Stellar Orbital Studies in Normal Spiral Galaxies II: Restrictions to Structural and Dynamical parameters on Spiral Arms
aa r X i v : . [ a s t r o - ph . GA ] J un Stellar Orbital Studies in Normal Spiral Galaxies II: Restrictions to Structuraland Dynamical parameters on Spiral Arms
A. P´erez-Villegas , B. Pichardo , E. Moreno Max-Planck-Instit¨ut f¨ur Extraterrestrische Physik, Gießenbachstraße, 85748 Garching,Germany;[email protected] Instituto de Astronom´ıa, Universidad Nacional Aut´onoma de M´exico, A.P. 70–264, 04510,M´exico, D.F.; Universitaria, D.F., M´exico
ABSTRACT
Making use of a set of detailed potential models for normal spiral galaxies, weanalyze the disk stellar orbital dynamics as the structural and dynamical parameters ofthe spiral arms (mass, pattern speed and pitch angle) are gradually modified. With thiscomprehensive study of ordered and chaotic behavior, we constructed an assemblage oforbitally supported galactic models and plausible parameters for orbitally self-consistentspiral arms models. We find that, to maintain orbital support for the spiral arms, thespiral arm mass, M sp , must decrease with the increase of the pitch angle, i ; if i is smallerthan ∼ ◦ , M sp can be as large as ∼ ∼ ∼
5% of the disk mass, for Sa, Sb, andSc galaxies, respectively. If i increases up to ∼ ◦ , the maximum M sp is ∼
1% of thedisk mass independently in this case of morphological type. For values larger than theselimits, spiral arms would likely act as transient features. Regarding the limits posed byextreme chaotic behavior, we find a strong restriction on the maximum plausible valuesof spiral arms parameters on disk galaxies beyond which, chaotic behavior becomespervasive. We find that for i smaller than ∼ ◦ , ∼ ◦ , ∼ ◦ , for Sa, Sb, andSc galaxies, respectively, M sp can go up to ∼ i is around ∼ ◦ , ∼ ◦ , ∼ ◦ , M sp is ∼ ∼ ∼
3% of the massof the disk. Beyond these values, chaos dominates phase space, destroying the mainperiodic and the neighboring quasi-periodic orbits.
Subject headings:
Chaos – galaxies: evolution – galaxies: kinematics and dynamics –galaxies: spiral – galaxies: structure
1. Introduction
With the advent of the new extended and profound surveys of the Galaxy and other galaxies, wewill likely understand much more of spiral galaxy morphology and kinematics with unprecedented 2 –detail. At this moment however, our understanding of splendid structures in galaxies, such asthe spiral arms, is quite limited; including for example, its very nature and origin, how they aresupported, whether they are long lasting or transient, if they exhibit noticeable (observable) effectson the stellar and gaseous dynamics behavior, what are their orbital effects in different types ofspirals, etc.The first firm step into the path of understanding the Milky Way and spiral galaxies in generalwas given by morphological classifications, that have provided important statistical informationabout their structural parameters, such as luminosity ratios of their main components (bulge, disk,and nonaxisymmetric large scale features: bars, spiral arms, rings, etc.), rotation curve, spiral-arms pitch angles, etc. The first morphological classification that tried to taxonomize galaxies wasthe Hubble sequence (Hubble 1926, 1936). This represents the simplest classification scheme andwithin it, the normal spiral galaxies, in which we focus this work, range from ‘early’ to ‘late’ (Sa toSc), mainly based on two criteria: the pitch angle of spiral arms and the bulge-to-disk luminosityratio. An Sa galaxy possesses smoother closed spiral arms and a conspicuous central bulge, an Scgalaxy has open and remarkably structured spiral arms and a small central bulge, an Sb galaxy isintermediate between both types. Although the Hubble morphological classification is satisfactoryfor galaxies with redshift z < . ◦ to 50 ◦ (Kennicutt 1981; Ma et al. 2000; Davis et al. 2012) for late type galaxies, for instance.Some recent studies present spiral arms as likely transient features from simulations (D’Onghiaet al. 2013; Baba et al. 2013; Grand et al. 2012a,b; Roˇskar et al. 2012; Wada et al. 2011; Sellwood2011; Fujii et al. 2011; Dobbs & Bonnell 2006), or transient as a product of overlapping of multiplespiral modes coupled together through resonances at the corotation radius (Sellwood & Carlberg2014; Roˇskar et al. 2012; Quillen et al. 2011). Spiral arms have also been found to corotate withthe disk (i.e. winding up, therefore transient ; Roca-F`abrega et al. 2013; Kawata et al. 2014). Also,from the observational point of view there seem to be evidences of transient spiral arms (Speights& Westpfahl 2011, 2012; Ferreras et al. 2012; Foyle et al. 2011; Meidt et al. 2008; Meidt et al.2009; Merrifield et al. 2006). On the other hand, observational and theoretical evidences of theopposite, i.e., long-lived spiral arms, has been presented (Scarano & L´epine 2013; Cedr´es et al.2013; Mart´ınez-Garc´ıa & Gonz´alez-Lopezlira 2013; Law et al. 2012; Scarano et al. 2011; S´anchez-Gil et al. 2011; Egusa et al. 2009; Grosbøl & Dottori 2009; Zhang 1998; Donner & Thomasson1994; Efremov 1985). Whether spiral arms are all transient features, or in some cases they couldbe long-lasting, remains still a polemic matter in modern astrophysics.In previous studies (P´erez-Villegas et al. 2013, hereafter Paper I, and P´erez-Villegas et al.2012), we employed the ideal Hubble classification scheme as the base to construct the axisymmetricbackground gravitational potential models for spiral galaxies (i.e. bulge, disk and halo). Withthis set of models, we analyzed the stellar orbital dynamics on disks, produced by spiral arms in 3 –different galactic morphological types. The constructed galactic potential models then follow thetypical morphology that characterizes the Hubble sequence in terms of the rotation curve, bulge-to-disk mass ratios and scale-lengths. These studies were performed on steady, realistic potentialsfor spiral galaxies. These type of potentials can not follow the galactic evolution, but they are ableto provide some restrictions of potentials based on the detailed structure of orbital chaos, and onthe existence and structure of periodic orbits as the dynamical support to shape stellar systems.The main purpose of our study has been to disentangle all possible details of the orbital structure,which are not straightforward to discern yet on N-body simulations.Using these models, we performed an extensive study of the pitch angle in normal spiralgalaxies. We run thousands of orbits for different timescales depending on the specific problem.Two restrictions to the spiral arms structure were imposed theoretically; one on their steady ortransient nature and the other on their maximum pitch angle prior to destruction. The firstrestriction is based on the orbital ordered behavior, where we found a maximum pitch angle of ∼ ◦ , ∼ ◦ and ∼ ◦ for Sa, Sb and Sc spiral galaxies, respectively. Up to these limits thedensity response supports closely the imposed spiral arms at all radii, the spiral arms are stable,and