How the bar properties affect the induced spiral structure
L. Garma-Oehmichen, L. Martinez-Medina, H. Hernández-Toledo, I. Puerari
MMNRAS , 1–15 (2021) Preprint 3 February 2021 Compiled using MNRAS L A TEX style file v3.0
How the bar properties affect the induced spiral structure
L. Garma-Oehmichen, ★ L. Martinez-Medina, H. Hernández-Toledo, and I. Puerari. Instituto de Astronomía, Universidad Nacional Autónoma de México, Apartado Postal 70-264, CDMX, 04510, México Instituto Nacional de Astrofísica, Optica y Electrónica, Apdo. Postal 51 y 216, 72000 Puebla, Puebla, México
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Stellar bars and spiral arms co-exist and co-evolve in most disc galaxies in the local Universe.However, the physical nature of this interaction remains a matter of debate. In this work, wepresent a set of numerical simulations based on isolated galactic models aimed to explorehow the bar properties affect the induced spiral structure. We cover a large combination of barproperties, including the bar length, axial ratio, mass and rotation rate. We use three galacticmodels describing galaxies with rising, flat and declining rotation curves. We found that thepitch angle best correlates with the bar pattern speed and the spiral amplitude with the barquadrupole moment. Our results suggest that galaxies with declining rotation curves are themost efficient forming grand design spiral structure, evidenced by spirals with larger amplitudeand pitch angle. We also test the effects of the velocity ellipsoid in a subset of simulations. Wefound that as we increase the radial anisotropy, spirals increase their pitch angle but becomeless coherent with smaller amplitude.
Key words: galaxies: disc – galaxies: evolution – galaxies: kinematics and dynamics –galaxies: structure
Stellar bars inhabit a large fraction of disc galaxies in the localUniverse, in a great variety of shapes, sizes, and environments.They promote the galaxy secular evolution by exchanging mass,energy and angular momentum with stars and gas across the disc(Weinberg 1985; Kormendy & Kennicutt 2004; Sellwood 2014;Díaz-García et al. 2016b), transporting angular momentum fromthe inner bar resonances to those outside corotation (Lynden-Bell& Kalnajs 1972; Tremaine & Weinberg 1984; Athanassoula 2003)and redistributing stars in the disc by radial heating and radialmigration (Monari et al. 2016; Martinez-Medina et al. 2017). Barscan also induce gas inflow to the central regions via shock waves,forming gaseous structures like dust-lanes and rings (Athanassoula1992; Hernquist & Mihos 1995; Martinet & Friedli 1997; Kimet al. 2012; Sormani et al. 2018; Seo et al. 2019). Some movinggroups in the solar neighbourhood have their dynamical origin underthe influence of the Galactic bar (Pérez-Villegas et al. 2017) andthrough its induced resonances shape the stellar velocities in thesolar neighbourhood (Fux 2001).The importance of these phenomena depends on the mor-phological and dynamical characteristics of the bar. In late-typegalaxies, bars tend to be smaller in relation to their discs (Méndez-Abreu et al. 2012; Díaz-García et al. 2016a; Erwin 2018), moreoblate shaped (Méndez-Abreu et al. 2012; Díaz-García et al. 2016a)and prone to having an exponential density profile (Elmegreen &Elmegreen 1985; Kim et al. 2015). In contrast, bars in early-type ★ E-mail: [email protected] galaxies tend to be larger, prolate shaped, and with a flat density pro-file. Such co-relations with the galaxy type have not been observedin measurements of the bar pattern speed (hereafter Ω 𝐵𝑎𝑟 ), neitherwith the dimensionless rotation rate (hereafter parameter R ) definedas the ratio between the bar length (hereafter 𝑎 ) and the corotationresonance (hereafter 𝑅 𝐶𝑅 ) (Aguerri et al. 2015; Guo et al. 2019;Cuomo et al. 2019; Garma-Oehmichen et al. 2020). Nonetheless, ithas been observed that the Ω 𝐵𝑎𝑟 seems to correlate with the galaxyluminosity and total stellar mass (Garma-Oehmichen et al. 2020;Cuomo et al. 2020).Spiral arms also play an important role in the secular evolutionof galactic discs. By exchanging angular momentum, spirals churnstars and gas in the disc (Sellwood & Binney 2002). There is alarge body of evidence, theoretical and observational, that radialmigration is an ubiquitous process in galaxies with spiral arms,central bar, or both (Minchev & Famaey 2010; Vera-Ciro et al. 2014;Hayden et al. 2015; Loebman et al. 2016; Martinez-Medina et al.2017; Daniel & Wyse 2018). Spiral arms also induce dynamicalheating in the disc, increasing the velocity dispersion with time(Holmberg et al. 2009; Roškar et al. 2013; Martig et al. 2014)and modifying the velocity ellipsoid, an effect that depends on thenature and morphological properties of the spiral pattern (Jenkins& Binney 1990; Gerssen & Shapiro Griffin 2012; Martinez-Medinaet al. 2015).In most barred galaxies, bars co-evolve with spiral arms, and insome cases, they could drive the spiral structure (Salo et al. 2010).This is especially apparent in two-armed grand design galaxies,where most spirals appear to be connected to the ends of the bar.How these structures interact remains a matter of debate. Kormendy © a r X i v : . [ a s t r o - ph . GA ] F e b L. Garma-Oehmichen et al. & Norman (1979) first suggested that strong bars or galaxy com-panions can lead to the formation of spiral density waves. Thereare different theoretical scenarios on how the pattern speed of bothstructures is related (see e.g. Dobbs & Baba 2014). If bars and spiralsare strongly coupled, the pattern speed and amplitude of both struc-tures should be strongly correlated. This strong coupling is expectedfrom the spiral density theory (Lin & Shu 1964), and the manifoldtheory where spirals are formed by stars in escaping orbits aroundthe unstable Lagrangian points at the end of the bar (Romero-Gómezet al. 2006, 2007, 2015; Athanassoula et al. 2009). Observationalevidence of a strong coupling comes from the correlation betweenthe torques and density amplitudes of both structures (Buta et al.2003; Block et al. 2004; Buta et al. 2005; Salo et al. 2010; Bittneret al. 2017; Díaz-García et al. 2019). Another scenario proposes thatthe pattern speeds are related by a non-linear mechanism (Taggeret al. 1987), as suggested through indirect measurements of the barpattern speed (Font et al. 2017). Finally, a third scenario where barand spirals are decoupled structures, is supported by observations ofthe galaxy NGC 1365, where Speights & Rooke (2016) found resultsconsistent with bar and spiral patterns being dynamically distinctfeatures. The observational evidence to distinguish these scenariosis limited though, and hard to obtain due to the great uncertainties inestimating the pattern speed of both structures (Garma-Oehmichenet al. 2020).The vast majority of N-body simulations show that spirals aretransient or multi-arm features, that can be formed recurrently intime (Sellwood 2011; Grand et al. 2012; D’Onghia et al. 2013;Mata-Chávez et al. 2019). Numerical simulations with steady barpotentials have found that the gas particles settle in steady-statetrailing spirals (Sanders & Huntley 1976; Athanassoula 1992; Wada1994; Rodriguez-Fernandez & Combes 2008). Collisionless starparticles can also produce prominent spirals and rings near theOuter Lindblad Resonance (Schwarz 1981; Bagley et al. 2009).In this paper we use a set of numerical simulations based ongalactic potential models with one million test particles to explorehow the bar properties affect the response spiral arms. Our sim-ulations cover a large bar parameter space, tailored to explore alarge number of combinations of bar properties (size, shape, massand pattern speed). We also test the effect of discs with differentshapes of rotation curves and velocity ellipsoids. To characterisethe response spiral arms, we identify the spiral particles using thedensity-based clustering algorithm
DBSCAN . This let us obtain cleanmeasurements of the spiral amplitude and pitch angle.The paper is organised as follows. In Section 2 we describe thegalactic potential models, and the space of parameters to explore. InSection 3 we describe how we use the algorithm
DBSCAN to detectthe spiral over-densities, and how we estimate the properties of theinduced spiral arms. In Section 4 we show how the bar parametersaffect the spiral arms properties. In Section 5 we explore the effectsof different galactic models and the rotation curve shape. In Section6 we present the effects of the velocity ellipsoid. In Section 7 wediscuss the relative importance of the bar and disc properties inpredicting the spiral properties. Finally, in Section 8 we discuss ourresults and present our conclusions.
