Transient Spiral Arms from Far Out-of-equilibrium Gravitational Evolution

Transcription

1 The Astophysicl Jounl, 851:19 (10pp), 017 Decembe The Ameicn Astonomicl Society. Tnsient Spil Ams fom F Out-of-equilibium Gvittionl Evolution Dvid Benhiem 1, Michel Joyce, nd Fncesco Sylos Lbini 3,1,4 1 Istituto dei Sistemi Complessi, Consiglio Nzionle delle Riceche, Vi dei Tuini 19, I Rom, Itli Lbotoie de Physique Nucléie et de Hutes Énegies, UPMC INP3 CNRS UMR 7585, Sobonne Univesités, 4, plce Jussieu, F-755 Pis Cedex 05, Fnce 3 Cento Studi e Riceche Enico Femi, Vi Pnispen I-00184, Rom, Itli 4 INFN Unit Rome 1, Diptimento di Fisic, Univesitá di Rom Spienz, Pizzle Aldo Moo, I Rom, Itli Received 017 August 7; evised 017 Octobe 7; ccepted 017 Octobe 5; published 017 Decembe 6 Abstct We descibe how simple clss of out-of-equilibium, otting, nd symmeticl mss distibutions evolve unde thei self-gvity to poduce qusi-pln spil stuctue suounding viilized coe, qulittively esembling spil glxy. The spil stuctue is tnsient, but cn suvive tens of dynmicl times, nd futhe epoduces qulittively noted fetues of spil glxies such s the pedominnce of tiling two-med spils nd lge pitch ngles. As ou models e highly idelized, detiled compison with obsevtions is not ppopite, but geneic fetues of the velocity distibutions cn be identified to be the potentil obsevtionl signtues of such mechnism. Indeed, the mechnism leds geneiclly to chcteistic tnsition fom pedominntly ottionl motion, in egion outside the coe, to dil bllistic motion in the outemost pts. Such dil motions e excluded in ou Glxy up to 15 kpc, but could be detected t lge scles in the futue by GAIA. We exploe the ppent motions seen by extenl obseves of the velocity distibutions of ou toy glxies, nd find tht it is difficult to distinguish them fom those of otting disk with sub-dominnt dil motions t levels typiclly infeed fom obsevtions. These simple models illustte the possibility tht the obseved ppent motions of spil glxies might be explined by non-tivil non-sttiony mss nd velocity distibutions without invoking dk mtte hlo o modifiction of Newtonin gvity. In this scenio the obseved phenomenologicl eltion between the centipetl nd gvittionl cceletion of the visible byonic mss could hve simple explntion. Key wods: glxies: fomtion glxies: spil Glxy: fomtion Glxy: kinemtics nd dynmics methods: numeicl 1. Intoduction The ms of spil glxies e one of the most stiking nd emkble fetues of the visible univese eveled by stonomy. They hve been the subject of much study, both obsevtionl nd theoeticl, ove mny decdes. Sevel competing theoies hve been dvnced to explin thei physicl oigin, but no single one hs emeged definitively s the coect fmewok (see, e.g., Dobbs & Bb 014). Undestnding thei motions is of pticul impotnce becuse it is the obseved ppent (i.e., on the line of sight LOS) motions in the oute pts of spil glxies tht hve led to the supposition tht much of the gvitting mtte in them is not visible (Rubin 1983). These sme motions hve lso led to ltentive scenios involving stong modifictions of Newtonin gvity (Milgom 1983). In this ppe we show how mss distibutions qulittively esembling those of the visible components of spil glxies cn esult fom the f out-of-equilibium dynmics of puely self-gvitting systems, stting fom clss of vey simple idelized initil conditions. We study in pticul the geneic fetues of the velocity distibutions of the stuctues poduced by this mechnism, nd conside thei qulittive comptibility with obsevtions of motions in spil glxies. Ou ppoch is diffeent fom stndd theoeticl ones, in which spil stuctue ises by petubtion (intenl o extenl) of n equilibium system, nd the lge-scle motions Oiginl content fom this wok my be used unde the tems of the Cetive Commons Attibution 3.0 licence. Any futhe distibution of this wok must mintin ttibution to the utho(s) nd the title of the wok, jounl cittion nd DOI. e modeled ssuming sttiony mss distibution. Indeed, ou study illusttes how, fo intinsiclly non-sttiony models, the eltion between ppent motions nd the ssocited mss distibution cn be completely diffeent fom tht in sttiony models. In pticul, we show tht the obsevtion of non-keplein ottion cuve in the oute pt of such stuctue does not necessily equie the existence of n extended dk mtte hlo o modifiction of Newtonin gvity, nd could insted be consistent with non-xisymmetic dil motion of wekly bound nd unbound mss. We note tht, becuse ou models involve only Newtonin gvity, the physics we descibe could potentilly be pplicble to stophysicl systems of vey diffeent ntues nd sizes to dwf glxies tht e infeed fom thei motions to be even moe dk-mtte-dominted thn spils (see, e.g., Combes 00); to potoplnety disks, which hve been eveled in obsevtions in the lst couple of yes to hve spil-like stuctue (see, e.g., Chistiens et l. 014); o even possibly to cicumplnety disks, whose existence is still inconclusive (see, e.g., Wd & Cnup 010). In fothcoming wok (D. Benhiem et l. 017, in peption) tht is complementy to this ppe, we will descibe the physicl mechnism in much gete detil, using both fo bod nge of initil conditions nd numeicl simultions with lge pticle numbes. The clss of models we conside s initil conditions consists of symmeticl nd isolted self-gvitting clouds with some ngul momentum. The dynmics of isolted self-gvitting systems fom out-of-equilibium initil conditions hs been extensively studied fo sevel decdes (Hénon 1973; vn Albd 198; Aseth et l. 1988; Dvid & Theuns 1989; 1

