N-BODY SOLUTIONS AND COMPUTING GALACTIC MASSES

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1 The Astoomical Joual, 49:74 (6pp), 05 May 05. The Ameica Astoomical Society. All ights eseved. doi:0.088/ /49/5/74 N-BODY SOLUTIONS AND COMPUTING GALACTIC MASSES Doald G. Uivesity of Califoia Ivie, CA , USA; Received 0 Jauay ; accepted 05 Febuay 0; published 05 Apil 8 ABSTRACT A classical appoach used to detemie the mass distibutio of a galaxy i tems of obseved otatioal velocities is applied to aalytic solutios of Newtoia systems of N discete bodies (so mass distibutios ae kow). Pedictios sigificatly exaggeate the amout of mass distibuted at lage distaces fom the cete; e.g., athe tha the actual 5% of mass o the oute edges, the method could pedict ove 80%. Explaatios ae give fo the diffeeces. Key wods: celestial mechaics galaxies: kiematics ad dyamics methods: aalytical. INTRODUCTION A classical way to detemie galactic mass distibutios uses the cicula velocities of stas. With obsevatios of spial galaxies, whee stella cicula velocities ted to flatte o eve icease with distace fom the galactic cete, the method pedicts much lage M()values (the total mass to distace ) tha ae kow to exist, paticulaly o galactic edges (e.g., Ostike et al. 974; Rubi 98, 99). This suppots the belief that huge amouts of uobseved mass must exist i a halo. Howeve, this mathematical appoach has ot bee evaluated to detemie whethe pedictios could be exaggeated. The validatio issue is examied hee by applyig the method to aalytic N-body solutios. Appoaches (theoetical ad computatioal) developed to elate mass values with cicula velocities ted to be based o Newto s fist ad secod laws (e.g., Sectio.. of Biey & Temaie 008). I a symmetic cotiuum settig, fo istace, a body iside a spheical shell expeieces o et gavitatioal foce fom the shell; the gavitatioal foce o a body outside a spheical shell behaves as though the shell s matte is cocetated at the cete. Thee ae also othe appoaches, so heeafte cotiuum efes to all models adoptig these assumptios (e.g., Newto s fist ad secod laws) to aive at Equatio (). The discete tem is eseved fo the stadad Newtoia N-body poblem. As depicted i Figue, discete N-body systems eed ot satisfy these assumptios. Istead, with a close appoach of two bodies, each stogly iflueces the gavitatioal pull of the othe, whee the ie body (movig i a couteclockwise diectio) tugs o the oute body. This madatoy Newtoia tuggig geeates lage otatioal velocities, which, usig cotiuum-type pedictio appoaches, would be eoeously itepeted as maifestig a added level of mass. The daggig would ted to affect bodies fathe out, which suggests the possibility of exaggeated mass pedictios fo lage distaces fom the cete. With Newto s equatios this pullig occus, but does it matte? Ca it be safely igoed as icosequetial, o could the daggig sigificatly distot the pedicted mass distibutios, paticulaly i egios fathe fom the cete? To exploe this questio, a stadad method is applied to aalytic solutios of Newtoia N-body systems. As show (Sectio 4), pedictios ca seiously distot actual mass distibutios by sigificatly exaggeatig the pecetage of the total mass that is located o the edges of the solutio. As show, the moe shaply the actual mass values decease with iceasig, the lage the pedictio eos. It is atual to test a method by applyig it to discete N-body solutios, but this has ot bee peviously doe because of a absece of aalytic solutios with suitable symmety popeties. That lacua is patly filled hee by ceatig, fo abitaily lage values of N, classes of solutios whee each class cotais a cotiuum (based o mass choices fo idividual bodies) of aalytic N-body solutios. These solutios ae desiged to esemble (as closely as possible with discete systems) the cetal popeties used to develop the pedictio method: solutios have highly symmetic mass distibutios, obits ae cicula, ad each body s gavitatioal attactio is diected towad the cete of mass.. STANDARD APPROACH I symmetic settigs whee Newto s two laws apply, the mass up to adius fom the cete of mass, M(), has the fom (e.g., Sectio.. of Biey & Temaie 008) vot M() = () G whee v ot is the cicula velocity of a sta at distace. With the Equatio () classical appoach, the obseved flatteig of otatioal velocities implies fo lage values that M() D, () whee D is a positive costat. As extesively descibed (ofte usig Equatio ()), this M()value is much lage tha ustified fom the lumiosity of stas ad othe methods. Thee ae moe sophisticated ways to estimate M(),but Equatio () is a stadad appoach.. CENTRAL CONFIGURATIONS To apply Equatio () to Newtoia systems with N discete bodies, aalytic solutios ae developed usig cetal cofiguatios. Afte itoducig these cofiguatios ad thei popeties, they ae used to ceate ay umbe of N-body solutios.