could be of long-lasting nature. Galaxies with spiral arms having pitch angles beyond theselimits would rather be explained as transient features. The second restriction is based on chaoticorbital behavior; in this case the limits for the pitch angle are ∼ ◦ , ∼ ◦ and ∼ ◦ for Sa, Sband Sc spiral galaxies, respectively. Beyond these limits, chaos becomes pervasive wiping out thespiral arms.In the present analysis we continue our extensive stellar dynamical studies in normal spiralgalaxies, from early to late types, but now we cover all the most important spiral arms parametersthat include: arms total mass (relative to the disk), angular velocity, and pitch angle interrelated.We produce experiments exploring the statistical effect on stellar orbits on the galactic plane dueto the variations of these parameters. The main objectives of this work are (1) to elucidate theinfluence of spiral arms on different morphological types of galaxies, as going from early to latetypes, (2) to provide some restrictions to structural and dynamical parameters of galaxies, and (3)to produce a set of parameters for ‘allowed spiral models’, which are self-consistent from an orbital(periodic orbits) point of view, with good probabilities of being long-lasting structures, and withmild or quiet chaotic nature. With these parameters, steady models can be constructed that resultin likely robust and persisting entities.This paper is organized as follows. The galactic potential and the methodology are describedin Section 2. The effect on the disk dynamics, due to the variation of the spral arms mass and theirangular velocity in different morphological types (Sa, Sb and Sc galaxies) is presented in Section 3.In Section 4, we present a valid set of structural and dynamical parameters for plausible long-lastingspiral arms nature, and also their maximum values before chaos dominates. In Section 5 we discussthe effect of structural and dynamical parameters of spiral arms (pitch angle, angular speed, andmass) in normal spiral galaxies and present our conclusions. 4 –
2. Numerical Implementation and Methodology
With the use of the observationally motivated family of models for normal spiral galacticpotentials presented in Paper I, we performed a comprehensive stellar orbital dynamics study. Themain tools employed for this task are periodic orbits, density response calculations, and phase-space (Poincar´e) diagrams. The potential of each galaxy is formed by an axisymmetric part, plusa nonaxisymmetric potential represented by a detailed model of the spiral arms. In the following,we summarize some properties of the galactic models and the employed tools.
The galactic models consist of axisymmetric and nonaxisymmetric parts. The axisymmetricpart is formed by a bulge and disk of the form proposed by Miyamoto-Nagai (1975), and a massivespherical halo (Allen & Santill´an 1991). With these components, in Paper I we fit the differentgalaxy types considering the typical rotation curves for Sa, Sb and Sc galaxies, and the bulge-to-diskmass ratios (see Figure 1 of Paper I). The nonaxisymmetric part is the three-dimensional modelof spiral arms given by Pichardo et al. 2003, called PERLAS. This model is a mass distributionformed by a set of inhomogeneous oblate spheroids lying on a logarithmic spiral locus; it has beentested and compared with other theoretical models (Martos et al. 2004; Antoja et al. 2009, 2011).In Table 1 we present the observational and theoretical parameters employed to fit the galacticmodels and their respective references (data taken from Paper I). In this table, D, B, H, refer tothe disk, bulge, and halo components, respectively; M sp is the total mass of the spiral arms, i istheir pitch angle, and Ω p is their angular speed. In our models we take a clockwise rotation. Tosimplify the notation, in the following we call µ = M sp /M D , i.e., the ratio of the mass of the spiralarms to the mass of the disk.In our models, regarding the radial extent of the spiral arms, we consider as their initialand final galactocentric radii the inner Lindblad resonance (ILR) and the corotation resonance(CR), respectively. This is based on theoretical studies of orbital self-consistency of spiral arms(Contopoulos & Grosbøl 1986, 1988; Patsis et al. 1991; Pichardo et al. 2003). In Table 2 we presentthe positions of the main resonances ILR, 4/1, and CR for normal spiral galaxies (from early tolate types). With a clockwise rotation for the disk, we assume Ω p between 10 and 60 km s − kpc − ,independently of the Hubble type.For the mass of the spiral arms, we explore its orbital effect in the range from 1% to 10% ofthe total disk mass, i.e., µ between 0.01 and 0.1, independently of the Hubble type. On the otherhand, the spiral arms strength depends mainly on two parameters, their mass and pitch angle.To measure this strength and assure that our parameters are within observational limits, we haveemployed the Q T parameter (Combes & Sanders 1981). This parameter Q T is defined as 5 –Table 1. Parameters of the Axisymmetric Potential Parameter Value ReferenceSpiral ArmsSa Sb ScLocus Logarithmic 1,9,10Arms Number 2 2Pitch Angle i ( ◦ ) 7 – 20 10 – 20 15 – 30 3,7 µ = M sp /M D p 1 ( km s − kpc − ) 10 to 60 1,6Axisymmetric ComponentsM D /M H 2 B /M D ( km s − ) 320 250 170 7M D (10 M ⊙ ) 12.8 12.14 5.10 4M B (10 M ⊙ ) 11.6 4.45 1.02 M B /M D basedM H (10 M ⊙ ) 16.4 12.5 4.85 M D /M H basedDisk Scale-Length ( kpc) 7 5 3 4,5Constants of the Axisymmetric Components Bulge (M B , b ) D , a , b ) H , a ) The rotation of the spiral arms is clockwise. Up to 100 kpc halo radius. V max . In galactic units, where a galactic mass unit = 2 . × M ⊙ and a galactic distance unit = kpc. b , a , b , and a are scale lengths.References. — (1) Grosbøl & Patsis 1998; (2) Drimmel et al. 2000; Grosbøl et al. 2002; Elmegreen &Elmegreen 2014; (3) Kennicutt 1981; (4) Pizagno et al. 2005; (5) Weinzirl et al. 2009; (6) Patsis et al. 1991;Grosbøl & Dottori 2009; Egusa et al. 2009; Fathi et al. 2009; Gerhard 2011; (7) Brosche 1971; Ma et al. 2000;Sofue & Rubin 2001; (8) Block et al. 2002; (9) Pichardo et al. 2003; (10) Seigar & James 1998; Seigar et al.2006. Q T ( R ) = F max T ( R ) / |h F R ( R ) i| , (1)where F max T (R) = | (cid:0) R ∂ Φ(R , θ ) /∂θ (cid:1) | max , is the maximum amplitude of the tangential force atgalactocentric radius R, and h F R (R) i , is the average axisymmetric radial force at the same radius,derived from the m = 0 Fourier component of the gravitational potential.In Figure 1a we show the maximum value, (Q T ) max , of the parameter Q T for each galaxy type(Sa, Sb and Sc), as we increase the pitch angle i from 0 ◦ to 90 ◦ . For this figure we employed themaximum plausible value of µ (i.e. before chaos dominates the phase space surrounding the familiesof orbits that support the pattern; see Section 3), of 0.1, 0.07, and 0.05, for Sa, Sb and Sc galaxies,respectively. In our study, taking i between 7 ◦ and 30 ◦ (Table 1) and using the maximum limitsof the mass of the spiral arms, (Q T ) max is not larger than ∼ .