2 The Astophysicl Jounl, 851:19 (10pp), 017 Decembe 10 Aguil & Meitt 1990; Theuns & Dvid 1990; Boily et l. 00; Bnes et l. 009). Bodly speking, such systems elx quite efficiently to viil equilibium, i.e., on timescles of the ode of few times the chcteistic dynmicl time. Ely studies showed tht spheicl configutions with little isotopic velocity dispesion (i.e., sub-viil, with n initil viil tio b >-1) could poduce equilibted stuctues esembling ellipticl glxies, with sufce bightness notbly close to the obseved de Vucouleus lw (vn Albd 198). One geneic fetue of such sub-viil collpses (fo b -0.5) is tht they led to the ejection of some of the initil mss (see, e.g., Joyce et l. 009; Sylos Lbini 01, 013): the stong contction of the initil configution leds to pidly vying men-field, which cuses pticle enegies to lso pidly vy, leving some of them wekly bound nd othes with positive enegy. Two of us hve ecently studied (Benhiem & Sylos Lbini 015, 017) the evolution fom configutions tht e initilly ellipsoidl o of n iegul shpe nd found them to give ise to viilized centl coe suounded by vey flttened configutions mde by both wekly bound nd ejected pticles. These esults, combined, hve led us to the ide tht, with some initil ottionl motion, it might be possible to genete spil stuctue fom these kinds of initil conditions. Indeed, the lge dil velocities e geneted in smll egion (of the ode of the miniml size eched) in vey shot time (much less thn one dynmicl time), thus the dil distnce these pticles subsequently tvel once they e outside the coe cn be expected, given ppoximte consevtion of ngul momentum, to be coelted with the integted ngle they move though. The ppe is ognized s follows. In Section we pesent the detils of ou numeicl simultions. Section 3 is devoted to discussion of the thee-dimensionl nd two-dimensionl esults of ou simultions nd thei eltions with some key obsevtionl esults on spil glxies. Finlly, in Section 4 we dw ou min conclusions. In the Appendix we detil how we constucted the pojected velocity mps fom ou simulted mss distibutions.. Simultions We hve consideed vey simple set of initil conditions tht combines the chcteistics descibed bove: beking of the spheicl symmety of the initil mss distibution, velocity distibution tht is sub-viil (o, moe genelly, out of equilibium), nd some coheent ottion. Moe pecisely, we conside the following: N pticles distibuted ndomly, with unifom men density, inside n ellipsoidl egion, nd velocities tht coespond to coheent igid body-like ottionl motion bout the shotest semi-pincipl xis. Although these e d hoc nd clely too idelized to descibe physiclly elistic sitution, in the context of the theoy of glxy fomtion these kinds of initil conditions kind hve often been gued to be esonble (see, e.g., Eggen et l. 196). Nevetheless, they e vey diffeent fom those descibed in cuent scenios fo glxy fomtion in the context of cold dk-mtte-dominted cosmologicl models, which e chcteized by hiechicl collpse. We note, howeve, tht in cosmologicl scenios with vey suppessed initil fluctutions t vey smll scles (e.g., in models with wm dk mtte), monolithic collpse fom qusi-unifom initil stte my be moe esonble ppoximtion. In ny cse, ou gol hee is to identify nd study physicl mechnism nd its possible obsevtionl signtues, nd not to povide elistic modeling of get complexity. The pmetes we choose to chcteize ou initil configutions e then (i) the tios of the semi-xes of lengths 1 3: the ellipsoids cn be polte, oblte, o tixil nd they e specified by the fltness pmete i = ( 1 3 ) - 1; (ii) the initil viil tio bot = Kot W0, whee K ot is the kinetic enegy of the ottionl component of the motion, which hs n ngul velocity independent of dius (i.e., solid body ottion) nd pllel to the shotest semi-pincipl xis, nd W 0 is the initil gvittionl potentil enegy. 5 We hve exploed lge pmete nge in this fmily of initil conditions, extending down to bot =-1, which, lthough stictly viil, is well out of equilibium fo the chosen velocity distibution. We follow the evolution unde self-gvity until time t» ( ) t d whee t d is the chcteistic timescle fo thei men-field evolution defined s t d p =, ( 1) 8GM 3 3 Benhiem, Joyce, & Sylos Lbini whee M is the initil mss nd G is Newton s constnt. All simultions 6 e pefomed fo N = 10 5 pticles, using the gdget- code (Spingel et l. 001), dopting focesmoothing tht is ppoximtely one-tenth of the initil men intepticle seption. In this ppe we epot in detil esults fo just one chosen simultion, whose fetues e epesenttive of this clss of models. Gete detils on numeicl issues nd nlyses of the esults fo bod epesenttive nge of these initil conditions, nd lso fo nge of pticle numbes extending up to n ode of mgnitude lge, will be povided in septe ppe (D. Benhiem et l. 017, in peption). 3. Results 3.1. Thee-dimensionl Popeties We obseve, s expected given the chosen initil velocity distibution nd nomliztion, significnt contction nd subsequent e-expnsion of the system on timescle t ~ td. Associted with this behvio is, s nticipted, lso stong injection of enegy into significnt fction of the pticles, which e those initilly locted futhest fom the cente (i.e., close to the longest semi-pincipl xis) nd which pss though the cente of the stuctue ltest duing the collpse. Coespondingly, we obseve n mplifiction of the sptil symmety duing this phse (with, in pticul, moe pid contction long the shotest semi-pincipl xis). In ddition to these fetues, which hve been studied extensively in pevious woks (Benhiem & Sylos Lbini 015, 017), we find tht these systems e qulittively chcteized in thei oute pts by spil-like stuctue, with ich viety of foms see Figue 1 nging fom some qulittively esembling moe gnd design spils, nd othes esembling bed spils, 5 Becuse the foce-smoothing t smll scles is fcto of 10 smlle thn the initil intepticle distnce, the diffeence between W 0 (computed using the Newtonin potentil) nd the Clusius viil tem (computed using the exct foces cting on pticles) is negligible. 6 See Joyce et l. (009), Sylos Lbini (01), nd Benhiem & Sylos Lbini (015, 017) fo detils.

