# How the bar properties affect the induced spiral structure

L. Garma-Oehmichen,1 L. Martinez-Medina,1 H. Hernández-Toledo,1 and I.
Puerari.2
1Instituto de Astronomía, Universidad Nacional Autónoma de México, Apartado
Postal 70-264, CDMX, 04510, México
2Instituto Nacional de Astrofísica, Optica y Electrónica, Apdo. Postal 51 y
216, 72000 Puebla, Puebla, México E-mail<EMAIL_ADDRESS>
(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, we present a set of numerical simulations
based on isolated galactic models aimed to explore how the bar properties
affect the induced spiral structure. We cover a large combination of bar
properties, including the bar length, axial ratio, mass and rotation rate. We
use three galactic models describing galaxies with rising, flat and declining
rotation curves. We found that the pitch angle best correlates with the bar
pattern speed and the spiral amplitude with the bar quadrupole moment. Our
results suggest that galaxies with declining rotation curves are the most
efficient forming grand design spiral structure, evidenced by spirals with
larger amplitude and pitch angle. We also test the effects of the velocity
ellipsoid in a subset of simulations. We found that as we increase the radial
anisotropy, spirals increase their pitch angle but become less coherent with
smaller amplitude.

###### keywords:

galaxies: disc – galaxies: evolution – galaxies: kinematics and dynamics –
galaxies: structure

††pubyear: 2021††pagerange: How the bar properties affect the induced spiral
structure–How the bar properties affect the induced spiral structure

##  1 Introduction

Stellar bars inhabit a large fraction of disc galaxies in the local Universe,
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 from the 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 radial migration (Monari et al., 2016; Martinez-
Medina et al., 2017). Bars can also induce gas inflow to the central regions
via shock waves, forming gaseous structures like dust-lanes and rings
(Athanassoula, 1992; Hernquist & Mihos, 1995; Martinet & Friedli, 1997; Kim et
al., 2012; Sormani et al., 2018; Seo et al., 2019). Some moving groups in the
solar neighbourhood have their dynamical origin under the influence of the
Galactic bar (Pérez-Villegas et al., 2017) and through its induced resonances
shape the stellar velocities in the solar neighbourhood (Fux, 2001).

The importance of these phenomena depends on the morphological and dynamical
characteristics of the bar. In late-type galaxies, bars tend to be smaller in
relation to their discs (Méndez-Abreu et al., 2012; Díaz-García et al., 2016a;
Erwin, 2018), more oblate 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 galaxies
tend to be larger, prolate shaped, and with a flat density profile. Such co-
relations with the galaxy type have not been observed in measurements of the
bar pattern speed (hereafter $\Omega_{Bar}$), neither with the dimensionless
rotation rate (hereafter parameter $\mathcal{R}$) defined as the ratio between
the bar length (hereafter $a$) and the corotation resonance (hereafter
$R_{CR}$) (Aguerri et al., 2015; Guo et al., 2019; Cuomo et al., 2019; Garma-
Oehmichen et al., 2020). Nonetheless, it has been observed that the
$\Omega_{Bar}$ seems to correlate with the galaxy luminosity and total stellar
mass (Garma-Oehmichen et al., 2020; Cuomo et al., 2020).

Spiral arms also play an important role in the secular evolution of galactic
discs. By exchanging angular momentum, spirals churn stars and gas in the disc
(Sellwood & Binney, 2002). There is a large body of evidence, theoretical and
observational, that radial migration 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 dynamical heating 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 the nature and morphological properties of the spiral
pattern (Jenkins & Binney, 1990; Gerssen & Shapiro Griffin, 2012; Martinez-
Medina et al., 2015).

In most barred galaxies, bars co-evolve with spiral arms, and in some 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 & Norman (1979) first suggested that strong bars or
galaxy companions can lead to the formation of spiral density waves. There are
different theoretical scenarios on how the pattern speed of both structures is
related (see e.g. Dobbs & Baba, 2014). If bars and spirals are strongly
coupled, the pattern speed and amplitude of both structures should be strongly
correlated. This strong coupling is expected from the spiral density theory
(Lin & Shu, 1964), and the manifold theory where spirals are formed by stars
in escaping orbits around the unstable Lagrangian points at the end of the bar
(Romero-Gómez et al., 2006, 2007; Romero-Gómez et al., 2015; Athanassoula et
al., 2009). Observational evidence of a strong coupling comes from the
correlation between the torques and density amplitudes of both structures
(Buta et al., 2003; Block et al., 2004; Buta et al., 2005; Salo et al., 2010;
Bittner et al., 2017; Díaz-García et al., 2019). Another scenario proposes
that the pattern speeds are related by a non-linear mechanism (Tagger et al.,
1987), as suggested through indirect measurements of the bar pattern speed
(Font et al., 2017). Finally, a third scenario where bar and spirals are
decoupled structures, is supported by observations of the galaxy NGC 1365,
where Speights & Rooke (2016) found results consistent with bar and spiral
patterns being dynamically distinct features. The observational evidence to
distinguish these scenarios is limited though, and hard to obtain due to the
great uncertainties in estimating the pattern speed of both structures (Garma-
Oehmichen et al., 2020).

The vast majority of N-body simulations show that spirals are transient or
multi-arm features, that can be formed recurrently in time (Sellwood, 2011;
Grand et al., 2012; D’Onghia et al., 2013; Mata-Chávez et al., 2019).
Numerical simulations with steady bar potentials have found that the gas
particles settle in steady-state trailing spirals (Sanders & Huntley, 1976;
Athanassoula, 1992; Wada, 1994; Rodriguez-Fernandez & Combes, 2008).
Collisionless star particles can also produce prominent spirals and rings near
the Outer Lindblad Resonance (Schwarz, 1981; Bagley et al., 2009).

