Astronomy. Astrophysics. Novel explanation for the shape of the lenticular galaxy bulge and its implication for red spiral galaxy evolution

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1 A&A 55, ) DOI:.5/4-636/9 c ESO 9 Astronomy & Astrophysics Novl xplanation for th shap of th lnticular galaxy bulg and its implication for rd spiral galaxy volution R. A. Schachar,G.G.Liao,R.D.Kirby, F. Kamangar 3,Z.E.Musilak,4, and G. Rosnstl 5 Dpartmnt of Physics, Univrsity of Txas at Arlington, Arlington, TX 769, USA -mail: ron@ras.com Dpartmnt of Mathmatics, Univrsity of Txas at Arlington, Arlington, TX 769, USA 3 Dpartmnt of Computr Scinc and Enginring, Univrsity of Txas at Arlington, Arlington, TX 769, USA 4 Kipnhur-Institut für Sonnnphysik, Schönckstr. 6, Friburg, 794 Grmany 5 Dpartmnt of Physics, Tulan Univrsity, Nw Orlans, LA 78, USA Rcivd 8 March 9 / Accptd 8 July 9 ABSTRACT Aims. According to Hubbl s classification schm, lnticular galaxis rprsnt an intrmdiat volutionary stp btwn lliptical and spiral galaxis. This volutionary path prdicts that th aspct ratios of both lnticular and spiral galaxis should b smallr than th aspct ratios of thir E6 or E7 lliptical prdcssors. In contradiction to this prdiction, obsrvation has dmonstratd that th aspct ratio of lnticular galaxis is largr than its immdiat lliptical prdcssor. In this papr, w suggst a novl xplanation for this inconsistncy. Mthods. Th approach dscribd in this papr is primarily basd on analytical mthods; howvr, som numrical mthods ar also usd. Rsults. Our ida coms from thortical and xprimntal rsults, which show that a small incras in th quatorial diamtr of an oblat sphroid with an aspct ratio.6 surprisingly causs its minor axis to also incras. W dmonstrat that th sam phnomnon occurs in th isodnsity contours of lliptical galaxis givn by Miyamoto & Nagai Miyamoto M., & Nagai R., 975, PASJ, 7, 533) and in a Maclaurin sphroidal mass in rspons to th gravitational forc gnratd by a circumfrntial quatorial disk. Conclusions. Th rsult of this papr is our xplanation for th transformation of a disky lliptical galaxy into a lnticular galaxy which in rspons to rotation and quatorial diamtr xpansion volvs into a rd spiral galaxy. This volutionary path is consistnt with th common nvironmntal location of disky llipticals, lnticular and rd spiral galaxis and xplains why lliptical galaxis ar gnrally E4. Th proposd volutionary path is opposit to th gnrally accptd formation of lnticular galaxis from th mrgr of spiral galaxis. Ky words. galaxis: bulg galaxis: lliptical, lnticular, cd galaxis: volution. Introction Lnticular galaxis, labld S in Hubbl s classification schm, form a transition btwn lliptical and spiral galaxis Hubbl 96, 936). Th fact that this transition is smooth, which mans that som S galaxis might wll b classifid as llipticals and othrs as spirals, has ld to controvrsy as to whthr lnticular galaxis ar mor closly linkd to lliptical or spiral galaxis van dn Brgh 98). Jans 98) postulatd that lnticular galaxis volv from lliptical galaxis as a consqunc of rotation Sandag 5). Consistnt with Jans postulat, Miyamoto & Nagai 975) showd that a family of dcrasing isodnsity contours with dclining aspct ratios can dscrib th morphological diffrncs of lliptical and spiral galaxis s also Satoh 98; Zhnglu ; Bas 8). Howvr, thr ar two basic problms with this volutionary path: First, spiral galaxis ar blur in color, signifying that thy ar youngr in ag than th rddr lliptical and lnticular galaxis. Rcntly, Bamford t al. 9) studid th morphology, color, nvironmnt and stllar mass of galaxis using data from ovr 5 objcts from Galaxy Zoo, th largst sampl of visually classifid morphologis yt compild. Intrstingly, thy dmonstrat th xistnc of rd spiral galaxis locatd in th sam nvironmnt as lliptical and lnticular galaxis and show that thr is an nvironmntal dpndnc for color byond that for morphology. This finding implis that th volution of spiral galaxis may dpnd on thir nvironmnt. Young blu spiral galaxis ar routinly obsrvd in lss dns nvironmnts whr thr ar fwr lnticular galaxis. Rd spirals ar locatd in dnsr nvironmnts whr thr ar many lliptical and lnticular galaxis. Dos this imply diffrnt volutionary paths for young and old spiral galaxis? Scond, it has bn