3 The Astophysicl Jounl, 851:19 (10pp), 017 Decembe 10 Benhiem, Joyce, & Sylos Lbini Figue 1. Configutions esulting fom fou diffeent initil conditions. Figue. Configutions esulting fom fou diffeent times (see the lbels). The solid line in the uppe left pnel coesponds to the initil ellipsoid. nd even flocculent spils in some cses. 7 Detiled nlysis of the evolving configutions confims tht the emegence of this sptil ogniztion ssocited with velocity distibution with vey specific chcteistic popeties tht we will descibe below is indeed the esult of the injection of enegy into some of the mss ound the time of mximl contction, which gives it lge dil velocities in ddition to the initil ottionl motion. As nticipted bove, we focus hee, fo simplicity, on the detiled nlysis of just one specific initil condition, with i = 1 nd = 3 (i.e., polte initil ellipsoid),nd bot =-1.0.We choose this cse becuse, even if it coesponds to cse tht is not so f out of equilibium nd chcteized by less violent contction nd expnsion, it poduces stuctue tht is fily typicl of ll cses. Shown in Figue e configutions of the evolved configution t diffeent times 8 pojected on the plne othogonl to the initil shotest semi-pincipl xis, long which the stuctue is (s expected) vey flttened in extent comped to the obseved pojection: digonlizing the ineti tenso to detemine the pincipl xes nd eigenvlues, we find typicl offset of couple of degees fom the initil xes, but much lge tio fo the eigenvlues, coesponding to fltness pmete i» 3, while the coe is tixil with fltness pmete i» 1 nd coesponds to tixil ellipsoid. We note tht, once fomed, the spil-like ms expnd dilly, slowly chnging shpe. Indeed, the velocity field of the pticles in the oute pt of the object is lmost dil nd diected outwd (see Figue 3). Figue 4 shows the density pofile n(), the velocity pofile v (), nd the enegy pofile ( ) computed s veges in dil bins of constnt logithmic width. Duing the time evolution, the oute til of n() is stetched to lge nd lge distnces. In genel, when the system contction duing the collpse is stong enough to poduce lge chnge of the pticle enegy 7 Hee, nd in the following figues, we use units of length in which 3 = 1, nd units of time in which t d = 1; enegies e given in units in which GNm 3 = 1. 8 See goo.gl/l1frzz fo the full movie of the time evolution. Figue 3. Configution t t = 5: ows e popotionl to velocities. distibution, the til of the density pofile is well fit by powe-lw behvio with n() ~ -4 (Sylos Lbini 013). Coespondingly the velocity nd the enegy pofiles lso extend to lge nd lge scles. At the lgest dii, s indicted by the vege vlue ( ), pticles e unbound (with > 0), while in the coe egion pticles e stongly bound (i.e., ò below 1.5); thee is then n extended intemedite egion in which mny pticles e mginlly bound (i.e., 0 > > -0.5). The enegy distibution P( ) t two diffeent times (t = 5, 50), togethe with tht of the initil conditions, is shown in Figue 5:wenotethtlgechngeofP( ) hs occued duing the gvittionl collpse of the cloud t» t d while t lte times the shpe of the distibution emins ppoximtely the sme. 3

4 The Astophysicl Jounl, 851:19 (10pp), 017 Decembe 10 Benhiem, Joyce, & Sylos Lbini Figue 4. Fom top to bottom: (i) density pofile, (ii) velocity pofile, nd (iii) enegy pofile, t two diffeent times: t = 5 (blck dots) nd t t = 45 (ed dots). Figue 5. Enegy distibution t t = 0 (geen),tt = 5 (blck) nd t t = 50 (ed). The uppe pnel of Figue 6 shows the vege, in spheicl shells of dius, denoted á ñ, of the dil component v = v of the velocity, nd of the tnsvese velocity v = v t, defined pllel to the ngul momentum eltive to the oigin (t the cente of the stuctue). Thus, in pticul, coheent ottion of the shell in plne coesponds to á ñ = áv t ñ. v t Figue 6. Configution t t = 45. Uppe pnel: components of pticle velocities veged in spheicl shells s function of dius. Middle pnel: nisotopy pmete b ( ). Lowe pnel: mss estimted fom the velocity ssuming sttiony cicul obits, nd the ctul enclosed mss. The middle pnel of Figue 6 shows the velocity nisotopy b = - á vt () 1 ñ. áv ñ Finlly, the lowe pnel of Figue 6 shows v G, the mss tht would be enclosed inside this dius if the motions wee puely cicul nd the mss distibution spheiclly symmetic, nd the mss M( < ) ctully enclosed inside the dius. Accoding to the behvios obseved, we cn divide the stuctue into thee egions: (i) n inne pt (R1) in which, s á v ñ áv ñ, t thee is no significnt net ottion, nd given tht b» 0, the velocity distibution is close to isotopic; (ii) n intemedite nge of dii (R), extending ove bout decde, in which β devites stongly fom zeo s net coheent ottionl motion develops nd domintes t lge dii, i.e., á v ñ» ávñ á v ñ; t t coespondingly (lowe pnel of Figue 6), thee is good geement between the estimted nd ctul enclosed mss in this egion; nd (iii) n oute egion (R3) in which the ottionl motion of the pticles is still coheent, but dil motions, with lmost negligible dispesion, e now pedominnt, i.e., á v ñ»á v ñ» áv ñ. Region R3 is lso chcteized clely by the behvio of the estimted enclosed mss, which getly oveestimtes the ctul enclosed mss. This eflects the fct tht the mss is wekly bound o even unbound the thn bound on cicul obits. Mesuement of the pticle enegies (see Figue 4) shows tht the tnsition fom R to R3 is indeed ppoximtely tht fom unbound to bound pticles, nd tht in the oute pt of R3 ll pticles e unbound. Indeed, symptoticlly the knee between the two egions is pecisely the tnsition fom bound t 4