2 The Astoomical Joual, 49:74 (6pp), 05 May Figue. Discete body iteactios... Cetal Cofiguatios N bodies defie a cetal cofiguatio (e.g., Wite 94; 005) if thee is a commo scala λ (depedig o distaces, masses, ad othe N-body vaiables) so that each body s positio ad acceleatio vectos satisfy U = λ, which is λ =, m Gmimk U =, =,..., N, () i< k ik, ad ik, = i k is the distace betwee paticles. I 77, Lagage poved, fo ay mass values, that the vetices of a equilateal tiagle defie a N = cetal cofiguatio. A example is whee the Su ad Jupite defie oe leg, ad the Toa asteoids populate the egio aoud the emaiig vetex of the leadig ad followig equilateal tiagles. I a 859 pize wiig pape, Maxwell used cetal cofiguatios to aalyze the igs of Satu by placig equal masses i a evely spaced mae o a cicle. This symmety equies the combied foce actig o the th paticle to lie alog the lie defied by ; i.e., fo each thee is a scala λ so that = λ. As all masses ae equal ad the paticles ae symmetically spaced alog the cicle, a idetical aalysis holds fo each paticle, so the λ values agee ad Maxwell s costuctio foms a cetal cofiguatio. Maxwell eeded a massive cetal body to epeset Satu. Placig such a body at the cete of mass of this cetal cofiguatio chages the commo λ = λ value, but symmety equies that they still agee. The cetal body is at = 0 (the cete of mass), ad the othe paticles ae symmetically located about 0, so it follows that = = 0. Asλ 0 = 0, the cetal body satisfies Equatio () fo ay λ value. Thus, Maxwell s cofiguatio, whee a cetal body (with ay desied mass) is symmetically suouded by equal masses o a cicle, defies a N = + body cetal cofiguatio. Moulto (90) aalyzed colliea cetal cofiguatios. He poved that fo ay N, ay choice of masses m, ad ay odeig of the masses alog a lie, thee exist uique (up to a commo scala multiple) spacigs betwee adacet paticles that defie a cetal cofiguatio. Fo ay specified mass choices, the, thee exist N! colliea cetal cofiguatios. (To explai the deomiato, thee ae N! ways to ode the masses o the lie, but a eflectio idetifies each cofiguatio with its evesed copy.) The cofiguatios desiged ext combie Moulto s ad Maxwell s appoaches to ceate examples with cicula symmety. Fo ay positive iteges k ad, stat with k lies i the plae that pass though the oigi whee adacet lies ae k adias apat. Choose ay mass values m > 0, =,,...,. As depicted i Figue (fo k = 4, = ), daw Figue. k = 4, = spideweb cetal cofiguatio. cocetic cicles ad place mass m at each poit whee the th cicle itesects a lie, =,...,. Each cicle has k masses (with two masses o each lie), so this costuctio defies a N = k body cofiguatio that esembles a spideweb. As asseted ext (ad poved i the Appedix), fo ay choice of k, ad masses m, adii fo the cicles exist that satisfy Equatio () to ceate a spideweb cetal cofiguatio with N = k masses. I the same way Maxwell developed his model fo Satu, a associated N = k + body cetal cofiguatio is ceated by placig a mass of ay size at the oigi. Fo k =, the spideweb cofiguatio is a symmetic case of Moulto s costuctio; fo =, it is Maxwell s costuctio. Theoem. Fo positive iteges k ad ad ay choice of m > 0, =,...,, thee exist uique spacigs betwee the cocetic cicles so that the cofiguatio is a spideweb cetal cofiguatio fo these spacigs ad ay positive multiple of them. To idicate why Theoem must be expected, symmety