The mass distribution of a galaxy could play a significant role inthe properties of the spiral arms. Using a sample of 94 galaxies Biviano et al. (1991), observed that galaxies with steeper, risingrotation curves tend to host flocculent spirals, while galaxies withflat rotation curves have grand design spirals. Seigar et al. (2005,2006) found that the pitch angle strongly correlates with the rate ofshear in the disc defined as 𝑆 = (cid:18) − 𝑅𝑉 𝑑𝑉𝑑𝑅 (cid:19) (1)They suggested that spirals in rising rotation curves galaxieshave greater pitch angles compared to galaxies with flat or decliningrotation curves. However, this relationship has been questioned byKendall et al. (2015) and Yu et al. (2018), who failed to observesuch correlation (see also Díaz-García et al. (2019)).To test the effects of different rotation curves, we used threegalactic potential models with similar enclosed mass at ∼
20 kpc,but different mass distributions. We will refer to these models as flat , rising and declining , to figure out the shape of their rotationcurve. The flat model is the well-known Allen & Santillan (1991)potential comprising a Plummer bulge, a Miyamoto Nagai disc, anda spherical halo that reproduces the nearly flat Milky Way rotationcurve. The density profiles of the bulge, disc and halo (referred withthe sub-indexes B, D and H, respectively) are: 𝜌 𝐵 ( 𝑟 ) = 𝑏 𝐵 𝑀 𝐵 𝜋 (cid:16) 𝑟 + 𝑏 𝐵 (cid:17) / (2) 𝜌 𝐷 ( 𝑅, 𝑧 ) = 𝑏 𝐷 𝑀 𝐷 𝜋 (cid:18) 𝑅 𝑎 𝐷 + (cid:16) 𝑧 + 𝑏 𝐷 (cid:17) / (cid:19)(cid:32) 𝑅 + (cid:18) 𝑎 𝐷 + (cid:16) 𝑧 + 𝑏 𝐷 (cid:17) / (cid:19) (cid:33) / × (cid:18) 𝑎 𝐷 + (cid:16) 𝑧 + 𝑏 𝐷 (cid:17) / (cid:19) (cid:16) 𝑧 + 𝑏 𝐷 (cid:17) / (3) 𝜌 𝐻 ( 𝑟 ) = 𝑀 𝐻 𝜋𝑎 𝐻 𝑟 (cid:18) 𝑟𝑎 𝐻 (cid:19) . (cid:18) . + ( 𝑟 / 𝑎 𝐻 ) . ( + ( 𝑟 / 𝑎 𝐻 ) . ) (cid:19) (4)where 𝑟 is the spherical radius coordinate, ( 𝑅, 𝑧 ) are cylindricalcoordinates, 𝑀 𝐵 , 𝑀 𝐷 and 𝑀 𝐻 are the masses, and 𝑏 𝐵 , 𝑎 𝐷 , 𝑏 𝐷 , 𝑎 𝐻 are the characteristic scales of each component.For the rising model, we reduced the central mass concentra-tion by extending the bulge scale radius to match the spherical haloand doubled the disc scale radius. In the declining model, we reducethe disc scale radius by a factor of 3/4. All galactic models share thesame mass for each component. We show the parameters used forthe galactic models in Table 1. Figure 1 shows the rotation curvesof the three axisymmetric models. To simulate the bar potential, we use a Ferrers ellipsoid of index 𝑛 = 𝜌 ( 𝑥, 𝑦, 𝑧 ) = (cid:40) 𝜋𝑎𝑏𝑐 𝑀 𝐵𝑎𝑟 ( − 𝑚 ) , 𝑚 < , 𝑚 > MNRAS , 1–15 (2021) ow the bar affect the spiral structure Table 1.
Parameters of the galactic modelsGalactic Model 𝑀 𝐵 𝑏 𝐵 𝑀 𝐷 𝑎 𝐷 𝑏 𝐷 𝑀 𝐻 𝑎 𝐻 [10 𝑀 (cid:12) ] [kpc] [10 𝑀 (cid:12) ] [kpc] [kpc] [10 𝑀 (cid:12) ] [kpc](1) (2) (3) (4) (5) (6) (7) (8)Rising 1.406 12.0 8.561 10.636 0.25 10.709 12.0Flat (Allen-Santillan) 1.406 0.387 8.561 5.318 0.25 10.709 12.0Declining 1.406 0.387 8.561 3.988 0.25 10.709 24.0 Table 1.
Col. (1): Galactic model. Col. (2): Bulge mass. Col. (3): Bulge scale length. Col. (4): Disc mass. Col. (5): Disc radial scale length. Col. (6): Discvertical scale length. Col. (7): Halo mass. Col. (8): Halo scale length.
R [kpc] V c ( R )[ k m / s ] Flat (Allen-Santillan)RisingDeclining
Figure 1.
