (-^)/(+ Cos[t]) Cos[t],a f (-^)/(+ Cos[w t+s]) Sin[w t+s]+b f (-^)/(+ Cos[t]) Sin[t]},{-a f (-^)/(+ Cos[w t+s]) Cos[w t+s]+b f (- ^)/(+ Cos[t]) Cos[t],-a f (-^)/(+ Cos[w t+s]) Sin[w t+s]+b f (-^)/(+ Cos[t]) Sin[t]},b f (-^)/(+ Cos[t]) {Cos[t],Sin[t]},-b f4 (-^)/(+ Cos[t]) {Cos[t],Sin[t]}},{t,0,tf},PlotStylThick,IagSiz{500,500},PlotRang{{-5,5},{- 5,5}},AspctRatioAutoatic,PlotRangClippingTru,ColorFunctionFunction[{x,y},ColorData["Rainbow"][y]],Epi log{pointsiz[0 p ],Brown,Point[{ f (-^)/(+ Cos[w tf+s]) Cos[w tf+s]+a f (-^)/(+ Cos[w tf+s]) Cos[w tf+s]+b f (-^)/(+ Cos[tf]) Cos[tf], f (-^)/(+ Cos[w tf+s]) Sin[w tf+s]+a f (- ^)/(+ Cos[w tf+s]) Sin[w tf+s]+b f (-^)/(+ Cos[tf]) Sin[tf]}],PointSiz[p ],,Point[{- f (- ^)/(+ Cos[w tf+s]) Cos[w tf+s]+a f (-^)/(+ Cos[w tf+s]) Cos[w tf+s]+b f (-^)/(+ Cos[tf]) Cos[tf],- f (-^)/(+ Cos[w tf+s]) Sin[w tf+s]+a f (-^)/(+ Cos[w tf+s]) Sin[w tf+s]+b f (-^)/(+ Cos[tf]) Sin[tf]}],PointSiz[4 p],black,point[{a f (-^)/(+ Cos[w tf]) Cos[w tf]+b f (-^)/(+ Cos[tf]) Cos[tf],a f (-^)/(+ Cos[w tf+s]) Sin[w tf+s]+b f (-^)/(+ Cos[tf]) Sin[tf]}],PointSiz[ p ],R,Point[{-a f (-^)/(+ Cos[w tf+s]) Cos[w tf+s]+b f (-^)/(+ Cos[tf]) Cos[tf],-a f (-^)/(+ Cos[w tf+s]) Sin[w tf+s]+b f (-^)/(+ Cos[tf]) Sin[tf]}],PointSiz[ p 4],Blu,Point[{-b f4 (-^)/(+ Cos[tf]) {Cos[tf],Sin[tf]}}],PointSiz[05 p],black,point[{b f (-^)/(+ Cos[tf]) {Cos[tf],Sin[tf]}}],PointSiz[06 p],black,point[{-b f (-^)/(+ Cos[tf]) {0,0}}]}],{{,4},0,0,Apparanc"Labl "},{{a,8},0,0,apparanc"labl "},{{b,},0,00,apparanc"labl "},{{,0},0,,Apparanc"Labl "},{{,0},0,,Apparanc"Labl "},{{,0},0,,Apparanc"Labl "},{{,RGBColor[0,,0]},ColorSttr,IagSizSall},{{tf,Pi/,"aniat orbit"},0,4 Pi/,ControlTypTriggr,IagSizSall}] 9

a b aniat orbit Now w ar going to chang th valu of ass (brown) fro 50 to 4500 ass units in orr to s how otion of th biggst ass is attnuat an also that of th sub-cntr S Also, w can notic that plant s

10 a b aniat orbit Now w ar going to chang th valu of ass (brown) fro 50 to 4500 ass units in orr to s how otion of th biggst ass is attnuat an also that of th sub-cntr S Also, w can notic that plant s paths now ar or fin in th sns that th asss follow a sa lin of otion aroun biggst ass Givn that Sun ass is tis Earth ass, ths ffcts ar uch or visibl in th Solar Syst Clar[,,,f,f,f,f,a,,p] =4500 = =+ =4 =++ 4=5 4=+++4 a=5 b=000 = p=04 w=7 w=4 s= Pi/ s=- Pi/ f=/(+) f=/(+) f=(+)/(++) f=/(++) f4=(++)/(+++4) 0

$f=4/(+++4) Manipulat[ParatricPlot[{{ f (-^)/(+ Cos[w t+s]) Cos[w t+s]+a f (-^)/(+ Cos[w t+s]) Cos[w t+s]+b f (-^)/(+ Cos[t]) Cos[t], f (-^)/(+ Cos[w t+s]) Sin[w t+s]+a f (-^)/(+ Cos[w t+s]) Sin[w$