U equies paticles o the th cicle to satisfy λ =,soλ m is a smooth fuctio of the masses ad the positioig of paticles (whee citical tems ae the distaces betwee cicles). With fixed m values, the foces, which defie the λ values, ca be vaied by alteig the distaces betwee cicles; deceasig distaces icease the λ magitudes. I this mae, spacigs ca be selected so that λ =... = λ to defie a cetal cofiguatio. Covesely, fo cetai distaces betwee cicles, the λ =... = λ costait defies liea equatios i the masses that ca be solved (up to a commo multiple). While the pimay itet of Figue is to depict the costuctio, this cofiguatio equies ealy equal m, m, m masses... Cetal Cofiguatios ad Aalytic Solutios A otatio o a scala chage of a cetal cofiguatio is agai a cetal cofiguatio (e.g., Wite 94; 005), so a cetal cofiguatio defies a class of cofiguatios. Fo istace, specified mass choices defie pecisely fou classes of thee-body cetal cofiguatios: thee colliea choices (whee the elative distaces deped o the mass values) ad the equilateal tiagle (ad its eflectio). Each cofiguatio ca be otated ad scaled by ay positive multiple; e.g., oe

3 The Astoomical Joual, 49:74 (6pp), 05 May equilateal tiagle descibes the Su Jupite Toa cofiguatio, aothe idetifies potetial Eath Moo satellite dockig locatios, ad a thid descibes cofiguatios fomed by bodies appoachig a thee-body collisio ( 005). A thid popety (fom Equatio ()) is that, with appopiate iitial coditios, ay coplaa N-body cetal cofiguatio ca be placed ito a cicula o elliptic obit that peseves the cofiguatio; the motio keeps the shape while assumig diffeet scale o otatio values. Appopiate iitial coditios fo the thee-body equilateal tiagle cofiguatio, the, ceate eithe a cicula o elliptic obit that maitais the equilateal tiagle cofiguatio; e.g., this motio eflects the obit behavio of the Toa asteoids. To aalyze the igs of Satu, Maxwell placed his cetal cofiguatio model ito a cicula obit. I the ext sectio, spideweb cetal cofiguatios ae placed i cicula obits. 4. PREDICTIONS OF MASS VALUES Fo ay choices of, k, let the aalytic N-body solutio be eithe the N = k o N = k + body spideweb cetal cofiguatio placed i a cicula obit. By costuctio, each paticle s acceleatio is poited towad the cete of mass, each body has a cicula obit, ad, with lage k values, the solutio has a highly symmetic mass distibutio. A scale chage of a cetal cofiguatio is agai a cetal cofiguatio, so the miimal distace betwee adacet cocetic cicles is set equal to oe uit. 4.. Pedictig Mass Values Idepedet of the m values, each of these aalytic Newtoia N-body solutios behaves like a otatig igid body. Fo each solutio, the, a costat S > 0 exists so that the th paticle s otatioal velocity is vot, = S, =,..., N. (4) Icidetally, igid body behavio ca be expected to occu i potios of spial galaxies as maifested by the ealy staight lie stuctue of the fist pat of the obseved otatioal velocity cuves. With these obseved values, the otatioal stuctue coveys the flavo of a igid body motio whee sepaatig paticles o the galactic edges ae beig pulled alog. While the value of S is ot eeded, it is deived fo completeess of the expositio. The cicula motio equies N the kietic eegy to equal T = = mv N = S = m = SI N, whee I = = m is the pola momet of ietia. As I is a costat, it follows fom the Lagage-Jacobi equatio ( I = ( T U)) that I 0 ad