# Differences in star formation activity between tidally triggered and
isolated bars: a case study of NGC 4303 and NGC 3627

Elizabeth J. Iles,1 Alex R. Pettitt1,2 and Takashi Okamoto1
1Department of Physics, Faculty of Science, Hokkaido University, Sapporo
060-0810, Japan
2Department of Physics and Astronomy, California State University, Sacramento,
6000 J Street, Sacramento, CA 95819-6041, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Galactic bars are important drivers of galactic evolution, and yet how they
impact the interstellar medium and correspondingly star formation, remains
unclear. We present simulation results for two barred galaxies with different
formation mechanisms, bars formed in isolation or via a tidal interaction, to
consider the spatially and temporally varying trends of star formation. We
focus on the early ($<1$Gyr) epoch of bar formation so that the interaction is
clearly identifiable. The nearby NGC 4303 (isolated) and NGC 3627 (interaction
history) are selected as observational analogues to tailor these simulations.
Regardless of formation mechanism, both models show similar internal dynamical
features, although the interaction appears to promote bar-arm disconnection in
the outer disc velocity structure. Both bars trigger similar boosts in star
formation (79%; 66%), while the interaction also triggers an earlier 31%
burst. Significant morphological dependence is observed in the relation
between surface gas and star formation rate. In both cases, the bar component
is notably steepest; the arm is similar to the overall disc average; and the
inter-arm clearly the shallowest. A distinguishable feature of the tidal disc
is the presence of moderately dense, inefficiently star forming gas mostly
confined to tidal debris outside the optical disc. The tidal disc also
exhibits a unique trend of radially increasing star formation efficiency and a
clear dearth of star formation which persists along the bar between the centre
and bar ends. These are potential signatures for identifying a barred system
post-interaction.

###### keywords:

galaxies: bar – galaxies: interactions – stars: formation – galaxies:
kinematics and dynamics – ISM: structure – methods: numerical.

††pubyear: 2021††pagerange: Differences in star formation activity between
tidally triggered and isolated bars: a case study of NGC 4303 and NGC 3627–A

##  1 Introduction

Understanding the formation and evolution of galaxies is important for
determining the properties and processes that govern physics in the Universe.
In modern astrophysics, barred-spiral galaxies in particular are of prime
interest due in part to the likely similarity in morphology with the Milky
Way. In addition, such galaxies have been shown to make up a large portion of
all observed spiral galaxies, although the exact fraction ($\sim 25$–$75$%)
depends heavily on bar classification criteria (Schinnerer et al., 2002;
Aguerri et al., 2009; Masters et al., 2011). It has been shown that a range of
formation mechanisms can produce such a bar-like feature in the central region
of galaxies. For example, it is possible for bars to form in kinematically
cold, sufficiently massive, isolated stellar discs (Hohl, 1971; Ostriker &
Peebles, 1973). Under this regime, an initial instability drives a fast growth
phase where the bar emerges and subsequently buckles out of the disc-plane,
before slowing down and growing gradually in a phase of secular evolution
(Raha et al., 1991; Sellwood, 2014).

Under the $\Lambda_{\rm CDM}$ cosmology, wherein structure forms
hierarchically, morphology can be influenced by the associated galaxy-galaxy
interaction (Miwa & Noguchi, 1998; Lang et al., 2014; Romano-Díaz et al.,
2008; Pettitt & Wadsley, 2018). Historically, it was shown that these
interactions can induce bar formation in isolated, bar-free galaxies or slow
the formation of a bar in an isolated disc galaxy that was already likely to
form a bar; although, this was likely to have negligible effect on the
properties of the bar itself (Noguchi, 1987; Salo, 1991). Conversely,
subsequent studies have proved that some interaction conditions can instead
dampen or completely halt bar formation, rather than induce it (Athanassoula,
2002; Kyziropoulos et al., 2016; Gajda et al., 2017; Moetazedian et al., 2017;
Zana et al., 2019); and that bars which form in interacting systems instead
appear to rotate slower than those formed in isolation (Miwa & Noguchi, 1998;
Martinez-Valpuesta et al., 2017; Łokas, 2018). It is also argued that the
presence of a bar must have some environmental dependence, which could imply a
subsequent co-dependence for bar formation on interaction (Méndez-Abreu et
al., 2012; Skibba et al., 2012; Pettitt & Wadsley, 2018).

Regardless of which factors influence the formation of the central bar, it is
widely accepted that the presence of such a feature has significant effect on
a range of properties within the host galaxy (Sheth et al., 2002; Fujii et
al., 2018; Zana et al., 2019). For example, star formation rates for regions
within morphological features, such as the bar and arms, are consistently
observed to differ considerably, often showing significantly lower star
forming efficiency in the bar compared to other areas of the disc (Downes et
al., 1996; Sheth et al., 2002; Watanabe et al., 2019). This variation is often
attributed to the non-circular motions of stars and gas in the bar, usually
induced by a non-axisymmetric bar potential that generates strong shock or
shear motions along the bar and disrupts the bar-located molecular clouds,
thus suppressing star formation in that region (Roberts et al., 1979;
Athanassoula, 1992; Schinnerer et al., 2002; Dobbs et al., 2014; Beuther et
al., 2018). As such, barred-spiral galaxies are often considered when probing
the relationship between star formation on small molecular cloud scales
(10–100 pc) and galactic dynamics or gas kinematics at $\sim$kpc scales (Kuno
et al., 2000; Warren et al., 2010; Watanabe et al., 2011; Hirota et al.,
2014).

While the presence of a bar clearly affects the star formation properties
within a single galaxy, interactions between galaxies have also been shown to
strongly affect the gas dynamics and gas fractions of each involved galaxy.
This will consequently also affect the star formation properties of these
galaxies. One such effect is the so-called starbursts – accelerated star
formation triggered by the inflowing gas provided through the interaction
forces (Mihos & Hernquist, 1994; Barnes & Hernquist, 1996; Hopkins et al.,
2008). These driven starburst events are mostly observed in the central region
of the galaxy; however, there is evidence to suggest that this effect is not
limited to the central region but also observed well into the disc of the
galaxy, or even only in the disc in some cases (Di Matteo et al., 2007;
Bournaud, 2011; Pettitt et al., 2017). Interactions in the form of minor
mergers, also appear to significantly impact star formation statistics at low
redshift (Darg et al., 2010; Kaviraj, 2014; Taylor et al., 2017). Such
interaction driven effects on the star formation properties of galaxies are
not necessarily distinguishable from similar effects which can be attributed
to bars in isolated, unperturbed galaxies. Hence, tracers of the mechanisms
which drive and impact bar formation, such as tidally driven bars, may not be
entirely separable from those galactic properties which are influenced by the
presence of the bar itself.

Increasingly detailed, high resolution, large-scale observational surveys
continue to improve our capability to analyse star formation rates and
efficiencies, as well as document the wide range of observed morphological
features in galaxies –i.e. the COMING–CO multi-line imaging of nearby galaxies
(Sorai et al., 2019), PHANGS—Physics at High Angular resolution in Nearby
GalaxieS (Lee et al., 2021) survey results utilised in the following sections.
However, regardless of the breadth and depth of modern observational
capabilities, it is simply impossible for observational surveys to capture
long term evolution of an individual galaxy.

High resolution numerical simulations have long been identified as important
in this area, as these provide the capability to track changes in properties
over long time-scales and at varying length scales. The formative work of
Toomre & Toomre (1972) and their contemporaries have cemented N-body
simulations as an indisputably effective tool for determining precise
mechanisms and kinematics which drive the formation of structure in observed
galaxies. Since then, due to constant scientific and technological
advancement, it has become possible to produce simulations which can even
replicate the evolution of a galaxy under self-gravitation at a resolution
comparable to a true number of stars (Fujii et al., 2018). The mechanisms
which may affect bar and arm features have been studied extensively in this
manner, including but not limited to internal properties of individual
galaxies, such as mass-fractions and internal forces (Hohl, 1971; Ostriker &
Peebles, 1973; Friedli & Martinet, 1993; Baba, 2015; Wu & Jiang, 2015;
Sellwood & Gerhard, 2020), and all manner of interactions constraining size-
ratios, orbits and internal structures (Noguchi, 1987; Elmegreen et al., 1991;
Fiacconi et al., 2012; Dobbs, 2013; Oh et al., 2015; Pettitt & Wadsley, 2018).