5 The Astophysicl Jounl, 851:19 (10pp), 017 Decembe 10 to unbound obits, with the shell with á v ñ» 0 coesponding to pticles with zeo enegy. At lge distnce in R3 we hve, coespondingly, line gowth with distnce of the dil velocity tht is simply eflection of the bllistic dil motion. Thus, when we study these cuves s function of time, egion R1 nd the inne pt of R e sttiony to vey good ppoximtion, while the boundy with R3 popgtes out pogessively nd the size of R3 itself gows linely in time, with the mximl velocity emining fixed. These ngle-veged dt do not give infomtion bout the ngul dependence of the dil velocities in pticul, which e vey non-tivil: the pesence of the spil-like stuctue visible in Figue is eflection of the fct tht the pticles tnsvese motions become coelted with thei dil velocities becuse of the ppoximte consevtion of ngul momentum (nd enegy) of the ejected pticles, which, once outside the coe, move in n ppoximtely sttiony centl potentil. The pticles foming the spil stuctue pefeentilly hve dil velocity oiented long diections close to the initil longest semi-pincipl xis, nd the stuctue is elongted the most long the diections in which the dil velocities e mximl. Clely, the pecise fom of the spil stuctue depends diectly on the dispesion of the enegies of the high-enegy pticles t the time of collpse comped to thei tnsvese velocity t this time (nd thus, in pticul, on the pmete b ot ). The non-sttiony ntue of the stuctue lso mnifests itself in the evolution of the fom of the spil stuctue. In pticul, it becomes moe elongted (nd less xisymmetic) in time. 3.. Estimtion of Typicl Length/Time Scles Even if ou models e too simple nd idelized to be meningfully confonted with obsevtions in ny get detil, we cn conside the qulittive comptibility of the fetues of the mss distibutions geneted with the obseved popeties of el stophysicl systems. In pticul, we focus hee on the pimy stophysicl motivtion fo ou study spil glxies lthough, s noted, sevel othe pplictions could lso be exploed. Fo ny such compison we evidently need to ppoximtely elte the scles of ou toy model to physicl scles. Being in mind tht the typicl scle of obseved ottion velocities in disk glxies is 00 km s 1,wedefine the dimensionless pmetes s follows: 3 td v = km s 1. We cn then wite 00 v00 3» tgy kpc, n whee n is the numbe of dynmicl times in ou simultion nd t Gy is its dution given in billion of yes. Thus, fo n» 50, s in Figue, nd tking tgy ~ 1, which coesponds to mss (by using Eqution (1)) p M =» 1011M, 8Gt 3 3 d we hve tht egion R 1 extends to ~ kpc, nd egion R extends to ~50 kpc. Thus, in ode to hve stuctue tht would possibly be comptible with the typicl size of spil glxies, we need to ssume tht the collpse pocess tht Benhiem, Joyce, & Sylos Lbini geneted the disk nd ms occued much moe ecently thn the fomtion of the oldest sts in these glxies (with n ge 10 Gy). This is vey diffeent fom the usul hypothesis tht the disk, nd its spil stuctue, e t lest s old s the oldest sts. Fom the obsevtionl point of view, howeve, thee is no definitive evidence estblishing the ge of spil ms; sevel obsevtionl studies hve suggested tht spil ms e not long-lived (Elmegeen et l. 1989; Vogel et l. 1993; Tully & Veheijen 1997; Heny et l. 003) Chcteistic of Spil Ams We note tht the ms fomed in ou models e lwys tiling. This is simple consequence of the ppoximte consevtion of ngul momentum fo the outgoing pticles, which mens tht the tnsvese components of thei velocities decese with thei dil distnce. Although, s mentioned, ich viety of foms of the ms cn be obtined with diffeent initil conditions, two dominnt ms s in ou chosen simultion e vey esily poduced, with pitch ngles of the ode of tens of degees. Thus, ou model ntully epoduces vey common fetues of spil glxies, which e vey difficult to explin within the much exploed fmewok of density wve theoy (Dobbs & Bb 014), lthough density vitions ssocited with spil ms in ou models e lge thn they e in elity Appent Velocity Mps Let us now conside the comptibility of the lge-scle motions of ou geneted mss distibutions with the obseved ppent motions in disk glxies. Depending on the initil conditions we choose, the detils of the kinemtic popeties will chnge (e.g., the exct dil dependence of the velocities), but it is geneic popety of this mechnism of genetion of the spil stuctue tht thee is cle tnsition fom pedominntly ottionl motion to pedominntly dil motion, the ltte being in the outemost pts the bllistic motion of feed pticles. This is the cse simply becuse the pticles tht e futhest fom the cente t long times e unbound o vey loosely bound outgoing pticles tht hve lost lmost ll thei tnsvese velocity becuse of ngul momentum consevtion. Let us focus on this chcteistic fetue. Decdes of study of vious diffeent obsevtionl tces of the velocity fields povide stong evidence fo pedominntly ottionl motions in disk glxies (Sofue & Rubin 001). Fo wht concens ou Glxy, in which ppent motions hve been mesued ove fou decdes in scle (Sofue 017), the ngul dependence of the pojected velocities, infeed fom HI emission in pticul, shows convincingly tht the motion of the disk is vey pedominntly ottionl up to scle of ode 15 kpc (Klbel & Dedes 008; Sofue 017): s mentioned bove, such coheent ottion is lso chcteistic of the egion R in ou models. Fo this eson, the key obsevtion thus concens the ntue of the motion t lge distnce, i.e., >15 kpc, in ou Glxy. In this espect it is inteesting to note tht thee is nevetheless lso evidence fo significnt coheent dil motions beyond few kpc nd incesing with dius (López-Coedoi & González- Fenández 016). Beyond this scle the constints e much weke, but in the ne futue mesuements fom the GAIA stellite (Gi Collbotion et l. 016) will mke it possible 5

