Spin-up of low mass classical bulges in barred galaxies
aa r X i v : . [ a s t r o - ph . C O ] D ec Mon. Not. R. Astron. Soc. , 1– ?? (2009) Printed 30 November 2018 (MN L A TEX style file v2.2)
Spin-up of low mass classical bulges in barred galaxies
Kanak Saha ⋆ , Inma Martinez-Valpuesta & Ortwin Gerhard Max-Planck-Institut f¨ur Extraterrestrische Physik, Giessenbachstrae, D-85748 Garching, Germany
Accepted xxxx Month xx. Received xxxx Month xx; in original form 2011 Nov. 24
ABSTRACT
Secular evolution is one of the key routes through which galaxies evolve along theHubble sequence. Not only the disk undergoes morphological and kinematic changes,but also a preexisting classical bulge may be dynamically changed by the secularprocesses driven primarily by the bar. We study the influence of a growing bar on thedynamical evolution of a low mass classical bulge such as might be present in galaxieslike the Milky Way. Using self-consistent high resolution N -body simulations, we studyhow an initially isotropic non-rotating small classical bulge absorbs angular momentumemitted by the bar. The basic mechanism of this angular momentum exchange isthrough resonances and a considerable fraction of the angular momentum is channeledthrough Lagrange point (-1:1) and ILR (2:1) orbits. In the phase of rapid dynamicalgrowth, also retrograde non-resonant orbits absorb significant angular momentum.As a result of this angular momentum gain, the initially non-rotating classical bulgetransforms into a fast rotating, radially anisotropic and triaxial object, embedded inthe similarly fast rotating boxy bulge formed from the disk. Towards the end of theevolution, the classical bulge develops cylindrical rotation. By that time, its innerregions host a ”classical bulge-bar” whose distinct kinematics could serve as directobservational evidence for the secular evolution in the galaxy. Some implications ofthese results are discussed briefly. Key words: galaxies: bulges – galaxies: structure – galaxies: kinematics and dynamics– galaxies: spiral – galaxies: evolution
In the hierarchical structure formation scenario(White & Rees 1978; Fall & Efstathiou 1980) mergershave played a strong role in forming and shaping galaxies.One of the common product of major mergers are theclassical bulges (Kauffmann et al. 1993; Baugh et al. 1996;Hopkins et al. 2009) which are the central building blocksin spiral galaxies. There have been a couple of othermechanisms suggested for the formation and growth ofclassical bulges e.g., monolithic collapse of primordialgas clouds (Eggen et al. 1962), the coalescence of giantclumps in gas-rich primordial galaxies (Immeli et al.2004; Elmegreen et al. 2008), multiple minor mergers(Bournaud et al. 2007; Hopkins et al. 2010), accretion ofsmall companions or satellites (Aguerri et al. 2001). Theclassical bulges formed via these processes seem to havelittle rotation as compared to the random motion. Onthe other hand, various observational measurements haveconfirmed that classical bulges in spiral galaxies possessrotation (Kormendy & Illingworth 1982; Cappellari et al.2007) about their minor axis and in most cases in the same ⋆ E-mail:[email protected] sense as the disk rotates. It is also known that classicalbulges rotate faster than elliptical galaxies and that oftentheir rotation velocities are comparable to that of anisotropic oblate rotator model (Binney 1978). So the originof the systematic rotational motion observed in the classicalbulges remains unclear.The photometric and kinematic properties of classicalbulges as well as their origin are quite distinct from thoseof the other class of bulges, the boxy/peanut and disk-likebulges. It is well known that the surface brightness profilesin classical bulges follow a sersic law µ ( r ) ∼ r /n with sersicindex n ∼
4. Whereas the sersic indices in boxy and disk-likebulges are, in general, low with n
2; so that their surfacebrightness profiles follow roughly an exponential distribu-tion, see Kormendy & Kennicutt (2004); Combes (2009) forextensive reviews. The kinematics of bulges are well illus-trated in the v/σ − ǫ plot (Kormendy & Illingworth 1982;Kormendy 1982) which clearly demonstrates the distinctionin the kinematic properties of ellipticals, classical bulges andboxy bulges and brings out the fact that in terms of theirrotational support, classical bulges fall in between ellipticalsand boxy/disk-like bulges. Since the boxy as well as disk-likebulges are thought to have formed from disk material, thesource of their angular momentum is known. c (cid:13) Saha et al.
The classical bulges formed early through major merg-ers and violent relaxation can subsequently accrete mate-rial quiescently as a result of which a disk grows inside-out (Mo et al. 1998; Katz et al. 2003; Springel & Hernquist2005; Kereˇs et al. 2009). The gas accretion may facilitate thedisk to grow sufficiently for the disk self-gravity to dominatethe internal dynamics. Eventually, a bar and/or spiral armsform in the disk and initiate secular processes in the galaxy.Indeed, bars are quite common in disk galaxies, about 2 / ∼
8% of the disk mass) hasbeen set by modelling the kinematics from the Bulge Ra-dial Velocity Assay (BRAVA) data (Shen et al. 2010). Butthere is evidence for a metallicity gradient above the Galac-tic plane (Zoccali et al. 2008; Zoccali 2010) which is takenas an indication for the existence of a classical bulge in ourGalaxy. It is therefore important to understand the dynami-cal interaction between preexisting classical bulges and barsin barred galaxies.In this paper, we investigate in considerable detail theinteraction of a bar and a low mass classical bulge via a highresolution N -body simulation of a galaxy consisting of a livedisk, bulge and dark matter halo, and follow the evolution ofthe dynamical structure and kinematics of the small classicalbulge. We find that its dynamical evolution is strongly con-nected to the growth of the bar which forms spontaneouslyin the disk. During the secular evolution, the structure andkinematics of the bulge are altered significantly, developingan interesting and complex rotation structure; in particular,cylindrical rotation (which is considered as a typical proxyof boxy bulge) appears in the inner region of the classicalbulge.The paper is organized as follows. Section 2 summarizesthe basics of bar-bulge interaction. Section 3 outlines the ini-tial galaxy model and set up