In this paper we use a set of numerical simulations based on galactic
potential models with one million test particles to explore how the bar
properties affect the response spiral arms. Our simulations cover a large bar
parameter space, tailored to explore a large number of combinations of bar
properties (size, shape, mass and pattern speed). We also test the effect of
discs with different shapes of rotation curves and velocity ellipsoids. To
characterise the response spiral arms, we identify the spiral particles using
the density-based clustering algorithm DBSCAN. This let us obtain clean
measurements of the spiral amplitude and pitch angle.

The paper is organised as follows. In Section 2 we describe the galactic
potential models, and the space of parameters to explore. In Section 3 we
describe how we use the algorithm DBSCAN to detect the spiral over-densities,
and how we estimate the properties of the induced spiral arms. In Section 4 we
show how the bar parameters affect the spiral arms properties. In Section 5 we
explore the effects of different galactic models and the rotation curve shape.
In Section 6 we present the effects of the velocity ellipsoid. In Section 7 we
discuss the relative importance of the bar and disc properties in predicting
the spiral properties. Finally, in Section 8 we discuss our results and
present our conclusions.

##  2 Simulations

###  2.1 Galactic models

The mass distribution of a galaxy could play a significant role in the
properties of the spiral arms. Using a sample of 94 galaxies Biviano et al.
(1991), observed that galaxies with steeper, rising rotation curves tend to
host flocculent spirals, while galaxies with flat rotation curves have grand
design spirals. Seigar et al. (2005, 2006) found that the pitch angle strongly
correlates with the rate of shear in the disc defined as

$S=\frac{1}{2}\left(1-\frac{R}{V}\frac{dV}{dR}\right)$ (1)

They suggested that spirals in rising rotation curves galaxies have greater
pitch angles compared to galaxies with flat or declining rotation curves.
However, this relationship has been questioned by Kendall et al. (2015) and Yu
et al. (2018), who failed to observe such correlation (see also Díaz-García et
al. (2019)).

To test the effects of different rotation curves, we used three galactic
potential models with similar enclosed mass at $\sim$20 kpc, but different
mass distributions. We will refer to these models as flat, rising and
declining, to figure out the shape of their rotation curve. The flat model is
the well-known Allen & Santillan (1991) potential comprising a Plummer bulge,
a Miyamoto Nagai disc, and a spherical halo that reproduces the nearly flat
Milky Way rotation curve. The density profiles of the bulge, disc and halo
(referred with the sub-indexes B, D and H, respectively) are:

$\rho_{B}(r)=\frac{3b_{B}^{2}M_{B}}{4\pi\left(r^{2}+b_{B}^{2}\right)^{5/2}}$
(2)

$\displaystyle\rho_{D}(R,z)$
$\displaystyle=\frac{b_{D}^{2}M_{D}}{4\pi}\frac{\left(R^{2}a_{D}+3\left(z^{2}+b_{D}^{2}\right)^{1/2}\right)}{\left(R^{2}+\left(a_{D}+\left(z^{2}+b_{D}^{2}\right)^{1/2}\right)^{2}\right)^{5/2}}$
$\displaystyle\times\frac{\left(a_{D}+\left(z^{2}+b_{D}^{2}\right)^{1/2}\right)^{2}}{\left(z^{2}+b_{D}^{2}\right)^{3/2}}$
(3)

$\rho_{H}(r)=\frac{M_{H}}{4\pi
a_{H}r^{2}}\left(\frac{r}{a_{H}}\right)^{1.02}\left(\frac{2.02+(r/a_{H})^{1.02}}{(1+(r/a_{H})^{1.02})^{2}}\right)$
(4)

where $r$ is the spherical radius coordinate, $(R,z)$ are cylindrical
coordinates, $M_{B}$, $M_{D}$ 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 concentration by extending
the bulge scale radius to match the spherical halo and doubled the disc scale
radius. In the declining model, we reduce the disc scale radius by a factor of
3/4. All galactic models share the same mass for each component. We show the
parameters used for the galactic models in Table 1. Figure 1 shows the
rotation curves of the three axisymmetric models.

Table 1: Parameters of the galactic models Galactic Model | $M_{B}$ | $b_{B}$ | $M_{D}$ | $a_{D}$ | $b_{D}$ | $M_{H}$ | $a_{H}$
---|---|---|---|---|---|---|---
| [$10^{10}M_{\odot}$] | [kpc] | [$10^{10}M_{\odot}$] | [kpc] | [kpc] | [$10^{10}M_{\odot}$] | [kpc]
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
Rising | 1.406 | 12.0 | 8.561 | 10.636 | 0.25 | 10.709 | 12.0
Flat (Allen-Santillan) | 1.406 | 0.387 | 8.561 | 5.318 | 0.25 | 10.709 | 12.0
Declining | 1.406 | 0.387 | 8.561 | 3.988 | 0.25 | 10.709 | 24.0
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): Disc
vertical scale length. Col. (7): Halo mass. Col. (8): Halo scale length.
Figure 1: Rotation curves of the three axisymmetric galactic models used in
this work.

###  2.2 Bar model

To simulate the bar potential, we use a Ferrers ellipsoid of index $n=2$
(Binney & Tremaine, 1987). The density distribution in Cartesian coordinates
is:

$\rho(x,y,z)=\begin{cases}\frac{105}{32\pi abc}M_{Bar}(1-m^{2})^{2},&m<1\\\
0,&m>1\\\ \end{cases}$ (5)

where $M_{Bar}$ is the bar mass and $m^{2}=(x/a)^{2}+(y/b)^{2}+(z/c)^{2}$. 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 by this distribution are
described in Pfenniger (1984).

We introduce the bar adiabatically as a smooth function of time by
transferring mass from the bulge to the bar. We used the fifth-degree
polynomial described in equation 4 of Dehnen (2000), which guarantees a smooth
transition to the barred state. We set the time growth of 500 Myr, similar to
Romero-Gómez et al. (2015).