Numerical methods are additionally indispensable in the effort to completely
resolve star formation traits and properties of galaxies (Di Matteo et al.,
2007; Springel, 2010; Federrath & Klessen, 2012; Hopkins et al., 2013; Okamoto
et al., 2015). The advent of detailed numerical studies of galactic discs has
made significant advancements to the global understanding of the impact that
specific disc features, such as the bar and spiral arms, may have on star
formation processes at a range of scales from molecular clouds to region
averaged star formation rates and efficiencies (Bournaud et al., 2010; Renaud
et al., 2013; Cole et al., 2014; Fujimoto et al., 2016; Baba & Kawata, 2020).
For example, through simulation, shear in different disc environments has been
shown to affect cloud formation conditions and cloud-cloud collisions,
producing differing likelihoods for the formation of massive stars or star
formation efficiency in different morphological features, such as the bar ends
or feathered and spurred offsets to the arms (Dobbs et al., 2006; Emsellem et
al., 2014; Renaud et al., 2015; Takahira et al., 2018). Additionally, it has
been possible to explore the limits of the so-called Kennicutt-Schmidt
relation (Schmidt, 1959; Kennicutt et al., 1987) in terms of both very high
spatial resolution over a complete galactic disc as well as morphologically
distinct features such as bars and arms (Fujimoto et al., 2014). However, in
most previous numerical studies of star formation in disc galaxies, the focus
has been predominantly on either the differing affects inherent to various
observed morphological features (Bournaud et al., 2010; Fujimoto et al., 2014;
Renaud et al., 2015; Baba & Kawata, 2020) or the influences of the external
drivers governing the formation of such features (Tan, 2000; Di Matteo et al.,
2007; Inoue & Fukui, 2013; Pettitt & Wadsley, 2018). Few have attempted to
look at whether different origins for morphological features, such as the
galactic bars studied herein, may impact the affect of these features on the
star-forming ISM.

In this work, we employ numerical hydrodynamics to produce simulations for two
specific case studies: a bar formed in isolation by random perturbation and a
bar triggered by the tidal forces of an interaction. These case studies are
determined to be consistent with real, observed target galaxies: NGC 4303
(M61), noting it has been determined to be an isolated disc galaxy on the
outskirts of the Virgo cluster with no HI gas depletion (Yajima et al., 2019);
and NGC 3627, a counterpart determined to have obvious traces of past
interaction through asymmetry and distorted HI gas (Haynes et al., 1979). A
brief summary of the observationally determined properties of these targets is
presented in Section 2. Section 3 outlines the simulation initial conditions
and methods. The relevant findings are presented in Section 4, specifically
those pertaining to general galaxy attributes, star formation features and the
evolution of morphologically dependent properties. This is accompanied by a
discussion of the relevant scientific implications for such results in Section
5. Conclusions and avenues for future work are summarised in Section 6.

##  2 Targets

These target galaxies were selected primarily due to the wealth of available
observational survey data and results, both previous and ongoing. They have
long been determined to be of barred-spiral type with relatively face-on
inclinations and local distances, conducive for taking relatively detailed
observational measurements. Thus, observations of these nearby galaxies were
used to situate the following simulations into a context which is physically
relevant.

NGC 4303 is a nearby barred-spiral galaxy which is likely to be associated
with the Virgo cluster of galaxies (Binggeli et al., 1985; Ferrarese et al.,
1996). Due to the relatively straightforward geometry (almost completely face-
on to the line-of-sight), the active galactic nuclei of LINER/Seyfert 2 type
(Ho et al., 1997), and potential double bar, NGC 4303 has been the target of a
number of observational studies, particularly focused on the central region
and constraining its dynamics or morphology (Sofue, 1997; Helfer et al., 2003;
Kuno et al., 2007; Momose et al., 2010). It has been determined to be an Sbc-
AB type galaxy with an approximate distance of 16.1–17.6 Mpc with an
inclination angle of 25.0 degrees (Schinnerer et al., 2002; Utomo et al.,
2018). The observationally derived rotation curve is generally flat with
deviation only in the very central region where it peaks sharply (Guhathakurta
et al., 1988; Sofue, 1997; Lang et al., 2020). Egusa et al. (2009) attempted
to determine a pattern speed, $\Omega_{P}$ , from CO–H$\alpha$ offsets but,
due to insufficient measurements, the uncertainty dominates ($\Omega_{P}\sim
24\pm 29$ km s-1 kpc-1). Other studies, both through observation and
simulation, have also sought to derive pattern speeds for the central bar
region, although these are considerably inconsistent based on the
determination of what radius constitutes this region, ranging from $R_{\rm
centre}=2.8\sim 9.2$ kpc (Colina & Wada, 2000; Rautiainen et al., 2005; Egusa
et al., 2009).

Utomo et al. (2018) derive the stellar mass as $\sim 7.943\times 10^{10}$ M⊙
and SFR of 5.248 M⊙ yr-1 from CO measurements at 120 pc resolution. NGC4303
also has an estimated average time for massive star formation from molecular
clouds in the spiral arms ($t_{\rm SF}$) of 10.8 $\pm$ 5.7 Myr (Egusa et al.,
2009) which is consistent with the age of young clusters in H$\alpha$
determined by Koda & Sofue (2006) to be approximately 10 Myr. The gas mass
derived was $5.3\times 10^{9}$ M⊙ with average surface density of 36 M⊙ pc-2
across $160^{\prime\prime}$ ($\sim 12$ kpc) in the molecular disc (Momose et
al., 2010). From this, NGC 4303 is determined to have an overall average SFR
surface density of $8.3\times 10^{-2}$ M⊙ yr-1 pc-2 with the bar contributing
$7.6\times 10^{-2}$ M⊙ yr-1 pc-2, approximately 10% lower than the overall
disc average, and arms $9.8\times 10^{-2}$ M⊙ yr-1 pc-2, approximately 10%
higher than the overall disc average (Momose et al., 2010). Subsequent studies
confirm this observation that NGC 4303 shows a lower SFE in the bar than in
the spiral arms (Yajima et al., 2019; Muraoka et al., 2019). In general, when
compared to the distribution of galaxies assessed by Kennicutt (1998), NGC
4303 appears to have higher than average star formation activity (factor of
$\sim 5$) according to observations (Momose et al., 2010; Yajima et al.,
2019).

Comparatively, NGC 3627 is a barred-spiral galaxy of type SABb according to
the Third Reference Catalogue of Bright Galaxies (de Vaucouleurs et al.,
1991). Like NGC 4303, it has an active nucleus of the LINER/Seyfert 2 type AGN
(Ho et al., 1997). It is widely considered to have experienced some
interaction with nearby NGC 3623 and NGC 3628 at some point in the
evolutionary history of both galaxies due in particular to the slight
asymmetry in arm structure observable at optical wavelengths and a distorted
HI gas distribution (Haynes et al., 1979). Additionally, significant
distortion is also evident in the disc in polarised maps soft X-ray emission
(0.2–1 keV), as well as a large asymmetry in hot gas temperature on the bar
edges which implies a recent collision with a dwarf companion galaxy
(Wezgowiec et al., 2012). The inclination angle of approximately 60 degrees
has made NGC 3627 a relatively popular target for CO mapping and the study of
molecular gas properties, as well as star formation activity (Reuter et al.,
1996; Helfer et al., 2003; Kuno et al., 2007). Hirota et al. (2009) determined
it to have pattern speed of $\Omega_{P}=39$ kms-1kpc-1 from measurements of
12CO (1-0) data and the method prescribed by Kuno et al. (2000). Law et al.
(2018) estimate NGC 3627 to have a total dynamical mass of $4.94\pm 0.7\times
10^{10}$ M⊙ at a galactocentric radius of $\sim 6.2$ kpc (121′′) from CO (2-1)
emission measurements.

Additionally, there is also postulated to be a correlation between the kinetic
temperature of gas in NGC 3627 and star formation efficiency (Law et al.,
2018). Considering the relationship between molecular gas and star formation
activity, NGC 3627 shows significant differences in the SFR from region to
region but the bar itself has very low observable star formation efficiency
(Watanabe et al., 2011). The bar ends, however, show the most intense star
formation of any region with SFE for the three regions Spiral Arm, Bar-End,
and the Nuclear Region determined by Watanabe et al. (2019) to be $1.3\pm
0.4$, $5.7\pm 1.7$ and $1.8\pm 1.0\times 10^{-9}$ yr-1 respectively. Despite
this, the chemical composition in these regions appears to be quite similar,
indicating that the characteristic chemistry on the observed scale is
generally insensitive to physical conditions and local effects, such as the
star formation rate (Watanabe et al., 2019). Many studies of NGC 3627 are
concerned by this significant variation between regions, particularly at the
bar ends and have endeavoured to attribute physical effects to such results
(Watanabe et al., 2011; Casasola et al., 2011; Law et al., 2018; Beuther et
al., 2018; Watanabe et al., 2019).

Figure 1: Rotation curves from initial conditions set for each case. IsoB is
identified by blue lines and TideB by orange. The different component
contributions are denoted by various line styles (solid–total, dashed–dark,
dot-dashed–stellar, dotted–gas). An example of observational measurements for
the two targets used to constrain appropriate initial conditions is also
represented by triangle points from the PHANGS survey data (Lang et al.,
2020).

##  3 Methods

###  3.1 Initial Conditions

To compare the effect of different formation mechanisms on the properties of
barred-spiral galaxies, initial conditions which would evolve to form a bar
feature were developed for the two case studies: the first, which relied on an
initial instability within the disc to trigger bar formation; and the second,
which instead required the external impetus of a passing companion. These
initial conditions were also constrained to specifically align with surface
density profiles and kinematic data of the observed target galaxies (NGC 4303,
NGC 3627).