for the N -body simulation. Thebar evolution and boxy bulge formation are described in Sec-tion 4. Section 5 describes, in detail, the angular momentumexchange between the bar and the classical bulge. The evo-lution of the classical bulge, its structure and kinematics arepresented in Section 6. Discussion and conclusions are con-tained in Sections 7 and 8 respectively. In the text, by bulge,we mean a classical bulge unless mentioned otherwise. As we have seen (in Section 1), the possible co-existenceof a bar and a classical bulge might be rather common inpresent day disk galaxies, and thus they are bound to inter-act gravitationally. In fact, a preexisting classical bulge inthe disk has a strong influence on the formation and growthof the bar itself. For the swing amplification to work, oneneeds to keep alive the feedback loop through which a set oftrailing waves traveling through the center are transformed into leading waves. This is possible as long as there is noinner Lindblad resonances (ILRs). A highly centrally con-centrated bulge can shield the center by putting an ILRbarrier and thus cutting the feedback loop which in turncould hinder the growth of the bar (Sellwood & Evans 2001).However, various non-linear processes are probably active inreal galaxies which would destroy the ILR barrier and even-tually lead to the formation of a bar (Widrow et al. 2008;Dubinski et al. 2009).Once a bar is formed, it takes over the dynamics in thecentral region of the disk and starts interacting with the stel-lar bulge and dark matter halo through exchange of angularmomentum. Based on the work of Lynden-Bell & Kalnajs(1972), hereafter LBK72, it has been emphasized by severalauthors (Tremaine & Weinberg 1984; Weinberg 1985;Hernquist & Weinberg 1992; Debattista & Sellwood2000; Weinberg & Katz 2002; Athanassoula 2002;Sellwood & Debattista 2006; Dubinski et al. 2009) that theresonant interaction plays a significant role in the angularmomentum transfer between the bar and the dark matterhalo. It has been suggested by Hernquist & Weinberg(1992); Athanassoula (2003); Weinberg & Katz (2007a)that the same underlying mechanism could as well applybetween the bar and the spheroid and in particular,Athanassoula & Misiriotis (2002) have studied how theshape of a bulge would change in response to a growing bar.Although the dynamical interaction between a growingbar and a bulge and their subsequent evolution can be beststudied via N -body simulations, an analytic understandingis required to complement this. Following LBK72, it can beshown that during the bar-bulge interaction the time rateof change of angular momentum of a classical bulge, whosedistribution function ( F b ) is descibed by a King model, isalways positive and can be written as˙ L z,b ∼ Ω B × | ψ lmn | × F b /σ b > , (1)where ψ lmn and Ω B are the Fourier amplitude and pat-tern speed of a non-responsive bar potential and σ b is thevelocity dispersion of the bulge stars. So at a given reso-nance, the angular momentum gained by the bulge dependson the strength of the bar and is inversely proportional tothe square of bulge velocity dispersion, implying a hotterbulge will absorb less angular momentum provided otherconditions remain unchanged. However, in real galaxies, theangular momentum transfer is more difficult to determine,because the time rate of change of bar’s angular momentuminvolves the change in its pattern speed, moment of inertia,and in the angular momentum associated with any inter-nal circulation (Villa-Vargas et al. 2009) within the bar. InSection 5, we show the angular momentum transfer betweenthe bar and the bulge in our simulation using orbital spectralanalysis. N -BODYSIMULATION An equilibrium model for a disk galaxy is con-structed using the self-consistent bulge-disk-halo modelof Kuijken & Dubinski (1995). Their prescription providesnearly exact solutions of the collisionless Boltzmann and c (cid:13) , 1– ?? pin-up of low mass classical bulges Poisson equations which are suitable for studying disk sta-bility related problems. All the components in our modelare live (i.e., the gravitational potential of each componentcan respond to an external or internal perturbation) and,hence, provide a realistic representation for the structureand evolution of the galaxy. Below we briefly describe eachcomponent of the model. For more details, the reader is re-ferred to Kuijken & Dubinski (1995).The disk distribution function is constructed using theapproximate third integral given by E z = v z + Ψ( R, z ) − Ψ( R, ρ d ( R, z ) = M d πh z R d e − R/R d erfc (cid:18) R − R out √ R out − R trun ) (cid:19) f d ( z ) , (2)where f d ( z ) = exp( − . z ( R, z ) / Ψ z ( R, h z )) withΨ z ( R, z ) = Ψ(
R, z ) − Ψ( R,
0) governs the vertical structureof the disk, erfc is the complementary error function. In theabove equation, M d is the disk mass, R d is the scale lengthand h z is the scale height.A spherical live classical bulge is constructed from theKing model (King 1966) and the corresponding distributionfunction (DF) is given by (Binney & Tremaine 1987) f b ( E ) = ρ b (2 πσ b ) − / e (Ψ b − Ψ c ) /σ b ×{ e − ( E − Ψ c ) /σ b − } if E < Ψ c , . (3)Here, the bulge is specified by three parameters, namelythe cut-off potential (Ψ c which determines the bulge tidalradius), central bulge density ( ρ b ) and central bulge velocitydispersion ( σ b ). The gravitational potential at the centre ofthe bulge is measured by Ψ b .An axisymmetric live dark matter halo is constructedusing the distribution function of a lowered Evans model(Evans 1993) and is given as f dm ( E, L z ) = [( AL z + B ) e − E/σ h + C ] × ( e − E/σ h −
1) if
E < , . (4)The halo is parameterized by a potential depth (Ψ ), veloc-ity ( σ h ) and density scales ( ρ ), a core radius R c and theflattening parameter q . The factors A, B, and C are func-tions of these parameters (see Kuijken & Dubinski 1995 andreferences therein). The halo has a tidal radius specified by E = 0.The total mass and the outer radii of both the bulgeand halo are calculated in an iterative procedure. The po-tential is computed self-consistently by solving the Poissonequation for the combined three component system in aniterative fashion. First, the densities for the bulge and haloare obtained from their respective distribution function andthen the disk density is added to it and the correspondingpotential for the combined mass distribution is used as a Figure 1.