Orbital and dynamical studies have shown that self-consistent bars cannot
extend beyond their co-rotation radius $R_{CR}$, where the bar and the disc
rotate at the same angular speed. Beyond $R_{CR}$ stellar orbits 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, the dimensionless parameter $\mathcal{R}=R_{CR}/a$ is used to
parametrize the bar rotation rate. Bars are classified kinematically as “slow"
if $\mathcal{R}$ >1.4, and “fast" if $\mathcal{R}$<1.4. The theoretically
impossible case of $\mathcal{R}$<1 is referred as “ultra-fast".

In this work, we explore the following bar parameter space: (i) Three bar
radius corresponding to $a=0.5,1.0,1.5$ times the disc radial scale radius of
the Allen-Santillan model. (ii) Two axial ratios between the minor and major
axis $b/a=0.3,0.6$, corresponding to prolate and oblate bars, respectively.
(iii) Two bar masses that correspond to a complete and a half mass transfer
between the bulge and the bar. (iv) Three values for the dimensionless
parameter $\mathcal{R}$, exploring the scenario of slow, fast and ultra-fast
bars. We do not explore the effects of the vertical axial ratio, which was set
to $c/a=0.3$ for all our simulations. Table 2 shows a summary of the bar
parameter space. Our simulations explore 36 combinations for the bar model,
and 3 galactic potential models, yielding a total number of 108 simulations.
All our simulations use 1 million test particles.

Table 2: Bar parameter space Parameter | Values
---|---
(1) | (2)
a [kpc] | 2.659, 5.318, 7.977
b/a | 0.3, 0.6
$\mathcal{R}$ | 0.9, 1.2, 1.5
$M_{Bar}$ [$10^{10}M_{\odot}$] | 0.703, 1.406
c/a | 0.3
Col. (1): Bar parameters. From top to bottom: Length, axial ratio, rotation
rate, mass and vertical axial ratio. Col. (2): Values explored.

Given the bar length $a$ and the parameter $\mathcal{R}$, the pattern speed
$\Omega_{Bar}$ is estimated by evaluating axisymmetric angular velocity curve
at the corotation resonance $R_{CR}=\mathcal{R}\cdot a$. However, as the bar
is being introduced, the mass distribution and the location of the resonances
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 $L1$ as a proxy for the location of $R_{CR}$(Binney &
Tremaine, 2008). If the relative difference between $L1$ and the axisymmetric
$R_{CR}$ is greater than 10%, we correct the value of $\Omega_{Bar}$ until
both distances match. From hereafter, when we refer to $\mathcal{R}$ we are
using the $L1/a$ ratio, accounting for the bar mass redistribution.

###  2.3 Initial conditions and orbit integration

A natural space to study spirals is the $(\theta,\ln R)$ plane, where $\theta$
is the azimuthal angle and $R$ the cylindrical radius. In this space, a
logarithmic spiral can be described with the straight line equation (Lin &
Shu, 1964):

$R(\theta)=R_{0}\times e^{\theta\tan\alpha}$ (6)

where, $R=R_{0}$ at $\theta=0$ and $\alpha$ is the pitch angle.

However, a disc populated with test particles uniformly distributed in the
$(\theta,\ln R)$ space would have few particles in the inner disc. Instead, we
choose to set the initial spatial distribution to be uniform in the
$(\theta,\log_{10}(R+1))$ space, which keeps the logarithmic spacing in the
radius, populates the inner disc and highlights the spiral over-densities over
a uniform background. The resulting density distribution resembles that of a
single exponential disc. For the vertical dimension, we choose a usual
$sech^{2}(z/z_{0})$ law distribution (van der Kruit & Searle, 1981) with
scale-height $z_{0}=0.25$ kpc.

We generated the initial velocities using the Hernquist (1993) moments method.
This procedure constrains the shape of the velocity ellipsoid, by assuming the
radial and vertical dispersion are proportional and constant throughout the
disc $(\sigma_{R}^{2}\propto\sigma_{z}^{2})$. For simplicity, we set the
velocity dispersion ratio to $\sigma_{z}/\sigma_{R}=1$. In Section 6 we
explore the effect of other normalisation constants with greater radial
velocity dispersion for a subset of simulations.

Before introducing the bar, we first integrate the test particles orbits in
the axisymmetric potential for 3 Gyr so they relax and reach the statistical
equilibrium (Romero-Gómez et al., 2015). We performed 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
vertical velocity distribution for a subset of test particles to confirm the
stability.

Figures 2 and 3 show snapshots of a simulation using the flat model and bar
parameters $a=5.318$ kpc, $b/a=0.3$, $\mathcal{R}=1.2$, $M_{Bar}=1.406\times
10^{10}M_{\odot}$ in the $(x,y)$ and $(\theta,\log_{10}(R+1))$ planes,
respectively. Both figures are in the bar rotating reference frame. The
snapshots are separated in time steps of 250 Myr. The first snapshot
corresponds to the relaxed disc distribution. The bar stops growing in mass at
500 Myr. We show the corotation resonance with a white solid line. The Inner
and Outer Lindblad Resonances (ILR and OLR from hereafter) are shown with
white segmented lines. These resonances were estimated using the axisymmetric
angular velocity curve and the epicyclic frequency curve.

Figure 2: Snapshots of a simulation in the flat model and bar properties
$a=5.318$ kpc, $b/a=0.3$, $\mathcal{R}=1.2$, $M_{Bar}=1.406\times
10^{10}M_{\odot}$. We masked particles inside 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 introduced adiabatially by transferring
mass from the bulge component up to 500 Myr. The spiral amplitude reaches a
maximum at 1000 Myr, and decreases afterwards. Figure 3: Same as Figure 2, but
in the $(\theta,\log_{10}(R+1))$ plane. The white segmented line highlights
the location of the OLR. Notice that in the first snapshot, the particles are
uniformly distributed.