To develop an isolated disc with a bar (hereafter IsoB), the galic package of
Yurin & Springel (2014) was used to produce an initial condition which would
evolve into a barred-spiral galaxy without external influence. Defining
initial parameters with galic, a spherical dark halo and stellar bulge were
characterised by a Hernquist profile with an axisymmetric velocity structure
of distribution function $f(E,L_{z})$ and a specified net rotation value. A
stellar disc component was defined to be a thin disc of thickness 0.2 times
the radial scale length. An axisymmetric velocity structure was also enforced
in the disc with a distribution function of the form $f(E,Lz,I_{3})$ and a set
disc dispersion ratio between the radial and vertical velocities ($\langle
v_{z}\rangle^{2}/\langle v_{r}\rangle^{2}$) was enforced, as well as a net
rotation value. Within these general descriptions, it was possible to refine
an initial condition which produced a disc galaxy physically comparable to the
desired target by simply modifying initial stellar mass ratios ($m{{}_{\rm
disc}},m{{}_{\rm bulge}}$) and the disc velocity dispersion. Through an
iteration of test initial conditions evolved at low resolution, these
parameters were selected to be $m{{}_{\rm disc}}=0.065,m{{}_{\rm
bulge}}=0.01$, as a fraction of total mass; and a disc dispersion ratio of
$\langle v_{z}\rangle^{2}/\langle v_{r}\rangle^{2}=1.5$.

Table 1: Initial component masses and scale lengths for the two models IsoB and TideB. Masses are given in units of $10^{10}$ M⊙ while distances are reported in kpc. The distance parameter for the companion is defined by the closest approach ($b$). |  $M_{{}_{\rm gas}}$ |  $M_{*_{\rm disc}}$ |  $M_{*_{\rm bulge}}$ |  $M_{{}_{\rm halo}}$ |  $M_{{}_{\rm companion}}$
---|---|---|---|---|---
IsoB | 0.522 | 2.611 | 0.402 | 37.16 | -
TideB | 0.759 | 2.441 | 0.072 | 42.57 | 2.401
| $a_{{}_{\rm gas}}$ | $a_{*_{\rm disc}}$ | $a_{*_{\rm bulge}}$ | $a_{{}_{\rm halo}}$ | $b_{{}_{\rm companion}}$
IsoB | 3.090 | 2.060 | 2.057 | 20.57 | -
TideB | 3.705 | 2.470 | 0.405 | 20.26 | 10

The appropriate parameters for these tests were selected by matching the
presence of morphological features, such as the presence of a distinguishable
bar, with an elementary assessment of the galaxy rotation properties. The
rotation curve for a range of appropriate initial conditions was compared with
observed rotation curves for the target isolated galaxy NGC 4303 (Guhathakurta
et al., 1988; Sofue, 1997; Yajima et al., 2019; Lang et al., 2020). The
rotation curve of the initial conditions used for each final case is shown in
Figure 1. These are decomposed by component and are shown alongside an example
of the observationally measured values of Lang et al. (2020). In both cases,
as this figure shows only the initial condition rotation curves, these
profiles are generally flat, dominated by the stellar component within the
disc radius and dark component outside of this. The overall shape is
considered consistent with the observation at first order before developing
the small scale features, such as the peaks and wiggles seen in real galaxies
as they evolve. The minimum resolution to ascertain the quality of these
initial conditions was set at a threshold of $10^{6}$ disc particles. This
limit has been shown to be a requirement in order to properly capture the
spiral and bar features in similar N-body galaxy simulations (Fujii et al.,
2011).

For the purpose of studying star formation, a gas component is also required
in the initial condition, however this is not included with galic
determination. To introduce a gas disc, additional particles were subsequently
added. This component was comprised of the existing stellar disc particles
produced by galic, rotated in the disc plane 180 degrees out of alignment with
their stellar counterparts and expanded to 1.5 times the stellar radius. The
gas particle mass was also modified to be comparable with the observed gas-
mass fraction and derived values for the total gas mass of the target galaxy
from observations. This component is singular and does not differentiate
between different kinds of gas contributions, for example, molecular or atomic
gas components. The complete initial condition for the isolated disc case can
be described in terms of a combined array of mass and scale length parameters
which are listed in Table 1. With $5\times 10^{6}$ gas particles, the mass
resolution of this model is 1044 M⊙. This resolution is sufficient to assess
the coarse-scale properties of GMCs, though as we lack a treatment of
molecular chemistry we defer such analysis to future works with a more
detailed stellar physics and chemical prescription. An assessment of the
likelihood for this disc to form a bar was also performed in the form of
Efstathiou et al. (1982) where;

$\epsilon_{\mbox{bar}}=\frac{V_{\mbox{max}}}{\sqrt{GM_{d}/a_{d}}}$ (1)

Here, $G$ is the gravitational constant and $V_{\mbox{max}}$ the maximum
rotation velocity. The IsoB disc produces a value of
$\epsilon_{\mbox{bar}}\sim 0.8$ which is an indication that this disc will be
distinctly unstable to bar formation.

For the second case, a bar triggered by a tidal interaction (hereafter TideB),
we use a modified version of the Rise-S10 simulation of Pettitt & Wadsley
(2018). While Pettitt & Wadsley (2018) study a number of tidal interactions,
Rise-S10 is the clearest case of a bar being tidally induced (e.g. the Mid and
Fall models tended to still form bars in isolation, the interaction merely
modified the bar structures). The original Rise-S10 initial condition was
appropriately rescaled in both mass and scale length to be comparable to both
the isolated initial condition and observational values for the interacting
target. Briefly, this corresponds to scaling by factors 0.5, 0.4 and 2 for
distances, stellar mass and gas mass respectively. Similar to the IsoB case,
this initial condition was constrained to align with surface density profiles
of the observed target galaxy. The specific orbit of the companion is not
intentionally constrained to any specific observational values of the target
galaxy or its neighbours. In this case, the orbit is as per the original
system in Pettitt & Wadsley (2018) and is defined purely to trigger an
interaction which will drive the desired bar formation. The TideB initial
rotation curve decomposed into components can also be seen in Figure 1.
Similarly, the rotation curve is generally flat but is notably less dominated
by stellar mass in the central region than the IsoB set of conditions. The
maximum rotation speed is also marginally higher between the two cases. As
this is the initial condition rotation curve, it is as yet unperturbed by the
companion and so is expected to differ more significantly from the observed
rotation curve particularly in the central region. After the interaction, this
curve undergoes a boost in rotation speed, particularly in the central region.
The specific evolution of the rotation curve for both cases is presented in
Appendix A.

The same mass and scale length parameters are used to describe the initial
conditions for the tidally affected disc, including the companion (see Table
1). In this case, the gas mass resolution is 1084 M⊙ which is comparable to
the isolated model ($\sim$ 1000 M⊙). The Efstathiou et al. (1982) metric for
bar stability was also calculated for this disc without influence from the
companion to produce an $\epsilon_{{}_{\mbox{bar}}}\sim 1.0$ which is
significantly greater than the isolated disc, indicating that this model would
otherwise be relatively more stable against bar formation. Thus, two initial
conditions are produced corresponding to an unstable isolated disc and a
tidally driven disc which should each generate a morphologically distinct
central bar feature.

###  3.2 Simulation Parameters

Figure 2: A face-on representation of stellar (top) and gas (bottom)
populations, setting the disc plane into the $xy$-plane for both the IsoB and
TideB cases. disc rotation is counter-clockwise and the columns step in time
by 100 Myr (one time pre-bar and three post bar formation) showing
morphological evolution. The companion can also be seen in passing at the far
left of the first TideB window in the sterllar map (100 Myr).

The subsequent simulation process was performed using the gasoline2 smoothed
particle hydrodynamics (SPH) code (Wadsley et al., 2004, 2017). We use the
standard hydrodynamical treatment advocated by Wadsley et al. (2017): using
200 SPH nearest neighbours and the Wendland C4 kernel (Dehnen & Aly, 2012).
Gravitational softening lengths of $[0.1,0.05,0.01]$ kpc were used for halo,
stars and gas respectively in both simulations. The star formation conditions
were prescribed by upper temperature and density thresholds of 100 atoms/cc
(minimum density for forming stars) and 300 K (temperature limit for forming
stars), respectively. The star forming efficiency was defined by a value of
$C_{*}=0.1$ ($\sim 10\%$ efficiency) following the standard gasoline2 sub-grid
prescriptions (Katz et al., 1996; Stinson et al., 2006); with a Chabrier
(2003) IMF and a convergent flow requirement for star formation. These values
are similar to previous studies (Saitoh et al., 2008; Tasker & Bryan, 2008;
Pettitt et al., 2017). The effects of UV and photoelectric heating, as well as
metal cooling were accounted for with a tabulated cooling function (Shen et
al., 2010). From an initial isothermal gas ($10^{4}$ K), this cooling produces
a two-phase thermal profile comparable to the ISM (Wolfire et al., 2003).
Stellar feedback is implemented using the Keller et al. (2014) supernova
super-bubble feedback with $10\%$ feedback efficiency. This feedback is
generated from clusters of young stars instead of individual supernovae and
appears more efficient at describing gas motion, regulating star formation and
producing the expected strong outflows (Keller et al., 2014). Both models were
integrated until it was possible to identify a clear bar feature in each case.
The integration was then allowed to progress for a further period of $\sim
200$ Myr (approximately another full galactic rotation) before undergoing the
analysis presented in the following sections. This corresponds to a period of
evolution where the effects of the tidal perturbation are most prominent in
the interacting case, thus highlighting any fundamental differences which
arise and may be particularly attributed to the variation between these two
types of bar formation mechanisms.