T U,so U S =. As is I aleady kow (e.g., Wite 94, 005), λ = U,soλ I also satisfies λ =S. Applyig Equatios () to (4) leads to the pedictio that S λ M() = = G G D (5) fo positive costat D. (Relative compaisos ae used i the followig to avoid poblems caused by D chagig value with ad k.) This meas that the pedicted mass distibutio is always give by a cubic equatio (Equatio (5)) eve though mass values have yet to be selected. Theefoe, fo all m choices whee the actual M()is fa emoved fom beig a cubic expessio, it is impossible fo Equatio (5) to descibe the actual mass distibutio. Ideed, the icemetal icease of Equatio (5) is M D, so this egative assetio must hold wheeve m values do ot sigificatly icease fo iceasig. Also, fuctios ae compaed ove domais athe tha a few poits, so the umbe of igs,, must be sufficietly lage. To illustate, coside a = 00 ig cofiguatio whee each ig has a mass equal to uity. To compute M(), detemie how may igs (s of them) ae equied to each distace : M()equals the sum of the masses of these s igs. The miimal spacig assumptio (whee the smallest distace is ) meas that the distace s equied to each the sth ig satisfies s s; e.g., fo this example with M( ) =, the distace is that of the fist two igs, so. Moe geeally, if is the distace to the example s th ig, the M( ) =. The esultig iequality equies M() ; i othe wods, athe tha beig cubic (Equatio (5)), the pecise M()distibutio is, at best, liea. The moe the actual M()diffes fom, the lage the 00 value; e.g., should computatios pove that M(), the 00 00, but should computatios pove that M(), the This diffeece affects the pedictio eos. A cubic equatio caot eve cudely esemble a liea equatio ove this age of a huded sepaated values, which meas that the cubic expessio seiously distots the tue M() distibutio; this is paticulaly tue fo lage values o the oute figes of the solutio. This assetio is illustated by the fact that the pedicted mass up to distace elative to the mass M() D of the fist ig is = ( ) ; with the exteme choice M( ) D of = 00, this meas that the pedicted total mass is at least 00 = 0 6 times the pedicted mass of the fist ig, whe, i eality, the multiple is oly 00. (As Equatio (5) might udeestimate the mass of, this statemet does ot asset that the pedicted total mass is 0 6 ; it meas that Equatio (5) sigificatly distots the actual mass distibutio by eoeously pedictig lage elative mass distibutios fo egios fathe fom the cete.) Similaly, pecisely 50% of the mass esides betwee the 50th ad 00th igs, but, fo k = ad the 600 body poblem, (fom the appoximatios followig the poof of Theoem ), which meas that Equatio (5) eoeously pedicts that appoximately 87.5% D D (give by 50 = 7 () = ) of the mass D is located hee. Similaly, M () = D idicates the mass of a ig at distace, so the pedictio of the ig mass at distace elative to that of some othe ig, say the oe at, is M () M ( ) (). With the above example ad agai usig the exteme = 00, this expessio eoeously pedicts that the mass of the last ig is 0 4 times lage tha that of the fist ig, while the tue multiple is uity. Smalle, moe ealistic mass choices, whee the m values decease as the distace fom the cete of mass iceases (so the km mass value of the th M () ig diffes extemely fom M ( ) () ), exacebate diffeeces betwee pedicated ad actual distibutios. Choosig, With M (), it would be 6 04 lage.