##  3 Estimating the properties of the spiral arms

###  3.1 Detecting over-densities using DBSCAN

To detect the spiral over-densities, we use the density-based clustering
algorithm DBSCAN (Ester et al., 1996). The algorithm works by classifying each
point in a given space as “core", “member" or “noise", based on two
parameters: (i) $\epsilon$ which specifies the distance between two points to
be considered “neighbours", and (ii) $mincnt$ which specifies the minimum
number of neighbours a point must have to be classified as a “core" of a
cluster. All points in the $\epsilon$-neighbourhood of a core, that do not
satisfy the $mincnt$ condition are classified as “members" of the cluster.
Finally, all points that do not inhabit a cluster are classified as “noise".

DBSCAN is especially useful to find arbitrarily shaped structures and does not
require knowing a priori the number of clusters in the data. The use of the
algorithm has been increasing in astronomy in recent years. To mention a few
applications: identifying lensed features in residual images (Paraficz et al.,
2016), classifying eclipsing binaries light curves (Kochoska et al., 2017),
detecting low surface brightness galaxies in the Virgo cluster (Prole et al.,
2018) and finding open clusters in the Gaia data (Castro-Ginard et al., 2018,
2020).

Choosing meaningful parameters $(\epsilon,mincnt)$ is challenging when the
size and density of the clusters are unknown. The parameter $\epsilon$ is
related to the spatial resolution of the clusters and can be determined from
their expected size. The spiral arms produced in our simulations typically
have widths of $\sim 0.6$ kpc, so we choose to fix the value of $\epsilon$ to
0.3 kpc. The parameter $mincnt$ is related to the expected density, or in this
case, the spiral amplitude. However, the amplitude changes with time and
between simulations. We performed the following steps to determine an
appropriate value of $mincnt$ at any given snapshot. We show an example of the
procedure in Figure 4:

  1. [i]

  2. 1.

We mask all particles between $R_{CR}$ and OLR + 4 kpc, where we expect the
spiral arms to be located. We re-scale the data, so the distances in
$\log(R+1)$ and $\theta$ are the same in an Euclidean metric. We also re-scale
the value of $\epsilon$. The first panel of Figure 4 shows a snapshot of a
simulation where the spiral arms have already formed, and the data has been
re-scaled.

  3. 2.

A second mask is used to select particles that lie in the radial range
$R=[OLR-2\epsilon,OLR]$. In the first panel of Figure 4, the two red
horizontal lines delimit this range.

  4. 3.

All our simulations produced two symmetrical spirals arms. We bin the masked
particles in 64 bins and fit a simple cosine wave function
$f(\theta)=A\cos(2\theta+\phi)+C$. This is equivalent to the $m=2$ Fourier
component of the azimuthal light profile near the OLR. We use the normalised
amplitude of the fit, $A/C$, as a proxy of the spiral amplitude. In the second
panel of Figure 4, we show the masked particles distribution and the cosine
fit.

  5. 4.

We count the number of particles around the peaks $\pm\epsilon$, illustrated
with two red segmented lines in the second panel of Figure 4. The resulting
count is the expected number of particles in two squares of size $2\epsilon$
at the peak of the spirals. We show these squares in the first panel of Figure
4.

  6. 5.

We multiply the resulting count by $\pi/8$ which is the ratio between the area
of a circle with radius $\epsilon$ and two squares of side $2\epsilon$. This
results in our estimation for $mincnt$.

Once we have estimated $(\epsilon,mincnt)$, DBSCAN can detect the spiral over-
densities as shown in the third and fourth panels of Figure 4. In this
example, the algorithm detected 3 clusters, shown in different colours. We
plot the core points with slightly bigger dots, so they appear as a solid
coloured region. Member points surround the core with an envelope of radius
$\epsilon$. Notice that the pink cluster is just a continuation of the blue
spiral, but displaced by $2\pi$ radians. In such cases, we join the separated
clusters by manually adding the $2\pi$ rotation.

Figure 4: Detecting spiral over-densities with DBSCAN. We use a snapshot from
the simulation with bar parameters $a=5.318$ kpc, $b/a=0.3$,
$M_{bar}=1.406\times 10^{10}M_{\odot}$ , $\mathcal{R}=1.2$, in the flat
galactic model. First panel: We select particles between $R_{CR}$ and the OLR
+ 4 kpc where the spiral arms are formed. We re-scale the data so distances in
radius and angle are the same. We mask the particles in the range [OLR
$-2\epsilon$, OLR] shown with two red horizontal lines. At the peak spiral
density we draw the two squares of side $2\epsilon$. Second panel:
Distribution of the masked particles vs. the scaled angle. We fit a cosine
function to get the amplitude and peaks of the spiral over-density. We count
the number of test particles around the peaks $\pm\epsilon$, shown with red
vertical lines. We determine $mincnt$ by multiplying this count by $\pi/8$.
Third panel: Using the estimated parameters $(\epsilon,mincnt)$, DBSCAN
identifies three clusters in the data. The points classified as “core” are
shown with slightly bigger dots, so they appear as a solid coloured area.
“Member” points surround the core area in an $\epsilon$ size envelope. Notice
this plot is in the $(\theta,\ln R)$ space. We use a piecewise linear function
to fit the identified clusters shown with two white lines. We measure the
pitch angle using the greatest slope in the piecewise fit. Fourth panel: Same
as the third panel, but in the (x,y) plane.

Not every simulation produces spirals as clearly as in the example shown in
Figure 4. Some models produce very weak spiral arms that are almost
indistinguishable from the background. In those cases, we reduce the value of
$mincnt$ manually until the algorithm can detect the underlying spiral
structure.