##  4 Results

###  4.1 General Results

####  4.1.1 Morphological Structures

We first present the general morphological features of our two models for bar
formation. Figure 2 shows a face-on view of the stellar and gas populations
with the disc plane set onto the $xy$-plane for both the IsoB and TideB cases.
The colour weighting is held constant over both cases. In this figure, the top
row of each case corresponds to the stellar component and the second row to
the gas component. The columns correspond to steps of 100 Myr, commencing one
period immediately preceding a clearly identifiable bar forming. Both cases
produce an essentially two-armed structure with a central bar, although there
are obvious visual differences within this general morphology.

The IsoB case, which formed a bar through some initial disc instability, seems
to evolve from a more flocculent type galaxy into a two-armed structure as the
bar grows. Conversely, the TideB case is strongly affected by the interaction
occurring at approximately 100 Myr (closest approach $\sim 94$ Myr), producing
clear, crisp two-arm features. These features however, are significantly
impacted by gravitational effects, both from the inner disc and the external
companion, constantly strengthening and decoupling over the period of
interest. As the simulation evolves, the bar forms and the disc stabilises.
Due to the decoupling of the arms in the tidally driven case the bar length
intermittently appears to extend. This may also affect an assessment of bar
orientation.

Additionally, it is possible to see that the tidally driven case produces a
pronounced bar feature significantly faster than the isolated disc. For
convenience, we present all subsequent results in terms of a shifted time
scale ($t^{\prime}$), relative to the bar formation time (rounded to the
nearest 100 Myr) such that $t^{\prime}=0$ at the approximate period of
distinct bar formation. This revised time of $t^{\prime}=0$ corresponds to the
$t=400$ Myr snapshot of IsoB and the $t=200$ Myr snapshot for TideB which can
be seen in Figure 2. At a given evolutionary period (within each column of
Figure 2) when the bar is well developed, the large-scale structural features
of each case do not appear substantially different. The bar lengths and
strength appear visually similar within each disc. The number and prominence
of the arms, as well as their pitch angles, are also similar.

Figure 3 shows the stellar surface density profile for each case, both along
the bar (major axis) and perpendicular to it (minor axis). Each model exhibits
very similar profiles both parallel and perpendicular to the bar orientation.
Stellar surface densities within the central region clearly decrease
exponentially and in a manner which is noticeably steeper than the surrounding
regions that are solely composed of the non-barred portion of the disc. This
sharply exponential shape is noticeably prevalent in a much narrower section
of the bar minor axis as would be expected from the elliptical shape of the
bar morphology that can be seen in Figure 2.

Figure 3: Stellar surface density profiles along the major (left) and minor
(right) axis of the bar for each case in the time periods where the bar is
well defined ($t^{\prime}=100,200$ Myr). The region shaded in grey indicates
the bar extent calculated as in Section 4.2.2 (IsoB: $R_{\rm bar}=2.10$ kpc;
TideB: $R_{\rm bar}=3.12$ kpc).

Both cases can be categorically described as hosting very similar bars of
exponential (late) type, rather than the alternative flat (early) type
(Elmegreen & Elmegreen, 1985). Both NGC 4303 and NGC 3627 have been determined
to have later type Hubble classifications (de Vaucouleurs et al., 1991).
Hence, this is in line with expectations for the target galaxies as well as
the exponential disc profile used to produce the initial simulation
conditions. It may thus be naively expected that these two bars should
generally impact the disc structure in a similar manner.

####  4.1.2 Velocity Structure

Non-axisymmetric morphological features are known to act as perturbations to
the galactic velocity field (e.g. de Blok et al. 2008). The two target systems
in particular are known to show clear non-circular motions, both from moment
maps and undulations in the rotation curves (Law et al., 2018; Schinnerer et
al., 2002).

Figure 4: The velocity of gas projected into the same $xy$\- (disc-aligned)
plane. For each case, radial (top) and tangential (bottom) velocities are
shown. The colour scale is set to be consistent over both cases to highlight
the difference in magnitude between the results. The columns show the
evolution through time at the same periods as Figure 2 denoted instead by the
$t^{\prime}$ scale.

As a first order metric of the non-circular velocities in these discs, we show
the azimuthal and radial streaming motions of the gas component in a face-on
projection in Figure 4 (also see Appendix A for the evolution of the rotation
curves). The top two rows show the IsoB disc velocity features evolving with
time, and the bottom two rows show the TideB disc. A significant difference
between the two cases is immediately obvious. It is clear that the tidal
forces in the interacting case increase the overall degree of non-axisymmetric
motion in the TideB disc. Both velocity components show clear asymmetries that
are strongest during closest approach ($t^{\prime}=-100$ Myr) which are
especially clear in the outer disc whereas the isolated case is relatively
unperturbed outside the bar region. These asymmetries correspond with a
migration of the system centre of mass, which has been corrected for in Figure
4. While these strong outer asymmetries in TideB do decay over time as the
companion moves away, they remain many times larger than those in the isolated
disc. The azimuthal response between the two arms of TideB is also asymmetric,
with the northern (southern) arm appearing to rotate somewhat faster (slower)
than the disc average, which will result in asysmmetries in galactic shear
experienced by gas in each arm region. Only the central regions of the IsoB
case seem to show similarly high non-circular velocities which are comparable
to the values that fill the entire disc regions of the post-interaction TideB
results.

A clear central quadrupole velocity signature is evident in the central region
of the radial velocity plots for both discs beyond $t^{\prime}\geq 0$ Myr.
This feature is indicative of a bar being present in the inner disc, as
material streams radially to follow elliptical orbits aligned with the bar
(e.g. Bovy et al. 2019). The bar kinematics appear strikingly similar between
the two models, also showing clear similarities in the azimuthal motion (lower
$V_{t}$ along the bar axis, with two “lobes" of high $V_{t}$ perpendicular to
the bar axis in the galactic centre, see also Renaud et al. 2015). The extent
of the quadrupole-like region acts a good visual approximation of the bar
extent. It can be seen that the radial size of the bar remains relatively
constant once it forms. By qualitatively defining the bar region (as outlined
in Section 4.2), it was determined that this approximation of the bar extent
from the velocity quadrupole was consistent with the values determined from a
Fourier method and that the bar extent does indeed remain generally constant
once it stabilises over the period of $t^{\prime}=200$ Myr, although the
expected bar shortening does become more significant over much longer time-
scales. This is also consistent with the assumptions drawn from Figure 2,
despite the clearly strong differences in the magnitude of non-axisymmetric
velocities between the two cases.

One particular point of comparison between the two models is how the velocity
features propagate radially. In the IsoB case the positive or negative radial
motion traces a single feature from the galactic centre to the outer disc,
following from the quadrupole to along the arms and outwards. In the TideB
case the feature is not so continuous, with the quadrupole radial velocity
features being engulfed by the strong radial motions exhibited by the tidal
arms. Such a signature is evidence of the rapid decoupling between the tidal
arms and inner bar of the TideB model. Bar and spiral decoupling is a known
phenomena in isolated discs (e.g. Sellwood & Sparke 1988; Baba 2015) but has
not been discussed in the context of tidal interactions where arms are driven
by an external agent. We will revisit this topic in Section 5.

###  4.2 Star Formation

Figure 5: A projection of surface star formation rate ($\Sigma_{\rm SFR}$)
into the same $xy$\- (disc-aligned) plane for the isolated (top) and tidally
driven (bottom) bar models. $\Sigma_{\rm SFR}$ is calculated by counting young
stars particles with ages $<\,10$ Myr and then binning over a 0.1 $\times$ 0.1
kpc grid.

####  4.2.1 Star Forming Morphology

We begin by formally constraining the morphology of active star forming
regions in the two cases as a measure of the number of new stars formed in a
given time period ($\Delta t_{\rm SF}$):

${\rm SFR}(t_{i})=\frac{1}{\Delta t_{\rm SF}}\sum_{t_{i}}^{t_{i}-\Delta t_{\rm
SF}}M_{*}$ (2)

Here, star formation rate (SFR) can be calculated at the current time
($t_{i}$) by summing new stellar mass ($M_{*}$) in the star forming period
($\Delta t_{\rm SF}$).This rate is then converted to a surface star formation
rate ($\Sigma_{\rm SFR}$) in order to be more comparable to the quantities
measured in observations by binning over a 0.1 $\times$ 0.1 kpc grid. The
look-back time for star formation is the age range $\Delta t_{\rm SF}=10$ Myr
which is comparable to the star forming time-scale determined observationally
for NGC 4303 (Koda & Sofue, 2006; Egusa et al., 2009). A projection of this
$\Sigma_{\rm SFR}$ is shown in Figure 5. This figure is similarly set into the
$xy$\- disc plane with columns advancing in time for the same time period as
the previous projections of the stellar, gas and velocity features (see
Figures 2, 4).