Rotation curves of the three axisymmetric galactic models usedin this work. where 𝑀 𝐵𝑎𝑟 is the bar mass and 𝑚 = ( 𝑥 / 𝑎 ) +( 𝑦 / 𝑏 ) +( 𝑧 / 𝑐 ) .The parameters 𝑎, 𝑏, 𝑐 are the semi-axes length in the 𝑥, 𝑦 and 𝑧 directions respectively, with 𝑎 > 𝑏 > = 𝑐 . The forces produced bythis distribution are described in Pfenniger (1984).We introduce the bar adiabatically as a smooth function oftime by transferring mass from the bulge to the bar. We used thefifth-degree polynomial described in equation 4 of Dehnen (2000),which guarantees a smooth transition to the barred state. We set thetime growth of 500 Myr, similar to Romero-Gómez et al. (2015).Orbital and dynamical studies have shown that self-consistentbars cannot extend beyond their co-rotation radius 𝑅 𝐶𝑅 , where thebar and the disc rotate at the same angular speed. Beyond 𝑅 𝐶𝑅 stellarorbits change their orientation, becoming perpendicular to the bar(Athanassoula 1980). Also, the density of resonances increases near 𝑅 𝐶𝑅 leading to chaotic behaviour in the phase space (Contopoulos& Papayannopoulos 1980). Because of its physical importance, thedimensionless parameter R = 𝑅 𝐶𝑅 / 𝑎 is used to parametrize thebar rotation rate. Bars are classified kinematically as “slow" if R >1.4, and “fast" if R <1.4. The theoretically impossible case of R <1is referred as “ultra-fast".In this work, we explore the following bar parameter space: (i)Three bar radius corresponding to 𝑎 = . , . , . 𝑏 / 𝑎 = . , .
6, correspondingto prolate and oblate bars, respectively. (iii) Two bar masses thatcorrespond to a complete and a half mass transfer between thebulge and the bar. (iv) Three values for the dimensionless parameter R , exploring the scenario of slow, fast and ultra-fast bars. We donot explore the effects of the vertical axial ratio, which was set to 𝑐 / 𝑎 = . Table 2.
Bar parameter spaceParameter Values(1) (2)a [kpc] 2.659, 5.318, 7.977b/a 0.3, 0.6 R 𝑀 𝐵𝑎𝑟 [10 𝑀 (cid:12) ] 0.703, 1.406c/a 0.3 Table 2.
Col. (1): Bar parameters. From top to bottom: Length, axial ratio,rotation rate, mass and vertical axial ratio. Col. (2): Values explored. for the bar model, and 3 galactic potential models, yielding a totalnumber of 108 simulations. All our simulations use 1 million testparticles.Given the bar length 𝑎 and the parameter R , the pattern speed Ω 𝐵𝑎𝑟 is estimated by evaluating axisymmetric angular velocitycurve at the corotation resonance 𝑅 𝐶𝑅 = R · 𝑎 . However, as the baris being introduced, the mass distribution and the location of theresonances changes. This is especially important in the rising model,where the bar formation increases the central mass concentration.In those cases, we use the unstable Lagrange point 𝐿 𝑅 𝐶𝑅 (Binney & Tremaine 2008). If the relativedifference between 𝐿 𝑅 𝐶𝑅 is greater than10%, we correct the value of Ω 𝐵𝑎𝑟 until both distances match.From hereafter, when we refer to R we are using the 𝐿 / 𝑎 ratio,accounting for the bar mass redistribution. A natural space to study spirals is the ( 𝜃, ln 𝑅 ) plane, where 𝜃 isthe azimuthal angle and 𝑅 the cylindrical radius. In this space, alogarithmic spiral can be described with the straight line equation(Lin & Shu 1964): 𝑅 ( 𝜃 ) = 𝑅 × 𝑒 𝜃 tan 𝛼 (6)where, 𝑅 = 𝑅 at 𝜃 = 𝛼 is the pitch angle.However, a disc populated with test particles uniformly dis-tributed in the ( 𝜃, ln 𝑅 ) space would have few particles in the innerdisc. Instead, we choose to set the initial spatial distribution to beuniform in the ( 𝜃, log ( 𝑅 + )) space, which keeps the logarithmicspacing in the radius, populates the inner disc and highlights thespiral over-densities over a uniform background. The resulting den-sity distribution resembles that of a single exponential disc. For the MNRAS , 1–15 (2021)
L. Garma-Oehmichen et al. vertical dimension, we choose a usual 𝑠𝑒𝑐ℎ ( 𝑧 / 𝑧 ) law distribution(van der Kruit & Searle 1981) with scale-height 𝑧 = .
25 kpc.We generated the initial velocities using the Hernquist (1993)moments method. This procedure constrains the shape of the ve-locity ellipsoid, by assuming the radial and vertical dispersion areproportional and constant throughout the disc ( 𝜎 𝑅 ∝ 𝜎 𝑧 ) . For sim-plicity, we set the velocity dispersion ratio to 𝜎 𝑧 / 𝜎 𝑅 =
1. In Section6 we explore the effect of other normalisation constants with greaterradial velocity dispersion for a subset of simulations.Before introducing the bar, we first integrate the test particlesorbits in the axisymmetric potential for 3 Gyr so they relax andreach the statistical equilibrium (Romero-Gómez et al. 2015). Weperformed the integration with a time adaptive fifth-order Runge-Kutta integrator using the
Fortran subroutines odeint and rkqs (Press et al. 2007). We followed the Jacobi energy and verticalvelocity distribution for a subset of test particles to confirm thestability.Figures 2 and 3 show snapshots of a simulation using the flat model and bar parameters 𝑎 = .
318 kpc, 𝑏 / 𝑎 = . R = . 𝑀 𝐵𝑎𝑟 = . × 𝑀 (cid:12) in the ( 𝑥, 𝑦 ) and ( 𝜃, log ( 𝑅 + )) planes,respectively. Both figures are in the bar rotating reference frame.The snapshots are separated in time steps of 250 Myr. The firstsnapshot corresponds to the relaxed disc distribution. The bar stopsgrowing in mass at 500 Myr. We show the corotation resonance witha white solid line. The Inner and Outer Lindblad Resonances (ILRand OLR from hereafter) are shown with white segmented lines.These resonances were estimated using the axisymmetric angularvelocity curve and the epicyclic frequency curve. To detect the spiral over-densities, we use the density-based clus-tering algorithm
DBSCAN (Ester et al. 1996). The algorithm worksby classifying each point in a given space as “core", “member" or“noise", based on two parameters: (i) 𝜖 which specifies the distancebetween two points to be considered “neighbours", and (ii) 𝑚𝑖𝑛𝑐𝑛𝑡 which specifies the minimum number of neighbours a point musthave to be classified as a “core" of a cluster. All points in the 𝜖 -neighbourhood of a core, that do not satisfy the 𝑚𝑖𝑛𝑐𝑛𝑡 conditionare classified as “members" of the cluster. Finally, all points that donot inhabit a cluster are classified as “noise". DBSCAN is especially useful to find arbitrarily shaped structuresand does not require knowing a priori the number of clusters in thedata. The use of the algorithm has been increasing in astronomy inrecent years. To mention a few applications: identifying lensed fea-tures in residual images (Paraficz et al. 2016), classifying eclipsingbinaries light curves (Kochoska et al. 2017), detecting low surfacebrightness galaxies in the Virgo cluster (Prole et al. 2018) and find-ing open clusters in the Gaia data (Castro-Ginard et al. 2018, 2020).Choosing meaningful parameters ( 𝜖, 𝑚𝑖𝑛𝑐𝑛𝑡 ) is challengingwhen the size and density of the clusters are unknown. The param-eter 𝜖 is related to the spatial resolution of the clusters and can bedetermined from their expected size. The spiral arms produced inour simulations typically have widths of ∼ . 