###  3.2 Estimating the pitch angle

Although we identify the spiral over-densities in the $(\theta,\log_{10}R+1)$
plane, the measurement of the pitch angle is done in the $(\theta,\ln R)$
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 kind of 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 snapshot shown in Figure 4 is an example of such behaviour. To
separate the rings from the spiral, we fit a piecewise linear function and use
the slope of the spiral segment to estimate the pitch angle. The fitted
piecewise function is shown with white lines in the third and fourth panels of
Figure 4. Notice how the change in pitch angle cannot be distinguished ‘by
eye’ in the $(x,y)$ plane. We measure the pitch angle in both spirals using
the whole cluster and only core points. From hereafter, when we refer to the
pitch angle we are using the average from these measurements.

###  3.3 Estimating the spiral amplitude

We use the normalised amplitude of the fitted cosine wave function (second
panel in Figure 4) as a proxy of the spiral amplitude. In general, all our
simulations showed the same trend: As the bar is being introduced, the spiral
amplitude increases as a function of time until it reaches a maximum and
slowly decreases. In Figure 5 we show the spiral amplitude as a function of
time of four simulations in the flat galaxy model that share the same bar
length $a$ and parameter $\mathcal{R}$ (and thus, the same $\Omega_{Bar}$),
but vary in mass and axis ratio. In most simulations, the amplitude peaks
between 1 or 2 dynamical times at the OLR after the bar has formed. After 3 to
4 dynamical times the spiral structure becomes more diffuse. Simulations that
share $a$ and $\mathcal{R}$, usually have spirals that peak in the same
snapshot, except for some simulations in the rising model, that are more
sensitive to the bar potential.

In the next section we discuss which bar parameters produce stronger spirals,
but from this figure is clear that the mass and shape of the bar are strongly
co-related to the spiral amplitude.

Figure 5: Spiral arm amplitude as a function of time, for the family of
simulations with bar parameters $a=5.318$ kpc and $\mathcal{R}=1.2$ in the
flat model. The solid black line shows the bar formation (500 Myr), the
segmented and dotted lines show 1 and 2 dynamical times at the OLR after the
bar formation, respectively.

##  4 Relations between the bar parameters and the spiral properties

The spiral arms produced in our simulations are the collective result of the
bar perturbing the stellar epicyclic orbits. Thus, our spiral arms are
produced by the density-wave mechanism and do not account for the effects of
self-gravity. Our results should be interpreted as the initial spiral
perturbation that arises from the presence of different kinds of bars. Spiral
arms produced this way, are expected to form near the OLR (Athanassoula,
1980), and be tightly wound with small pitch angles.

Because of the great variety of parameters we are exploring, the spirals arms
form at different times, amplitudes and locations throughout the disc. To make
the comparison between different simulations as fair as possible, we use the
measurements of the snapshot where spirals are at their greatest amplitude.

###  4.1 Observed correlations

In Figures 6, 7, and 8 we show the average pitch angle and the spiral
amplitude vs. the bar parameters of our simulations in the rising, flat, and
declining galactic models, respectively. We join with grey dotted lines the
simulations that share the same bar parameters except for the one that is
being plotted. The Spearman correlation coefficient $r_{S}$ and the
corresponding statistical significance $p$ are shown at the top of each panel.
We colour the results with the bar pattern speed. To facilitate the
interpretation of our results, Table 3 summarises all Spearman correlation
coefficients that we mention through the text.

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 spiral amplitude. The grey segmented lines join simulations that share the
same bar parameters, except for the one that is being plotted. The Spearman
correlation coefficient and the corresponding p-value are shown at the top of
each panel. Figure 7: Same as Figure 6, but in the flat model. Figure 8: Same
as Figure 7 and 6, but in the declining model.

In all galactic models we found a strong anti-correlation of the pitch angle
with the bar length $(r_{S}\sim-0.6)$ and the $\mathcal{R}$ parameter
$(r_{S}\sim-0.65)$. At a first approximation, it would appear that the mass
and shape of the bar do not correlate with the pitch angle, with a small
$r_{S}$ coefficient and p-values $>0.05$. However, the connected models
display a clear downward trend with $b/a$ and an upper trend with $M_{Bar}$.
We performed a Student’s t-test to see if these slight differences between
connected simulations could happen by random chance. We could reject the null
hypothesis in both cases in all galactic models ($p\sim 1\times 10^{-4}$ for
$b/a$ and $p\sim 1\times 10^{-5}$ for $M_{Bar}$).

The relations with the spiral amplitude does seem to depend more on the
galactic models. The correlation with the bar length is non-existent in the
rising model ($r_{S}=0.16$), but becomes strong in the flat and declining
models ($r_{S}=0.64$ and $r_{S}=0.85$, respectively). In contrast, the
relation with the $\mathcal{R}$ parameter does not seem significant in the
declining model ($r_{S}=-0.14$) but becomes a weak anti-correlation in the
rising and flat models ($r_{S}\sim-0.40$ in both cases). The connected
simulations in all three models show a clear upward 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 the null-hypothesis. The strong trends with the bar
mass and axis ratio can also be observed in amplitude vs. time plot in Figure
5.

Some of the observed correlations are related in the form of a third
relationship. For example, the bar length $a$ and parameter $\mathcal{R}$ are
intimately related with $\Omega_{Bar}$ and the disc rotation curve V(R):

$\Omega_{Bar}=V(R_{CR})/R_{CR}=V(R_{CR})/(\mathcal{R}\cdot a)$ (7)

Thus, the strong trends we observe among the pitch angle, the bar length and
$\mathcal{R}$ are probably a consequence of the much stronger correlation
between the pitch angle and the bar pattern speed, $(r_{S}\sim 0.85)$ in all
three galaxy models. The relation between the spiral properties and
$\Omega_{Bar}$ is shown in Figure 9 for the three galactic models. The colours
and shape of the dots are used to distinguish models in $\mathcal{R}$ and
$M_{Bar}$ respectively. The relation between the spiral amplitude and
$\Omega_{Bar}$ depends on the galactic model, as this relation is non-existent
in the rising model, but becomes a strong anti-correlation in the declining
model. 0’p0’po

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 the right). Simulations are colour-coded according to their $\mathcal{R}$
parameter (blue for slow, red for fast and green for ultra-fast bars). The
more massive bars are shown with diamonds, while their less massive
counterpart are shown with crosses.