obsrvd that th man aspct ratio, minor axis/major axis ratio, of lnticular galaxis is.5 ±.5 calculatd from Tabl of Michard & Marchal 994), which is gratr than E6 and E7 lliptical galaxis, that hav aspct ratios of.4 and.3, rspctivly. According to th Hubbl - Jan s schm E6 and/or E7 galaxis should volv by rotation into lnticular galaxis. But how can this occur whn th aspct ratio and thrfor th minor axis of lnticular galaxis is gratr than thir immdiat lliptical prdcssors? Th lattr qustion is addrssd by th gnral blif that lnticular galaxis form as a consqunc of th collision of spiral galaxis Abraham & van dn Brgh ; Bdrgal t al. 6); howvr, this hypothsis has not yt bn confirmd by N-body simulations Hrnandz & L 4; Mallr t al. 6). In addition, th complx rlationship btwn morphology,

2 64 R.A. Schachar t al.: Bulg of lnticular galaxis Fig.. Photograph of a stp profil incd by applying traction in th 8 dgr mridian to an oil film floating on watr. Th tractional pins xtnd into th oil film and not into th watr blow. color, stllar mass and th nvironmntal location of galaxis implis a gntl volutionary path, not violnt major mrgrs Bamford t al. 9). Th purpos of this papr is to suggst, consistnt with Bamford s t al. 9) data, that thr may b sparat pathways for th dvlopmnt of blu youngr) and rd oldr) spiral galaxis. This diffrntiation is basd on our xplanation for th incras of th minor axis of an lliptical galaxy as it transforms into a lnticular galaxy. Th ida is basd on th chang in shap that occurs whn th major axis of an ncapsulatd biconvx objct with a long oval cross sctional profil is incrasd 5%, countrintuitivly, th lngth of its minor axis also incrass, its cntral surfacs stpn, and its priphral surfacs flattn; i.., th objct morphologically obtains a stp profil. It has bn dmonstratd that this morphological stp profil occurs as a consqunc of minimization of surfac curvatur, a surrogat for bnding nrgy Schachar t al. 8). Th morphological stp profil has a rmarkabl rsmblanc to th dg-on isophotal shap of a lnticular galaxy bulg as shown by th stp profil incd in a floating oil film in rspons to quatorial traction Fig. ). Th floating oil film xprimnt is shownasanxamplofthffct of minimization of nrgy on objct shap and not to imply that th hydrodynamics, molcular forcs and surfac tnsion of th floating oil film ar dirctly applicabl to a rotating body of stars controlld by gravity. To dmonstrat that th sam phnomnon can b important for galaxy volution, w considr th shap changs that occur to isodnsity contours of lliptical galaxis givn by Miyamoto & Nagai 975) and a Maclaurin sphroidal mass in rspons to th gravitational forc of a circumfrntial quatorial disk. In both cass, similar morphological stp profils ar obsrvd. Th obtaind rsults ar usd to justify our suggstion that disky lliptical galaxis transform into lnticular galaxis, which thn volv into rd spiral galaxis as a consqunc of an incras in th quatorial diamtr of th lnticular galaxy s Scts. and 3). Our discussion and conclusions can b found in Scts. 4 and 5. Obviously, our goal is not to dvlop a dtaild modl of spiral galaxy volution, but only to dmonstrat that th phnomnon may b rlvant.. Transformation of galaxis with Miyamoto & Nagai s isodnsity contours into stp profils Whn a circumfrntial quatorial disk is addd to an lliptical galaxy, th disk applis a gravitational forc that xpands th galaxy in its quatorial plan. Th rdistribution of th galaxy s mass changs th ovrall shap of th galaxy so that it is in a minimal nrgy stat. To dtrmin th nw shap of th galaxy, th minimal nrgy stat of th mass distribution of th galaxy is simulatd by minimizing th curvatur of th galaxy s isodnsity contour. Using solutions of Poisson s quation, Miyamoto & Nagai 975) calculatd th isodnsity contours of a rotating lliptical galaxy with dclining dnsity distribution. Thy show that th incd dnsity chang can account for th morphological diffrncs btwn lliptical and spiral galaxis; howvr, thir analysis dos not prdict th xistnc of a lnticular galaxy btwn an lliptical and spiral galaxy with an aspct ratio gratr than both typs of galaxis. It has bn wll stablishd that most lliptical galaxis hav disks Bndr t al. 989; Rix & Whit 99; Jørgnsn & Franx 994; Magorrian 999; Cappllari 8; Krajnović