𝜖 to 0.3 kpc. The parameter 𝑚𝑖𝑛𝑐𝑛𝑡 is relatedto the expected density, or in this case, the spiral amplitude. How-ever, the amplitude changes with time and between simulations. Weperformed the following steps to determine an appropriate value of 𝑚𝑖𝑛𝑐𝑛𝑡 at any given snapshot. We show an example of the procedurein Figure 4:(i) We mask all particles between 𝑅 𝐶𝑅 and OLR + 4 kpc, where weexpect the spiral arms to be located. We re-scale the data, so thedistances in log ( 𝑅 + ) and 𝜃 are the same in an Euclidean metric.We also re-scale the value of 𝜖 . The first panel of Figure 4 shows asnapshot of a simulation where the spiral arms have already formed,and the data has been re-scaled.(ii) A second mask is used to select particles that lie in the radial range 𝑅 = [ 𝑂𝐿𝑅 − 𝜖, 𝑂𝐿𝑅 ] . In the first panel of Figure 4, the two redhorizontal lines delimit this range.(iii) All our simulations produced two symmetrical spirals arms. Webin the masked particles in 64 bins and fit a simple cosine wavefunction 𝑓 ( 𝜃 ) = 𝐴 cos ( 𝜃 + 𝜙 ) + 𝐶 . This is equivalent to the 𝑚 = 𝐴 / 𝐶 , as a proxy of the spiralamplitude. In the second panel of Figure 4, we show the maskedparticles distribution and the cosine fit.(iv) We count the number of particles around the peaks ± 𝜖 , illustratedwith two red segmented lines in the second panel of Figure 4. Theresulting count is the expected number of particles in two squaresof size 2 𝜖 at the peak of the spirals. We show these squares in thefirst panel of Figure 4.(v) We multiply the resulting count by 𝜋 / 𝜖 and two squares of side 2 𝜖 . Thisresults in our estimation for 𝑚𝑖𝑛𝑐𝑛𝑡 .Once we have estimated ( 𝜖, 𝑚𝑖𝑛𝑐𝑛𝑡 ) , DBSCAN can detect thespiral over-densities as shown in the third and fourth panels ofFigure 4. In this example, the algorithm detected 3 clusters, shownin different colours. We plot the core points with slightly bigger dots,so they appear as a solid coloured region. Member points surroundthe core with an envelope of radius 𝜖 . Notice that the pink cluster isjust a continuation of the blue spiral, but displaced by 2 𝜋 radians.In such cases, we join the separated clusters by manually adding the2 𝜋 rotation.Not every simulation produces spirals as clearly as in the ex-ample shown in Figure 4. Some models produce very weak spiralarms that are almost indistinguishable from the background. In thosecases, we reduce the value of 𝑚𝑖𝑛𝑐𝑛𝑡 manually until the algorithmcan detect the underlying spiral structure. Although we identify the spiral over-densities in the ( 𝜃, log 𝑅 + ) plane, the measurement of the pitch angle is done in the ( 𝜃, ln 𝑅 ) space, as described by equation 6 (i.e arctan of the slope of the over-densities). In some cases, the spirals wind up at the outer radius,forming a ring-like structure just outside of the OLR. These kindof rings are common in numerous simulations with test particles(Schwarz 1981, 1984; Bagley et al. 2009) and barred galaxies (Buta& Crocker 1991; Buta & Combes 1996; Buta 2017). The snapshotshown in Figure 4 is an example of such behaviour. To separate therings from the spiral, we fit a piecewise linear function and use theslope of the spiral segment to estimate the pitch angle. The fittedpiecewise function is shown with white lines in the third and fourthpanels of Figure 4. Notice how the change in pitch angle cannotbe distinguished ‘by eye’ in the ( 𝑥, 𝑦 ) plane. We measure the pitchangle in both spirals using the whole cluster and only core points.From hereafter, when we refer to the pitch angle we are using theaverage from these measurements. MNRAS , 1–15 (2021) ow the bar affect the spiral structure Y [ k p c ]
10 0 10
X [kpc] Y [ k p c ]
10 0 10
X [kpc]
10 0 10
X [kpc]
10 0 10
X [kpc] N / ( p c ) Figure 2.
Snapshots of a simulation in the flat model and bar properties 𝑎 = .
318 kpc, 𝑏 / 𝑎 = . R = . 𝑀 𝐵𝑎𝑟 = . × 𝑀 (cid:12) . We masked particlesinside corotation 𝑅 𝐶𝑅 to make spirals more visible. The OLR is shown with a white segmented line. The bar is shown with a black ellipse. The bar is introducedadiabatially by transferring mass from the bulge component up to 500 Myr. The spiral amplitude reaches a maximum at 1000 Myr, and decreases afterwards. l o g ( R + ) [rad] l o g ( R + ) [rad] [rad] [rad] N Figure 3.
Same as Figure 2, but in the ( 𝜃, log ( 𝑅 + )) plane. The white segmented line highlights the location of the OLR. Notice that in the first snapshot,the particles are uniformly distributed. We use the normalised amplitude of the fitted cosine wave function(second panel in Figure 4) as a proxy of the spiral amplitude. Ingeneral, all our simulations showed the same trend: As the bar isbeing introduced, the spiral amplitude increases as a function of timeuntil it reaches a maximum and slowly decreases. In Figure 5 we show the spiral amplitude as a function of time of four simulations inthe flat galaxy model that share the same bar length 𝑎 and parameter R (and thus, the same Ω 𝐵𝑎𝑟 ), but vary in mass and axis ratio. In mostsimulations, the amplitude peaks between 1 or 2 dynamical timesat the OLR after the bar has formed. After 3 to 4 dynamical timesthe spiral structure becomes more diffuse. Simulations that share 𝑎 MNRAS000
Same as Figure 2, but in the ( 𝜃, log ( 𝑅 + )) plane. The white segmented line highlights the location of the OLR. Notice that in the first snapshot,the particles are uniformly distributed. We use the normalised amplitude of the fitted cosine wave function(second panel in Figure 4) as a proxy of the spiral amplitude. Ingeneral, all our simulations showed the same trend: As the bar isbeing introduced, the spiral amplitude increases as a function of timeuntil it reaches a maximum and slowly decreases. In Figure 5 we show the spiral amplitude as a function of time of four simulations inthe flat galaxy model that share the same bar length 𝑎 and parameter R (and thus, the same Ω 𝐵𝑎𝑟 ), but vary in mass and axis ratio. In mostsimulations, the amplitude peaks between 1 or 2 dynamical timesat the OLR after the bar has formed. After 3 to 4 dynamical timesthe spiral structure becomes more diffuse. Simulations that share 𝑎 MNRAS000 , 1–15 (2021)
L. Garma-Oehmichen et al. S c a l e d l o g ( R + ) O L R O L R - l n ( R [ k p c ])
10 5 0 5 10 X [kpc]1050510 Y [ k p c ]) Figure 4.
Detecting spiral over-densities with DBSCAN. We use a snapshot from the simulation with bar parameters 𝑎 = .