25. From the literature, reasonablemaximum values for Q T are ∼ . ∼ . T ) max for each galaxy type as we increase µ from 0.01 to 0.1. In thisfigure we have considered the value of i before chaos dominates the available phase space, whichcorresponds to ∼ ◦ , ∼ ◦ and ∼ ◦ , for Sa, Sb and Sc galaxies, respectively (see Section 3).For Sa and Sb models, (Q T ) max increases much slower than for the Sc ones. In early galaxy typemodels very massive spiral arms are allowed within this observational restriction without exceedingobservational limits, even for spiral arms with µ = 0.1; see for example (Weiner & Sellwood 1999).For the Sc type, (Q T ) max increases much faster with mass, and for approximately µ > T ) max > .
3) more than what observations indicate for this typeof galaxies.
For the orbital dynamics analysis we employed periodic orbits and Poincar´e diagrams. Wehave also calculated the density response as in Paper I, using the method of Contopoulos & Grosbøl(1986), which quantifies the support of spiral arms with periodic orbits. This method to estimatethe density response has been widely used in literature (Contopoulos & Grosbøl 1988; Amaral &Lepine 1997; Yano et al. 2003; Pichardo et al. 2003; Voglis et al. 2006; Tsoutsis et al. 2008;P´erez-Villegas et al. 2012, 2013; Junqueira et al. 2013). We computed between 40 and 60 periodicorbits for each galactic model. The density response is defined as the regions where periodic orbitscrowd producing a density enhancement. The position of the maximum density response along eachperiodic orbit is calculated, and with these positions the locus formed by the maxima of densityresponse is found and compared with the position of the imposed spiral locus (i.e. PERLAS).The method implicitly considers a small and variable dispersion since it studies a region wherethe flux is conserved. Additionally, we estimated the average density response around each one of 7 –
Fig. 1.— Maximum value, (Q T ) max , of the parameter Q T ( R ). The continuous, dotted, and dashedlines give (Q T ) max for an Sa, Sb, and Sc galaxy, respectively. In a ), we show (Q T ) max vs. pitch angleof the spiral arms, i , where µ = M sp /M D = 0.1, 0.07, and 0.05 for Sa, Sb and Sc, respectively. In b ),we show (Q T ) max vs. µ , where the pitch angle is 30 ◦ , 40 ◦ and 50 ◦ for Sa, Sb and Sc, respectively. 8 –these maxima response. In order to do that, we took a circular vicinity, and compared the densityresponse with the imposed density (this is the sum of the axisymmetric disk density on the galacticplane and the central density of the spiral arms).Regarding the Poincar´e diagrams, these are constructed in the plane ( x ′ , v ′ x ), in the non-inertialreference system that rotates with the spiral arms. The x ′ axis points toward the direction of theline where the spiral arms begin in the inner galactic region. Poincar´e diagrams have two regions:the prograde region, where the stars move in the same direction of rotation of the spiral arms,and the retrograde region, where the stars move in opposite direction to the spiral arms rotation.These regions (prograde and retrograde) were defined in the non-inertial frame where the spiralarms are at rest. In our models the rotation of the spiral arms is clockwise, thus, the left side ofthe diagram is prograde (launching orbits with x ′ < , v ′ y > x ′ > , v ′ y > X periodic orbits regions). Large-scalestructures are not expected to arise from systems fully dominated by chaos (Voglis et al. 2006).The most interesting part of the phase space diagrams is the prograde region where the greatmajority of stars are moving in spiral galaxies. Chaos is generated in this region, mostly due toresonance overlapping (Martinet 1974; Athanassoula et al. 1983 and references therein). In theretrograde region, the resonances are very separated, thus the production of chaos is almost null.Each Poincar´e diagram contains 50 orbits, distributed between the prograde and retrograde regions,with 300 points each (points correspond to the numbers of periods). For more details about thismethodology (periodic orbits, density response and phase-space diagrams), see Paper I.