The bar length $a$, mass $M_{Bar}$ and axial ratio $a/b$ are related to the
bar quadrupole moment (hereafter $Q_{Bar}$). For a Ferrers ellipsoid of index
$n$ $Q_{Bar}$is:

$Q_{Bar}=M_{Bar}a^{2}\left(1-\frac{b^{2}}{a^{2}}\right)/\left(5+2n\right)$ (8)

Same as with other bar properties, in Figure 10 we show the correlations
between the spiral properties and the quadrupole moment in the three galactic
models. Our results show a remarkably strong correlation between the spiral
amplitude and $Q_{Bar}$ in the flat and declining models. Thus, the observed
correlations between spiral amplitude and $a$, $b/a$ and $M_{Bar}$ could be a
consequence of the more stronger correlation with $Q_{Bar}$. Nonetheless, this
correlation 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 more complex. The
general trend is negative $(r_{S}\sim-0.45$ in the three models). However, if
we look only at simulations with the same pattern speed as shown by the colour
code (i.e. simulations with same bar length and $\mathcal{R}$ parameter) the
positive relation with the mass (and anti-correlation with $b/a$) becomes
visible.

Figure 10: Pitch angle and spiral amplitude vs. bar quadrupole moment in the three galactic models. Table 3: Spearman correlation coefficient and statistical significance with the spiral properties |  | Pitch Angle |  |  | Amplitude |
---|---|---|---|---|---|---
Parameter | Rising | Flat | Declining | Rising | Flat | Declining
(1) | (2) | (3) | (4) | (5) | (6) | (7)
$a$ |  $-0.56$ $(4\times 10^{-4})$ |  $-0.65$ $(2\times 10^{-5})$ |  $-0.60$ $(1\times 10^{-4})$ |  $0.16$ $(4\times 10^{-1})$ |  $0.64$ $(3\times 10^{-5})$ |  $0.85$ $(3\times 10^{-11})$
$b/a$ |  $-0.16$ $(4\times 10^{-1})$ |  $-0.13$ $(4\times 10^{-1})$ |  $-0.17$ $(3\times 10^{-1})$ |  $-0.23$ $(2\times 10^{-1})$ |  $-0.24$ $(2\times 10^{-1})$ |  $-0.20$ $(2\times 10^{-1})$
$M_{Bar}$ |  $0.18$ $(3\times 10^{-1})$ |  $0.26$ $(1\times 10^{-1})$ |  $0.22$ $(2\times 10^{-1})$ |  $0.45$ $(7\times 10^{-3})$ |  $0.55$ $(6\times 10^{-4})$ |  $0.36$ $(3\times 10^{-2})$
$\mathcal{R}$ |  $-0.64$ $(4\times 10^{-5})$ |  $-0.65$ $(1\times 10^{-5})$ |  $-0.67$ $(6\times 10^{-6})$ |  $-0.39$ $(2\times 10^{-2})$ |  $-0.40$ $(1\times 10^{-2})$ |  $-0.14$ $(4\times 10^{-1})$
$\Omega_{Bar}$ |  $0.83$ $(1\times 10^{-9})$ |  $0.88$ $(1\times 10^{-12})$ |  $0.84$ $(1\times 10^{-10})$ |  $0.02$ $(9\times 10^{-1})$ |  $0.39$ $(2\times 10^{-2})$ |  $0.71$ $(1\times 10^{-6})$
$Q_{Bar}$ |  $-0.42$ $(1\times 10^{-2})$ |  $-0.48$ $(3\times 10^{-3})$ |  $-0.45$ $(6\times 10^{-3})$ |  $0.36$ $(4\times 10^{-2})$ |  $0.82$ $(1\times 10^{-9})$ |  $0.95$ $(2\times 10^{-19})$
Shear |  $-0.44$ $(1\times 10^{-2})$ |  $-0.82$ $(8\times 10^{-10})$ |  $-0.75$ $(1\times 10^{-7})$ |  $0.00$ $(1\times 10^{0})$ |  $0.44$ $(7\times 10^{-3})$ |  $0.73$ $(4\times 10^{-7})$
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.

##  5 Influence of the rotation curve

We study the effects of the rotation curve in the spiral properties, by
comparing simulations with the same bar parameters, but different galaxy
model. In Figure 11 we show the pitch angles of the flat models vs. the rising
and declining models. As in Figure 9 the colours and shape are used to
distinguish the $\mathcal{R}$ and $M_{Bar}$ respectively. 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 also appears to be more significant with the ultra-fast
bars, as those differ more significantly from the 1:1 relation. We include the
best linear fit for reference.

Figure 11: Comparison of the pitch angle between the three galactic models.
For reference, the slopes of the best fits are $m=0.92$ in the top panel
(excluding the outlier) and $m=1.26$ in the bottom one.

Similarly, in Figure 12 we compare the spiral amplitude between the three
galactic models. The rising model tends to produce weaker spirals. The
difference becomes more significant with the more massive bars. On the other
hand, the declining and flat models have similar spiral amplitudes for all the
bar models.

Figure 12: Comparison of the spiral amplitude between the three galactic
models. For reference, the slopes of the best fits are $m=0.39$ in the top
panel and $m=0.90$ in the bottom one.

We also estimate the shear rate $S$ (equation 1) at the OLR in all our
simulations. Values of $S<0.5$ correspond to a rising region in the rotation
curve, $S=0.5$ to a flat region and $S>0.5$ to a declining region. The
presence of the bar changes the shape of the rotation curve from the
axisymmetric model. We estimate the perturbed rotation curve by averaging
$V(R)=\sqrt{Rd\phi/dR}$ over 10 angles uniformly distributed between 0 and
$\pi/2$ radians.