6 The Astophysicl Jounl, 851:19 (10pp), 017 Decembe 10 to distinguish the ntue of the motions t much lge scles. In ddition the GAIA stellite will be ble to shed light on the ntue of hype-velocity sts tht e unbound fom the Milky Wy nd shows supising nisotopic distibution (Bown 015). A popultion of such sts might possibly coespond to ejected pticles in ou models. Let us now conside constints on velocity fields fom extenl disk glxies. In this cse ppent (LOS) velocities e pobed obustly out to scles of sevel tens of kilopsecs, nd in some cses even lge (Sofue 017), but the stength of evidence fo ottion depends on the scle nd wekens t lge scles. These mesuements e both one-dimensionl (i.e., long the mjo xis of the obseved glxy) nd twodimensionl (mpping out the full pojected velocity field). The fome mesuements povide diect mesue of ottionl velocities, but only on the ssumption tht the glxy is in fct otting disk: in this cse the mjo xis of the pojection (which is n ellipse) is othogonl to the LOS, nd thus motions pllel to the LOS must be ottionl. In ou models, the mss distibution is not disk indeed it is clely non-xisymmetic t lge dii nd, futhemoe, s we hve noted, thee is intinsiclly stong coeltion of the diection of the oute dil velocities with the intinsic longest semi-pincipl xis. As esult, we show below tht thee is geneiclly contibution, which my be vey lge, long the pojected mjo xis. In ou models, s esult, even t length scles whee the motion is puely dil, we will infe non-tivil ottion cuve fom one-dimensionl mesuement. Fo two-dimensionl dt the evidence fo pedominnt ottion (nd the infeed ottion cuves) is bsed on the qulity of best fits to otting xisymmetic disk models povided by two-dimensionl dt. In pticul, two-dimensionl velocity mps of numeous glxies show the ptten distinctive of otting disk: the lignment of the kinemtic xis long which thee is mximl vition of the pojected velocities with the pojected mjo xis. Such lignment is, howeve, f fom pefect nd vey significnt ngul offsets e fequently obseved (nd ttibuted to the beking of xisymmety by bs). Futhemoe, vey significnt esiduls e typiclly mesued in such fitting pocedues typiclly of the ode of 30% o even lge nd these e ttibuted to dil motions (see, e.g., Eoz-Fee et l. 015). To qulittively evlute whethe the dil motions tht e dominnt in the oute pts of the spil stuctue in ou models cn be stongly excluded by obsevtions, s one might nively expect, we hve thus exmined whethe the pojected motions of ou toy glxies cn povide fits to otting disk models of compble qulity to those povided by the obseved glxies. To do so, s detiled in the Appendix, using ou distibutions, we hve geneted the pojected LOS velocity mps v los of ndom obseves, chcteized by two ngles: the inclintion ngle i, defined s the ngle between the vecto u o giving the oienttion of the obseve s LOS nd the vecto u g in the diection of the shotest semi-pincipl xis of the model glxy, nd n ngle j defined s the ngle between the pojection of the LOS in the plne of the glxy nd its longest semi-pincipl xis. To fit the esulting two-dimensionl mp with otting disk model, we detemine the velocity s function of distnce long the xis of mximl vition, nd use it s the input ottionl velocity fo otting disk, fo which we nlyticlly detemine the pojection. Shown in Figue 7 e the pojected velocity mps fo the sme Benhiem, Joyce, & Sylos Lbini simultion nlyzed bove, fo n obseve with i = 30 nd j = 30. The mps hve been veged on gid of size 64 (mimicking the finite esolution of mesued mps); the diffeent pnels show the following: (i) the two-dimensionl pojection of the mss distibution, with the kinemtic xis nd the mjo xis of the pojection indicted; in this cse the ngle between the two xes is bout 40 ; (ii) the two-dimensionl LOS velocity dispesion mp; the lgest dispesion is in the coe whee the velocities e isotopic; (iii) the two-dimensionl LOS velocity mp; (iv) the two-dimensionl LOS velocity mp in which the dil velocities hve been emoved, illustting tht the motions e indeed vey pedominntly dil; (v) the two-dimensionl LOS velocity mp of the best-fit otting disk model (this is obtined using the onedimensionl LOS velocity pofile long the estimted kinemtic xis); (vi) the two-dimensionl LOS velocity esidul mp. Figue 8 shows, espectively, the one-dimensionl LOS velocity pofile long the kinemtic xis nd long the xis othogonl to it (uppe pnel) nd (lowe pnel) the mss estimted by ssuming tht the velocities e cicul, nd the ctul mss (i.e., using Eqution (7)). We then exploed (see Figue 9) the full nge of i nd j. Only fo j vey close to p (i.e., n obseve with n LOS lmost exctly othogonl to the xis long which dil velocities e mximl) do we fil to obtin fit to otting disk model with