4 The Astoomical Joual, 49:74 (6pp), 05 May fo istace, m =, the mass o the th cicle is, so oly k oe ig is eeded to each a mass of oe uit but it takes fou igs to have a mass ove two uits. A total mass of 0 equies s s igs whee = l( s) 0, o s e 0. Because s, 0 the M() = 0value equies a distace of e. Because the tue M()is bouded by l( ), ad because a cubic equatio (Equatio (5)) diffes dastically fom a logaithmic cuve, seious pedicted distibutio eos (demostatig the iadequacy of Equatios () ad (5)) must be aticipated. As a illustatio, Equatio (5) would idicate that the total mass is expoetially lage tha the mass of, say, the e e mass of the fist fou igs (give by ( ) > ( ) 0 0 ), whe i 4 4 fact the coect multiple is less tha 0. Also, oly 5% of the total mass lies betwee the e 9 th ad e 0 th igs. While the distibutio of igs fo this settig is still beig examied, a estimate give by the M() l()expessio suggests that e9, which would equie Equatio (5) to icoectly e0 e pedict that about 95% of the total mass is located i this fige egio; ote how this eo is cosistet with the exaggeatios of ig masses. Othe examples follow by selectig othe m values; e.g., = defies the moe exteme m k l( ) m M() l(l())while = equies M() M*. k Rathe tha beig esticted to M() D, poblems aise wheeve the pedicted M()fom sigificatly diffes fom the actual epesetatio. This icludes the Equatio () choice of M() D.With m = ad the actual distibutio give k by M() l(), the liea M() D must yield seiously distoted pedictios that exaggeate the elative amout of mass at a geate distace fom the cete. Accodig to the piciples of scietific validatio, uless ew suppotig agumets ca be foud, Equatio () is ot a appopiate way to pedict M() values fo systems of N discete bodies. Somethig i additio to, but pobably diffeet fom, Equatio () is eeded. (Elsewhee a ew appoach is developed to pedict ad explai the full stuctue of otatioal velocity cuves fo N discete bodies icludig the shap, almost liea iitial gowth; it pedicts sigificatly smalle M() values fo flatteed otatioal velocity cuves.) I paticula, it follows fom the above that with discete dyamics, expessios such as Equatio () that do ot captue tuggig effects ae iappopiate. 4.. Moe Geeal Systems Although the above suffices fo validatio puposes, it is easoable to wode about othe settigs; e.g., with theedimesioal N-body solutios o othe symmety settigs. The dimesioality of physical space is ot a issue; it is whethe solutios (with cicula otatioal values) exist that cause eoeous pedictios of mass distibutios, such as D o eve D. They exist; as oe of may thee-dimesioal aalytic N-body solutios that lead to adically icoect mass pedictios, stat with the k + spideweb cofiguatio ad eplace the body at the oigi with two bodies of equal mass that peiodically oscillate (i opposite diectios) alog the z-axis, while the cofiguatio otates as descibed above. Should this ot be tue, the e 0 has a much lage value tha used with the fist examples of this paagaph, which could cause eve lage eo multiples. Similaly, cetal cofiguatios ca have a wide assotmet of symmety popeties, agig fom colliea to spideweb settigs, so elated agumets apply to a wide spectum of symmety ad asymmety assumptios about mass distibutios. The easo Equatio () has these distoted pedictios is captued by the diffeece i dyamics i Figue betwee discete systems ad cotiuum appoximatios whee the tuggig equied by discete systems is ot icluded, o eve ecogized, by the appoximatios. I paticula, a otatig cetal cofiguatio equies a tuggig o each body to keep it i a fixed positio elative to the othe bodies. Howeve, the assumptios leadig to Equatio () fail to iclude these lage pullig velocities; this allows the Equatio () pedictios fo M()to adically diffe fom the pecise values. Moe geeally, aticipate exaggeated Equatio () pedictios wheeve idividual otatioal velocities expeiece tuggig effects. This is due to the obseved cicula velocity beig detemied by actual M() values plus the tuggig of othe bodies; the iceased velocity caused by the