In Figure 13 we show the correlations between the spiral properties and the
local shear rate. We found a strong anti-correlation with the pitch angle, but
only in the flat and declining models. The relations with the spiral amplitude
are weaker, except on the declining models, where we observe a strong
correlation.

Figure 13: Pitch angle and spiral amplitude vs. Shear rate measured at the OLR
in the three galactic models.

##  6 Influence of the velocity ellipsoid

Disc heating mechanisms play an important role in the secular evolution of
galactic discs, altering the stellar kinematics and increasing the random
motion of stars. It has been shown that the shape of the velocity ellipsoid,
i.e., the ratio of the vertical and radial velocity dispersion
$\sigma_{z}/\sigma_{R}$ is strongly correlated with the Hubble type, with
late-type galaxies being more anisotropic and early-types being more isotropic
(van der Kruit & de Grijs, 1999; Gerssen & Shapiro Griffin, 2012). However
these results have been questioned by Pinna et al. (2018), who found a wide
range of dispersion ratios from the literature around
$\sigma_{z}/\sigma_{R}=0.7\pm 0.2$. In the solar neighbourhood
$\sigma_{z}/\sigma_{R}$ ranges from $\sim 0.4$ to $\sim 0.9$ depending on the
stellar age, mean orbital radii, or even the dynamical modelling (Mackereth et
al., 2019; Nitschai et al., 2020).

To study the effects of the velocity ellipsoid on the formation of spiral
arms, we re-simulate a subset of 8 galaxies in the flat galactic model. We
build new initial conditions where the velocity dispersion relation is set to
$\sigma_{z}/\sigma_{R}=0.5$ and a more extreme case
$\sigma_{z}/\sigma_{R}=0.33$. We chose the subset of simulations to cover a
wide range of pitch angles and spiral amplitudes produced by the original
simulations with isotropic velocity ellipsoid.

The spiral arms produced by the new simulations are still located near the
OLR, and peak in amplitude at the same time as the $\sigma_{z}/\sigma_{R}=1$
simulation. In Figure 14 we show a side-by-side comparison of the snapshot we
presented in Figure 4, but simulated with different velocity ellipsoids. As we
decrease the $\sigma_{z}/\sigma_{R}$ ratio, the response spirals increase
their pitch angle, but result in a less coherent, wider structure with smaller
amplitude.

$\sigma_{z}/\sigma_{R}=1$

$\sigma_{z}/\sigma_{R}=0.5$

$\sigma_{z}/\sigma_{R}=0.33$

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
$(\theta,\log_{10}(R+1))$plane. Test particles classified by DBSCAN as part of
a clusters are coloured.

In Figure 15 we show the pitch angle and spiral amplitude as a function of the
velocity dispersion ratio. We connect with a grey segmented line simulations
that share the same bar model. Is clear, from this small subset that the
effects the velocity ellipsoid has on the spiral arms are substantial. The
more radially heated discs produced very low amplitude spirals. In one
simulation, we were not able to detect any spiral structure from the
background.

Figure 15: Pitch angle and spiral amplitude as a function of the velocity
dispersion ratio $\sigma_{z}/\sigma_{R}$. Simulations that share the same bar
model are connected with the grey segmented lines.

##  7 Bar vs Disc properties

So far, we have explored how different bar and disc parameters relate with the
induced pitch angle and spiral amplitude independently. Using solely their
Spearman correlation coefficient and the statistical significance, the
parameters that best predict the pitch angle and amplitude are the
$\Omega_{Bar}$ and $Q_{Bar}$, respectively. Nonetheless, the differences
observed between galactic models suggest the shear or galactic model itself,
could also be important features for predicting the spiral properties. Also,
some of these parameters are related to each other and cannot be treated as
independent variables.

To rank the relative importance of each parameter compared to the rest, we
used a random forest regressor (Breiman, 2001), trained to predict the spiral
properties of our simulations using all bar parameters, the shear rate and the
galactic model. The model works by averaging the prediction of multiple
decision trees, which try 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 maximum number 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
$\Omega_{Bar}$ is considered an important variable to predict the pitch angle,
a decision tree that does not train with this feature should weight
$\mathcal{R}$ more heavily, as these two are highly correlated.

As a proof of concept, we trained several models using an 80/20 train-test
split which consistently achieved a coefficient of determination $r^{2}>0.80$
for both the pitch angle and the spiral amplitude. Since the purpose of these
models is to get the relative importance of each feature, we re-train the
models using all our simulation data. We used the permutation feature
importance technique; where the relevance of each feature is estimated by the
difference between the $r^{2}$ score of the original data ($r^{2}_{baseline}$)
and the score after randomly shuffling such feature ($r^{2}_{permuted}$)
(Breiman, 2001). In Figure 16 we show the feature importance estimated after
shuffling each feature 50 times.

Figure 16: Permutation feature importance of two random forest regressors
trained to predict the pitch angle (top panel) and spiral amplitude (bottom
panel). Each feature was permuted 50 times, to produce a distribution of
$\Delta r^{2}$ scores. The orange line shows the median value of the
distribution. The box limits are the 25 and 75 percentiles. The lines extend
to the maximum and minimum values. Outliers of the distribution are shown with
open circles.

The most important features for predicting the pitch angle are related to the
bar rotation rate. This was expected, as $\Omega_{Bar}$ and $\mathcal{R}$
showed the strongest correlations with the pitch angle. In comparison all
other parameters are not consider as important, as these only change the
$r^{2}$ score of the prediction by 0.1 or less. This shows that the relative
strong correlation with the bar length was actually a consequence of the much
more stronger relations with the bar frequency. Notice in particular, that the
”Declining" feature is considered more important than the Flat and Rising
features. This is probably due the declining simulations having consistently
higher pitch angles as we showed in Figure 11.