tal. 8) and ths disky lliptical galaxis ar distinguishabl from lnticular galaxis Jaff t al. 994; Caon & Einasto 995; Katsiyannis t al. ; Kormndy t al. 9). W propos that th gravitational forc gnratd by th circumfrntial quatorial disk incrass th quatorial diamtr major axis) of th lliptical galaxy causing it to volv into a lnticular galaxy. To assss this possibility, w valuat th morphological changs that will occur to an lliptical galaxy whn its major axis is incrasd. W bgin by calculating th shap of an lliptical galaxy using cylindrical coordinats as givn by Miyamoto & Nagai s 975) ρr, z) = b M 4π ar + [q + 3 z + b ][a + z + b ] {R + [a +, ) z + b ] } 5 z + b ) 3 whr ρ is dnsity, M is total mass, and a and b ar positiv constants with dimnsions of lngth. W modify this quation by introcing dimnsionlss vrtical and horizontal coordinats, Z = z a and X = R a, rspctivly, and ltting q = b a so that ρx, Z) = q M X + [ + 3 Z + q ][ + Z + q ] 4πa 3 {X + [ +, ) Z + q ] )} 5 Z + q ) 3 whr ρ, M, a 3 and 4π ar incorporatd into a singl paramtr, P = 4πρa3 M and thn PX, Z) = q {X + [ + 3 Z + q ][ + Z + q ] } {X + [ +. 3) Z + q ] )} 5 Z + q ) 3 Sinc th man aspct ratio of lnticular galaxis is.5 and th Jans-Hubbl schm prdicts an E6 or E7 lliptical galaxy will convrt into a lnticular galaxy, shap paramtrs ar slctd to covr this rang of lliptical galaxis; i.., E7 to E4.3 aspct ratio.6). For ach lliptical galaxy th dnsity paramtr, P, is varid from. to.. Th curvatur of th isodnsity contour is givn by Z κx, Z) =. 4) + Z ) 3

3 To simulat th gravitational ffct of a circumfrntialquatorial disk, th major axis of th isodnsity contour of th lliptical galaxy is incrasd.5%. To nsur th nw galaxy shap is in th lowst nrgy stat, th calculus of variation is usd to minimiz th squar of th diffrnc of th curvaturs btwn th isodnsity contour of th lliptical galaxy and th nw galaxy contour [κzx)) κz o X))] ds. 5) Th problm is solvd numrically using standard optimization softwar with th following constraints: R.A. Schachar t al.: Bulg of lnticular galaxis 65 Arc lngth + Z o dx + Z dx n + Z o dx. 6) Ara Z o dx Z dx n Z o dx. 7) Surfac ara X + Z o dx X + Z dx n X + Z o dx, 8) Fig.. Simulation of th ffct of a circumfrntial quatorial disc on th shap of galaxy isodnsity contours solid curv) with aspct ratios =.6 and dnsitis P =.., which all chang into stp profils dashd curvs). whr. n.3 and volum is unconstraind. Th boundary conditions ar Z.5X o ) =, 9) dx dz =, ) Z= dz dx =, ) X= whr X o, ) is th intrcpt of th isodnsity contour with th major axis of th lliptical galaxy. Th obtaind rsults ar givn in Figs. 5, which show that in ach cas th minor axis of th galaxy isodnsity contour incrass as a consqunc of an incras in its major axis transforming th galaxy isodnsity contour into a stp profil. 3. Formation of a stp profil in a Maclaurin sphroid with a toroidal disk Th shap changs of a Maclaurin sphroidal mass in rspons to incrasing angular momntum hav bn usd to prdict th shaps of galaxis Sandag 5). With incrasing angular momntum th Maclaurin sphroidal mass bcoms mor oblat until it has an ccntricity of.8, which is quivalnt to an aspct ratio of.6. At this point, th Maclaurin sphroid is unstabl and with a dcras in angular momntum will bcom a Jacobi llipsoid Eriguchi & Hachisu 98; Dankova & Rosnstl 998). Altrnativly, w propos that if a Maclaurin sphroid mass has an aspct ratio.6 and has a circumfrntial quatorial disk, th Maclaurin sphroid will morphologically form a stp profil whil angular momntum rmains constant. Fig. 3. Simulation of th ffct of a circumfrntial quatorial disc on th shap of galaxy isodnsity contours solid curv) with aspct ratios =.5 and dnsitis P =.., which all chang into stp profils dashd curvs). 3.. Configuration Considr th symmtry axis of a surfac of rvolution alignd with th z-axis. Th surfac bounds a connctd rgion Ω that contains a total mass M in a volum V = volω). Th mass is distributd uniformly insid th surfac. Using cylindrical coordinats, th profil of th surfac is dfind by a function r = f z) for z m z z m. For ach point z along th symmtry axis insid th surfac, th polar coordinat rangs from r =, th symmtry axis, to r = f z), th boundary of th surfac. Th surfac of rvolution is symmtrical: f z) = f z)for z m z z m.