From Figure 5 it is possible to assess the location of current (or very
recent) star formation, as well as the intensity of such formation. For
example, the centre-most region in all the barred snapshots ($t^{\prime}\geq
0$ Myr) for both cases is dominated by an almost circular, bulge-centred
region with the highest star formation rate. This circular region expands as
the bar evolves in both cases, a likely by-product of gas inflow to this
central circumnuclear disc. However, while it is approximately the same size
in both cases in the $t^{\prime}=100$ Myr snapshot, by $t^{\prime}=200$ Myr
the circular region in the TideB case has become significantly larger than the
IsoB case, showing the TideB case has a less concentrated but no less intense
region of star formation at the centre. This central nuclear disc of star
formation is commonly seen in barred systems, where it is fueled by the inflow
of gas from the torques present in the bar region (Athanassoula, 1992; Wang et
al., 2012; Cole et al., 2014; Baba & Kawata, 2020). Interactions are also
known to be drivers of enhanced star formation (e.g. Patton et al. 2013; Pan
et al. 2019), with the interaction event itself driving inflows that help fuel
central star formation (Torrey et al., 2012; Pettitt et al., 2016). As such,
the combined effect of the bar evolution and the interaction provides an
increase in fueling of the central nuclear disc in TideB, and the larger
central star forming region.

The bar is also a clear feature in these SFE maps, with pockets of intense
star formation flaring along the bar and in clumps along the spiral arms.
However, star formation for the tidally driven bar appears to be predominantly
contained within the bar and arm features in all barred time periods
($t^{\prime}>0$). In contrast, each of the IsoB maps in Figure 5 show wide
areas of strong star formation throughout the disc which cannot be clearly
associated with either the bar or any obvious arm feature. This seems to
indicate the isolated IsoB case has a greater amount of inter-arm star
formation, whereas the tidally driven case concentrates gas more strongly into
the arms and core, limiting star formation outside of these areas.

Considering time dependent features, the shape of the SFR projection in the
bar region changes in both cases dramatically over a relatively short time-
scale ($\Delta t^{\prime}\approx 200$ Myr). Figure 5 shows distinctly
different star forming structures within the bar region, and that the
evolution process of this structure can be seen to differ between the two
cases. For example, in the $t^{\prime}=200$ Myr snapshot of the IsoB case,
very clear arms comprised of star forming regions can be seen to be somewhat
disconnected from a dense, almost triangular central region within the bar.
Comparatively, in the same snapshot for the TideB case, the central region is
also clumpy but is mostly connected to the arms along a straight central axis
of star formation that spans the full bar length. This axis is accompanied by
a curved envelope of similarly intense star formation encasing the central
axis from both above and below, which seemingly connects each arm to the
other. Additionally, the bar in the TideB model at $t^{\prime}=100$ Myr shows
a wide, long clump along the bar axis in the centre which tapers off toward
the middle of the bar, before once again intensifying into a thicker bar at
the end just before the arms connect. This is dissimilar to any of the other
snapshots for either case. These star forming structures vary broadly, even
between evolutionary periods in a single simulated case which make it
difficult to produce any sweeping, general statements to describe definitive
features. However, this difficulty itself supports the range of observations
that indicate star formation within galaxies seems to be consistently
inconsistent.

####  4.2.2 Star Forming History

To consider more specifically the time evolution of star formation in the
simulations, we take the disc-averaged star formation history for the two
simulations up to $t^{\prime}=200$ Myr and present Figure 6. The axis in this
figure is set in the original simulation time, and for each case a vertical
line denotes $t^{\prime}=0$. The solid blue line is the star formation history
for the isolated IsoB case, whereas the orange dotted line indicates the
history of the tidally driven case. The time of closest approach for the
companion in this case is at approximately 94 Myr.

Figure 6: Star formation history for the two cases for the evolutionary period
until $t^{\prime}=200$ Myr. The axis is set at original simulation time so,
for each case, a vertical line denotes $t^{\prime}=0$ Myr. The solid blue line
represents IsoB. The orange dotted line represents TideB. The large peak at
$t<100$ Myr is an artefact of the simulation process. Vertical dotted lines
approximately indicate bar formation times.

It is apparent that both cases seem to show relatively similar star formation
rates. There are also some similarities in the shape of the star formation
profile as it evolves. For example, there is evidence of an extended rise in
star formation at around the time the bar forms in both cases (indicated by
the vertical dotted lines). The relative shape (height and duration) of this
increase in both cases appears to be similar: IsoB $\sim\Delta\mbox{\rm
SFR}=3.08$ M⊙ yr-1, $\Delta t=145$ Myr; TideB $\sim\Delta\mbox{\rm SFR}=3.79$
M⊙ yr-1, $\Delta t=127$ Myr. The first obvious difference between the two
cases is in the earlier stages of evolution (pre-bar formation $t^{\prime}\leq
0$ Myr). While the IsoB case seems to have a relatively constant level of star
formation before the bar forms (after the initial boost on start-up), the
TideB case has an additional, earlier period with an extended rise and fall of
star formation intensity which is similar to the boost feature previously
linked to the bar formation. This occurs just after the closest approach of
the companion. Although this early boost is smaller in amplitude than the bar
effect, it appears to last for a similar duration before returning to what
seems to be the original baseline rate just before the advent of the bar
formation, where it steeply rises again. This can be considered to be a double
peaked trend of star formation history and is likely attributable to a
starburst triggered by the interaction. As the bar has not yet formed and the
density and SFR projections both show the two arm features are dominant in
that period, a logical assumption would be for this boost to be mostly limited
to increased star formation in the strong tidally induced arms, rather than
within the central region which later becomes the bar. The magnitude of the
bar-triggered boost in SFR is similar for both the isolated and perturbed disc
(79% and 66% respectively), while the interaction triggered burst is milder at
31%.

To study the morphological dependence of these star formation features in more
depth, three key morphological regions are defined to be the bar, arm and
inter-arm components. Using Fourier decomposition, it was possible to identify
analytically relevant traits of the bar and arm components. The inter-arm
region could then be defined as neither bar nor arm within a certain radius of
interest. The extent of the bar region was determined from the amplitude of
the Fourier $|A_{2}|$ mode. Identifying the radial extent of features in this
mode allowed for the bar component to be defined with a simple radial
criterion to form a central circular region encasing the full bar length.

Figure 7: The fraction of total young stellar mass produced by each region
(bar, arm, inter-arm) over the period of interest
($-100\,<\,t^{\prime}\,<\,200$ Myr) for each bar model. Each region is only
defined on steps of 10 Myr, corresponding to the previously assumed star
formation time. The bar is denoted by a blue line with crosses at each
calculation point; the arms by a red line with filled-in circles; and the
inter-arm with magenta line and open circles. The axis is set in the
$t^{\prime}$ scheme with a dotted vertical line at $t^{\prime}=0$ to indicate
the approximate time of bar formation.

As mentioned previously, it was found that the bar extent was relatively
constant throughout the considered duration, decreasing only a small fraction
by $t^{\prime}=200$ Myr. Hence, a single value was adopted for the bar extent
for the following analysis (IsoB: $R_{\rm bar}=2.10$ kpc; TideB: $R_{\rm
bar}=3.12$ kpc). From the population outside of this $R_{\rm bar}$, the arm
component was defined by fitting a log spiral of the form $r=a\exp{b\theta}$
with a standard width of $\pm 1$ kpc to the peaks of the polar angle $\theta$
set to lie in the disc plane at a given radius. The inter-arm region was
correspondingly defined as the remainder within a set radius of 10 kpc
determined to contain the majority of significant disc features for both
cases.

In Figure 7, a similar star formation history is plotted to Figure 6, however,
this is a measure of the fractional contribution from each region (bar, arm
and inter-arm) to the total number of young stars produced, shown over the
simulation time. These specific regions were defined and analysed at every 10
Myr, with a look-back time consistent with the previous star formation
analysis ($\Delta t_{\rm SF}=10$ Myr). The blue line with crosses at each
calculation point denotes the contribution from the defined bar region; the
filled-in red circles indicate the contribution from within the arm regions;
and finally, the open circles in magenta show the inter-arm contribution. The
axis is set in the $t^{\prime}$ scheme with a vertical line at $t^{\prime}=0$
to indicate the approximate time of bar formation.

Figure 8: The Kennicutt-Schmidt relation between surface star formation rate
($\Sigma_{\rm SFR}$) and surface gas mass density ($\Sigma_{\rm gas}$ ) for
both cases during the clearly barred periods $t^{\prime}=100,200$ Myr. These
surface densities are determined as in Figure 5 and smoothed with a spline
interpolation. The literature relations: Kennicutt (1998) orange dotted line;
Bigiel et al. (2008) cyan dot-dashed line; and Momose et al. (2010) purple
dashed line–are each shown for comparison.