11 f=4/(+++4) Manipulat[ParatricPlot[{{ f (-^)/(+ Cos[w t+s]) Cos[w t+s]+a f (-^)/(+ Cos[w t+s]) Cos[w t+s]+b f (-^)/(+ Cos[t]) Cos[t], f (-^)/(+ Cos[w t+s]) Sin[w t+s]+a f (-^)/(+ Cos[w t+s]) Sin[w t+s]+b f (-^)/(+ Cos[t]) Sin[t]},{- f (-^)/(+ Cos[w t+s]) Cos[w t+s]+a f (-^)/(+ Cos[w t+s]) Cos[w t+s]+b f (-^)/(+ Cos[t]) Cos[t],- f (-^)/(+ Cos[w t+s]) Sin[w t+s]+a f (-^)/(+ Cos[w t+s]) Sin[w t+s]+b f (-^)/(+ Cos[t]) Sin[t]},{a f (- ^)/(+ Cos[w t+s]) Cos[w t+s]+b f (-^)/(+ Cos[t]) Cos[t],a f (-^)/(+ Cos[w t+s]) Sin[w t+s]+b f (-^)/(+ Cos[t]) Sin[t]},{-a f (-^)/(+ Cos[w t+s]) Cos[w t+s]+b f (- ^)/(+ Cos[t]) Cos[t],-a f (-^)/(+ Cos[w t+s]) Sin[w t+s]+b f (-^)/(+ Cos[t]) Sin[t]},b f (-^)/(+ Cos[t]) {Cos[t],Sin[t]},-b f4 (-^)/(+ Cos[t]) {Cos[t],Sin[t]}},{t,0,tf},PlotStylThick,IagSiz{500,500},PlotRang{{-5,5},{- 5,5}},AspctRatioAutoatic,PlotRangClippingTru,ColorFunctionFunction[{x,y},ColorData["Rainbow"][y]],Epi log{pointsiz[00 p ],Brown,Point[{ f (-^)/(+ Cos[w tf+s]) Cos[w tf+s]+a f (-^)/(+ Cos[w tf+s]) Cos[w tf+s]+b f (-^)/(+ Cos[tf]) Cos[tf], f (-^)/(+ Cos[w tf+s]) Sin[w tf+s]+a f (-^)/(+ Cos[w tf+s]) Sin[w tf+s]+b f (-^)/(+ Cos[tf]) Sin[tf]}],PointSiz[p ],,Point[{- f (-^)/(+ Cos[w tf+s]) Cos[w tf+s]+a f (-^)/(+ Cos[w tf+s]) Cos[w tf+s]+b f (-^)/(+ Cos[tf]) Cos[tf],- f (-^)/(+ Cos[w tf+s]) Sin[w tf+s]+a f (-^)/(+ Cos[w tf+s]) Sin[w tf+s]+b f (-^)/(+ Cos[tf]) Sin[tf]}],PointSiz[4 p],black,point[{a f (-^)/(+ Cos[w tf]) Cos[w tf]+b f (-^)/(+ Cos[tf]) Cos[tf],a f (-^)/(+ Cos[w tf+s]) Sin[w tf+s]+b f (-^)/(+ Cos[tf]) Sin[tf]}],PointSiz[ p ],R,Point[{-a f (-^)/(+ Cos[w tf+s]) Cos[w tf+s]+b f (-^)/(+ Cos[tf]) Cos[tf],-a f (-^)/(+ Cos[w tf+s]) Sin[w tf+s]+b f (-^)/(+ Cos[tf]) Sin[tf]}],PointSiz[ p 4],Blu,Point[{-b f4 (-^)/(+ Cos[tf]) {Cos[tf],Sin[tf]}}],PointSiz[05 p],black,point[{b f (-^)/(+ Cos[tf]) {Cos[tf],Sin[tf]}}],PointSiz[06 p],black,point[{-b f (-^)/(+ Cos[tf]) {0,0}}]}],{{,4},0,0,Apparanc"Labl "},{{a,8},0,0,apparanc"labl "},{{b,},0,00,apparanc"labl "},{{,0},0,,Apparanc"Labl "},{{,0},0,,Apparanc"Labl "},{{,0},0,,Apparanc"Labl "},{{,RGBColor[0,,0]},ColorSttr,IagSizSall},{{tf,Pi/,"aniat orbit"},0,4 Pi/,ControlTypTriggr,IagSizSall}] a b aniat orbit

12 Conclusion Visualization of attracting gravitational bois otion unr nst two-boy configurations givn by our quations in prvious xapls asy xplains th stability an prforanc of th whol Solar Syst in a consistnt way Rfrncs: [] Franco-Roriguz, Jorg Aalbrto, March, 04: A Sipl Classic Solution to th N-Boy Probl Th Gnral Scinc Journal Roriguz [] Franco-Roriguz, Jorg Aalbrto, Jun, 04: Stabl Configurations of th N-Boy Probl Th Gnral Scinc Journal Astrophysics/Downloa/550

The second condition says that a node α of the tree has exactly n children if the arity of its label is n.

The second condition says that a node α of the tree has exactly n children if the arity of its label is n. CS 6110 S14 Hanout 2 Proof of Conflunc 27 January 2014 In this supplmntary lctur w prov that th λ-calculus is conflunt. This is rsult is u to lonzo Church (1903 1995) an J. arkly Rossr (1907 1989) an is