pullig foces the exaggeatios i Equatio (). Covesely, this effect ca be miimal with a limited umbe of bodies ad masses so small that the system ca be decoupled ito idividual two-body poblems, such as i ou plaetay system whee a tuggig effect (e.g., as captued by the chage i the peihelio of Mecuy) is miimal. (Two body systems ae cosistet with Equatio ().) Ideed, a expessio simila to Equatio () eoys success i detemiig the Su s mass based o plaetay motios. Howeve, wheeve a decouplig is iappopiate (e.g., with lage mass values ad may bodies), expect Newtoia coectios to exaggeate M() pedictios. A emaiig coce is whethe thee ae settigs (othe tha decoupligs) of N discete bodies whee Equatio () might be valid. This suggests seachig fo N-body solutios with the geeal popeties used to deive Equatio (), but without the tuggig effects. As idicated ext, they caot be expected to exist. Cetal to Equatio () is that all bodies have cicula obits; i.e., each body s foce is diected towad the cete of mass to defie = λ () t. (6) The cicula obit assumptio equies the momet of ietia, I = m, to have a costat value. A logstadig coectue ( 970, 005) is that a costat I equies the system to otate as a igid body, which esues Equatio () pedictios of D. A simple coectue (a special case) is that if each obit i a system has a uifom cicula motio, the the system otates as a igid body. Both coectues have bee veified fo ay colliea N-body poblem (Diacu et al. 005; 005 (Sectio.4.)), ad fo the geeal thee-body poblem (Moeckel 005, 008). Moeove, the coectues ca be expected to hold i geeal; e.g., the fist oe holds fo geeic settigs of N-body systems (Schmah & Stoica 007). Ituitio suppotig the simple coectue is that Equatio (6) equies the paticles to be symmetically located with espect to each othe. Howeve, cicula obits destoy this symmety if the agula otatio ates do ot agee. If agula otatio ates agee, the paticles ae i a igid body otatio. 4

5 The Astoomical Joual, 49:74 (6pp), 05 May Figue. Compaig λ values. (Fo istace, fo moe geeal N-body modeligs that iclude petubatios.) Thus, eithe the assumptios that ae cetal i developig Equatio () do ot apply to discete systems of N bodies, o the associated N-body solutio ca be expected to be a otatig igid body causig exaggeated M() pedictios. If the basic assumptios used to deive Equatio () fail to hold fo discete systems, the ew agumets ae equied to ustify usig somethig esemblig Equatio () with these systems. Fo istace, while these agumets cast seious doubts about the deivatio of Equatio (), the equatio may be easoably coect. Howeve, to take esults based o Equatio () seiously equies developig a alteative validatio agumet. This eseach is pat of a pogam suppoted by NSF DMI ad CMMI My thaks fo commets made afte seveal colloquia, geeal public, ad cofeece pesetatios of these esults ad to G Hazeligg fo his cotiued iteest i this poect. Also, my thaks to a efeee fo useful suggestios. APPENDIX Poof of Theoem. The theoem is poved by Maxwell fo = ad Moulto (90) fo k =. The followig supplies details fo the agumets developed i Sectio.7. of (005) ad (0). Symmety equies the paticles o each cicle to satisfy U λ = whee λ m is the same fo all paticles o the th cicle; λ is a cotiuous fuctio of masses ad the distaces betwee paticles ad cicles; its sig is detemied by the foce diectio. As descibed i the commets followig the statemet of Theoem, fo give masses, the goal is to show that thee ae distaces betwee cicles whee all λ agee. Let be fixed. Maxwell s case is = whee λ = λ* < 0. Fo =, let = + x. To pove thee is a uique x = x whee λ( x) = λ( ) (defiig a cetal cofiguatio), popeties of the λ( x) ad λ( x) cuves (oughly depicted by the solid Figue (a) cuves) ae used; they esue that the two cuves coss at a uique poit whee the λ ae equal. All paticles o the secod cicle ae attacted to the cete of mass, so λ ( x) < 0 fo all x > 0. The citical foce tems i the aalysis ae detemied by the distace betwee cicles. As x 0, the secod cicle appoaches the fist, ceatig a ifiitely lage attactio of paticles