3. Orbital Study of Ordered and Chaotic Behavior
With the methodology described in the previous section, we have carried out a detailed orbitalstudy with periodic orbits, density response calculations, and phase space diagrams. With all thesetools we try to determine whether limiting values to different structural and dynamical parametersof normal spiral galaxies can be established. In Paper I we found two limits to the pitch angle i for normal spiral galaxies (Sa, Sb and Sc); in the present study we set limits to combinations ofthe spiral arms-to-disk mass ratio, µ , the angular velocity of the spiral arms, Ω p , combined withthe pitch angle, i , in order to seek for plausible long-term spiral galactic models. By plausible andlong-lasting in this context, we mean spiral arms fully supported by periodic orbits and moderateproduction of chaotic behavior. 9 –To measure observationally both the mass and angular velocity of spiral arms is not an easytask. We have tried instead to constrain these parameters through the orbital support of the spiralarms with periodic orbits (ordered behavior), and with the study of chaotic behavior with Poincar´ediagrams, by searching for a limit before chaos dominates the available phase space and destroysperiodic orbits.We present in this section a family of orbitally plausible long-lasting spiral galactic potentials,and provide optimal ranges for the parameters i , Ω p , and µ . We performed an exhaustive study of periodic orbits for different morphological galactic types.With the maps of periodic orbits we found the position of the maximum density response alongeach periodic orbit, and in order to analyze some orbital self-consistency of the spiral arms, thesepositions were compared with the center of the imposed spiral pattern. If the imposed spiralarms are supported by the maxima of density response, then these arms are stable and are of along-lasting nature. If this condition is not satisfied, the spiral arms might be rather explained astransient structures.For each morphological type, we used the corresponding axisymmetric background potential,based on the parameters presented in Table 1. In order to dilucidate the relative importance ofthe different parameters, we present in this section several examples first. µ has been varied from0.01 to 0.1, and the representative employed pitch angles i are taken as 10 ◦ , 15 ◦ and 20 ◦ , for Sa,Sb and Sc galaxies, respectively. Additionally, in our computations we have varied slightly Ω p ineach galactic type.In Figure 2, for an Sa galaxy with i = 10 ◦ , we show in the galactic plane x ′ , y ′ the maxima ofdensity response, with filled squares, which correspond to crowding regions of periodic orbits (blackcurves) that produce density enhancements; the center of the imposed spiral arms potential (PER-LAS model) is shown with open squares and the dotted lines mark the width of spiral arms. Fromtop to bottom panels (the x ′ axis is at the bottom), Ω p lies in the interval [20 ,
40] km s − kpc − ,and the values of µ are marked at the top. In each panel we show with red, blue, and yellow circlesthe positions of the resonances ILR, 4/1 and CR; see Table 2. This figure shows that the maximaof density response coincide well (within 3 ◦ difference) with the imposed spiral arms, but the extentof this density support reduces significantly for the largest values of µ , reaching only up to the 4/1resonance if µ is around 0.1. Also, the density support diminishes strongly if Ω p increases.In Figure 3 we consider i = 20 ◦ in an Sa type galaxy. In this case the orbital dynamics isstrongly affected. The density response systematically forms spiral arms with a smaller pitch anglethan the imposed 20 ◦ , and with a reduced radial extent compared with the case i = 10 ◦ . Thepartial density support is destroyed when Ω p increases. Spiral arms with this strong forcing in agalaxy, might be better explained as transient structures. In Paper I we found that the regime 10 –where the spiral arms of Sa galaxies are transient occurs when i & ◦ .Figures 4 and 5 show our results for an Sb galaxy with pitch angles i = 15 ◦ and i = 20 ◦ ,respectively. In these cases Ω p lies in the interval [15 ,
35] km s − kpc − . These figures show asimilar behavior to the case of an Sa galaxy, but it was harder to obtain a reasonable densitysupport for larger values of µ . With the greater value i = 20 ◦ , the resulting response pitch angle isslightly smaller than the imposed 20 ◦ .Our results for an Sc galaxy are shown in Figures 6 and 7, with pitch angles i = 20 ◦ and i =30 ◦ , respectively. Ω p lies in the interval [10 ,
30] km s − kpc − . We do not find a density supporttoward the greatest values of µ and Ω p , and for their smallest values the response density shows apitch angle smaller than the imposed one. The radial extent of this response shortens comparedwith that obtained in Sa and Sb types.In order to complement and to reinforce the results obtained by the construction of periodicorbits, in Figures 8 -13, we have compared the spiral arm density response (filled squares) with thespiral arms imposed density (PERLAS, open squares). Each mosaic of density response (Figures 8-13) corresponding to each periodic orbit mosaic (Figures 2 -7). In Figure 8, we present densitiesfor an Sa galaxy with i = 10 ◦ . As the maximum density response was shown in Figure 2, this figurepresents that for µ up to ∼ .
05, the density response fits well with the imposed density. Figure 9shows also a Sa galaxy, but with i = 20 ◦ , in this case the spiral arms are stronger, and we see thatdensity response fits to imposed density with a smaller µ . Therefore, if the pitch angle increases,the allowed mass in spiral arms should be smaller, in order to maintain the orbital support and thedensity response fits better to the imposed density.Figures 10 and 11 show density response for an Sb galaxy with pitch angles i = 15 ◦ and i = 20 ◦ , respectively. In these figures we see a similar behavior than for Sa galaxies. The densityresponse in this case, fits the imposed density up to µ ∼ .
03, but if the pitch angle increases, thevalue of µ is affected.Figures 12 and 13 show densities for an Sc galaxy with pitch angles i = 20 ◦ and i = 30 ◦ ,respectively. In these figures we see a similar behavior than in Sa and Sb galaxies. We can noticethat the density response fits to imposed density up to µ ∼ .
03. For larger values of µ and Ω p ,the density support is not found.Figures 8 -13, show that the response is compatible with the imposed densities up to a certainlimit in mass. The larger the force of the spiral arms, the stronger the response relative to theimposed one. It is worth noticing that, this over-response does not indicate that the model isinconsistent as long as the response is in phase. Rather, it would indicate a growing mode, whichwould be probably damped by an increase in velocity dispersion, if feedback were included in atotally self-consistent model.Now, in order to study the chaotic behavior, we produced a comprehensive study of Jacobienergy ( E J ) families in phase space, from very bounded orbits (the inner part of galaxy) to the 11 –Fig. 2.— Periodic orbits (black curves), density response maxima (filled squares), and the imposed spiralarms locus (open squares and dotted lines mark the width of spiral arms) for the three-dimensional spiralarms model of an Sa galaxy with a pitch angle i = 10 ◦ . The values of µ = M sp /M D and the angular speedof the spiral arms, Ω p in units of km s − kpc − , are given at the top and left, respectively.
12 –Fig. 3.— As in Figure 2, here with i = 20 ◦ . 13 –Fig. 4.— Periodic orbits (black curves), density response maxima (filled squares), and the imposed spiralarms locus (open squares and dotted lines mark the width of spiral arms) for the three-dimensional spiralarms model of an Sb galaxy with a pitch angle i = 15 ◦ . The values of µ and Ω p are given at the top andleft, respectively.
14 –Fig. 5.— As in Figure 4, here with i = 20 ◦ . 15 –Fig. 6.— Periodic orbits (black curves), density response maxima (filled squares), and the imposed spiralarms locus (open squares and dotted lines mark the width of spiral arms) for the three-dimensional spiralarms model of an Sc galaxy with a pitch angle i = 20 ◦ . The values of µ and Ω p are given at the top andleft, respectively.