318 kpc, 𝑏 / 𝑎 = . 𝑀 𝑏𝑎𝑟 = . × 𝑀 (cid:12) , R = .
2, in the flat galactic model. First panel: We select particles between 𝑅 𝐶𝑅 and the OLR + 4 kpc where the spiral arms are formed. Were-scale the data so distances in radius and angle are the same. We mask the particles in the range [OLR − 𝜖 , OLR] shown with two red horizontal lines. Atthe peak spiral density we draw the two squares of side 2 𝜖 . Second panel: Distribution of the masked particles vs. the scaled angle. We fit a cosine function toget the amplitude and peaks of the spiral over-density. We count the number of test particles around the peaks ± 𝜖 , shown with red vertical lines. We determine 𝑚𝑖𝑛𝑐𝑛𝑡 by multiplying this count by 𝜋 /
8. Third panel: Using the estimated parameters ( 𝜖 , 𝑚𝑖𝑛𝑐𝑛𝑡 ) , DBSCAN identifies three clusters in the data. The pointsclassified as “core” are shown with slightly bigger dots, so they appear as a solid coloured area. “Member” points surround the core area in an 𝜖 size envelope.Notice this plot is in the ( 𝜃, ln 𝑅 ) space. We use a piecewise linear function to fit the identified clusters shown with two white lines. We measure the pitchangle using the greatest slope in the piecewise fit. Fourth panel: Same as the third panel, but in the (x,y) plane. Time [Myr] Sp i r a l A m p li t u d e Bar Formation1 Dynamical Time2 Dynamical Time b / a =0.3 M Bar =0.703 b / a =0.3 M Bar =1.406 b / a =0.6 M Bar =0.703 b / a =0.6 M Bar =1.406
Figure 5.
Spiral arm amplitude as a function of time, for the family of sim-ulations with bar parameters 𝑎 = .
318 kpc and R = . flat model.The solid black line shows the bar formation (500 Myr), the segmentedand dotted lines show 1 and 2 dynamical times at the OLR after the barformation, respectively. and R , usually have spirals that peak in the same snapshot, exceptfor some simulations in the rising model, that are more sensitive tothe bar potential.In the next section we discuss which bar parameters producestronger spirals, but from this figure is clear that the mass and shapeof the bar are strongly co-related to the spiral amplitude. The spiral arms produced in our simulations are the collective re-sult of the bar perturbing the stellar epicyclic orbits. Thus, our spiralarms are produced by the density-wave mechanism and do not ac-count for the effects of self-gravity. Our results should be interpretedas the initial spiral perturbation that arises from the presence of dif-ferent kinds of bars. Spiral arms produced this way, are expected toform near the OLR (Athanassoula 1980), and be tightly wound withsmall pitch angles.Because of the great variety of parameters we are exploring, the spirals arms form at different times, amplitudes and locationsthroughout the disc. To make the comparison between differentsimulations as fair as possible, we use the measurements of thesnapshot where spirals are at their greatest amplitude.
In Figures 6, 7, and 8 we show the average pitch angle and the spiralamplitude vs. the bar parameters of our simulations in the rising , flat , and declining galactic models, respectively. We join with greydotted lines the simulations that share the same bar parametersexcept for the one that is being plotted. The Spearman correlationcoefficient 𝑟 𝑆 and the corresponding statistical significance 𝑝 areshown at the top of each panel. We colour the results with the barpattern speed. To facilitate the interpretation of our results, Table 3summarises all Spearman correlation coefficients that we mentionthrough the text.In all galactic models we found a strong anti-correlation of thepitch angle with the bar length ( 𝑟 𝑆 ∼ − . ) and the R parameter ( 𝑟 𝑆 ∼ − . ) . At a first approximation, it would appear that themass and shape of the bar do not correlate with the pitch angle, witha small 𝑟 𝑆 coefficient and p-values > .
05. However, the connectedmodels display a clear downward trend with 𝑏 / 𝑎 and an uppertrend with 𝑀 𝐵𝑎𝑟 . We performed a Student’s t-test to see if theseslight differences between connected simulations could happen byrandom chance. We could reject the null hypothesis in both casesin all galactic models ( 𝑝 ∼ × − for 𝑏 / 𝑎 and 𝑝 ∼ × − for 𝑀 𝐵𝑎𝑟 ).The relations with the spiral amplitude does seem to dependmore on the galactic models. The correlation with the bar length isnon-existent in the rising model ( 𝑟 𝑆 = . flat and declining models ( 𝑟 𝑆 = .
64 and 𝑟 𝑆 = .
85, respec-tively). In contrast, the relation with the R parameter does not seemsignificant in the declining model ( 𝑟 𝑆 = − .
14) but becomes a weakanti-correlation in the rising and flat models ( 𝑟 𝑆 ∼ − .
40 in bothcases). The connected simulations in all three models show a clearupward trend with 𝑀 𝐵𝑎𝑟 and a downward trend with the ratio 𝑏 / 𝑎 ,which we were able to confirm with the Student t-test, rejecting thenull-hypothesis. The strong trends with the bar mass and axis ratiocan also be observed in amplitude vs. time plot in Figure 5. MNRAS , 1–15 (2021) ow the bar affect the spiral structure Bar length [kpc] A v e r a g e P i t c h A n g l e [ ° ] r s = -0.56, p = 4e-04 Axis ratio b / a r s = -0.16, p = 4e-01 Bar Mass [10 M ] r s = 0.18, p = 3e-01 Parameter r s = -0.64, p = 4e-05 Bar length [kpc] Sp i r a l A m p li t u d e r s = 0.16, p = 4e-01 Axis ratio b / a r s = -0.23, p = 2e-01 Bar Mass [10 M ] r s = 0.45, p = 7e-03 Parameter r s = -0.39, p = 2e-02 b a r [ k p c / k m / s ] Figure 6.
Spiral properties vs bar parameters in the rising galactic model. Top row: Relations with the average pitch angle. Bottom row: Relations with the spiralamplitude. The grey segmented lines join simulations that share the same bar parameters, except for the one that is being plotted. The Spearman correlationcoefficient and the corresponding p-value are shown at the top of each panel.
Bar length [kpc] A v e r a g e P i t c h A n g l e [ ° ] r s = -0.65, p = 2e-05 Axis ratio b / a r s = -0.13, p = 4e-01 Bar Mass [10 M ] r s = 0.26, p = 1e-01 Parameter r s = -0.65, p = 1e-05 Bar length [kpc] Sp i r a l A m p li t u d e r s = 0.64, p = 3e-05 Axis ratio b / a r s = -0.24, p = 2e-01 Bar Mass [10 M ] r s = 0.55, p = 6e-04 Parameter r s = -0.40, p = 1e-02 b a r [ k p c / k m / s ] Figure 7.
Same as Figure 6, but in the flat model.