o each cicle towad its eighbo o the othe cicle; i.e., λ( x) ad λ( x) as x 0. Because x epesets whee the secod cicle is pushed away fom the fist, the magitude U of appoaches zeo, so λ m ( x) 0 i a mootoic mae as x. The gavitatioal effect that the secod cicle has o the fist deceases to zeo, educig the poblem to Maxwell s settig, so λ( x) λ * < 0. The cuves popeties (depicted i Figue (a) as solid cuves) equie them to coss at a uique poit to shae a commo λ < 0 value; this completes the poof fo =. Amodificatio of this agumet hadles ay umbe of cicles. With,let = + x, =,...,. The above is epeated to show, fo ay choice of positive x,..., x, that thee exists a poit x( x,..., x) whee λ( x( x,..., x)) = λ( x( x,..., x)). These ( ) exta cicles exet a outwad pull o the fist two, so λ ( x; x,... x) > λ ( x ), =, (dashed cuves i Figue (a)). Fo fixed x,..., x, λ( x; x,..., x) is mootoically deceasig, it appoaches as x 0 (the secod cicle comes abitaily close to the fist), ad it appoaches λ * < 0 as x (all cicles ae pushed ifiitely fa fom the fist cicle). Fofixed x,..., x ad the easos specified fo λ( x ), λ( x; x,..., x) appoaches as x 0 ad zeo as x. Fuctio U ( + x) λ( x; x,..., x) = is mootoically iceasig m ad appoaches a positive value (because, fo fixed x,thethid cicle pulls paticles outwad o the secod cicle). Thus λ( x; x,..., x) is mootoic to a positive value ad the deceases to appoach zeo. The stuctue of the cuves equies them to itesect to defie a poit x( x,..., x) whee λ( x( x,..., x) = λ( x( x,..., x) fo all x,..., x. I paticula, the commo λ value is positive fo sufficietly small x values (due to the domiace of the thid cicle causig λ( x; x,..., x) to have a lage positive value) ad appoach the egative λ( x) = λ( x) value as x (the thid ad othe cicles become ifiitely fa fom the fist two). Because of the smooth foliatio fomed by the λ ( x( x,..., x)) cuves ad the aalytic stuctue of the equatios, x( x,..., x) is a smooth fuctio. If x * > x (the oute cicles ae pushed fathe away fom the fist two), λ ( x( x,..., x) > λ ( x( x *,..., x)). I Figue (a), this λ cuve is betwee the coespodig solid ad dashed lies, so the itesectio defies a smalle commo λ value; that is, the commo λ ( x( x,..., x)) value is mootoically deceasig i x. Fo iductio, assume that up to the ( )th cicle, thee is a x ( x; x+,..., x) o which the fist ( ) λ fuctios agee, the commo λ ( x ( x; x+,..., x)) has a positive value fo sufficietly small x values, ad it is mootoically deceasig to a egative value as x. The λ ( x ( x; x+,..., x)) fuctio (with fixed x+,..., x) has the same popeties as λ( x; x,..., x) ;itappoaches as x 0 ad (uless = is the last cicle) mootoically iceases to a positive value ad the deceases to zeo as x.fosufficietly lage x + values (so the oute cicles ae pushed fa away), λ ( x ( x; x+,..., x)) appoaches a egative valued fuctio that appoaches zeo. The popeties of these cuves foce them to itesect at a uique poit. Uless =, the commo itesectio poit has a abitay lage positive value as x 0 (whee the oute cicles appoach the ie oes) ad a egative value as.fo =, the poof is completed. x 5

6 The Astoomical Joual, 49:74 (6pp), 05 May Distaces betwee cicles. Iceasig a mass value, such as m * > m, iceases the foce (with the stoge attactio of othe bodies o the same cicle), which moves the λ cuve dowwad as epeseted by the dashed lie i Figue (b). Similaly, this lage mass value has a stoge attactio o the ie cicle, which causes lage values fo the λ cuve as depicted i Figue (b). Thus the sepaatio value (whee the dashed λ cuves coss) betwee cicles has a lage value. This captues the expected fact that smalle masses decease sepaatio distaces while lage masses icease them. The distaces betwee cicles fo specified m values ae difficult to compute, but they ca be estimated by followig the above poof; that is, compute λ i values fo specified m ad, ad the adust m ad values so that the λ i values come close to ageeig. With the symmety, the paticles alog ay specified ay ca be used to detemie the λ i values; the