16 –Fig. 7.— As in Figure 6, here with i = 30 ◦ . 17 – I L R / CR Fig. 8.—
Filled squares are the density response of spiral arms for an Sa galaxy, and open squaresrepresent the imposed density with a pitch angle i = 10 ◦ . The values of µ and Ω p are given at the top andleft, respectively. The dotted, dashed and dot-dashed lines show the ILR position, 4/1 resonance positionand CR position, respectively.
18 – I L R / CR Fig. 9.— As in Figure 8, here with i = 20 ◦ . 19 – I L R / CR Fig. 10.—
Density response diagrams. Filled squares are the density response of spiral arms for an Sbgalaxy, and open squares represent the imposed density with a pitch angle i = 15 ◦ . The values of µ and Ω p are given at the top and left, respectively. The dotted, dashed and dot-dashed lines show the ILR position,4/1 resonance position and CR position, respectively.
20 – I L R / CR Fig. 11.— As in Figure 10, here with i = 20 ◦ . 21 – I L R / CR Fig. 12.—
Density response diagrams. Filled squares are the density response of spiral arms for an Scgalaxy, and open squares represent the imposed density with a pitch angle i = 20 ◦ . The values of µ and Ω p are given at the top and left, respectively. The dotted, dashed and dot-dashed lines show the ILR position,4/1 resonance position and CR position, respectively.
22 – I L R / CR Fig. 13.— As in Figure 12, here with i = 30 ◦ . 23 –outer galaxy, even in some cases passing the corotation barrier. In the Poincar´e diagrams we variedthe spiral arms mass, and their angular velocity and pitch angle, as we did with the periodic orbitalstudy. The analysis of the chaotic behavior is relevant because it can provide constraints to themaximum values of some important parameters of galaxies (pitch angle or spiral arms masses,for example). With this study, we find a limit to the spiral arms mass, for which chaos becomespervasive dominating the available phase space and destroying all periodic orbits as well as theordered orbits surrounding them.We present a set of Poincar´e diagrams for each morphological type. In our experiments,assorted spiral arms masses, pitch angles and angular velocities are tested. We explored a compre-hensive set of E J families, from energies representing the most bounded orbits (galactic centers)to the corotation barrier and beyond to cover the total extension of the spiral arms. The valuespresented in the mosaics of Poincar´e diagrams correspond approximately to the CR position ineach case (that represent the most extreme and clear cases, regarding chaos). For energies morebounded the presence of chaos diminishes, but the general behavior is similar, i.e., if the pitchangle (or mass) increases chaos increases in the different energies. However, when chaos becomespervasive and the main periodic orbits are destroyed, the chaotic behavior dominates in boundedenergies as much as closer to corotation.Figures 14 to 19 show phase-space diagrams for Sa, Sb, and Sc galaxies, considering in eachtype two values of the pitch angle. The common trend in all these diagrams is that the chaoticregion which appears in the prograde (left) sides increases as µ and Ω p increase. This chaotic regionextends toward the inner galactic region, destroying periodic orbits that could support the spiralarms. In each galactic type the chaotic region is more extended for the larger employed value of i ,and it is also markedly stronger for an Sc galaxy.In summary, analyzing orbital self-consistency through periodic orbits, we find that in orderto produce long-lasting spiral arms, the ratio µ in early spiral galaxies can be much larger thanin late spiral galaxies without compromising the stability of the arms. Consequently, when thepitch angle is smaller, the limit for µ can be considerably larger. Approximately, the intervals in i and µ to obtain long-lasting spiral arms in this scheme are the following: for an Sa galaxy withΩ p ∼
30 km s − kpc − , i . ◦ and µ . .
07; for an Sb galaxy with Ω p ∼
25 km s − kpc − , i . ◦ and µ . .
05; and for an Sc galaxy with Ω p ∼
20 km s − kpc − , i . ◦ and µ . .
03. For greatervalues than these, the spiral arms would be rather explained as transient structures. The limits for µ are only examples that depend on the values of i and Ω p ; this means that these parameters aredeeply interrelated.Regarding the chaotic behavior, with phase-space studies we also found a maximum value for µ , before chaos becomes pervasive destroying the main periodic orbits which give support to spiralarms. As we mentioned in the ordered case, the maximum limit of µ is mainly linked to i and lessto Ω p . Therefore, when i is smaller, the limit for µ can be larger, this is due to both parametersare related to the spiral arm force (or amplitude of the force). An example of the limits for µ
24 –Fig. 14.— Phase-space diagrams for an Sa galaxy with i = 10 ◦ . The values of the Jacobi energy, µ , and Ω p , are given at the right, top, and left, respectively. 25 –Fig. 15.— As in Figure 14, here with i = 20 ◦ . 26 –Fig. 16.— Phase-space diagrams for an Sb galaxy with i = 15 ◦ . The values of the Jacobi energy, µ , and Ω p , are given at the right, top, and left, respectively. 27 –Fig. 17.— As in Figure 16, here with i = 20 ◦ . 28 –Fig. 18.— Phase-space diagrams for an Sc galaxy with i = 20 ◦ . The values of the Jacobi energy, µ , and Ω p , are given at the right, top, and left, respectively. 29 –Fig. 19.— As in Figure 18, here with i = 30 ◦ . 30 –depending on i and Ω p are: for an Sa galaxy, µ . .
1, with i . ◦ and Ω p ∼
40 km s − kpc − ; foran Sb galaxy, µ . .
07, with i . ◦ and Ω p ∼
35 km s − kpc − ; and for an Sc galaxy, µ . . i . ◦ and Ω p ∼
20 km s − kpc − . For grater values of µ the spiral arms are destroyed bychaotic behavior.This analysis are some selected examples to clarify the general orbital behavior. In Section 4we will summarize in a set of plots the ordered and chaotic behavior, taking a significant increasein the number of values of the parameters µ and i employed. In Section 3.1, we have analyzed the effect of the mass of the spiral arms on the orderedand chaotic stellar dynamics on the equatorial plane of normal spiral galaxies; for this purpose weemployed assorted masses, pitch angles and angular velocities of the spiral arms. In this Sectionwe present a similar orbital study, analyzing the effect in the ordered and chaotic stellar dynamicsas we vary Ω p in an extended interval, from 10 to 60 km s − kpc − for each morphological type.As in the case of the spiral-arms-mass analysis, in order to dilucidate their relative importance, wealso slightly change other parameters; for the pitch angle we take respectively the values 7 ◦ , 18 ◦ ,and 25 ◦ , in Sa, Sb, Sc galactic types, and in all these galactic types µ takes the values 0.01, 0.03,and 0.05. As in the previous subsection, these are only some examples to obtain a perception ofthe dynamical behavior exerted by changes in the spiral arms parameters. In the next section wesummarize the results.In Figure 20 we present periodic orbits for an Sa galaxy with i = 7 ◦ . This figure shows thatthe amount of periodic orbits which give support to the spiral arms decrease with Ω p and µ . ForΩ p .