Some of the observed correlations are related in the form of athird relationship. For example, the bar length 𝑎 and parameter R are intimately related with Ω 𝐵𝑎𝑟 and the disc rotation curve V(R): Ω 𝐵𝑎𝑟 = 𝑉 ( 𝑅 𝐶𝑅 )/ 𝑅 𝐶𝑅 = 𝑉 ( 𝑅 𝐶𝑅 )/(R · 𝑎 ) (7)Thus, the strong trends we observe among the pitch angle, the bar length and R are probably a consequence of the much strongercorrelation between the pitch angle and the bar pattern speed, ( 𝑟 𝑆 ∼ . ) in all three galaxy models. The relation between the spiralproperties and Ω 𝐵𝑎𝑟 is shown in Figure 9 for the three galacticmodels. The colours and shape of the dots are used to distinguishmodels in R and 𝑀 𝐵𝑎𝑟 respectively. The relation between the spiralamplitude and Ω 𝐵𝑎𝑟 depends on the galactic model, as this relation
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Bar length [kpc] A v e r a g e P i t c h A n g l e [ ° ] r s = -0.60, p = 1e-04 Axis ratio b / a r s = -0.17, p = 3e-01 Bar Mass [10 M ] r s = 0.22, p = 2e-01 Parameter r s = -0.67, p = 6e-06 Bar length [kpc] Sp i r a l A m p li t u d e r s = 0.85, p = 3e-11 Axis ratio b / a r s = -0.20, p = 2e-01 Bar Mass [10 M ] r s = 0.36, p = 3e-02 Parameter r s = -0.14, p = 4e-01 b a r [ k p c / k m / s ] Figure 8.
Same as Figure 7 and 6, but in the declining model. is non-existent in the rising model, but becomes a strong anti-correlation in the declining model. 0’p0’poThe bar length 𝑎 , mass 𝑀 𝐵𝑎𝑟 and axial ratio 𝑎 / 𝑏 are related tothe bar quadrupole moment (hereafter 𝑄 𝐵𝑎𝑟 ). For a Ferrers ellipsoidof index 𝑛 𝑄
𝐵𝑎𝑟 is: 𝑄 𝐵𝑎𝑟 = 𝑀 𝐵𝑎𝑟 𝑎 (cid:18) − 𝑏 𝑎 (cid:19) /( + 𝑛 ) (8)Same as with other bar properties, in Figure 10 we show thecorrelations between the spiral properties and the quadrupole mo-ment in the three galactic models. Our results show a remarkablystrong correlation between the spiral amplitude and 𝑄 𝐵𝑎𝑟 in the flat and declining models. Thus, the observed correlations betweenspiral amplitude and 𝑎 , 𝑏 / 𝑎 and 𝑀 𝐵𝑎𝑟 could be a consequence ofthe more stronger correlation with 𝑄 𝐵𝑎𝑟 . Nonetheless, this corre-lation does not seem to be as important in the rising model (as with 𝑎 , 𝑏 / 𝑎 and 𝑀 𝐵𝑎𝑟 ).The relationship between 𝑄 𝐵𝑎𝑟 and the pitch angle is morecomplex. The general trend is negative ( 𝑟 𝑆 ∼ − .
45 in the threemodels). However, if we look only at simulations with the samepattern speed as shown by the colour code (i.e. simulations withsame bar length and R parameter) the positive relation with themass (and anti-correlation with 𝑏 / 𝑎 ) becomes visible. We study the effects of the rotation curve in the spiral properties,by comparing simulations with the same bar parameters, but dif-ferent galaxy model. In Figure 11 we show the pitch angles of the flat models vs. the rising and declining models. As in Figure 9 thecolours and shape are used to distinguish the R and 𝑀 𝐵𝑎𝑟 respec-tively. We do not observe a significant difference between the rising and flat models, except for one outlier that corresponds to an ultra-fast, small, massive bar. In comparison, the declining simulations produce spirals with consistently larger pitch angles. The effect alsoappears to be more significant with the ultra-fast bars, as those differmore significantly from the 1:1 relation. We include the best linearfit for reference.Similarly, in Figure 12 we compare the spiral amplitude be-tween the three galactic models. The rising model tends to produceweaker spirals. The difference becomes more significant with themore massive bars. On the other hand, the declining and flat modelshave similar spiral amplitudes for all the bar models.We also estimate the shear rate 𝑆 (equation 1) at the OLRin all our simulations. Values of 𝑆 < . 𝑆 = . 𝑆 > . 𝑉 ( 𝑅 ) = √︁ 𝑅𝑑𝜙 / 𝑑𝑅 over 10angles uniformly distributed between 0 and 𝜋 / flat and declining models.The relations with the spiral amplitude are weaker, except on the declining models, where we observe a strong correlation. Disc heating mechanisms play an important role in the secular evo-lution of galactic discs, altering the stellar kinematics and increasingthe random motion of stars. It has been shown that the shape of thevelocity ellipsoid, i.e., the ratio of the vertical and radial veloc-ity dispersion 𝜎 𝑧 / 𝜎 𝑅 is strongly correlated with the Hubble type,with late-type galaxies being more anisotropic and early-types beingmore isotropic (van der Kruit & de Grijs 1999; Gerssen & ShapiroGriffin 2012). However these results have been questioned by Pinnaet al. (2018), who found a wide range of dispersion ratios from theliterature around 𝜎 𝑧 / 𝜎 𝑅 = . ± .
2. In the solar neighbourhood 𝜎 𝑧 / 𝜎 𝑅 ranges from ∼ . ∼ . MNRAS , 1–15 (2021) ow the bar affect the spiral structure
20 40 60 80 100
Bar [km/s/kpc] A v e r a g e P i t c h A n g l e [ ° ] r s = 0.83, p = 1e-09 Rising
20 40 60 80
Bar [km/s/kpc] r s = 0.88, p = 1e-12 Flat
20 40 60 80
Bar [km/s/kpc] r s = 0.84, p = 1e-10 Declining
20 40 60 80 100
Bar [km/s/kpc] Sp i r a l A m p li t u d e r s = 0.02, p = 9e-01
20 40 60 80
Bar [km/s/kpc] r s = -0.39, p = 2e-02
20 40 60 80
Bar [km/s/kpc] r s = -0.71, p = 1e-06 SlowFastUltra-fast M Bar = 0.703 M Bar = 1.406
Figure 9.
Pitch angle and spiral amplitude vs. bar pattern speed in the three galactic models (
Rising in the left column,
Flat in the middle, and
Declining in theright). Simulations are colour-coded according to their R parameter (blue for slow, red for fast and green for ultra-fast bars). The more massive bars are shownwith diamonds, while their less massive counterpart are shown with crosses. Table 3.