esiduls comptible with the level epoted in the litetue fo such fits pplied to obsevtionl dt. These esiduls e smll in ll cses, i.e., of the ode of 10% 30%, except fo j 90, in which they cn be s high s 50% 70%. In these imges one cn discen clely tht ou model glxies e non-xisymmetic t lge dii, nd s we hve noted, thee is stong coeltion between the diection of the oute dil velocities with the intinsic longest semi-pincipl xis: the velocities in the oute pts of the stuctue e dil nd vey pefeentilly oiented long the xis, which is significntly elongted in the stuctue. As esult thee is geneiclly contibution fom these dil velocities long the pojected mjo xis. In ddition, except fo vey smll inclintion ngles, the pojection of the theedimensionl longest semi-pincipl xis is typiclly vey close to the mjo xis of the pojected imge, nd the lge dil velocities poject out thei components long this ltte xis. Thee is thus in pctice ough degenecy between otting disk models with significnt but sub-dominnt dil motions nd nonxisymmetic models with specific ptten of dil velocities like the one in ou models. This is vey clely illustted by comping in ech cse the mp in which the thee-dimensionl dil velocity is set to zeo nd the mp of the best-fit ottionl model: despite the fct tht most of the signl t lge scles comes fom the dil velocities, they cn be fit quite well by the ottionl model. The eson fo this supising esult is the stong coeltion in the lignment of the kinemtic xis nd the longest semipincipl xis of the pojected distibution, which is chcteistic of ou out-of-equilibium stuctues: s we hve seen, the velocities in the oute pts of the stuctue, which we e esolving in these mock mesuements, e dil nd vey pefeentilly oiented long the xis, which is significntly 6

7 The Astophysicl Jounl, 851:19 (10pp), 017 Decembe 10 Benhiem, Joyce, & Sylos Lbini Figue 7. Pojection fo i = 30, j = 30 (fom top to bottom). Left pnel: pojection of the object on the obseve s sky; the kinemtic xis (ed) nd the mjo xis (blck) e shown. Middle pnel: two-dimensionl LOS velocity dispesion mp. Right pnel: two-dimensionl LOS velocity mp. Left pnel: two-dimensionl LOS velocity fo the cse in which the thee-dimensionl dil velocity hs been set to zeo. Middle pnel: two-dimensionl ottionl mp deived fom the LOS velocity pofile. Right pnel: two-dimensionl esidul mp. elongted in the stuctue. In pojection the mjo xis typiclly emins vey close to this xis othe thn fo vey specific obseves, looking long the xis with vey smll inclintion ngles nd the lge dil velocities poject out thei components long this xis. A much moe detiled nd sophisticted nlysis of obseved pojected velocity mps of spil glxies would evidently be equied to estblish o exclude thei possible comptibility with velocity distibutions qulittively simil to those in ou models, i.e., non-xisymmetic distibutions with pedominntly dil velocities vey non-tivilly coelted with the sptil distibution. As we hve illustted with ou models, the motions in the oute pts of glxies e in fct pedominntly dil; thee is no need to invoke dk mtte hlo to explin them. Indeed, s illustted in the lowe pnels of Figue 6 nd of Figue 8, the mss estimte using the hypothesis of ottionl motions leds to n infeed mss tht gows stongly with dii, while the ctul enclosed mss does not gow t ll Flt Rottion Cuves nd Coeltion between the Centipetl nd Gvittionl Acceletion We conclude by speculting on two futhe impotnt obsevtionl esults bout velocity fields, nd thei possible explntion within the fmewok suggested by ou models. One of the noted popeties of ottion cuves of spil glxies is tht they e typiclly flt s function of scle t the lgest scles pobed by obsevtion, lthough get viety of behvios e in fct obseved in individul glxies (see, e.g., Sofue 017). Ou models e not pedictive in this espect: we cn obtin vey diffeent behvios depending on the nge of scle consideed, nd notbly whethe we ssume the egion obseved coesponds to R o R3. Futhemoe, the pecise functionl dependence on scle my be vey diffeent if we modify, fo exmple, the dil dependence of the initil ngul velocity. We note, howeve, tht, if we conside the egion R3, in which dil motions dominte, the ottion cuve (infeed by supposing the pojected motions to ise fom otting disk) will pogessively fltten in time: s the velocities e essentilly bllistic the sme velocity nge extends ove nge of scle, which gows monotoniclly with time. In this hypothesis of puely dil velocities with n ppoximtely constnt (i.e., vey slowly incesing) mplitude, we note finlly tht one my lso obtin, vey tivilly in models like ous, the obseved phenomenologicl eltion, c µ g (), whee c is the centipetl cceletion infeed fom the ppent motions, nd g() is the gvittionl cceletion of the visible byonic mss (see e.g., McGugh et l. 016),which lso undelies the so-clled Modified Newtonin Dynmics (Milgom 1983, 016). Indeed, scle-independent dil motions would give n infeed scle-independent ottion cuve, nd vmx thus c» whee v mx is the infeed constnt velocity of ottion, while g ()» GMc,wheeM c is the mss in the viilized coe. Thus, whee c vmx»» 0 g() 0 4 vmx = GM. c The obseved ppoximte constncy of 0 fo diffeent 4 systems then coesponds to v mx µ M c, i.e., the Tully Fishe eltion. 7