30 km s − kpc − , the density response follows the imposed spiral arms potential almost tothe CR position; for Ω p ∼
40 km s − kpc − , the density support extends slightly beyond the 4/1resonance position. This behavior is obtained with µ ∼ .
01. If µ increases between 0.03 and 0.05the density support extends beyond the 4/1 resonance position. For Ω p >
40 km s − kpc − , thereis no density support. If we increase the pitch angle to 18 ◦ , the density support extends almost tothe CR position only if µ < ∼ µ there is a density support up to the 4/1resonance. If Ω p >
30 km s − kpc − and µ < ∼ i = 18 ◦ . For Ω p .
30 km s − kpc − and µ = 0.01, the density support extends approximately up to the CR position. For Ω p =40 km s − kpc − and µ = 0.01 this support extends only up to the 4/1 resonance. If µ > Periodic orbits (black curves), density response maxima (filled squares), and the imposed spiralarms locus (open squares and dotted lines mark the width of spiral arms) for the three-dimensional spiralarms model of an Sa galaxy with a pitch angle i = 7 ◦ . The values of µ and Ω p are given at the top and left,respectively.
32 –imposed arms. For cases where Ω p >
40 km s − kpc − , there is no density support.In Figure 22 we show periodic orbits for an Sc galaxy with i = 25 ◦ . For Ω p .
30 and µ = 0.01,the density support extends not far from the CR position. For µ = 0.03, 0.05, this support extendsup to the 4/1 resonance position (in some cases slightly beyond) forming a smaller pitch angle thanin the imposed arms. With µ = 0.05 there is no density support if Ω p ∼
30 km s − kpc − or larger.As we did in the case where we analyzed the effect of spiral arms mass, we have comparedthe spiral arms density response (filled squares) with the spiral arms imposed density (PERLAS,open squares). We constructed a mosaic of density response corresponding to each periodic orbitmosaic. Figures 23, 24 and 25 show the densities for Sa, Sb and Sc galaxies, respectively. Withthese mosaics we reinforce the results presented with periodic orbits and maxima density responsein Figures 20 - 22.Now, in order to analyze the chaotic behavior, as was done varying the mass of spiral arms(Section 3.1), in this part we also present a detailed study of Poincar´e diagrams varying the angularspeed of the spiral arms in an extended interval. We change slightly the mass of the spiral armsand the pitch angle. With this analysis we found a limit to the angular speed for which chaosbecomes pervasive, dominating the available phase space and destroying all the periodic orbits aswell as the ordered orbits surrounding them.Figure 26 shows Poincar´e diagrams for an Sa galaxy with i = 7 ◦ . For Ω p = 10 and 20 km s − kpc − the orbital behavior is ordered; however the chaotic regions emerge when µ increases from 0.01 to0.05. For Ω p = 30 and 40 km s − kpc − , the ordered orbits dominate, but even with µ = 0 . µ . If i = 15 ◦ thechaotic behavior increases with the mass and angular speed of the spiral arms.Figure 27 shows Poincar´e diagrams for an Sb galaxy with i = 18 ◦ . The ordered orbits dom-inate the prograde region and there is a small region of chaos, which increases slightly with Ω p .Additionally, the orbits are more complex and resonant islands appear. A severe increment ofthe chaotic region towards the main periodic orbits supporting the spiral arms is related with anincrement of µ ; for example, when µ = 0.05 and Ω p = 40 km s − kpc − , the chaotic behavior coversan important part of the prograde region.Figure 28 shows Poincar´e diagrams for an Sc galaxy with i = 25 ◦ . The majority of orbitsare ordered, but the chaotic region slowly increases with Ω p . The chaotic behavior is more proneto emerge when µ is larger; for example, if µ = 0.05 and Ω p = 40 km s − kpc − chaos dominatesan important region of the available phase-space, covering practically all the prograde region anddestroying the periodic orbits.In summary, regarding ordered behavior, we constrain the angular speed of the spiral armsthrough the existence of periodic orbits. We found that the orbital support for these arms dependson three parameters: the pitch angle, mass, and angular speed, but the orbital support seems tobe much more sensitive first to the pitch angle, and second to the mass of spiral arms, and almost 33 –Fig. 21.— As in Figure 20, here for an Sb galaxy with i = 18 ◦ .
34 –Fig. 22.—
As in Figure 20, here for an Sc galaxy with i = 25 ◦ .
35 –
ILR 4/1 CR
Fig. 23.—
Density response diagrams. Filled squares are the density response of spiral arms for an Sagalaxy, and open squares represent the imposed density with a pitch angle i = 7 ◦ . The values of µ and Ω p are given at the top and left, respectively. The dotted, dashed and dot-dashed lines show the ILR position,4/1 resonance position and CR position, respectively.
36 –
ILR 4/1 CR
Fig. 24.—
As in Figure 23, here for an Sb galaxy with i = 18 ◦ .
37 –
ILR 4/1 CR
Fig. 25.—
As in Figure 23, here for an Sc galaxy with i = 25 ◦ .
38 –Fig. 26.— Phase-space diagrams for an Sa galaxy with i = 7 ◦ . Here a more extended interval ofvalues for Ω p is considered, compared with that employed in Section 3.1. The values of the Jacobienergy, µ , and Ω p , are given at the right, top, and left, respectively. 39 –Fig. 27.— As in Figure 26, here for an Sb galaxy with i = 18 ◦ . 40 –Fig. 28.— As in Figure 26, here for an Sc galaxy with i = 25 ◦ . 41 –insensitive to the angular speed (although it is important because this defines the extension ofthe spiral arms). With our analysis, we set a limit to Ω p for each morphological type taking intoaccount the three parametes: for an Sa galaxy, i . ◦ , Ω p ∼
40 km s − kpc − , and µ . . i . ◦ , Ω p ∼
30 km s − kpc − , and µ . .