Spearman correlation coefficient and statistical significance with the spiral propertiesPitch Angle AmplitudeParameter Rising Flat Declining Rising Flat Declining(1) (2) (3) (4) (5) (6) (7) 𝑎 − . ( × − ) − . ( × − ) − . ( × − ) . ( × − ) . ( × − ) . ( × − ) 𝑏 / 𝑎 − . ( × − ) − . ( × − ) − . ( × − ) − . ( × − ) − . ( × − ) − . ( × − ) 𝑀 𝐵𝑎𝑟 . ( × − ) . ( × − ) . ( × − ) . ( × − ) . ( × − ) . ( × − )R − . ( × − ) − . ( × − ) − . ( × − ) − . ( × − ) − . ( × − ) − . ( × − ) Ω 𝐵𝑎𝑟 . ( × − ) . ( × − ) . ( × − ) . ( × − ) . ( × − ) . ( × − ) 𝑄 𝐵𝑎𝑟 − . ( × − ) − . ( × − ) − . ( × − ) . ( × − ) . ( × − ) . ( × − ) Shear − . ( × − ) − . ( × − ) − . ( × − ) . ( × ) . ( × − ) . ( × − ) Table 3.
Col. (1): Parameter. Col. (2): Pitch angle in the rising model. Col. (3): Pitch angle in the flat model. Col. (4): Pitch angle in the declining model. Col.(5): Spiral amplitude in the rising model Col. (6): Spiral amplitude in the flat model Col. (7): Spiral amplitude in the declining model. mean orbital radii, or even the dynamical modelling (Mackerethet al. 2019; Nitschai et al. 2020).To study the effects of the velocity ellipsoid on the formationof spiral arms, we re-simulate a subset of 8 galaxies in the flat galactic model. We build new initial conditions where the velocitydispersion relation is set to 𝜎 𝑧 / 𝜎 𝑅 = . 𝜎 𝑧 / 𝜎 𝑅 = .
33. We chose the subset of simulations to cover a widerange of pitch angles and spiral amplitudes produced by the originalsimulations with isotropic velocity ellipsoid. The spiral arms produced by the new simulations are still lo-cated near the OLR, and peak in amplitude at the same time asthe 𝜎 𝑧 / 𝜎 𝑅 = 𝜎 𝑧 / 𝜎 𝑅 ratio,the response spirals increase their pitch angle, but result in a lesscoherent, wider structure with smaller amplitude.In Figure 15 we show the pitch angle and spiral amplitude asa function of the velocity dispersion ratio. We connect with a grey MNRAS000
33. We chose the subset of simulations to cover a widerange of pitch angles and spiral amplitudes produced by the originalsimulations with isotropic velocity ellipsoid. The spiral arms produced by the new simulations are still lo-cated near the OLR, and peak in amplitude at the same time asthe 𝜎 𝑧 / 𝜎 𝑅 = 𝜎 𝑧 / 𝜎 𝑅 ratio,the response spirals increase their pitch angle, but result in a lesscoherent, wider structure with smaller amplitude.In Figure 15 we show the pitch angle and spiral amplitude asa function of the velocity dispersion ratio. We connect with a grey MNRAS000 , 1–15 (2021) L. Garma-Oehmichen et al.
Quadrupole moment Q A v e r a g e P i t c h A n g l e [ ° ] r s = -0.42, p = 1e-02 Rising
Quadrupole moment Q r s = -0.48, p = 3e-03 Flat
Quadrupole moment Q r s = -0.45, p = 6e-03 Declining
Quadrupole moment Q Sp i r a l A m p li t u d e r s = 0.36, p = 4e-02 Quadrupole moment Q r s = 0.82, p = 1e-09 Quadrupole moment Q r s = 0.95, p = 2e-19 b a r [ k m / s / k p c ] Figure 10.
Pitch angle and spiral amplitude vs. bar quadrupole moment in the three galactic models. segmented line simulations that share the same bar model. Is clear,from this small subset that the effects the velocity ellipsoid hason the spiral arms are substantial. The more radially heated discsproduced very low amplitude spirals. In one simulation, we werenot able to detect any spiral structure from the background.
So far, we have explored how different bar and disc parameters re-late with the induced pitch angle and spiral amplitude independently.Using solely their Spearman correlation coefficient and the statisti-cal significance, the parameters that best predict the pitch angle andamplitude are the Ω 𝐵𝑎𝑟 and 𝑄 𝐵𝑎𝑟 , respectively. Nonetheless, thedifferences observed between galactic models suggest the shear orgalactic model itself, could also be important features for predictingthe spiral properties. Also, some of these parameters are related toeach other and cannot be treated as independent variables.To rank the relative importance of each parameter comparedto the rest, we used a random forest regressor (Breiman 2001),trained to predict the spiral properties of our simulations using allbar parameters, the shear rate and the galactic model. The modelworks by averaging the prediction of multiple decision trees, whichtry to predict the target variable using a flowchart-like decision,ranking the input variables.We set the number of decision trees to 1000 and the maximumnumber of features used by each tree to 4. This prevents over-fitting, and increases the relative importance of features that are highly correlated. For example, if Ω 𝐵𝑎𝑟 is considered an importantvariable to predict the pitch angle, a decision tree that does not trainwith this feature should weight R more heavily, as these two arehighly correlated.As a proof of concept, we trained several models using an 80/20train-test split which consistently achieved a coefficient of determi-nation 𝑟 > .
80 for both the pitch angle and the spiral amplitude.Since the purpose of these models is to get the relative importanceof each feature, we re-train the models using all our simulation data.We used the permutation feature importance technique; where therelevance of each feature is estimated by the difference between the 𝑟 score of the original data ( 𝑟 𝑏𝑎𝑠𝑒𝑙𝑖𝑛𝑒 ) and the score after randomlyshuffling such feature ( 𝑟 𝑝𝑒𝑟𝑚𝑢𝑡𝑒𝑑 ) (Breiman 2001). In Figure 16 weshow the feature importance estimated after shuffling each feature50 times.The most important features for predicting the pitch angleare related to the bar rotation rate. This was expected, as Ω 𝐵𝑎𝑟 and R showed the strongest correlations with the pitch angle. Incomparison all other parameters are not consider as important, asthese only change the 𝑟 score of the prediction by 0.1 or less. Thisshows that the relative strong correlation with the bar length wasactually a consequence of the much more stronger relations with thebar frequency. Notice in particular, that the ”Declining" feature isconsidered more important than the Flat and Rising features. This isprobably due the declining simulations having consistently higherpitch angles as we showed in Figure 11.For the spiral amplitude the most important features are related MNRAS , 1–15 (2021) ow the bar affect the spiral structure Flat (Allen-Santillan) [°] R i s i n g [ ° ] Average Pitch Angle [°]
SlowFastUltra-fast0 5 10 15 20
Flat (Allen-Santillan) [°] D e c li n i n g [ ° ] M Bar = 0.703 M Bar = 1.4061:1 RelationBest fit
Figure 11.
Comparison of the pitch angle between the three galactic models.For reference, the slopes of the best fits are 𝑚 = .
92 in the top panel(excluding the outlier) and 𝑚 = .