8 The Astophysicl Jounl, 851:19 (10pp), 017 Decembe 10 Benhiem, Joyce, & Sylos Lbini esiduls ttibuted to dil motion tht e vey significnt but of the ode typiclly found in fitting otting disk models to obsevtions. This suggests the possibility tht these motions could be explined without invoking eithe dk mtte o modifiction of Newtonin gvity, which e unvoidble if these glxies e modeled s sttiony nd otting. Rthe, these motions might be consistent with the puely Newtonin gvittionl dynmics of the visible mss if the oute pts of the glxy e f fom sttiony nd the motions e pedominntly dil nd sptilly coelted in nonxisymmetic distibution, the thn ottionl. Insted of poviding single pedictive model, we hve opened Pndo s box of models, diffeent fmewok of completely non-sttiony mss distibutions tht must be comped in much gete detil with obsevtions. Any such model is obviously lso vey simplistic, not just becuse of the ideliztion of the initil conditions but lso in tht it neglects eveything but gvittionl dynmics. Any detiled quntittive model will of couse necessily need to conside moe complex initil conditions nd lso incopote non-gvittionl physics. Thee e othe obvious ppent shotcomings of the toy model. Fo exmple, (i) spil ms coespond to modest vitions in mss density, nd (ii) the timescle fo collpse, s we hve discussed, must be ssumed to be shot comped to the ges of old sts. The fome my plusibly be elted to the low mss esolution we hve used, while the ltte constint my chnge in moe complex initil conditions. Nevetheless we believe it is emkble nd tntlizing tht the simple fmewok we hve discussed poduces stuctues being so much qulittive esemblnce to stophysicl objects, nd suggesting the possibility of diffeent nd simple explntion fo thei obseved pojected motions. Figue 8. Pojection fo i = 30, j = 30. Uppe pnel: LOS velocity pofile long the kinemtic xis nd long the xis pependicul to it. Bottom pnel: tio between the mss estimted fom the LOS velocity (ssuming it to be cicul nd sttiony) nd the ctul mss. 4. Discussion In summy, we hve shown, using simultions of evolution fom vey simple toy initil conditions, tht tnsient spil-like stuctue my be geneted in the f out-of-equilibium evolution of elxing self-gvitting system. As will be detiled futhe in fothcoming wok (D. Benhiem et l. 017, in peption), the sptil ogniztion in spil-like stuctue ises dynmiclly s pticles tht gin significnt enegy duing n initil collective contction nd expnsion of the system move outwd, with the moe enegetic pticles losing thei tnsvese motion fste. The mechnism is completely diffeent in ntue fom the usul petubtive mechnisms widely studied to explin such stuctue. Despite the unelisticlly simplified ntue of the models, we hve gued tht qulittive compison with obsevtionl dt is possible: ou models show tht the mechnism will genete stuctues velocity fields tht hve vey chcteistic behvio. This is tnsition to pedominntly dil motion with vey smll dispesion in the outemost pts. Supisingly, we hve found tht the pojected motions of these egions cn typiclly be quite comptible with otting disk model, up to The utho of this wok wee gnted ccess to the HPC esouces of The Institute fo Scientific Computing nd Simultion finnced by Region Ile de Fnce nd the poject Equip@Meso (efeence ANR-10-EQPX- 9-01) oveseen by the Fench Ntionl Resech Agency (ANR) s pt of the Investissements d Aveni pogm. Appendix We detil hee how we constuct the pojected velocity mps epoted in Section 3 fom ou simulted mss distibutions. This pojection is defined fo ndom obseve t infinity. It is convenient, in ode to undestnd the dependence on the oienttion of the obseve s LOS, to define this oienttion with espect to the pincipl xes of the mss distibution. Hving done so, it then stightfowd to detemine the pojected velocities s function of this oienttion nd the components of the position nd velocity in the pincipl xes. A.1. Pincipl Axes We compute the ineti mtix of the mss distibution eltive to n oigin tken t the minimum of the gvittionl potentil. We then detemine its eigenvlues l i, whee l1 l l3, nd coesponding eigenvectos l i. The longest semi-pincipl xis is then designted by unit vecto u 1 pllel to l 1, the intemedite semi-pincipl xis by unit vecto u pllel to l, nd the shotest semi-pincipl xis by l 3. The plne of the glxy is then othogonl to l 3. We then 8

9 The Astophysicl Jounl, 851:19 (10pp), 017 Decembe 10 Benhiem, Joyce, & Sylos Lbini Figue 9. Residuls fo i = 60 nd diffeent vlues of j fom 0 to 80 in steps of 10. otte fom ou oiginl Ctesin xes (of the simultion) to detemine the components of the pticle positions, x i, nd thei velocities, v i, in the new bsis { u i }. A.. Oienttion of the Obseve Following stndd conventions (see, e.g., Beckmn et l. 004) we define the inclintion ngle i of the obseve s the ngle between his LOS nd vecto othogonl to the plne of the glxy, which we tke to be u 3. Futhemoe, s the glxy is non-xisymmetic bout this xis, we define n zimuthl ngle j s the ngle between the pojection into the glxy plne of the LOS nd the mjo xis. Thus, we wite the unit vecto pllel to the LOS s u = sin() i cos( j) u + sin() i sin( j) u + cos () i u, ( ) o 1 3 A.3. Detemintion of Pojected Velocities To define the xes giving the obseve s plne of pojection it is convenient fist to define the set of xes ux = sin( j) u1 - cos ( j) u, u = cos( j) u + sin( j) u ( 3) y 1 in the plne of the glxy. The vecto u y is thus pllel to the xis of the pojection in the plne of the glxy of the obseve LOS, while u x is the xis in the plne of the glxy othogonl to the obseve LOS. The pojected plne, othogonl to the LOS, is then spnned by the unit vectos u = u, u = u u. ( 4) x x y o x Using the expessions bove, little lgeb gives u1 = sin( j) ux + cos( j) cos( i) uy + cos( j) sin( i) uo u =- cos( j) ux + sin( j) cos( i) uy + sin jsin( i) uo u =- sin() i u + cos () i u. ( 5) 3 y 0 The position coodintes ( x, y ) of the pticles in the plne of pojection, nd pojected velocity v los = v u 0, cn then be clculted, fo ny given obseve (i, j), s x = x1sin( j) - xcos( j) y = x1cos() i cos( j) + xcos() i sin( j) - x3sin() i v = v sin() i cos( j) + v sin() i sin( j) + v cos () i. ( 6) los 1 3 9