4 66 R.A. Schachar t al.: Bulg of lnticular galaxis Fig. 6. Rfrnc llips solid curv) and stp profil dashd curv). Fig. 4. Simulation of th ffct of a circumfrntial quatorial disc on th shap of galaxy isodnsity contours solid curv) with aspct ratios =.4 and dnsitis P =.., which all chang into stp profils dashd curvs). rigidly rotating mass M within th rgion Ω. Th kintic nrgy and slf-nrgy of th toroidal mass ar considrd constants. 3.. Volum of th Maclaurin sphroid To dtrmin th volum of th mass, imagin th mass is dividd into narrow cylindrical slics prpndicular to th symmtry axis. Th slic at position z along th axis has width dz and radius r = f z). Th slic s volum is π f z) dz and th mass s volum is th intgral zm zm V = π f z) dz = π f z) dz. ) z m For an oblat sphroid, th profil is f sph z) = a z /c, whr c = z m dnots th minor smi-axis lngth and a > c is th major smi-axis lngth. Th volum is V = 4πa c/3. Fig. 5. Simulation of th ffct of a circumfrntial quatorial disc on th shap of galaxy isodnsity contours solid curv) with aspct ratios =.3 and dnsitis P =.., which all chang into stp profils dashd curvs). Anothr mass M xt is distributd uniformly along th circumfrnc of a circl of radius R in th z = plan. This circl is cntrd at th origin, which is th gomtrical cntr of th symmtrical surfac. Th radius R >> f ) > z m >. Th xtrnal mass modls a toroidal galactic disk. Th total potntial nrgy U is th sum of two trms: ) th gravitational intraction nrgy U xt btwn th mass M and th xtrnal mass M xt and ) th gravitational slf-nrgy U slf of th mass. Th total kintic nrgy is just th kintic nrgy of th 3.3. Two-dimnsional stp profil and volum corrctions Using a rfrnc llips with a smimajor axis lngth a = arbitrary units) and smiminor axis lngth c = / and th stp profil gnratd by incrasing th major axis of th rfrnc llips %, th diffrnc Δ f btwn th stp profil s function, r = f z), and th llips s function, r = f sph z), ar plottd in Fig. 6. Th diffrnc function is approximatd by thr sgmnts Fig. 7). Δ f =.8.593z.5336z, r r sph = Δ f =.57z.664)/.956z), 3) Δ f 3 =.6.58 z).493, whr Δ f, Δ f and Δ f 3 ar valid in th rang z.53,.53 z.5 and.5 z.58, rspctivly. Th ara of on quadrant of th llips is πac/4 = π/8 =.393. Th diffrnc btwn th singl quadrant aras boundd by th stp profil and llips is zm f z)dz.5 + Δ f dz +.53 c f sph z)dz = Δ f dz Δ f 3 dz =.67. 4)

5 R.A. Schachar t al.: Bulg of lnticular galaxis 67 Fig. 7. Diffrnc function r r sph btwn th stp profil curv and th rfrnc llips, which is approximatd by thr sgmnts. Fig. 8. Diffrnc Δg btwn th squar of th stp profil s function and th squar of th llips s function, which is approximatd by thr sgmnts. Th fractional chang in th quadrant s ara is.%, a ngligibl shift Thr-dimnsional surfac of rvolution Th diffrnc btwn th volum of th stp profil and th volum of th llipsoid is ΔV = π zm f z) dz π c f sph z) dz. 5) Th diffrnc Δg btwn th squar of th stp profil s function and th squar of th llips s function is approximatd by thr sgmnts Fig. 8) Δg z) = z z z 6, 6) Δg z) = z z z 6, 7) Δg 3 z) = z, 8) whr z I = [,.39) for Δg z), z I = [.39,.5) for Δg z), and z I 3 = [.5,.58] for Δg 3 z). Th function Δg is Δgz) =Δg z)χ I z) +Δg z)χ I z) +Δg 3 z)χ I3 z), 9) Volum corrction Th volums V ɛ of th surfacs of rvolution, indxd by ɛ> and dfind by th cylindrical radial function rɛ z) = r sph z) + ɛδgz), ) whr th llipsoid s radial function is r sph z) = 4z ) χ Isph z), I sph = [,.5], 3) ar lss than th volum of th rfrnc sphroid V sph = π/3: V ɛ =.375 ɛ. 4) V sph To rst th volums qual to th llipsoid s valu, dfin th dilation r ɛλ z) = λr ɛ z/λ) 5) for λ. Th volum of th nw surfac of rvolution is V ɛλ = λ 3 V ɛ which is st qual to V sph Fig. 9). Thrfor, th volumcorrcting factor λ is λ =.375 ɛ) /3 = +.58ɛ +. 6) whr th charactristic function is χ I z) = {, z I,, z I. ) Th volum of th llipsoid is 4πa c/3 = π/3 =.94. Th diffrnc btwn th volums boundd by th stp profil and llips is.39.5 ΔV = π Δg dz + π Δg dz +π Δg 3 dz =.665. ) Th fractional chang in th volum is 3.% Kintic nrgy of th Maclaurin sphroid and th stp profil Whn th surfac of rvolution rotats rigidly about its symmtry z-axis, th momnt of inrtia is I = dz πr dr dr = πd dz rz) 4, 7) whr d = M/V is th uniform dnsity. Whn th surfac is a sphroid with smi-axis lngths, a = b and c, th momnt of inrtia is I sph = I a/r) = I ) /3, I = 5 MR, 8)