26 in the bottom one. to the bar strength. Again, this was expected from the strong corre-lations of 𝑄 𝐵𝑎𝑟 , 𝑀 𝐵𝑎𝑟 and 𝑎 . In contrast with the pitch angle, theamplitude model considers the bar frequency ( R and Ω 𝐵𝑎𝑟 ) andthe disc rotation curve ( 𝑆 and Rising) to be as important as the barmass and length. This could be because of the rising simulationsforming consistently spirals with lower amplitudes as we showed inFigure 12.Our random forest models do not take into account the velocitydispersion ratio of the stellar disc. However, it is clear from thesimulations we studied in Section 6 that the dynamical temperatureof the disc is an important feature to consider when predictingthe spiral properties. Perhaps equally or more important than theperturber properties or the shape of the velocity curve. How these Flat (Allen-Santillan) R i s i n g Average Spiral Amplitude
SlowFastUltra-fast0.0 0.2 0.4 0.6 0.8
Flat (Allen-Santillan) D e c li n i n g M Bar = 0.703 M Bar = 1.4061:1 RelationBest fit
Figure 12.
Comparison of the spiral amplitude between the three galacticmodels. For reference, the slopes of the best fits are 𝑚 = .
39 in the toppanel and 𝑚 = .
90 in the bottom one. effects complement or affect the relations discussed, will be exploredin a future work.
Since our simulations lack the effects of self-gravity, our resultscannot predict how the system would evolve after the initial pertur-bation. Our spirals have typical lifetimes between 2 or 3 dynamicaltimes at the OLR after the bar formation. However, we point outthat self-gravity and star formation could increase and highlight thespiral amplitude. It is expected that these spirals would eventually
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Shear rate S A v e r a g e P i t c h A n g l e [ ° ] r s = -0.44, p = 1e-02 Rising
Shear rate S r s = -0.82, p = 8e-10 Flat
Shear rate S r s = -0.75, p = 1e-07 Declining
Shear rate S Sp i r a l A m p li t u d e r s = 0.00, p = 1e+00 Shear rate S r s = 0.44, p = 7e-03 Shear rate S r s = 0.73, p = 4e-07 Figure 13.
Pitch angle and spiral amplitude vs. Shear rate measured at the OLR in the three galactic models. disappear and fragment into smaller substructures as several N-bodysimulations have shown.The evolution of the pitch angle is unclear. In the Lin-Shudensity wave picture, the pitch angle remains constant in time anddepends on the global galaxy properties. Several N-body simula-tions have shown that the pitch angle decreases with time (Grandet al. 2013; Pettitt & Wadsley 2018), while others have shown it re-mains roughly constant, independently of their origin (Mata-Chávezet al. 2019). Nonetheless, Mata-Chávez et al. (2019) also observedan increase in the pitch angle after the buckling of the bar, proba-bly because of the mass redistribution. Recently, Pringle & Dobbs(2019) proposed the pitch angle evolves as a decreasing functionof time, from an initial measurable maximum 𝛼 𝑚𝑎𝑥 to a minimum 𝛼 𝑚𝑖𝑛 , as evidenced from the uniform distribution of cot 𝛼 of galax-ies observed by Yu et al. (2018). Our results suggest that, when thespirals are induced by a bar perturber, the initial pitch angle is notrandom nor has a fixed maximum for all galaxies. It mainly dependson the bar pattern speed.In the majority of our simulations the spirals are tightly wound.In order to produce wide open spirals in barred galaxies like NGC1365, NGC 1672 and NGC 2903 other interaction mechanismsshould be considered. For example, the different nature of spiralarms has also been explained by invariant manifolds (Athanassoula2012; Romero-Gómez et al. 2015), galaxy encounters (Pettitt et al.2016), evolution of the mass distribution (Mata-Chávez et al. 2019)or a shearing spiral pattern (Speights & Westpfahl 2011).Although the shear and the rotation curve shape did not appearto be powerful predictors for the spiral properties (in comparison with the bar) they seem to be important on how the disc responds tothe perturbations. Our simulations show that bars in galaxies with rising rotation curves are less efficient forming a grand design spiralstructure. We found evidence in the weaker relationship between thespiral amplitude and 𝑄 𝐵𝑎𝑟 , and the 1:1 comparison of the spiralamplitude between galaxy models with same bar parameters. Incontrast, bars in galaxies with declining rotation curves appear tobe the most efficient, having the strongest response with 𝑄 𝐵𝑎𝑟 , andconsistently producing spirals with higher pitch angles comparedwith the other two galactic models.These results are consistent with observations. Using galaxiesfrom the Spitzer Survey of Stellar Structure in Galaxies ( 𝑆 𝐺 ), Bit-tner et al. (2017) showed that the distribution of flocculent galaxiesis statistically different from the multi-arm and grand design galax-ies. Flocculent spirals are more common in late-type galaxies, withweaker bars and less concentrated bulges. The low mass concen-tration suggests this galaxies should have a slowly rising rotationcurve. N-body simulations have shown that galaxies with rapidlyrising rotation curves form strong flat bars, whereas exponentialbars are more typical of slowly rising rotation curves (Combes &Elmegreen 1993; Athanassoula & Misiriotis 2002).Díaz-García et al. (2019) measured the pitch angle, spiralstrength and bar strength of galaxies in the 𝑆 𝐺 survey. Theyobserved the same strong relationship between the spiral and barstrength, even in flocculent galaxies, independently of the methodused for the measurement (see also Buta et al. 2003; Block et al.2004; Salo et al. 2010). The relationship with flocculent galaxies isconsistently weaker (i.e. bars in flocculent galaxies are associated MNRAS , 1–15 (2021) ow the bar affect the spiral structure 𝜎 𝑧 / 𝜎 𝑅 =
10 5 0 5 10 X [kpc]1050510 Y [ k p c ]) 𝜎 𝑧 / 𝜎 𝑅 = .
10 5 0 5 10 X [kpc]1050510 Y [ k p c ]) 𝜎 𝑧 / 𝜎 𝑅 = .
10 5 0 5 10 X [kpc]1050510 Y [ k p c ]) l n ( R [ k p c ]) l n ( R [ k p c ]) l n ( R [ k p c ]) Figure 14.
Bar induced spiral arms in discs with different velocity ellipsoids. As in Figure 4, this is the view of the test particles between 𝑅 𝐶𝑅 and the OLR+ 4 kpc at the snapshot where the spiral amplitude is maximum. The top row shows the view in the xy-plane, bottom row shows the ( 𝜃, log ( 𝑅 + )) plane.Test particles classified by DBSCAN as part of a clusters are coloured. with weaker spirals). They also observed a positive weak correlationbetween the spiral strength and pitch angle (see their Figs. 17 andE.1). However, the dispersion in the pitch angle measurements isquite large. We do not find any correlation between the spiral prop-erties independently of the galaxy model used ( 𝑟 𝑆 = . 𝑝 = . DATA AVAILABILITY
The data that support the findings of this study are available fromthe corresponding author, upon reasonable request.
ACKNOWLEDGEMENTS
We thank the referee C. Struck for his useful comments that im-proved this paper. We acknowledge DGTIC-UNAM for providingHPC resources on the Cluster Supercomputer Miztli. LGO andLMM acknowledge support from PAPIIT IA101520 grant. LGOacknowledge support from CONACyT scholarship.
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