Initial circular velocity curve for the model galaxy.Solid line represents the total circular velocity . Dotted line is forthe bulge, dashed line for the dark matter halo and dashed-dot-dash for the disc. starting point for carrying out the next iteration. We usea maximum of l = 10 in the potential harmonic expansionand the iteration is continued until the outer radii for thebulge and halo are unchanged between successive iterations.The outer radii of the bulge and the halo correspond to therespective tidal radii. The masses of the bulge and halo cor-respond to the total mass enclosed within their respectiveouter radii computed by integrating the density profiles.In this paper, we present the analysis of a particulargalaxy model hosting a low mass classical bulge. For histori-cal reasons, we call this model RCG004. The circular velocitycurve for the model is presented in Fig 1. The length, massand velocity units for this model are given by L = 4 . M = 2 . × M ⊙ and V = 157 kms − . The disk outerradius ( R out ) is fixed at about 6 . R d and a truncation width ∼ . R d is adopted within which the disk density smoothlydecreases to zero at the outer radius. The disk scale length( R d ) is fixed at 4 . M d = 4 . × M ⊙ . The central value of the radialvelocity dispersion is 78 . − . The Toomre Q profile isnearly flat in the radial range 0 . .
4. The bulge mass M b = 3 × M ⊙ .In table 1, we quote the outer radius for the classical bulge(denoted by R b ) in our galaxy model. The halo has a flat-tening of q = 0 . R c = 0 .
25 kpc and amass of M h = 1 . × M ⊙ within about 60 kpc.We evolve the galaxy model in isolation to examinethe evolution of the bulge shape, morphology and kine-matics. The simulation is performed using the Gadget code(Springel et al. 2001) which uses a variant of the leapfrogmethod for the time integration. The forces between theparticles are calculated using the Barnes & Hut (BH) treewith some modification (Springel et al. 2001) with a toler-ance parameter θ tol = 0 .
7. The integration time step is ∼ . . ∼
296 Myr. A to-tal of 1 . × particles is used to simulate the model galaxy.The softening lengths for the disk, bulge and halo particlesare 12, 40 and 36 pc respectively. The masses for the disk,bulge and halo particles are 1 . × M ⊙ , 0 . × M ⊙ and3 . × M ⊙ respectively. To examine the effect of unequal c (cid:13) , 1– ?? Saha et al.
Figure 2.
Surface density maps for the disk particles alone. Topleft: density map at T=0 Gyr, showing the axisymmetric disk.Top right: Same at T=0.56 Gyr. Bottom left: at T=1.1 Gyr andBottom right: at T=2.1 Gyr.
Table 1.
Initial parameters for the model galaxy.Galaxy
Q B/D B/T σ b R b model (kms − ) (kpc)RCG004 1.40 0.0666 0.01306 65.0 6.08B/D is the bulge-to-disk mass ratio, B/T is the bulge-to-total(including dark halo mass) mass ratio, σ b is the bulge centralvelocity dispersion , and finally R b is the outer radius of the bulge. softenings, we have re-run the simulation with new soften-ing parameters as prescribed by McMillan & Dehnen (2007).We note that with the new softenings, the bar growth is de-layed by ∼
90 Myr while the main results remain unchanged.The total energy is conserved within 0.2% till the end ofthe simulation. The total angular momentum is conservedwithin 3% at 2.2 Gyr for both the runs having different soft-ening parameters.
Although it is not clearly understood how bars are formedin real galaxies, swing amplification (Toomre 1981) plays asignificant role in making an initially axisymmetric, equilib-rium model of a disk galaxy bar unstable (Sellwood 1981).Once formed, N -body bars are found to be long-lived, domi-nate the disk dynamics, and are responsible for driving secu-lar evolution processes in the galaxy (Sellwood & Wilkinson1993). Fig. 2 depicts the formation and evolution of the barfrom the initially axisymmetric disk. Strong two-armed spi-rals are also formed along with the bar and last till 1.1 Gyrin our simulation. The ring-like structure at T = 0 .
56 Gyris intersecting the spiral arms indicating that it is probably
Figure 3.
Time evolution of the bar amplitude and the patternspeed (Ω B ). Red solid line is the result of a linear regressionanalysis done on the measured pattern speed values from the N -body snapshots. not real. Such a ring-like feature arises because of the galaxymodel not being in perfect equilibrium.In the upper panel of Fig. 3, we show the time evolutionof the bar amplitude measured as the maximum of m = 2Fourier coefficient ( A ) of the density perturbation normal-ized to the unperturbed axisymmetric component ( A ). Thebar reaches its first peak in amplitude at 0 .
28 Gyr and thesecond peak at 0 .
44 Gyr. The m = 1 vertical Fourier mode( | A ,z | ) in the r − z plane corotating with the bar patternspeed shows that the disk is undergoing buckling instabilityfrom ∼ . − . . c (cid:13) , 1– ?? pin-up of low mass classical bulges Figure 4.
Edge-on surface density (left) and velocity maps (right) for the disk particles alone at two different epochs during the secularevolution. From top to bottom, the panels are taken at T = 0 .
56 and 2 . in Debattista & Sellwood (2000). Using a linear regressionanalysis on the simulation data, we find the half-life, T / ,(the time period over which the bar pattern speed woulddecay to half its initial value) of the rotating bar to be ∼ .
09 Gyr. This indicates that the rate of angular momen-tum transfer from the bar is rather slow in our simulation;for an in-depth analysis on the bar slow down, readers arereferred to Weinberg (1985); Weinberg & Katz (2007a).As the bar grows stronger, its self-gravity increasesand it goes through the well-known buckling instabil-ity (Combes & Sanders 1981; Pfenniger & Norman 1990;Raha et al. 1991; Martinez-Valpuesta & Shlosman 2004)following which the bar transforms into a boxy/peanutbulge. In Fig. 4, we present the surface density (left pan-els) and velocity field (right panels) for the boxy bulge seenedge-on; i.e., only disk particles are shown. The cylindricalrotation is evident. The final boxy bulge contains approxi-mately 33% of the disk mass including the inner barred diskcomponent. Note that the density drops off sharply alongthe vertical direction in the boxy bulge region. In Section 6,we will compare the structure and kinematics of the boxybulge in Fig. 4 formed in our simulation with the classicalbulge undergoing the bar driven secular evolution.