6 68 R.A. Schachar t al.: Bulg of lnticular galaxis Fig. 9. Ellips solid curv), stp profil dashd curv) and volumcorrctd stp profil dottd curv). whr R 3 = a c and th ccntricity is = c /a.thllipsoid has smi-axis lngths a = b = andc = /, ccntricity = 3/, R = /3, I = 5 M4 /3,andI sph = 5 M. For th volum-corrctd surfac of rvolution, th momnt is I = π dλ 4 dz r ɛ z/λ) 4 = π dλ 5 r ɛ u) 4 = π dλ 5 r sph u) + ɛδgu) π dλ 5 [ rsph 4 u) + ɛ r sph u) Δgu)]. 9) Whn ɛ =, th momnt of inrtia simplifis to th llipsoidal valu I sph = πd = πd / dzr 4 sph z) 4z ) dz = M 5. 3) Th momnt of inrtia of th volum-corrctd stp profil is I ɛλ = I sph.34 ɛ). 3) Th kintic nrgy is th squar of th angular momntum L dividd by twic th momnt of inrtia. Lt U dnot a convnint nrgy unit chosn blow to giv th gravitational slfnrgy of a sphr of mass M and radius R) and dfin th dimnsionlss angular momntum λ = L/ I U. Th kintic nrgy, in units of U, of a sphroid is T sph = L I sph U = λ ) /3, 3) and for th stp profil T ɛλ = T sph +.34ɛ). 33) 3.5. Slf-nrgy of th stp profil Th stp profil forms as a rsult of a small incomprssibl dformation of th sphroid. Th mass dnsity dr) quals a constant d for a position vctor r insid th stp profil and quals zro for r outsid th rgion Ω. Th mass dnsity of th llipsoid, d sph r), quals d whn r is insid th llipsoid and quals zro for r outsid th llipsoid. Th diffrnc btwn ths dnsitis is a function δdr) = dr) d sph r) which is zro xcpt for a narrow domain around th surfac of th llipsoid. Mor gnrally, considr a family of stp profils indxd by a ral numbr ɛ with th dnsitis d ɛ r) = d sph r) + ɛδdr). 34) At ɛ = th stp profil s dnsity d ɛ quals th llipsoid s dnsity d sph. Th slf-nrgy of a stp profil is U slf = G dɛ r)d 3 rd ɛ r )d 3 r r r, 35) whr G is th univrsal gravitational constant and r r dnots th distanc btwn positions r and r.toordrɛ th slf-nrgy is U slf G dsph r)d 3 rd sph r )d 3 r r r dsph r )d 3 r δdr)d 3 r ɛg r r. 36) Th first trm is th Maclaurin sphroid s slf-nrgy U sph and is a known quantity Chandraskhar 969; Dankova & Rosnstl 998), U sph = U /6 arcsin) ), 37) whr th ccntricity = c /a. Th slf-nrgy U of a rfrnc sphr of radius R and mass M is dtrmind by U = 3/5)GM /R,whrR 3 = a c. Th scond intgral may b xprssd in trms of th function dsph r )d 3 r U surf r) = G r r. 38) Thrfor, th stp profil s slf-nrgy is, to ordr ɛ, U slf = U sph + ɛ δdr) U surf r)d 3 r. 39) Sinc δdr) is zro outsid of a narrow rgion nar th llipsoid s surfac, th potntial nrgy U surf r) to th llpsoidal mass must b dtrmind only for points r nar th llipsoid s surfac. For th stp profil s surfac of rvolution, th intgral is rɛλ z) U slf U sph = d dz πr dru surf r) πd r sph z) dz r ɛλ z) r sph z)) U surf z), 4) whr, to obtain ovrall accuracy of ordr ɛ, itissufficint to valuat th potntial nrgy U surf z) at th surfac, Eq. 38). Th potntial nrgy at th surfac is th sum of two trms, a constant

7 R.A. Schachar t al.: Bulg of lnticular galaxis 69 trm and a trm proportional to z. Th intgral involving th constant trm is zro to volum consrvation. Lt us dfin ΔU U slf U sph and writ ΔU = πd dz rɛλ ) a r sph πgd) A ) c A 3 z = πd) a G A ) c A 3 dz λ rɛ sph) r z. 4) Th first trm in this intgral, to ordr ɛ, is dz λ rɛ z λ ) z = λ 5 rsph u) + ɛδgu)) u = rsph u) u + ɛ.59 rsph u) +Δgu)) u, 4) sinc λ ɛ. Thrfor, to ordr ɛ, th chang in th slf-nrgy is ΔU = πd) a G A ) c A 3 ɛ.59/3.9) = πd) G.74) ɛ =.95)U ɛ, 43) a whr A A c 3 = Potntial nrgy of th Maclaurin sphroid Th surfac of th llipsoid is dfind by th quation, = x /a + x /a + x 3 /a 3. Th gravitational potntial nrgy for th intrior and xtrior of a homognous llipsoid with smiaxis lngths a, a, a 3 is dfind by two quantitis: th gravitational potntial nrgy V and th gravitational slf-nrgy W Chandraskhar 969) Intrior points At an intrior point r = x, x, x 3 ), th potntial nrgy Chandraskhar 969) is 3 V = πgd I A l x l, 44) l= whr I = a l A l. For an oblat sphroid a = a > a 3 ) with ccntricity = a 3 /a, th indx symbols Chandraskhar 969) ar A = A = arcsin). 45) 3 A 3 = arcsin). 