03; and for an Sc galaxy, i . ◦ ,Ω p ∼
25 km s − kpc − , and µ . .
01. For laeger values, there are not enough periodic orbits toprovide support to the spiral arms. These limits for i , Ω p , and µ are only examples. In the followingsection we provide a general analysis.Regarding chaotic behavior, with Poincar´e diagrams we also found a maximum value for µ ,before chaos becomes pervasive destroying all the main periodic orbits which give support to spiralarms. As we found for the ordered case, the maximum limit of Ω p is linked to i and µ . Anexample of the limits for Ω p depending on i with µ = 0 .
05 are: for an Sa galaxy with i . ◦ ,Ω p ∼
40 km s − kpc − ; for an Sb galaxy with i . ◦ , Ω p ∼
40 km s − kpc − ; and for an Sc galaxywith i . ◦ , Ω p ∼
30 km s − kpc − .
4. Limits to Parameters for Plausible Dynamical Models for Spiral Arms
Considering the analysis of the effect of the pitch angle that was performed in Paper I, andwith the examples obtained in Section 3 concerning the effects of the mass and angular speed of thespiral arms, in this section we present an extended analysis increasing the number of values studiedof the parameters i and µ in normal spiral galaxies. We study the ordered and chaotic behavior onthe galactic plane, through periodic orbits, maxima density response and Poincar´e diagrams. Wepresent two maximum limits for these parameters. The first of them is regarding periodic orbitsand density response, where the imposed spiral arms are supported by the density response; thiscould tell us about the nature of spiral arms: if they are transient or long-lasting. The second limitis a detailed analysis based on phase space diagrams (regardless the spiral arms nature), before thechaotic behavior becomes pervasive dominating all available phase space, and destroying all orbitalsupport.In Paper I and in P´erez-Villegas et al. (2012) we presented a deep study of the effects of thepitch angle on normal spiral galaxies. The purpose of that study was to provide the values of thepitch angle in galaxies that produced transient or long-lasting galactic models (assuming typicalmasses for the axisymmetric background potential for disk galaxies from Sa to Sc morphologicaltypes). In those studies we considered only the effect of the pitch angle, keeping fixed the massand angular speed of the spiral arms.In Figure 29 we summarize our results. First we present a 3 × µ to obtain long-lasting spiral arms, in Sa, Sb, Sc galaxies, depending on the pitch angle i and angular speed Ω p . Beyond these values the spiral arms would be considered as transientfeatures.Based in the detailed phase-space orbital study presented in P´erez-Villegas et al. (2012) and 42 –Fig. 29.— Dynamically plausible models for Sa (top line), Sb (middle line), and Sc (bottom line)galaxies. The shaded regions provide the parameters to construct long-lasting spiral arms modelsin the scheme presented in this work. Spiral arms with parameters outside the shaded regions,would most likely act as transient features. 43 –Paper I, concerning the restriction of the pitch angle given by the chaotic behavior in normalspiral galaxies, here we present maximum values for structural and dynamical parameters of spiralarms such as pitch angle, mass, and angular speed, before the chaotic behavior dominates theavailable phase-space destroying the main stable periodic orbits. Large-scale structures such asspiral arms are not expected to appear in galaxies where chaotic behavior dominates completely(Voglis et al. 2006); however, confined chaotic orbits may provide some support to spiral arms(Patsis & Kalapotharakos 2011; Kaufmann & Contopoulos 1996; Contopoulos & Grosbøl 1986), upto a certain point, prior to the destruction of all phase space surrounding the periodic orbits thatgive the shape to spiral arms.In Figure 30 we present a 3 × µ and pitch angle i , depending on morphological type and angular speed, before chaos dominates. For values of thespiral arm parameters smaller than those given by the continuous lines, the chaotic behavior maybe important, but it is still confined by stable quasiperiodic orbits. For values lager than theselimits, the chaotic behavior becomes pervasive destroying all the available prograde phase-space.
5. Discussion and Conclusions
With the use of a family of models observationally motivated to simulate typical Sa, Sb and Scspiral galaxies, that includes a bisymmetric density-based spiral arms potential model, we performan extensive analysis of the stellar dynamical effects of spiral arms on galactic disks. The spiralarms model is a self-gravitating three-dimensional potential constructed with individual oblatespheroids (as bricks in a building); this means that the model produces a density based force field(i.e. a more physical model, instead of an ad hoc mathematical fit), which means in turn thatthe potential responds for example to changes on the structure such as a larger pitch angle thatnaturally produces that disk particles feel a more aggressive effect (i.e. the attack angle for particlesis larger the larger the pitch angle is).In this work, we have extended the studies of Paper I (devoted only to pitch angle effects), tothe effects of the spiral arm strength (mass of the spiral arms), its angular speed and pitch angle alltogether. For all morphological types, we varied the mass of the spiral arms within approximately1 to 10% of the mass of the disk, and its angular speed from 10 to 60 km s − kpc − . In Sa, Sb andSc galaxies, pitch angle employed values were 7 ◦ to 20 ◦ , 10 ◦ to 20 ◦ , and 15 ◦ to 30 ◦ , respectively.As in paper I, we present two sets of restrictions different in nature for spiral arms parameters.One is based on ordered dynamical behavior and the second on chaotic behavior. Restrictions basedon ordered behavior, provide us a tool based on orbital support for the spiral arms that refers totheir transient vs. long-lasting nature. The second set of restrictions, based on chaotic behavior,represents the limits beyond which, spiral arms are no longer feasible.For the first limit, we produced an orbital study based on periodic orbits and computed themaxima density response comparing it with the imposed potential to produce a set of plausible 44 –Fig. 30.— Models for Sa (top line), Sb (middle line), and Sc (bottom line) galaxies. The solid lineis the maximum limit for the spiral arms models before the domain of chaotic behavior. Parametersfor spiral arms on the shaded regions, would be dynamically plausible (independently of their likelytransient nature). 45 –models for spiral galaxies with more probable long-lasting spiral arms. In this case we find thatthe mass of the spiral arms, M sp , should decrease with the increase of the pitch angle i ; if i issmaller than ∼ ◦ , M sp can be as large as ∼ ∼ ∼