We compute the specific angular momentum for each speciese.g., disk, bulge and halo particles in our simulation andre-confirm the already established fact that the inner re-gions of the disk loose angular momentum through the bar.While a significant fraction of the total angular momentumemitted by the bar is absorbed by the surrounding darkmatter halo, the angular momentum gained by the bulgeis non-negligible. In Fig. 5, we show the angular momentumtransfer amongst the disk, bulge and halo components in ourmodel. The total angular momentum is conserved within 3%at the end of 2.2 Gyr in our simulation. Initially both thebulge and halo have zero net angular momentum i.e., theystart as non-rotating objects. Note that the rate of gain of
Figure 5.
Evolution of the specific angular momentum of thebulge (green), disk (red) and halo (blue) components in ourmodel. Along the y-axis plotted are the specific angular momen-tum minus its value at T = 0 normalized by the disk angularmomentum ( L dz (0)) at T = 0. angular momentum by the classical bulge particles nearlysaturates towards the end of the simulation and closely fol-lows the growth of the bar (see Fig. 3). Using orbital spectralanalysis, we show below that the gain of angular momentumby the bulge occurs primarily through resonances (see alsoHernquist & Weinberg 1992).The simulation presented here shows an increase in thebulge rotation velocity (Section 6.2), and a correspondingincrease in bulge angular momentum. The transfer of angu-lar momentum from the bar to the bulge depends stronglyon the pattern speed which sets the resonance locations. Itis important to note that if the resonances are sparsely pop-ulated because of lack of particles in the simulation, the an-gular momentum transfer will be inefficient (Weinberg 1985;Weinberg & Katz 2007b). In our case, we have a total of10 particles with 10 particles in the classical bulge. So wecan test whether angular momentum transfer through reso- c (cid:13) , 1– ?? Saha et al.
Figure 6.
The top panels show the distribution of particles with frequency (Ω − Ω b ) /κ at five different times throughout the evolution ofthe bulge. The lower panels show the gain of angular momentum of the selected particles with respect to the previous time, as indicatedon the top of the panel. The vertical dotted lines in the figure indicate the most important resonances, -1:1, 2:1, 3:1 and 4:1. Note thatfrom time 1 . . − Ω b ) /κ =0.5 gain less angular momentum than earlier. nant interaction, is the mechanism for the angular momen-tum gain of the classical bulge. Here, we quantify its effectby using an orbital spectral analysis method described inMartinez-Valpuesta et al. (2006) based on that presented inBinney & Spergel (1982). In previous works, this methodwas applied to halo resonant orbits (Athanassoula 2003;Martinez-Valpuesta et al. 2006; Dubinski et al. 2009) to un-derstand the bar-halo interaction. We apply it to the classi-cal bulge particles to find out how many of them are trappedin resonances and what is the corresponding gain of angu-lar momentum. The potential is extracted from the N -bodysimulation at different snapshots using the grid code pro-vided by Sellwood & Valluri (1997) and then frozen to com-pute the orbits. We randomly select 100000 (10%) particlesout of 1 million in the bulge and compute their correspond-ing orbits. We calculate the azimuthal and radial epicyclicfrequencies Ω and κ respectively for each of the orbits byFourier analysis.We present the results of this orbital spectral analysis inFig. 6 at 5 different epochs. On the top panels, we present theclassification of classical bulge particles by their frequencyratio η = (Ω − Ω B ) /κ . The bar has an irregular evolutionand this can be seen in the time sequence of the top panel inFig. 6. Initially, the bulge particles are distributed half co-rotating with the disk and half counter-rotating. Therefore,when we study the orbital distribution in the very early stageof the bar growth at T = 0 .
17 Gyr, there still is an almostsymmetric distribution. When the bar is already formed,right after reaching the maximum, at T = 0 .
39 Gyr, manyparticles have been trapped around the 2:1 resonance. Tak-ing a careful look at these orbits, we have checked that theyare of x1-type. There is also a considerable group of particleswith η ∈ ( − . , . ); a look to the orbits allows to identifythem as mainly stochastic trajectories. In the lower panels, we show the angular momentum gain by the particles ateach η during the growth and evolution of the bar. Thereis a considerable gain of angular momentum by three maingroups in our diagram (Fig. 6, second bottom panel). Themain gain of angular momentum comes from those parti-cles at resonance with η = −
1, corresponding to particlesorbiting around the Lagrangian points. Since these particlesare at negative frequency it means that they are counter-rotating with the bar. Another gaining group correspondsto the particles with η ∈ ( − . , . ), the stochastic group.By gaining angular momentum their (counter) rotation de-creases. Amongst the low order resonances, the importantgaining group is around the ILR ( η = 0 . T = 0 . − .
39 Gyr, the bar goes through the buckling event, there-fore there is still some trapping of particles around the 2:1resonance. Although there not much gain or loss of angu-lar momentum, some angular momenta are gained through η = − T = 1 . η = 0 . η = − , −
2. The particles at the OLR ( η = − .
5) andthose with η ∈ ( − . , . ) are losing angular momentum. At T = 2 . η = − c (cid:13) , 1– ?? pin-up of low mass classical bulges Figure 7.
Edge-on surface density and velocity maps for the bulge particles alone at four different epochs during the secular evolution.From top to bottom, the panels are taken at T=0, 0.56, 1.1 and 2.1 Gyr. The left panels show the surface densities and the right panelsthe velocity fields. Initially the bulge is non-rotating and flattened by the disk potential. Similar maps for the boxy-bulge are shown inFig. 4. ing over time, their angular momentum gain does not followaccordingly. By comparing the last two panels (upper andlower), it is evident that the particles trapped at the ILRare now hardly gaining angular momentum. It is plausiblethat the inner bar-like structure in the classical bulge (seesection 6.3) is giving away angular momentum to the outerparts of bulge and perhaps to disk and halo.The gain of angular momentum by the bulge can thusbe explained by resonances during the slow secular evolu-tion of the bar, and by resonances together with stochasticorbits in the dynamical stage. During the dynamical phase, T = 0 . − .
17 Gyr, the net gain of angular momentum(computed by adding up the averaged angular momentumof each orbit) is 3 times larger than that gained in the rela-tively quiet secular phase ( T = 2 . − .