46) 3 For points intrior to th llipsoid s surfac, U surf = V Surfac points Th gravitational potntial at an llipsoid s surfac is a U surf = πgd A ) c + A 3 c a + A ) ) c A 3 z. 47) Points xtrior to th sufac At an xtrior point r = x, x, x 3 ), th potntial nrgy Chandraskhar 969) is 3 V = πgda a a 3 Δ x i a i + u, 48) λ i= whr Δ= a + u)a + u)a 3 + u). Th lowr limit λ of intgration is dtrmind implicitly by th quation 3 i= x i =. 49) a i + λ For points xtrior to th llipsoid s surfac, U surf = V. For oblat sphroids, th abov intgrals may b valuatd as follows: λ lim + λ λ lim + λ λ lim + λ π arccot Δ = Δ = a Δa + u) = Δa 3 + u) = π arccot / ) λ/a + a, 5), 5) λ/a + / )) λ/a + a 3 3 λ/a + a 5) 3 λ/a + ), Δa + u) = 3 a 3 λ/a + ), 53) 3 / )) π arccot λ/a + + Δa 3 + u) = 3 a 3 a 3 3 a 3 λ/a +, 54) λ/a + ). 55) Potntial nrgy to th xtrnal toroidal mass For a mass M xt distributd uniformly along a circl of radius R in th z = plan xtrnal to th llpsoid with linar mass dnsity M xt /πr, th gravitational potntial nrgy at a point r is U xt r) = GM xt πr π R dθ r r, 56) whr th intgration is ovr th circumfrnc of a circl of radius R, th points of which ar dnotd by r. In cylindrical coordinats r = r,θ,z), r = R,θ, ) and th Grn s function is r r = π rr imθ θ ) Q m / χ), 57) m=

8 6 R.A. Schachar t al.: Bulg of lnticular galaxis whr r + R χ = + z) 58) rr and Q m / is th half-intgr Lgndr function of th scond kind Cohl & Tohlin 999). Th intgrals for vry trm in th summation xcpt m = ar zro and th potntial simplifis to U xt r) = GM xt π Q / χ). 59) rr Th m = Lgndr function of th scond kind is Q / χ) = μkμ), whr K is th complt lliptic intgral of th first kind and μ = + χ = 4rR r + R ) + z. 6) Thrfor th potntial fild to th xtrnal mass is GM xt U xt r, z) = π Kμ). 6) r + R ) + z A usful approximation to th complt lliptic intgral of th first kind is π Kμ) μ) /4 π ) μ4 9μ μ μ ) For larg R th potntial fild may b approximatd by a Taylor sris in / R, w obtain U xt r, z) = GM xt + r 7 R R 3/ 6 r R +GM xt 7 3 r3/ + 5z 5r + R 5/ 4 R 3. 63) 3.8. Intraction nrgy btwn th Maclaurin sphroid and th xtrnal toroidal mass Th gravitational intraction nrgy btwn a sphroid and th toroidal mass is U sph tor = U xt r)dr)d 3 r 64) a c r /a = d πr dr dz[fr, z)] = 3GM Mxt ) a R 3 r dr[gr, u)], M whr GM xt Fr, z) = π Kμ), r + R ) + z 65) Gr, u) = sinh u) + rr and u = c r /a /R + r). tan u) R + r +, 66) Whn R is larg compard to th dimnsions of th llipsoid, th intraction nrgy U st U sph tor /U rcs to U st = GMM xt R U + 5π Mxt 6 γ /6 M.643 γ / Mxt ) ) 3/ R M R ) ) ) 5/ R R + O R = GMM xt R U ηγ / +, 67) whr γ = ) and th small prturbation paramtr for th xtrnal mass is η =.643 Mxt M R ) R R ) 3/. 68) 3.9. Intraction nrgy btwn th stp profil and th xtrnal toroidal mass Using th Taylor sris for th gravitational potntial, Eq. 63), th intraction nrgy U sph tor btwn a stp profil and th toroidal mass is rɛλ z) = d dz πr dru xt r) U sph tor = GMM xt R = GMM xt R GMM xt R π dgm xt R 3/ U M xt M U M xt M R rɛλ dz ) 3/ 7/6 R ) 3/ R 7/6 λ 7/ R [ r 5/ sph u) ɛr/ sph u)δgu) ] r 3/ dr + O ) R r 5/ ɛλ z)dz + GMM ) 3/ xt M xt R U / ɛ) R M R [ / 4z ) 5/4 dz + 5ɛ / ] 4z ) /4 Δg dz = GMM ) 3/ xt M xt R U / ɛ) R M R ɛ) = GMM xt ηu / ɛ).453ɛ) R = U s t +.9)ηU ɛ. 69) 3.. Equilibrium configurations of th rotating masss By minimizing th total nrgy, w dtrmin quilibrium configurations Total nrgy of th Maclaurin sphroid Th total nrgy E of an llipsoid of ccntricity is th sum of th kintic nrgy of rigid rotation and th gravitational slfnrgy, E = λ γ /3 /6 arcsin) γ, 7)

9 whr th nrgis ar in units of U and th angular momntum λ is in units of I U. Minimization of th nrgy dtrmins th Maclaurin sphroid xprssion for th angular momntum as a function of th ccntricity, λ Maclaurin = γ /6 [ 3 ) arcsin) 3 γ R.A. Schachar t al.: Bulg of lnticular galaxis 6 ]. 7) 3... Total nrgy of th Maclaurin sphroid + xtrnal toroidal mass Th total nrgy is E = λ γ /3 /6 arcsin) γ ηγ /. 