positive x-axis is chose. Fo the ith paticle at distace i, the cotibutio to λ i fom othe paticles o the x-axis is give by m m( i) γ i (0) whee iγ i (0) = = + [ + ] =, i ; the fist i i summatio ivolves paticles o the egative x-axis ad the secod ivolves bodies o the positive x-axis. With symmety, whee the foce compoet i the y diectio fo a paticle o the positive y-axis is caceled by its pate o the egative y-axis, the foce o the ith paticle by bodies o the y-axis is stictly alog the x-axis. The cotibutio to λ i is give by γ i ( ) whee m i iγ i ( ) = = [. Accompayig each lie defied by + i ] θ, 0 < θ <, is a lie defied by θ whee symmety foces compoets i the y diectio to cacel, so the cotibutio to λ i fom this pai of lies is γi ( θ) whee m( cos( θ) + i) i γi ( θ) = = + i + i cos( θ) m( cos( θ) i) +. (7) = + i i cos( θ) Fo k =, λi = γi (0); fo k =, λi = γi(0) + γi( ); ad fo k s k, λi = γi (0) + s = γi ( ). k To illustate with m =, if x = = fo =,...,, the λ = > 0, ad ( + ) λ i = fo i ; i k= i i( i k) fo i >. (8) ii ( k) k= i, k 0 This expessio cacels tems that ae α igs to the ight of i with tems that ae α igs to the left; Equatio (8) shows those tems without a cacelig mate. (With deceasig m > m+, the zeos fom cacellatios ae eplaced with egative values mi + such as α mi + α.) These λ α i values, paticulaly λ > 0, equie a wide spacig. Ideed, a lage x = leads to λ = s= + ( s ) i s = =, with λ = [( s ) ] ( ) i (i ) s=. i [( i+ s) ] The elative closeess of these λ i values, whee 0 > λi > λi + fo i =,..., =, meas that slightly lage sepaatios ceate a cetal cofiguatio. (Note the associated λ, ad hece D, value is small fo this cofiguatio.) The actual distibutio is M() a whee costat a ; the esultig lage iceases some of the discepacies descibed i Sectio 4.. This chage fom x = to x = shows how the λ i values ae easoably sesitive to chages i spacig distaces. Ideed, the equatios show that chages i λ i ae moe sesitive to chages i the distaces of cicles ea the ith cicle. To illustate Equatio (7) with k = (so θ = ) ad xi = =, the geeal expessio is i γ i = + k=+ i k= + k i ( k + i ki) i( k + i ki) ( k + i). i k + i + ki (9) k= i+ ( ) Each tem i each summatio is egative. The fist summatio combies the tem fom the secod summatio of Equatio (7) that is α igs beyod the ith with the tem i the fist summatio epesetig the αth ig. The secod ad thid summatios ae emaiig ot-matched tems. The λ i values fo the k = settig ad N = 6 body poblem is the sum of γ i ( ) ad λ i fom Equatio (8); all λ i values ae ow egative ad thei close values idicate that slight chages (e.g., m must come slightly iwads) i distace will geeate the cetal cofiguatio. That is, ad, while the spacig is ot k egula, a estimate fo lage values is that k, which is what oe would expect fom M() a fo costat 0 < a ; the smalle the a value, the lage the 00 value, which could cause lage pedictio discepacies. Note how chages i ad k alte the λ (ad D) value. Lage k values, fo istace, ceate a dese cocetatio of mass eae the oigi, which exets a effect that is simila to iceasig mass values fo paticles close to the cete i the colliea settig. Also, λ has a lage value. REFERENCES Biey, J., & Temaie, S. 008, Galactic Dyamics (d ed.; Piceto, NJ: Piceto Uiv. Pess) Diacu, R., Péez-Chavela, E., & Satopete, M. 005, Tas. Ame. Math. Soc., 57, 45 Moeckel, R. 005, Tas. Ame. Math. Soc., 57, 05 Moeckel, R. 008, Discete ad Cotiuous Dyamical Systems Seies S,, 6 Moulto, F. R. 90, AMat,, Ostike, J. P., Peebles, P. J. E., & Yahil, A. 974, ApJ, 9, Rubi, V. C. 98, Sci, 0, 9 Rubi, V. C. 99, PNAS, 90, 484, D. G. 970, i Peiodic Obits, Stability, ad Resoaces, ed. G. Giacaglia (Dodecht: Reidel), 76, D. G. 005, Collisios, Rigs, ad Othe Newtoia N-body Poblems (Povidece, RI: Ameica Mathematical Society), D. G. 0, i Expeditios i Mathematics, ed. D. Hayes, & T. Shubi (Bosto, MA: Mathematical Associatio of Ameica), 8 Schmah, T., & Stoica, C. 007, Tas. Ame. Math. Society, 59, 449 Wite, A. 94, The Aalytic Foudatios of Celestial Mechaics (Piceto, NJ: Piceto Uiv. Pess) 6