56 Gyr). While ap-proximately 70% of the net angular momentum gain comesfrom the resonances, stochastic orbits contribute to ∼ R b / = 0 . R d and0 . R d at T = 1 . . R bar = 0 . R d and 2 . R d at those times) inour simulation. The bar size is measured from the phase an-gle of the bar (Athanassoula & Misiriotis 2002). The phaseangle of the bar (i.e., the m = 2 Fourier component of thedisk surface density) remains approximately constant uptoa certain radius and starts varying beyond that. We mea-sure the length of the bar as the radius at which the phaseangle of the bar starts deviating from the contant value. Wealso note that the corotation resonance ( R cr = 0 . , . . R d at T = 0 . , . . R cr /R bar lies between 1 − . c (cid:13) , 1– ?? Saha et al.
Figure 8.
Normalized radial profile for the a coefficient for theclassical bulge alone at three different times. and the subsequent change of the orbital structure of theclassical bulge are indeed responsible for the transformationof the classical bulge as described below. From Sections 2 and 5, we learn that a classical bulge canabsorb a non-negligible fraction of the total angular momen-tum emitted by the bar through resonant interaction. Theangular momentum gained by the bulge (being a smallermass object than the dark matter halo) has a profound ef-fect on its structure, kinematics and dynamics. The result ofthe bar-bulge interaction in our simulation is the transfor-mation of an initially non-rotating low mass classical bulgeinto a highly rotating triaxial one . Below we describe, in con-siderable detail, various diagnostics which show that this isindeed true.
In the left panels of Fig. 7, we show the surface densitymaps for the classical bulge (viewed edge-on) at four dif-ferent epochs during the evolution. The classical bulge isshown edge-on ( i = 90 ◦ ) such that the major axis is along X -axis and the minor axis along Z -axis. Initially the bulgeis isotropic and flattened by the strong gravity of the diskpotential. At later phases of evolution, the inner regions ofthe bulge become rounder and the outer parts become disky.In order to understand the structure of the classi-cal bulge more quantitatively, we have also performed anisophotal analysis using the IRAF ellipse task on a setof edge-on images of the bulge including the ones pre-sented in Fig. 7, and compute the fourth-order Fourier co-sine coefficient a /a normalized to the semi-major axis a at which the ellipse was fit. Fig. 8 shows the normalized a prodiles at three different epochs in the simulation. Thevalues of a /a determine the degree of boxiness or diski-ness (Nieto & Bender 1989), with a /a < a /a > a /a ∼ X/R d . Figure 9.
Radial variation of the bulge rotational velocity nor-malized to the average velocity dispersion in the central region,for five snapshots from T=0 to T=2.1 Gyr. The long tick markon the x-axis denotes the initial value of R b , see Table 1. of the classical bulge becomes mildly boxy at T = 0 .
56 Gyrand the boxiness increases at 1 . T = 2 . a /a > a /a parameter of the boxybulge in a similar fashion as outlined above. It is found thatat T = 1 . a /a of the boxy bulge is negative inside X/R d < . . T = 1 . z = 0) andits midplane surface density is ∼ z = 0 . R d is higher than that of the boxybulge, and the classical bulge extends further. However, at T = 2 . z . R d and above this height, their densityprofiles are comparable. The influx of angular momentum to the initially non-rotating bulge enforces the bulge particles to have a netrotational motion. In Fig. 9, we show the radial profiles ofthe rotational velocity normalized to the average velocitydispersion in the central region ( R R b / ) of the classi-cal bulge at different epochs during the evolution. The ro-tational velocity profiles remain nearly unchanged at laterstages of evolution when the rate of angular momentum gainby the bulge also nearly saturates as can be seen from Fig. 5and Fig. 6. c (cid:13) , 1– ?? pin-up of low mass classical bulges Figure 10.
Parallel minor axis velocity profiles of the bulge atT=0.56 Gyr (upper panel) and T=2.1 Gyr (lower panel). Theupper panel shows no clear signature of cylindrical rotation. Butat later stages of the evolution, cylindrical rotation develops inthe inner region of the bulge, as indicated by the parallel shapesof the velocity profiles in the lower panel.
To illustrate the evolution further, we present four ve-locity maps of the classical bulge on the right panels ofFig. 7. During the initial phases of secular evolution, theangular momentum gained by the bulge particles is pri-marily converted into streaming motion and the classicalbulge starts rotating around the z-axis, with gradients in thestreaming velocity both along the radial and vertical direc-tions. Note that in the barred potential, the classical bulge isno longer axisymmetric and its inner regions becomes mod-erately boxy (see Sections 6.1 and 6.3). This could be a sig-nature of a thick bar formed inside the classical bulge (seeSection 6.3). Indeed as time progresses, mild signatures ofcylindrical rotation emerge in the inner regions of the bulgeand gradually become prominent (see Fig. 7).To have a clearer picture of the velocity structure, weshow parallel minor-axis (Fig. 10) and major-axis (Fig. 11)velocity profiles of the classical bulge at two different epochs T = 0 .
56 (upper panels) and T = 2 . X/R d = 0 .
31 and
X/R d = 0 .
71) on either side ofthe bulge center. At T = 0 .
56 Gyr, the minor axis rotationvelocity decreases along the vertical direction ( dV b dz <
0) in-dicating clearly a non-cylindrical rotation throughout thebulge (see the upper panel of Fig. 10). Note, the velocityprofiles in the outer parts are asymmetric which is probablyinfluenced by the on-going buckling instability of the bar.
Figure 11.
Major axis velocity profiles of the bulge at T=0.56Gyr (upper panel) and T=2.1 Gyr (lower panel). The upper panelshows no clear signature of cylindrical rotation. But at later stagesof the evolution, cylindrical rotation develops in the inner regionof the bulge. The long tick mark on the x-axis denotes the initialvalue of R b , see Table 1. The major-axis profiles are taken at four different slits (theslit positions are indicated in Fig. 11) parallel to the majoraxis of the classical bulge. The major-axis velocity profilesin the upper panel of Fig. 11, also indicate non-cylindricalrotation throughout the classical bulge.At later times, the inner regions of the classical bulgehave developed cylindrical rotation. However, the gradientof this cylindrical rotation in the classical bulge is shallowerthan that in the boxy bulge. The minor-axis velocity profilesin the bottom panel of Fig. 10, show clear indication for thatin the inner regions. The same is evident from the bottompanel of Fig. 11. The major axis velocity profiles at T = 2 . X/R d < .