In both cases, it is possible to see how star formation in the bar region
climbs steadily as the bar is formed and that the inter-arm region contributes
the least for the majority of times analysed. For the IsoB case, this inter-
arm contribution is consistently $\sim 10\%$ of the total stellar mass formed
and the fraction is even lower in the TideB case ($\sim 5\%$ or less once the
bar has formed). This is also a representation of the way that the star
formation seems to be more cleanly located in the apparent bar and arm
features in the TideB case, particularly as seen in Figure 5. Considering the
contributions from both bar and arm components however, there are a number of
very noticeable differences between the two cases. For example, for times when
$t^{\prime}\leq 0$ Myr, it can be seen that the arms in the TideB case show a
very different development of the fraction of stars produced. In the IsoB
case, the fraction of star formation over the disc features seems to be
initially similar for the first $\sim 50$ Myr ($t^{\prime}\leq-50$ Myr),
before becoming increasingly centrally dominated as the bar forms ($\sim
t^{\prime}=0$ Myr). Contrastingly, the highest period of intensity for star
formation in the TideB arms is very noticeably within $\pm\,50$ Myr either
side of the bar formation time ($-50\leq t^{\prime}\leq 50$ Myr). This
corresponds to the time in Figure 6 where the smaller peak of star formation
can be seen after closest approach and before the bar has completely formed.
Compared to the IsoB case, which has a steady baseline level of star formation
at that time, the interaction has clearly driven a burst of star formation
which is predominantly located in the arms. At later times in Figure 7 ($\sim
t^{\prime}>50$ Myr), a potential periodicity also appears in the contribution
from the bar region of the IsoB case which is not reflected in the bar of the
TideB case. The TideB case at this time shows a much smoother profile. While
overall star formation history may appear relatively indistinguishable once
the bar has formed, a number of significant differences seem to exist when
considering instead the contribution to the overall SFR from region to region.

Figure 9: As in Figure 8 but now decomposed by region with scattered data
points showing the bar (blue crosses), arm (red filled circles) and inter-arm
(magenta open circles) components. The Kennicutt (1998) relation is again
included for reference (dotted orange line), as well as power law fits for
each region: bar (blue solid line), arm (red dashed line) and inter-arm
(magenta dot-dashed line).

###  4.3 Star Formation Scaling Relations

To compare our measured star formation analytically with observations, as well
to consider these results in relation to the gas availability rather than
simply counting stars, we plot a range of the so-called Kennicutt-Schmidt
diagrams showing the relation between $\Sigma_{\rm SFR}$ and surface gas mass
density ($\Sigma_{\rm gas}$) (Schmidt, 1959; Kennicutt, 1998). These can be
seen in Figure 8 and Figure 9 matching the resolution used in Figure 5
(binning over a 0.1 $\times$ 0.1 kpc grid). For both simulated cases, Figure 8
shows this relation in a plot for each of the two clearly formed bar snapshots
($t^{\prime}=100,200$ Myr). These are compared with literature relations
derived from observational studies. The calculated data points in this
$\Sigma_{\rm SFR}$ vs. $\Sigma_{\rm gas}$ are shown as contours, whereas the
coloured lines are the literature relations: Kennicutt (1998) in orange
dotted; Bigiel et al. (2008) in cyan dot-dashed and Momose et al. (2010) in
purple dashed. Kennicutt (1998) use disc averaged values for many galaxies;
Momose et al. (2010) use only values from across NGC 4303; and Bigiel et al.
(2008) constrain their measurements to be ‘resolved’ star formation in many
galaxies.

The shape of the distribution is clearly different between the two cases. The
IsoB case has a narrow, arrowhead-like shape which seems to align relatively
well in orientation with the literature results. Comparatively, the TideB case
is much broader along the $\Sigma_{\rm gas}$ axis making more circular
distribution with less clear alignment. In TideB in general there are many
more components with high $\Sigma_{\rm gas}$ and low $\Sigma_{\rm SFR}$. This
indicates TideB appears to have noticeably more regions over the disc which
are relatively inefficient at forming stars, but can maintain higher gas
density. Both cases show there is clear evidence of evolution with time, in
terms of both shape and location of the distribution of simulated results with
respect to the literature results. This is particularly evident in the area of
the plot which corresponds to low efficiency, high gas density features
(around $\Sigma_{g}>10\,{\rm M_{\odot}pc^{-2}}$ and $\Sigma_{\rm
SFR}<10^{-9}\,{\rm M_{\odot}pc^{-2}yr^{-1}}$) which seem to become more
pronounced, spreading to higher gas densities for a given $\Sigma_{\rm SFR}$
value, as both cases evolve. However, while the TideB distribution in Figure 8
shows many more such high $\Sigma_{\rm gas}$ and low $\Sigma_{\rm SFR}$
results than IsoB, the significant difference between the two cases in this
region is conspicuously absent when these results are are deconstructed into
components of the three morphological regions of bar, arm and inter-arm (see
Figure 9). This indicates that many of these points in the TideB case were not
classified as either bar, arm or inter-arm through the classification scheme,
indicating the presence of some other component.

While these values trace star formation within the $10\times 10$ kpc field of
view that can be seen in the windows of the face-on galaxy maps, these points
must be outside the $R\,>\,10$ kpc radial limit used in the region
classification to define the extent of the optical disc. These results should
then be attributed to star formation occurring in high density gas which has
been stripped to the very outer edges of the disc or even into a trailing
tidal tail by the companion. This shows that the largest temporal changes in
the TideB case are likely to be a direct result of the interaction still
affecting the outer-reaches of the optical disc and beyond. Additionally, that
the definition of the disc limit may make significant impact on the shape of
the KS relation in tidally affected galaxies. In response, slightly changing
the outer radius of the disc for both cases was also tested to find that the
overall shape of the IsoB distribution is mostly unaffected while the TideB
distribution is clearly sensitive to the change. As we are predominantly
concerned with the visible morphological features within the disc (bar, arms,
inter-arm) this is not considered in great detail, however, studies of stellar
populations in the outer discs of galaxies or tidal features in interacting
systems may find such sensitivity particularly relevant.

In comparison, Figure 9 is similar to Figure 8 but the simulation points are
deconstructed into the components of bar, arm and inter-arm, as defined
previously. For the purpose of producing this relation, the definition of
morphological features is based on the $(x,y)$ positions determined previously
such that these regions in both gas and stellar components are spatially
consistent. These are presented through scattered points, identifiable with
the same symbols as in Figure 7 (bar with blue crosses, arms with filled red
circles, inter-arm with magenta open circles). The Kennicutt (1998) relation
is included again for reference as in Figure 8, now with a grey solid line.
This allows us to begin to understand how each region contributes to the
overall distribution, as well as to make an assessment of how the star
formation efficiency may differ from region to region. For example, in
observations of NGC 4303, Momose et al. (2010) found the overall star
formation efficiency to be higher than the average trend for all galaxies
derived by Kennicutt (1998), particularly in the arm regions. That is not
necessarily the case here, although a fraction of the arm results do show
values higher than the Kennicutt (1998) relation.

The most interesting result is that in this figure the bar in both cases has
the steepest slope, followed consistently by the arm and then inter-arm
regions. While visually this is clearer for some panels than others, with the
early IsoB data appearing to show similar slopes for the arm and bar, through
calculations of the gradients these trends are obvious. Table 2 shows the
$\alpha$ coefficient for a power law fit to the simulation data of the form
$\log\Sigma_{s}=\alpha\log\Sigma_{g}+\beta$ for each region component as well
as the total distribution over the disc ($R<10$ kpc as in Figure 9).
Considering these coefficients as an analogue for the linear gradient in the
log-log space of Figure 9, it can be confirmed that the bar region in both
cases certainly shows a significantly steeper relation than the arms, inter-
arm or even total distribution on average.

Table 2: Gradient coefficients for a power law fit to the previous $\Sigma_{\rm gas}$ vs. $\Sigma_{\rm SFR}$ data for the disc region of interest ($R<10$ kpc). This fit follows the form $\log\Sigma_{s}=\alpha\log\Sigma_{g}+\beta$, the $\alpha$ coefficient is listed. Model – $t^{\prime}$ | All | Bar | Arm | Inter-arm
---|---|---|---|---
IsoB-100 Myr | $2.7\pm 0.4$ | $3.0\pm 0.5$ | $2.5\pm 0.6$ | $1.3\pm 0.8$
IsoB-200 Myr | $2.3\pm 0.4$ | $3.1\pm 0.7$ | $2.2\pm 0.7$ | $1.0\pm 0.8$
TideB-100 Myr | $2.5\pm 0.4$ | $3.0\pm 0.9$ | $2.4\pm 0.7$ | $1.6\pm 0.9$
TideB-200 Myr | $2.4\pm 0.5$ | $3.2\pm 0.8$ | $2.4\pm 0.7$ | $1.7\pm 1.1$

The outlying point with highest $\Sigma_{\rm SFR}$ and $\Sigma_{\rm gas}$ can
be seen to always correspond to the bar component. In fact, this is the result
from the centre-most region which is defined in some studies as a fourth
morphological feature (a central nucleus) and is considered separate from bar
properties. For the subsequent analysis, care has been taken to test both
including and excluding this point in assessing the bar response. The results
included in Table 2 are for fits including the nucleus within the barred
region and excluding star formation at and beyond the defined disc limit of 10
kpc. However, the addition of the nucleus has been determined to make
negligible change with the bar slope without this extremely high value still
significantly steeper than all other results. The addition of tidal remnants
in the total average, however, saturates the overall fit to be more consistent
with the flatter inter-arm component rather than the arms. This is expected as
within the disc star formation in the arms spatially dominates the total star
formation, particularly in the TideB case as can be seen in Figure 5, whereas
the conditions in the outer regions are more consistent with the inter-arm
space. Hence, when these outer regions are included the influence of the arms
on the total average must be decreased.