7) Enrgy minimization dtrmins th angular momntum of th quilibrium stats λ = λ Maclaurin η. 73) γ5/ Not that th zro angular momntum quilibrium solution occurs at ccntricity > Total nrgy of th stp profil + xtrnal toroidal mass Whn th nrgy is minimizd for an llipsoid with ccntricity = 3/ plus an xtrnal toroidal mass, Eq. 73) dtrmins th angular momntum. If th llipsoid is dformd continuously into th stp profil whil th angular momntum is hld constant, th total nrgy changs from its sphroidal valu E Maclaurin to E stp profil = E Maclaurin.5.57η)U ɛ + 74) This xprssion is a Taylor sris which, for small η, dtrmins accuratly th drivativ of th nrgy of th stp profil as a function of ɛ. For larg valus of η, whichisa masur of th strngth of th intraction btwn th mass M and th xtrnal toroidal mass M xt, trms byond th first two in th xpansion of Eq. 63) must b includd in th drivation. Th stp profil has a lowr nrgy than th sphroid whn η<.8 transforming th disky lliptical galaxy into a lnticular galaxy. Th ngativ chang from th sphroidal nrgy is th rsult of an incras in th kintic nrgy that is offst by a slightly gratr dcras in th gravitational slf-nrgy. 4. Discussion Our rsults dmonstrat that th minor axs of both th isodnsity surfacs of lliptical galaxis with aspct ratios.6 and Maclaurin sphroidal masss will incras and morphologically form stp profils whn subjctd to a small incras in thir major axs in rspons to th gravitational forc of a circumfrntial quatorial disk. Th small displacmnt and consquntly minimal forc rquird to mak this chang may xplain th transformation of a disky lliptical galaxy into a lnticular galaxy. Disky lliptical galaxis can form as a consqunc of th collaps of an lliptical galaxy. During th collaps sufficint outward transfr of angular momntum is gnratd to proc a circumfrntial quatorial disk Larson 975). Altrnativly, disky lliptical galaxis may form from minor Fig.. A narly dg-on photograph of lnticular galaxy NGC 549 Courtsy of Sloan Digital Sky Survy) with suprposition of an oil film s stp profil outlin on th lnticular galaxy s bulg. mrgrs Bournaud t al. 7; Naab t al. 999) or rotation Jans 98). Disky lliptical galaxis appar to rprsnt a diffrnt track for galaxy volution Bndr t al. 989; Nito & Bndr, 989; Kormndy t al. 9). Sinc such a disky lliptical galaxy rotats rapidly Cappllari t al. 6, 7), its ccntricity will incras until it bcoms an E4, aspct ratio =.6. At that point, a small gravitational forc gnratd by th circumfrntial quatorial disk will incras both th major and minor axs of th lliptical galaxy transforming it into a lnticular galaxy Fig. ). This procss is mor fficint as th aspct ratio of th disky lliptical galaxy dclins blow.6 Schachar t al. 8). This may xplain why lliptical galaxis ar E7 and th majority ar E4 Crsci t al. 6; Hao t al. 6). According to th abov discussion, th transformation of a disky lliptical galaxy into a lnticular galaxy sms to b a plausibl procss. Morphological high-rsolution, multiwavlngth data from high-rsolution Hubbl Spac Tlscop HST) Advancd Camra Survys suggsts lnticular galaxy formation is a slow procss Wilman t al. 9). Th Sloan Digital Sky Survy SDSS) shows that disky lliptical and lnticular galaxis ar mostly found in isolatd high dnsity nvironmnts Bamford t al. 9). Lnticular galaxis xist whr th gas dnsity and tmpratur ar too low for thm to form by ram prssur striping or vaporation of a spiral galaxy and th bulgdisc ratio of lnticular galaxis is systmatically largr than thos of spiral galaxis Drsslr 98). Th distanc btwn lnticular, lliptical and rd spiral galaxis Bamford t al. 9) is smallr than th distanc btwn lnticular and blur spiral galaxis Drsslr t al. 997; Bamford t al. 9). Th ag of th bulg and th disc of lnticular galaxis Fishr t al. 996), kinmatics, and proprtis of lnticular and lliptical galaxis ar similar Sandag & Visvanathan 978; Fishr t al. 996; Jørgnsn t al. 996; D Propris t al. 5; Cappllari t al. 6, 7; Emsllm t al. 7a; Hlmboldt 7; Naab & Ostrikr 9); howvr, it is important to apprciat and diffrntiat btwn slowly rotating lliptical galaxis, which hav low llipticitis, ar gnrally massiv, mtal-rich, and hav shallow photomtric profils, and th rapidly rotating lliptical, disky lliptical and lnticular galaxis, which hav high llipticitis, tnd to b rotationally supportd, ar faintr