4) of the classical bulge rotate cylindrically while the outerregions beyond about twice the half-mass radius (2 × R b / ∼ . R d ) still maintain differential rotation both along theradial and vertical directions ( dV b dz < So in the later stagesof the secular evolution, the initially non-rotating classicalbulge has developed a mixed rotational state with the innerregion rotating cylindrically while the outer region rotatesdifferentially in z. c (cid:13) , 1– ?? Saha et al.
In order to achieve a deeper understanding of the complexnon-linear dynamical interplay of the bar and the bulge,we investigate the three dimensional structure of the classi-cal bulge using spherical harmonics analysis. In particular,we are looking for non-axisymmetric modes in the classicalbulge which could have been influential for producing someof the complex structure and kinematics as discussed in Sec-tions 6.1 and 6.2. The outcome of the bar-bulge interaction isnot only the transfer of angular momentum between the twoand changes in kinematics thereby, but a structural transfor-mation of the classical bulge, a prediction of which is proba-bly beyond the scope of the analytic/semi-analytic theories(Lynden-Bell & Kalnajs 1972; Tremaine & Weinberg 1984)briefly outlined in Section 2. From our analysis, it becomesclear that the interaction of a bar and a small classical bulgeis more vigorous than that between the bar and the massivedark halo as discussed in section 2. The primary reason be-ing the smaller mass and size of the bulge compared to thedark matter halo.To analyze the structural components developed in thesmall classical bulge after the evolution, we expand the fullthree dimensional bulge density distribution ( ρ ) in terms ofspherical harmonics: ρ ( r, θ, φ ) = ∞ X l =0 l X m = − l ρ lm ( r ) Y ml ( θ, φ ) , (5)where r, θ, φ are the usual spherical coordinates, and the Y ml are the spherical harmonics. ρ lm denotes the radial densityfunction. We bin the bulge particles into spherical shells andcompute B lm as function of the bin radius ( r k ), as follows: B lm ( r k ) = N lm X j m b P ml (cos θ j ) e imφ j , (6)where N lm = ((2 l +1) / π ) × ( l − m )! / ( l + m )!, m b is the massof each bulge particle, P ml are the Associated Legendre poly-nomials. The function B lm is directly related to the mass ofeach bin and thereby to ρ lm via the bin radius ( r k ). Thenusing the above formula (Eq. 6), we can derive the radialvariation of the amplitude of a particular l, m mode in thebulge as S lm ( r k ) = p ℜ B lm + ℑ B lm . The correspondingphase angle φ j can be used to derive the pattern speed ofthe l, m mode.In Fig. 12, we show the time evolution of the amplitudeof l = 2 , m = 2 mode. The classical bulge-bar (hereafter,denoted as ClBb) is weaker than the disk bar (see Fig. 3 forthe bar amplitude and pattern speed) but rotates nearly inphase with it. By analyzing the radial variation of the phaseangles, we conclude that the physical size of the ClBb ismuch smaller compared to the disk bar. Initially, the ClBband disk bar are not in phase, the ClBb seems to be laggingbehind the disk bar by about 2 ◦ − ◦ in angle. But soon,they start rotating in-phase with each other. After about 1Gyr, the pattern speed of the ClBb is also nearly the same asthat of the disk bar (see Fig. 12). A convenient way of view-ing the dynamics of the ClBb, is to think of it initially as adriven oscillation phenomenon where the disk bar is actingas a driver and the bar-like structure in the classical bulgeis its forced response. Later bulge particles are trapped bythe 2:1 resonance; i.e., both components populate the or- Figure 12.
The strength of the classical bulge-bar (l=2,m=2mode) and its pattern speed evolution. bits in their jointly rotating potential. It has been shownin previous simulations e.g., by Holley-Bockelmann et al.(2005), Col´ın et al. (2006), Athanassoula (2007) that a bar-like structure also forms in the inner regions of the darkmatter halo as a result of its interaction with the bar inthe disk. These studies have shown that such a bar in thehalo is rather weak and nearly corotates with the bar inthe disk. It turns out that some of the characteristics ofthe ClBb are quite similar to that of the halo-bar. How-ever, with the classical bulge being much less massive thanthe halo, the dynamical impact of the bar-like structure ismuch more pronounced in the classical bulge as we havealready demonstrated above. Beside the transfer of energyand angular momentum between the disk-bar and the clas-sical bulge, the stars in the classical bulge are also beingheated during the evolution and hence the inner bulge re-gion becomes moderately thicker and rounder (see Fig. 7).We have checked that the slow variation in the ellipticityof the classical bulge is consistent with the variation in thekinetic energy tensor in accordance with the tensor virialtheorem. A more detailed picture of the dynamics of thebulge hosting a bar and its observational properties will bepresented in a future paper.
From the misalignment of the photometric major axis of thedisk and the bulge and the isophotal twists, it is inferredthat many of the bulges in spiral galaxies are indeed triaxial(Stark 1977; Gerhard et al. 1989; Bertola et al. 1991; Ann1995; M´endez-Abreu et al. 2010). Here, we show the evolu-tion of the triaxiality and velocity anisotropy in the low massclassical bulge in our simulation. The global parameter for c (cid:13) , 1– ?? pin-up of low mass classical bulges Figure 13.
Time evolution of the shape of the classical bulge. Ini-tially the bulge is flattened by the strong disk potential and henceoblate. At later phases during the secular evolution it becomestriaxial. The solid diagonal line denotes a prolate configuration.The red open circles are the measured values of the axes ratiosat T = 0 , . , . , . , . , . , . , . b/a = 0 .
99 and0 .
877 at T = 0 and 2 . Figure 14.
Radial variation of the anisotropy parameter β rz forthe classical bulge. Beyond about 0 .
56 Gyr, the anisotropy pro-files remains nearly unchanged. the bulge triaxiality T b can be computed using the followingrelation (Franx et al. 1991; Jesseit et al. 2005): T b = 1 − ( b/a ) − ( c/a ) , (7)where a , b , and c are the semi-axes defining the shape of theclassical bulge (see Section 6.5 for the measurement of theaxis ratios). a = b > c denotes an oblate configuration i.e. T b = 0, and b = c < a is a prolate figure corresponding to T b = 1. a = b = c defines a triaxial configuration with peakvalues reaching T b = 0 .