5% of the disk mass, for Sa, Sb, andSc galaxies, respectively. If i increases to ∼ ◦ , M sp is around 1% of the mass of the disk for allmorphological types. For values larger than these limits, spiral arms would be transient features.For the second limit, we produced a phase-space study with Poincar´e diagrams, based onchaotic orbital behavior. We seek the parameters of the spiral arms prior to the domain of chaosthat destroys all orbital support for the arms. In this case we also found that M sp should decreasewith the pitch angle i . If i is smaller than ∼ ◦ , ∼ ◦ , and ∼ ◦ , for Sa, Sb, and Sc galaxies,respectively, then M sp can be up to ∼
10% of the mass of the disk. If the corresponding i is around ∼ ◦ , ∼ ◦ , and ∼ ◦ , then M sp is 1%, 2% and 3% of the mass of the disk. Beyond these values,chaos dominates all the available phase-space prograde region, destroying the main periodic andthe neighboring quasiperiodic orbits.All the structural and dynamical parameters of the spiral arms play an important role in theorbital dynamics. We found however that the parameter that seems to affect the most the stellardynamics is the pitch angle since this presents a wide range of possible observational values, unlikethe case of the mass (or density contrast), and angular velocity. Within the typical values of thespiral arms angular speed ( ∼ −
40 km s − kpc − ), obtained from observations and theory, thechaotic orbital dynamical response does not seem to be extremely sensitive.With all the performed simulations we summarize our results, which are separated accordingto the two restrictions based on ordered or chaotic orbital behavior: Restrictions based on ordered orbital behavior: The nature of spiral arms, transient orlong-lasting. • If the maxima density response (at all radii), produced by periodic orbital crowding, supportthe imposed potential of the spiral arms, i.e. if the arms are orbitally self-consistent at a firstapproximation, they are more prone to be long-lasting structures; otherwise, the spiral armswould be transient structures. • Considering the combination of all parameters studied in this work for the spiral arms, wepresent Table 3. The table shows for different types of galaxies, the spiral arms persistencebased on orbital support. • All the parameters that characterize the spiral arms combined, have to do with the orbitalsupport. From these, due to the wide range of values that can take in all galaxies, the onewith more effect on stellar and gas dynamics seems to be the pitch angle.
Restrictions based on chaotic orbital behavior: The destruction of spiral arms.
46 –Table 2. Resonance Positions
Sa Sb ScΩ p ILR 4/1 CR ILR 4/1 CR ILR 4/1 CR( km s − kpc − ) ( kpc) ( kpc) ( kpc)10 9.93 20.25 30.23 8.62 17.74 26.13 4.0 11.32 16.7815 6.46 13.83 20.53 5.13 12.18 17.88 2.71 7.44 11.4920 4.45 10.5 15.66 3.52 9.22 13.70 2.03 5.35 8.6325 3.0 8.44 12.69 2.29 7.34 11.14 1.5 4.11 6.9430 3.0 7.04 10.6 2.0 6.04 9.38 1.5 3.34 5.735 3.0 6.02 9.21 2.0 5.11 8.08 1.5 2.83 4.840 3.0 5.24 8.1 2.0 4.40 7.0 1.5 2.45 4.1250 3.0 4.07 6.51 2.0 3.42 5.64 1.5 1.92 3.1960 3.0 3.21 5.4 2.0 2.74 4.65 1.0 1.55 2.6 Table 3. Results based on ordered orbital behavior
Galactic type ParameterΩ sp Pitch angle µ = M sp /M D Spiral arm persistence( km s − kpc − ) ( o )Sa 20 . Ω sp . . . .
07 Long-lasting . . .
02 Long-lasting & & .
08 Transient & & .
03 TransientSb 15 . Ω sp . . . .
04 Long-lasting . . .
02 Long-lasting & & .
05 Transient & & .
03 TransientSc 10 . Ω sp . . . .
04 Long-lasting . . .
01 Long-lasting & & .
05 Transient & & .
02 Transient
47 – • The main parameters that determine the destruction of spiral arms are their pitch angle andmass, both directly related to the force amplitude. The destroying effect of their angularspeed is slight. Table 4 shows the combination of parameters for which chaotic behaviordominates and destroys the spiral arms.This study searches for the periodic orbits, that are expected to be the dynamical backbone ofa given system. We search for their presence or absence, as a condition for the long-lasting supportof large-scale structures in a galaxy. In the same manner, when chaos dominates the phase space(to such extent that even the main periodic orbits are fully destroyed), it is an indication of thedemolition of large scale structures, such as spiral arms. Although it is known that confined chaos(trapped between ordered orbits) is able to provide support to structures like spiral arms (Vogliset al. 2006), this can only be true as long as chaos does not become pervasive.With all the performed simulations we are able to provide a detailed set of plausible galacticmodels (transient or long-lasting), for normal spiral galaxies, and these idealized galactic modelsreproduce astrophysical properties of parameters of observed normal spiral galaxies, such as themaximum pitch angles observed in spirals. Although one might wonder about the effect of a bar,given the fact that bars and spiral arms are formed by disk instabilities, likely, even by similarphysical processes, the region where a bar grows up on a galaxy and the region where spiral armsgrow are dominated by different physical characteristics (e.g. strong differential rotation, mass ratiobetween spiral arms and the hosting disk, structures size and density etc.). In an ongoing work, weinclude a galactic bar potential (combined with the spiral arms). Some preliminary results however,show that the presence of a massive bar will change dramatically the orbital self-consistency studiesand new and different restrictions will be likely posed.We acknowledge the anonymous referee for an excellent review that helped to greatly improvethis work. We thank PAPIIT through grant IN114114. APV acknowledges the support of comple-mentary postdoctoral fellowship of Conacyt at Max Planck Institute for Extraterrestrial Physics.
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This preprint was prepared with the AAS L A TEX macros v5.2.
51 –Table 4. Results based on chaotic behavior
Galactic type ParameterΩ sp Pitch angle µ = M sp /M D Chaos predominates( km s − kpc − ) ( o )Sa 20 . Ω sp . . . .
09 No . . .
02 No & & .
07 Yes & & .
01 YesSb 15 . Ω sp . . . .
07 No . . .
02 No & & .
05 Yes & & .
02 YesSc 10 . Ω sp . . . .
05 No . . .
03 No & & .
04 Yes & & ..