For the spiral amplitude the most important features are related to the bar
strength. Again, this was expected from the strong correlations of $Q_{Bar}$,
$M_{Bar}$ and $a$. In contrast with the pitch angle, the amplitude model
considers the bar frequency ($\mathcal{R}$ and $\Omega_{Bar}$) and the disc
rotation curve ($S$ and Rising) to be as important as the bar mass and length.
This could be because of the rising simulations forming consistently spirals
with lower amplitudes as we showed in Figure 12.

Our random forest models do not take into account the velocity dispersion
ratio of the stellar disc. However, it is clear from the simulations we
studied in Section 6 that the dynamical temperature of the disc is an
important feature to consider when predicting the spiral properties. Perhaps
equally or more important than the perturber properties or the shape of the
velocity curve. How these effects complement or affect the relations
discussed, will be explored in a future work.

##  8 Discussion and Conclusions

Since our simulations lack the effects of self-gravity, our results cannot
predict how the system would evolve after the initial perturbation. Our
spirals have typical lifetimes between 2 or 3 dynamical times at the OLR after
the bar formation. However, we point out that self-gravity and star formation
could increase and highlight the spiral amplitude. It is expected that these
spirals would eventually disappear and fragment into smaller substructures as
several N-body simulations have shown.

The evolution of the pitch angle is unclear. In the Lin-Shu density wave
picture, the pitch angle remains constant in time and depends on the global
galaxy properties. Several N-body simulations have shown that the pitch angle
decreases with time (Grand et al., 2013; Pettitt & Wadsley, 2018), while
others have shown it remains roughly constant, independently of their origin
(Mata-Chávez et al., 2019). Nonetheless, Mata-Chávez et al. (2019) also
observed an increase in the pitch angle after the buckling of the bar,
probably because of the mass redistribution. Recently, Pringle & Dobbs (2019)
proposed the pitch angle evolves as a decreasing function of time, from an
initial measurable maximum $\alpha_{max}$ to a minimum $\alpha_{min}$, as
evidenced from the uniform distribution of $\cot\alpha$ of galaxies observed
by Yu et al. (2018). Our results suggest that, when the spirals are induced by
a bar perturber, the initial pitch angle is not random nor has a fixed maximum
for all galaxies. It mainly depends on 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 NGC 1365, NGC 1672 and NGC
2903 other interaction mechanisms should be considered. For example, the
different nature of spiral arms has also been explained by invariant manifolds
(Athanassoula, 2012; 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 appear to be powerful
predictors for the spiral properties (in comparison with the bar) they seem to
be important on how the disc responds to the perturbations. Our simulations
show that bars in galaxies with rising rotation curves are less efficient
forming a grand design spiral structure. We found evidence in the weaker
relationship between the spiral amplitude and $Q_{Bar}$, and the 1:1
comparison of the spiral amplitude between galaxy models with same bar
parameters. In contrast, bars in galaxies with declining rotation curves
appear to be the most efficient, having the strongest response with $Q_{Bar}$,
and consistently producing spirals with higher pitch angles compared with the
other two galactic models.

These results are consistent with observations. Using galaxies from the
Spitzer Survey of Stellar Structure in Galaxies ($S^{4}G$), Bittner et al.
(2017) showed that the distribution of flocculent galaxies is statistically
different from the multi-arm and grand design galaxies. Flocculent spirals are
more common in late-type galaxies, with weaker bars and less concentrated
bulges. The low mass concentration suggests this galaxies should have a slowly
rising rotation curve. N-body simulations have shown that galaxies with
rapidly rising rotation curves form strong flat bars, whereas exponential bars
are more typical of slowly rising rotation curves (Combes & Elmegreen, 1993;
Athanassoula & Misiriotis, 2002).

Díaz-García et al. (2019) measured the pitch angle, spiral strength and bar
strength of galaxies in the $S^{4}G$ survey. They observed the same strong
relationship between the spiral and bar strength, even in flocculent galaxies,
independently of the method used for the measurement (see also Buta et al.,
2003; Block et al., 2004; Salo et al., 2010). The relationship with flocculent
galaxies is consistently weaker (i.e. bars in flocculent galaxies are
associated with weaker spirals). They also observed a positive weak
correlation between the spiral strength and pitch angle (see their Figs. 17
and E.1). However, the dispersion in the pitch angle measurements is quite
large. We do not find any correlation between the spiral properties
independently of the galaxy model used ($r_{S}=0.01$, $p=0.88$ for all our
simulations). Nonetheless, this could be an effect of the spiral arm acting on
itself via self-gravity that we cannot capture in our simulations.

It is possible that our major results could be generalised to other kinds of
spiral-producing perturbations (galaxy encounters, tri-axial dark matter
halos, giant molecular clouds, etc). That is, the frequency of the perturber
relates to the spiral pitch angle, and the strength of the perturbation
relates to the spiral amplitude (see e.g. Pettitt et al., 2016; Pettitt &
Wadsley, 2018).

It is clear that the shape of the velocity ellipsoid and the dynamical state
of the disc have an important role in predicting the spiral properties and how
the disc responds to different gravitational perturbations. Simulations by
Athanassoula & Sellwood (1986) showed that bars delay their growth in
dynamically hotter discs. In our tests, spirals generated in discs with
isotropic ellipsoids had a greater amplitude, but a smaller pitch angle. As we
increased the anisotropy with greater radial dispersion, the spirals unwind,
but reduced their amplitude. Is possible that an optimal range of anisotropy
is required for discs to be able to form grand design spirals. How the
velocity ellipsoid affects the relations we discussed with the bar and the
rotation curve remains unclear, and will be explored in a future work.

## Data availability

The data that support the findings of this study are available from the
corresponding author, upon reasonable request.

## Acknowledgements

We thank the referee C. Struck for his useful comments that improved this
paper. We acknowledge DGTIC-UNAM for providing HPC resources on the Cluster
Supercomputer Miztli. LGO and LMM acknowledge support from PAPIIT IA101520
grant. LGO acknowledge support from CONACyT scholarship.

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