5. In Fig. 13, we show the evolution ofthe shape of the classical bulge. Initially the bulge is oblate( T b ∼
0, see Fig. 13); thereafter it evolves as a result of theangular momentum gain and change in the orbital structure.During the period of 0 . − .
56 Gyr (roughly the dynam-ical phase) a considerable fraction of angular momentum isgained at resonances η = − η = 0 . Figure 15.
Radial variation of velocity dispersion ratios com-puted from bulge particles trapped at ILR ( η = 0 .
5) and froma group of resonant and non-resonant particles in the frequencyrange − . η − . and thereby producing a disky structure (see Fig. 7) in theouter parts of the bulge. In this period, essentially only b/a changes while c/a remains nearly constant.Beyond ∼ .
56 Gyr, the ClBb forms in the bulge causingsubstantial changes in the bulge structure. As mentioned insection 6.3, the ClBb heats (Saha et al. 2010) the bulge starsmainly in the central region and makes it thicker. We thinkthat this heating due to ClBb is primarily responsible forsubsequent changes in the c/a . During the period from 1 . − . b/a changes more than c/a . At T = 2 . T b = 0 .
48. This suggeststhat more generally, fast rotating and triaxial bulges couldhave developed through the interaction of a strong bar anda small classical bulge in galaxies with low
B/D ratio. Amore comprehensive analysis focusing on the role of the mostimportant parameters such as the bulge mass and size willbe presented in a future paper.As shown in Section 3, the initial velocity distributionin the classical bulge in our simulation is isotropic repre-sented by a King model. As a result of the angular momen-tum influx and the readjustment of the orbits, the veloc-ity structure changes during the evolution. We measure thedeviation from isotropy in the velocity distribution by theanisotropy parameters defined as β rz = 1 − ( σ z /σ r ) and β rϕ = 1 − ( σ ϕ /σ r ) , where σ r , σ φ and σ z are the veloc-ity dispersions in the radial, azimuthal and vertical direc-tion. Then β rz > β rϕ < T = 0 .
56 Gyr. To understand the source of radialanisotropy, we have studied orbits in the classical bulge asclarified in Fig. 6. Fig. 15 shows radial variation of σ z /σ r and σ φ /σ r computed from bulge particles that are trappedat ILR ( η = 0 .
5) and from a group of resonant and non-resonant particles in the frequency range − . η − . c (cid:13) , 1–, 1–
5) and from a group of resonant and non-resonant particles in the frequency range − . η − . c (cid:13) , 1–, 1– ?? Saha et al.
Figure 16. V m /σ − ǫ relation for the small classical bulge alone.Initially the bulge is non-rotating but then it acquires angularmomentum emitted by the bar and evolves into a fast rotatingtriaxial bulge. Each open circle represents an epoch in the simu-lation beginning at T = 0 (the bottom most point) . Subsequentcircles are drwan at 0 . , . , . , . , . , . , . the inner region of the classical bulge which host the ClBb.Whereas in the outer region, definite contributions to theradial anisotropy comes from the bulge particles in the fre-quency range − . η − .
1. The particles that are atresonance with the bar e.g., at η = − , − V m /σ − ǫ relation To quantify the degree of ordered motion in bulges and el-lipticals and illuminate the difference between the two typesof stellar systems, the V m /σ − ǫ diagram relating the ratio ofrotational to random motions and the observed ellipticity ( ǫ )was introduced (Illingworth 1977). It was shown that bulgesare, in general, fast rotators compared to bright ellipticalgalaxies (Kormendy & Illingworth 1982; Davies et al. 1983;Cappellari et al. 2007; Morelli et al. 2008). Here, we focuson the relation between the shape and the kinematics of thesimulated low mass classical bulge that has been subject tothe secular evolution driven by a strong bar. We show explic-itly the evolutionary track of this particular classical bulgein the V m /σ − ǫ diagram below (Fig. 16).In observations, it is rather difficult to have an accuratemeasurement of the bulge rotation velocity due to possibledisk contamination. On the other hand, in simulations, itis rather straightforward to compute the velocity profile forthe classical bulge alone because it is possible to filter outthe disk and halo components of the model galaxy. We de-termine, V m as the maximum of the azimuthally averagedrotational velocity of the bulge particles measured in theequatorial plane of the bulge; σ is the mean velocity disper-sion in the central region (calculated at r ∼ . × R / ) ofthe bulge.For the bulge ellipticity, we use c/a for edge-on view.Measuring the ellipticity for a bulge is a bit tricky becausethere can be strong radial variation in the ellipticity pro- file ǫ ( r ). Triaxiality could add another degree of complexityto such a measurement. Below we describe how the bulgeellipticities are measured.In order to determine the intrinsic ellipticity, we firstcompute the moment-of-inertia tensor of the three dimen-sional mass distribution of the classical bulge and diagonal-ize it to obtain the principal moments and three orthogonaleigenvectors. The principal moments of inertia determine theintrinsic axis ratios of the inertia ellipsoid and the eigenvec-tors determine the orientation of the ellipsoid with respectto the co-ordinate space. Using the eigenvalues and eigen-vectors, we determine the two axis ratios namely b/a and c/a , where a > b > c are the three semi-axes of the inertiaellipsoid. We have done this at the bulge half-mass radiuscontaining 50% of the total bulge particles and at the radiuscontaining about 90% of the total bulge particles. The el-lipticity measurements for the classical bulge are nearly thesame in both cases. In the following, we use the ellipticityat radii enclosing ∼
90% of the total bulge particles and usethe definition of the ellipticity of the bulge as ǫ = 1 − c/a when viewed edge-on. We have also performed isophote anal-ysis using the ellipse-fitting routine from IRAF on a set ofsuitably rotated and inclined edge-on images (Fig. 7) of theclassical bulge at different epochs during the evolution. Wefind a good agreement between the two different types ofmeasurements of the bulge ellipticity.In Fig. 16, we show the V m /σ and ǫ values for the smallclassical bulge during the secular evolution. Each point inthis diagram corresponds to a particular epoch during itsevolution and when connected together they form its evo-lutionary track. This shows that the small classical bulgerotates significantly faster in the latter stages of evolutioncompared to the oblate isotropic rotator model, which canbe approximated by (Binney 1978; Kormendy 1982): V m /σ ∼ = r ǫ − ǫ . (8)The interpretation of Fig. 16 is complicated by the fact thatthe classical bulge is not an isolated stellar system, but inter-acts dynamically with the bar within a disk galaxy. The en-tire period of evolution of the classical bulge can be broadlydivided into two parts: one before the formation of the ClBb( ∼ .