10 The Astophysicl Jounl, 851:19 (10pp), 017 Decembe 10 A.4. One-dimensionl Appent Velocity Pofiles Most obsevtions of ppent velocities e not fully twodimensionl, but given long specific xis(coesponding to the oienttion of the slit used fo the spectogphic mesuements). In ode to obtin such one-dimensionl velocity pofiles we define two such slits: one ligned pllel to the kinemtic xis, i.e., the xis long which thee is the mximum gdient of the LOS velocity (detils below), nd one othogonl to this diection. We hve lso consideed pojections long the mjo xis nd mino xis of the pojected distibution (defined following pocedue nlogous to tht descibed bove fo the thee-dimensionl cse). The slit is ssumed to be ectngul, of width Δ whichissmllfctionoftheminoxis.fom these LOS velocity pofiles long the kinemtic xis v los (R), we estimte the mss M c (R) enclosed in the dius R ssuming tht pticles e in sttiony cicul obits s vlos( R) R Mc ( R) =, ( 7) sin() ig whee the inclintion ngle is estimted fom the pojection s descibed below. A.5. Velocities fo Rotting Disk Model If one models glxy s disk, the pojected LOS velocities cn be witten (see, e.g., Beckmn et l. 004) s v (, f) = v sin( i) cos( q) + v sin( i) sin ( q), ( 8) los q whee (, f) e pol coodintes in the plne of the pojection, with f defined eltive to the xis othogonl to the obseve LOS (i.e., pllel to the xis u x defined bove), ndv θ nd v R e the components of the velocity field given in pol coodintes ( R, q) in the plne of the glxy (with θ defined eltive to the sme xis u x, which is lso in the plne of the glxy). The pol coodintes e elted by the tnsfomtion tn( q) = tn( f) cos( i) R = cos( f) cos ( q). ( 9) Fo puely otting xisymmetic model, v R = 0 nd vq = vq( R ). The kinemtic xis is tht long which thee is mximl vition of the pojected velocity, i.e., q = f = 0. A.6. Fitting to Rotting Disk Model To fit ou pojected velocity dt to otting xisymmetic disk we fist estimte fom ou dt the oienttion of the xis, ssumed to coespond to the kinemtic xis. We detemine the kinemtic xis s the xis pssing though the cente of mss of the distibution nd long which the diffeence of the velocities t the two exteme points is mximl. While this xis must stictly be the mjo xis of the pojection if the undelying distibution is elly disk, this is genelly not the cse fo ou distibutions tht e not xisymmetic. Howeve, becuse in ou models the diections of the dil velocities e stongly coelted with the el thee-dimensionl mjo xis of the non-xisymmetic distibution (see below), the offset between the kinemtic xis nd the pojected mjo xis is, in fct, typiclly (i.e., fo lge fction of ndom obseves) quite smll. Such offsets e, indeed, typiclly seen in obsevtions (see, e.g., Eoz-Fee et l. 015). To find the best-fitting otting disk model, we need to detemine the inclintion ngle i: we do this by minimizing the R esiduls between the ottionl model, computed fo geneic i, nd the ctul dt on ech gid cell. To do so we compute fist, fo ech gid cell, lbeled by α nd centeed on pojected coodintes x, y, the pol coodintes s defined bove: = ( x ) + ( y ) f = ccos( x ) R = cos ( f) + sin ( f) cos ( i) q = c tn( tn( f ) cos ( i)). ( 10) Then, fo the given vlue of the inclintion ngle i, we use Eqution (8) (with v R = 0) to compute the LOS velocity of the ottionl model, denoted v los,model. Note tht in the cse whee the unpojected size of the glxy is lge thn the mximum distnce t which the LOS velocity pofile extends, we pefom line fit ove the lst five points of v los (R) nd then extpolte using this fit to highe dius. Finlly, in ode to get the best-fitting inclintion ngle, we minimize the sum of the esiduls in ll the cells with espect to i, i.e., å Residuls = v - v. ( 11) los los,model ORCID ids Fncesco Sylos Lbini https: /ocid.og/ Refeences Benhiem, Joyce, & Sylos Lbini Aseth, S., Lin, D., & Pploizou, J. 1988, ApJ, 34, 88 Aguil, L., & Meitt, D. 1990, ApJ, 354, 33 Bnes, E. I., Lnzel, P. A., & Willims, L. L. R. 009, ApJ, 704, 37 Beckmn, J. E., Zuit, A., & Veg Beltán, J. C. 004, LNEA, 1, 43 Benhiem, D., & Sylos Lbini, F. 015, MNRAS, 448, 634 Benhiem, D., & Sylos Lbini, F. 017, A&A, 598, A95 Boily, C., Athnssoul, E., & Koup, P. 00, MNRAS, 33, 971 Bown, W. 015, ARA&A, 5, 15 Chistiens, V., Csssus, S., Peez, S., vn de Pls, G., & Ménd, F. 014, ApJL, 785, L1 Combes, F. 00, NewAR, 46, 755 Dvid, M., & Theuns, T. 1989, MNRAS, 40, 957 Dobbs, C., & Bb, J. 014, PASA, 31, e035 Eggen, O. J., Lynden-Bell, D., & Sndge, A. R. 196, ApJ, 136, 748 Elmegeen, B. G., Seiden, P. E., & Elmegeen, D. 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