Comparing these results with related studies, it is possible to see that the
range of results is relatively similar. In the case of the isolated result,
Momose et al. (2010) also find a similar arrowhead shape for NGC4303, however
it is possible to see from Figure 8 that the alignment of the distribution is
steeper overall than the relation observed by Momose et al. (2010) with the
IsoB result showing consistently lower values for star formation in the low
gas region of the plot, causing a steeper alignment of the arrow shape
overall. Additionally, when considering the regional separation, observational
results appear to indicate a preference for a significant portion of the arm
component to have distinguishably higher SFE than the other regions; with arm
values independently occupying the top arrow-edge and showing a wider spread
of values to higher SFR for a given gas value than any other regional
components. The bar population is similarly shown to have lower SFR than
either the arm or inter-arm component for higher gas surface densities. Such
separable features are not necessarily evident in the IsoB result, firstly as
arm and inter-arm are indistinguishable, and secondly because while the arm
and inter-arm regions show higher SFR than the bar for similar gas density
they also show lower values at the same gas value. In the IsoB results, the
bar component forms a thin distribution almost bisecting the larger spread of
the other regions. Comparatively, in the TideB case, the bar response is more
dispersed throughout the distribution, particularly in the second time period
($t^{\prime}=200$ Myr). This is in line with the results of Watanabe et al.
(2011) where the region of the KS plot occupied by bar results is also
predominantly interspersed with the other region.

However, it is important to note that Watanabe et al. (2011) do not
specifically define an arm and inter-arm region in their sample, leaving these
populations somewhat ambiguously in a region defined as ‘Other (mostly arms)’.
There is also a difference in the bar region definition as Watanabe et al.
(2011) define both a bar region plus a separate classification for the bar
ends which neither Momose et al. (2010) or this work have elected to
differentiate from the bar on the whole. It is very likely that the difference
between region classification has introduced a certain level of ambiguity into
the possible comparison between regionally specific trends between studies. In
addition, Onodera et al. (2010) have shown that resolution in observational
studies can have a non-trivial affect on the shape and orientation of the KS
relation even within a single target galaxy. By nature, simulation studies can
achieve a significantly higher resolution than most observational results.
Even within simulations, for example, Fujimoto et al. (2014) have shown that
the KS relation is sensitive to the star formation models used to prescribe
the physics of the simulation. The differences are particularly evident when
considering the regionally dependent relation with standard star formation as
prescribed by mass and free-fall time constraints, GMC turbulence as per
Krumholz & McKee (2005) and cloud-cloud collision as per Tan (2000) each
producing a different orientation of the same bar, arm and inter-arm regions
in $\Sigma_{\rm gas}$–$\Sigma_{\rm SFR}$ parameter space. The results
presented here are likely not immune to such constraints. For example, the
IsoB results in Figure 9 are similar to the standard response in Fujimoto et
al. (2014)–unsurprising considering the star formation recipe used in this
simulation–whereas, the observational results of Momose et al. (2010) appear
to be more similar to the cloud-cloud collision result of Fujimoto et al.
(2014). Hence, it is possible that some discrepancy between these results and
the observational results from both target galaxies may arise due to inherent
resolution and star formation conditions, as well as the region classification
method which is consistently inconsistent in the literature.

Considering instead the mean values for $\Sigma_{\rm{SFR}}$ and
$\Sigma_{\rm{gas}}$ for each region, the corresponding values show an evident
decrease from bar to arm to inter-arm which is interestingly consistent with
the distinct flattening of the slope for the
$\Sigma_{\rm{gas}}$–$\Sigma_{\rm{SFR}}$ relation (as in Figure 9, Table 2).
This is true in all time periods. The gas means also reflect decreasing values
per region in order of bar – arm – inter-arm but the magnitude of the
difference between each region is less in the gas surface density than the
star formation. These mean values for each of the defined regions are shown in
Table 3 for each model. However, in this classification the high
$\Sigma_{\rm{SFR}}$, high $\Sigma_{\rm{gas}}$ outlying point attributed to the
central nucleus is included within the bar component. If this is removed and
the averages for the originally defined regions are calculated instead as four
regions, it can be seen that this nucleus saturates the response of the bar.
Table 3 shows the original bar component with nucleus as Bar and the bar
without the centre-most region as Bar (/cen.). The Centre result is exactly
the points removed from the original bar component. Of the four regions, the
nucleus is considerably the highest by two orders of magnitude, however, the
bar without centre still produces a value significantly higher than the
corresponding arm region which is in turn greater than the mean inter-arm
values. This is also consistent with the related analysis of simulated discs
by Fujimoto et al. (2014) who similarly find in the disc decomposed by region
that the average of bar values are generally higher than the average arm
region and then the inter-arm region with varying degrees of separation
depending on the star formation model implemented.

Table 3: Mean values of gas and star formation surface density in morphologically distinct regions in each model. Region | $\bar{\Sigma}_{\rm gas}\,{[\rm M_{\odot}pc^{-2}]}$ | $\bar{\Sigma}_{\rm SFR}\,{[10^{-2}\rm M_{\odot}yr^{-1}pc^{-2}]}$ |
---|---|---|---
| IsoB | TideB | IsoB | TideB |
Arm | $12.9\pm 0.6$ | $15.3\pm 0.2$ | $1.01\pm 0.1$ | $1.14\pm 0.3$ |
Bar | $20.1\pm 1.1$ | $23.6\pm 5.4$ | $16.2\pm 5.8$ | $19.6\pm 12$ |
Centre | $86.1\pm 14$ | $143\pm 13$ | $212\pm 79$ | $410\pm 193$ |
Bar (/cen.) | $15.0\pm 0.1$ | $18.4\pm 4.3$ | $1.13\pm 0.1$ | $2.17\pm 1.7$ |
Inter-arm | $10.0\pm 0.5$ | $12.0\pm 1.4$ | $0.42\pm 0.1$ | $0.56\pm 0.3$ |

Comparing these results to observationally determined values for the average
SFR surface density, as calculated by Yajima et al. (2019) for NGC4303 and
Watanabe et al. (2011) for NGC3627, indicates that $\Sigma_{\rm{SFR}}$ for the
main features of bar (not including the nucleus) and arms are at least of a
similar order. The simulation values are all slightly lower than the
observational counterparts for each region. Comparing the relative values
between the two main morphological regions, Yajima et al. (2019) find that the
arms in NGC4303 have a higher $\Sigma_{\rm{SFR}}$ than the bar by $\sim$2%.
For NGC3627, Watanabe et al. (2011) indicate that the bar should have $\sim$4%
higher $\Sigma_{\rm{SFR}}$ than the other region (classified as ‘mostly arms’
so, possibly also including an inter-arm component). This is a very small
difference between the two regions in observations of both targets. However,
both these studies define and calculated $\Sigma_{\rm{SFR}}$ while separating
the bar ends to a 3rd significant region which has significantly higher value
than either the arms or bar. In this study, the bar ends are not specifically
separated and it is difficult to know to which component (bar or arms) in our
classification that this 3rd region has been drawn from to compare with the
observational studies.

Assuming the bar and bar ends together make a complete bar component,
averaging the bar and bar end results from observations (Watanabe et al.,
2011; Yajima et al., 2019) instead indicates the bar+bar end component of
NGC4303 to have a $\Sigma_{\rm{SFR}}$ $\sim 27$% more than the arms whereas
this is $\sim$71% for NGC3627 with the bar+bar end component being higher than
the arms in both cases. This resolves the difference between the IsoB
simulation showing lower star formation in the bar rather than the higher star
formation rates in the arms than the bar of NGC4303 (Momose et al., 2010;
Yajima et al., 2019). The simulation results also show a significantly greater
difference between the bar and arm components in the TideB result than IsoB,
similar to the corresponding targets, although the calculated values are also
lower than observations in this respect. The IsoB case has the bar (/cen.)
greater the arm by 11% whereas the TideB case has a difference of 47% between
the two regions. However, as mentioned previously, the spatial definition of
regions is often inconsistent between studies (both observational and
theoretical) and this inconsistency can give rise to significant differences
in the trends observed. So, in the following section we attempt to quantify
how the star formation properties differ spatially throughout the bar and disc
in a method independent of a specific regional classifications.

###  4.4 A Directional Approach

To further investigate the sub-kpc star formation features of these galactic
bars, we perform an analysis of the SFE within the inner regions of each disc.
We apply a directional approach to consider how star formation varies and
evolves with disc radius, as well as along the primary axis of the bar–a long
contested topic in the literature (e.g. Roberts et al. 1979; Muraoka et al.
2019). Star formation efficiencies are used as a metric specifically to
account for the interrelation between gas availability and star formation
activity, with SFE $=\Sigma_{\rm SFR}/\Sigma_{\rm gas}$ [Myr-1].