56 Gyr) and the second after its formation. Before theformation of the ClBb, the bulge stars in the outer regiongain a significant fraction of angular momentum emitted bythe bar and move mainly outwards in radius. As a resultof this, the values of V m /σ increase till ∼ .
56 Gyr whilethe axis ratio c/a remains unchanged. In the second half ofthe evolution, the bulge stars are heated due to the ClBbby a factor of ∼ . c/a ratio increases, making theellipticity decrease considerably. The near saturation in the V m /σ towards the end of the simulation is connected withthe fact that the rate of angular momentum gain by thebulge nearly saturates at these epochs (see Fig. 5, Fig. 6). Amore detailed analysis on how the spinning up of the clas-sical bulge depends on the various parameters of the bulge(bulge-to-disk mass ratio, size of the bulge, its central veloc-ity dispersion) and disk (Toomre Q, bar strength, bar size)will be presented in a future paper. c (cid:13) , 1– ?? pin-up of low mass classical bulges The primary goal of this paper has been to describe a genericmechanism, the transfer of angular momentum from a barto an embedded classical bulge. We have shown that thismechanism is important for understanding the rotationalmotion of low mass classical bulges in the central regionsof barred galaxies. The growth rate and the strength of thebar are important factors for the mechanism to work effi-ciently. One very interesting outcome of this process is thecylindrical rotation in the small classical bulge in the modelstudied here. Some possible observational implications andother issues are addressed below.
During the secular evolution, a flat bar transforms into aboxy bulge. While the bar grows, buckles and evolves, afraction of the angular momentum emitted by the bar isabsorbed mainly in the outer parts of the embedded classi-cal bulge in the galaxy. As a result, streaming motions areinduced in the classical bulge (Fig. 9), and the orbital struc-ture changes, causing velocity anisotropies. After about halfa Gyr, the ClBb forms in the inner regions of the bulge. TheClBb introduces a pattern rotation in the classical bulgewhich transforms into a rotating triaxial object. A compar-ision of Fig. 3 and Fig. 12 shows that beyond ∼ . ∼ . It is widely accepted that cylindrical rotation is a character-istic feature of the kinematics of a boxy bulge formed outof disk material, via the vertical buckling instability of thebar. Thus the presence of cylindrical rotation in the centralregions of a galaxy may lead one to infer the presence of aboxy bulge that originated from the disk, without any needfor a classical bulge in this galaxy.The work presented in this paper adds a new aspect tothis simple picture. The cylindrical rotation could also in-clude the stars of the classical bulge whose rotational prop-erties have been modified by the interaction with the bar.One would measure the net cylindrical rotation of the starsin the combined boxy bulge and classical bulge. In absenceof strong photometric evidence, other information such asfrom stellar populations and metallicity gradients would beneeded to determine the presence of a small classical bulge.Although the pure kinematic modelling of the BRAVAdata (Shen et al. 2010) suggests only an upper limit onthe mass of a classical bulge in the Milky Way, the mea-surements of the metallicity gradient above the galacticplane (Zoccali et al. 2008) may indicate the presence of a classical bulge. The upper limit on the total mass of thebulge (boxy bulge +classical bulge) in our model, includingthe remaining disk component in the boxy bulge region, is ∼ . × M ⊙ . Of this, 0 . × M ⊙ is in the classicalbulge, and ∼ . × M ⊙ in the boxy bulge and the cen-tral disk. Since the classical bulge extends further above thegalactic plane than the boxy bulge, the metallicity composi-tion of the composite system would change with height. Weplan to investigate this further and use our model to look forsignatures of the classical bulge from the metallicity distri-bution in order to compare with observations of the MilkyWay. Previous studies mainly based on N -body simulationshave focused on the bar-halo interaction (Athanassoula2002; Weinberg & Katz 2007a; Villa-Vargas et al. 2009;Dubinski et al. 2009) and shown that a significant amountof angular momentum emitted by the bar is absorbedin the dark matter halo. The angular momentum gainedby the halo changes the internal structure of the darkmatter halo. Some authors have utilized this mecha-nism of angular momentum exchange to resolve the cusp-core issue (El-Zant et al. 2001; Weinberg & Katz 2002;Sellwood 2008) in galaxies while others have focused on thehalo-bar (Holley-Bockelmann et al. 2005; Col´ın et al. 2006;Athanassoula 2007). However, direct observational evidencefor the halo-bar, and hence direct observational verificationof the ongoing secular evolution and angular momentumtransfer can not be obtained unless dark matter is detected.Unlike the dark matter, it is possible to observe galacticbulge stars in detail, both the kinematics and stellar popu-lation parameters. It is thus possible to verify observation-ally the bar-bulge interaction and the resulting dynamicalproperties of the ClBb. Observational evidence for the ClBbcould be a direct confirmation of the angular momentumtransfer and secular evolution in the galaxy. The secular processes driven by the bar not only restruc-ture the disk, but also the other components in the galaxy.Since the classical bulge is less massive here compared withthe surrounding dark matter halo, the angular momentumgained by the classical bulge has a more significant effecton its evolution. The present work has shown in consider-able detail that the unavoidable gravitational interaction be-tween these two components can have profound implicationsfor the structure of a low mass classical bulge, as highlightedbelow.1. We have established that the main mechanism of an-gular momentum transport operating between the bar andthe classical bulge is through resonances. The bulge parti-cles gain angular momentum emitted by the bar through thebar’s ILR ( η = 0 . η = − , − η ∈ ( − . , . )during the dynamical phase when bar growth is rapid. Ap-proximately 3 / c (cid:13) , 1– ?? Saha et al. by the classical bulge during the dynamical phase wherestochastic orbits contribute ∼ Acknowledgements
K.S. acknowledges support from the Alexander von Hum-boldt Foundation. It is a pleasure to thank Lodovico Coccatofor his generous help with IRAF, Jerry Sellwood for lettingus use his potential solver code, and Francoise Combes forvaluable inputs. The authors thank the anonymous refereefor useful comments.
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