10 6 R.A. Schachar t al.: Bulg of lnticular galaxis objcts, and tnd to b mtal-poor Davis & Illingworth 983; Jaff t al. 994; Kormndy & Bndr 996; Erwin t al. 4; Bdrgal t al. 7; Cappllari t al. 6, 7). Th distribution of fast and slow rotating galaxis and th vidnc that boxy and disky llipticals rprsnt diffrnt volutionary paths justifis sparating fast and slow rotating galaxis Bndr t al. 989; Nito & Bndr 989; Emsllm t al. 7a; Kormndy t al. 9). Slowly rotating galaxis ar mor commonly classifid as massiv lliptical galaxis, whil fast rotators ar gnrally faintr and classifid as lliptical or lnticular galaxis Bndr 988; Emsllm t al. 7b). Francis & Andrson 9) hav proposd a nw thory for th formation of galaxy spiral arms that dos not rquir a significant incras in galaxy rotational vlocity. According to this thory, spiral arms ar formd as a consqunc of an incras in galaxy quatorial diamtr. Onc th spiral arms form thy nlarg and vntually form two nring spiral arms. This thory fits with our postulation of th transformation of a lnticular galaxy into a rd spiral galaxy. A small quatorial diamtr xpansion of a disky E4 in rspons to th gravitational forc of its circumfrntial quatorial disk transforms th disky E4 into a lnticular galaxy. If th mass of th circumfrntial quatorial disk incrass by accrtion or a minor mrgr, it will apply mor gravitational forc, which will furthr xpand th quatorial diamtr of th lnticular galaxy initiating th formation of spiral arms as proposd by Francis & Andrson 9). In addition, lnticular galaxis hav a rang of rotational vlocitis Emsllm t al. 4, 7a). Any incras in rotational vlocity will nhanc spiral arm formation Jans 98) and th transformation of a lnticular galaxy into a rd spiral galaxy. Th transition of a disky lliptical galaxy into a lnticular galaxy and thn into a rd spiral galaxy is consistnt with thir location with rspct to th galaxis groupcntr. A gratr prcntag of lliptical galaxis ar locatd closr to th group cntr than rd spiral galaxis, which hav a clar fractional pak that is locatd furthr from th group cntr. Approximatly 3% of rd spiral galaxis ar locatd at.4 R vir. At highr dnsitis or in th cors of groups th fraction of rd spiral galaxis dclins sharply Bamford t al. 9). This distribution of galaxy typ in rlationship to th group cntr suggsts a timlin that is consistnt with th proposd volutionary path for rd spiral galaxis. In contrast, blu spiral galaxis prfrntially inhabit low dnsity nvironmnts and thy diminish stadily from th last dns to th most dns nvironmnts Bamford t al, 9; Schawinski t al. 9). Th proposd volutionary pathway is opposit to th gnrally accptd formation of lnticular and lliptical galaxis from th mrgr of spiral galaxis. Although it is documntd that galaxy mrgrs occur, it is our contntion that this is not th major volutionary path for lnticular or rd spiral galaxis. That disky lliptical galaxis transform into lnticular galaxis which convrt into spiral galaxis implis that th tiology of Hubbl s tuning fork schm should b r-xamind. Jans 98) thorizd that th Hubbl s tuning fork was a consqunc of galaxy rotation. As th rotational vlocity of an lliptical galaxy incrasd it would progrssivly flattn and vntually form spiral arms. According to Jans 98) hypothsis rotational vlocity should prdict galaxy typ and lnticular galaxis should hav a smallr aspct ratio than thir immdiat lliptical prdcssors. Nithr of ths prdictions is tru. Jans 98) was not awar of th xistnc of disky lliptical galaxis. Thrfor h did not considr th implications of th gravitational forc of th circumfrntial quatorial disk on th shap of a galaxy and its ability to initiat th formation of spiral arms with a minimal chang in rotational vlocity. Although w contnd that blu spiral galaxis follow a diffrnt volutionary path than rd spiral galaxis, it is possibl that thy ar part of Hubbl s tuning fork schm. An E4 disky lliptical transforms into a lnticular galaxy that dvlops spiral arms convrting it into a rd spiral galaxy. Th formation of spiral arms causs gratr turbulnc and pockts of high dnsity gnrating molcular gas clouds from which protostars form Francis & Andrson 9) transforming th rd spiral galaxy into a blu spiral galaxy Kannappan t al. 9). 5. Conclusions Th main rsult of this papr is our novl xplanation for th obsrvation that th aspct ratio of lnticular galaxis is gratr than thir immdiat lliptical prdcssors. Th ida is that a small incras in th major axis of an oblat objct with an aspct ratio.6 causs its minor axis to incras Schachar t al. 8). Th circumfrntial quatorial disk of an E4 lliptical galaxy supplis th gravitational forc rquird to incras its quatorial diamtr. This causs th E4 to transform into a lnticular galaxy. 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