####  4.4.1 Radial Dependence of SFE

The radial dependence of SFE across the disc of each case is shown in Figure
10. Each radial bin is defined with a width of $\Delta R=1$ kpc and the SFR
and gas mass averaged over each annuli to calculate the SFE value in each bin.
This is determined every 10 Myr for each case over the barred period
($t^{\prime}>0$ Myr) with consistent star formation look-back time ($\Delta
t_{\rm SF}=10$ Myr). Saturation indicates time evolution with colour from dark
to light corresponding to increasing $t^{\prime}$. Additionally, the bin which
would coincide with the bar end is shaded in grey.

Figure 10: Azimuthally averaged SFE with radius for each bar model (IsoB top,
TideB bottom). The SFE across the disc is binned in 1 kpc steps of radius for
the barred period ($t^{\prime}>0$). The star formation time remains consistent
with previous analysis ($\Delta t_{\rm SF}=10$ Myr). For each radial bin, the
evolution of SFE can be seen with individually coloured bars corresponding to
the result at time steps of 20 Myr. The colour gradient shows time evolution,
with dark to light bins corresponding to forward time progression (over 200
Myr). The bin in which the bar extent would be located is shaded in grey.

Naturally, the centre-most bin shows the highest result at all times for both
cases of bar formation. This is in line with the noticeably high star forming
central region, or nucleus, which was identified in previous sections (see
Figure 5, 8, 9). Within each bin, there does not appear to be any immediately
obvious or consistent trend as the SFE varies with time in either case. It
neither continuously grows, declines, nor oscillates in any discernible
pattern. In considering the broader trend as SFE changes with radius across
all bins, there does appear to be a certain difference between the two cases.
Outside of the central bin, the average level of SFE in the disc appears to be
mostly constant in the IsoB case, whereas the TideB case shows more of an
increasing or sinusoidal shape with radius; showing lower SFE towards the
centre ($\sim R_{\rm bar}\pm 2$ kpc) and higher SFE towards the outer edge.
This may be evidence that the interaction has caused the inner regions to
suffer a depletion of gas compared to the relatively more stable conditions in
outer disc. However, these broad trends are not necessarily true at all times,
as the inner deficit is clearly not prevalent in the first 3-4 periods ($\sim
t^{\prime}<80$ Myr).

Figure 11: The time evolution of the SFE($R$) gradient as determined by linear
fit to the bins from Figure 10 plotted with $t^{\prime}$ time. For these fits,
the prevalent central bin containing contributions from the intensely star
forming nucleus region is excluded. The calculated gradient is marked by a
blue square for IsoB and an orange triangle for TideB. A positive value
indicates the overall SFE increases with radius while a negative value
indicates a decreasing SFE with radius.

To further understand any trends in this evolution of the radial dependence of
star formation efficiency, we consider the change of the gradient in Figure 10
with time for each case. To determine this gradient a linear least-squares fit
was applied for all histogram bins excluding the centre-most bin defining
$R<1$ kpc. The resulting first coefficient of this fit is plotted in Figure 11
showing the gradient change with time. A positive result indicates the SFE
increases overall with radius and, conversely, a negative result shows a
decreasing SFE with radius. In this figure blue squares correspond to the
calculated IsoB gradients, whereas TideB is denoted by orange triangles. It is
clear that there is significant variation in the SFE dependence with radius
over the evolution time for both cases. During the first 50-100 Myr after bar
formation, the trend for each case appears similar: a steep negative gradient
steadily begins to flatten and oscillate around zero, indicating average SFE
across the disc is generally constant at that time. However, while the
isolated case seems to remain mostly within $\pm 1$ Myr-1kpc-1 of zero, the
TideB results continue to climb, transitioning through zero to develop a
clearly positive gradient for the duration of the second 100 Myr and thus a
decreasing depletion time increasing with radius. In the last 50 Myr of the
period considered, the TideB gradient does reduce to values possibly
comparable to the steepest positive extent of the isolated case.There is one
period of the IsoB case in later times which seems noticeably inconsistent
with the other values. At $t^{\prime}=160$ Myr the gradient is almost the
steepest negative result of both cases after bar formation. It is unclear from
this analysis what kind of event caused such a strongly declining profile of
SFE just for this brief time. However, it can be seen clearly in Figure 13
that there is a uncommonly high value of SFE across the length of the bar for
this particular time period ($t^{\prime}=160$ Myr). Although, this is simply
an illustration of the mathematical cause for such a strongly declining trend
and does not necessarily explain the physical origin. More discussion of the
spatial and temporal variations of bar SFE is included in the following
section.

Figure 12: For each case, the SFE across the bar binning along the bar axis
for the barred period ($t^{\prime}>0$). The bar axis defined by a co-rotating
co-ordinate system to always align with the $x$-axis. The bar thickness in the
disc plane is set to be $y\pm 1$ kpc. As the bar length differs for each case,
the bar is divided into a consistent number of bins so each bin corresponds to
a fractional bar element ($\delta_{b}$). These are numbered outwards from the
centre making it possible to simply assess symmetry between equidistant bar
elements. The star formation time remains consistent with previous analysis
($\Delta t_{\rm SF}=10$ Myr). The time evolution of SFE can be seen with
individually coloured bars corresponding to the result at time steps of 20
Myr, with evolution from dark to light. The bin containing the bar extent is
shaded in grey.

In the context of observational studies, the results in Figure 10 and Figure
11 are generally consistent with observed SFE trends in galaxies, however the
TideB response is somewhat significant. Leroy et al. (2008) measure the radial
changes of SFE in the THINGS dataset (Walter et al., 2008) and observe that
most systems show that SFE in most galaxies tends to either maintain a
constant value or decay with increasing radius. Some systems however, such as
NGC 5194 (M51) and NGC3627 (the observational target for TideB), do conversely
display localised regions with increasing SFE with radius in the disc (Leroy
et al., 2008). This result is confirmed by Muraoka et al. (2019) with the
COMING (Sorai et al., 2019) observational survey data showing that, while
NGC3627 appears to have a generally constant SFE across the radii measured,
there is clearly a slightly increasing trend at outer radii. M51 was not
included in Muraoka et al. (2019) but the isolated target (NGC4303) was.
Comparatively, NGC4303 shows a more consistently flat profile with smaller
scale radial fluctuations in SFE across the disc (Muraoka et al., 2019). This
observed SFE behaviour for NGC4303 is indeed similar to the IsoB result
reflected in Figure 11 which is primarily constant with only slight variation
in both directions over time. M51 is a clearly interacting galaxy (Leroy et
al., 2008; Buta, 2019; Colbert et al., 2004; Karachentsev et al., 2013);
NGC3627 was specifically chosen for this study due to the possibility of an
interaction rich history; and, the TideB response can also be seen to decay on
a time-scale similar to the interaction time-scale. This may suggest that the
characteristic of a positive SFE gradient toward outer radii is indicative of
an ongoing, or at least recent, interaction. If this is the case, it would
mean that an increasing SFE with radius may be used as a possible metric of
tidally driven disc features. The relatively constant or negative
gradients—which dominate the IsoB result and other times of TideB—can then
also be considered compatible with more commonly observed trends, as instances
where this may be observed for either IsoB or TideB over the duration of the
simulation would indeed make it most common.

####  4.4.2 Dependence Along the Bar

To specifically highlight changes within the bar, Figure 12 shows a similar
histogram representation of SFE to Figure 10 this time binned along the bar
axis. Many studies have shown that SFE in barred-spiral galaxies is lower in
the bar than the associated arm features (Momose et al., 2010; Hirota et al.,
2014; Yajima et al., 2019), and in some cases that SFE even varies across the
length of the bar (Downes et al., 1996; Sheth et al., 2002; Muraoka et al.,
2019). For the target galaxy NGC 3627 specifically, it appears that there is
unexpectedly higher SFE at the bar ends and centre than elsewhere (Watanabe et
al., 2011; Law et al., 2018). Additionally, this work has shown in the
previous sections that there are discernible differences in the shape of
features within the bar region which can be seen in projections $\Sigma_{\rm
SFR}$ across the disc (see Figure 5).

Figure 13: The time evolution of SFE along the bar axis from Figure 12 is re-
oriented as a heat-map for SFE smoothed by a spline interpolant. Each resolved
$t^{\prime}$ corresponds to a strip of intensities along the $x$-axis which
correspond to the SFE values in each bar element. Time evolution is downward
along the $y$-axis. The colour weighting is such that brightest areas
correspond to high efficiency and dark values to low efficiency. The bright
central band corresponds to the nuclear disc, and the white dot-dashed lines
illustrate the previously defined bar length.

We define a bar axis and calculate the specific SFE in a polygon cut-out
running parallel to this axis with a width of 2 kpc. We consider the bar to be
a solid body rotating about the major axis, as bar slow-down tends to occur on
much longer time-scales than investigated here (Miwa & Noguchi, 1998;
Debattista & Sellwood, 2000; Kormendy, 2013). The bar orientation is
determined by identifying density peaks in the stellar population within the
bar radius. The orientation of the bar axis is then used to transform all the
simulated positional information to a co-rotating co-ordinate system wherein
the bar lies along the $x-$axis. Additionally, the bar length varies between
the two cases so, in order to produce comparable bin limits, the total bar
length was divided into a set number of bins which would correspond to a
fraction of the whole bar. These are labelled as dimensionless fractional bar
elements ($\delta_{b}$) and numbered outwards from the centre-most bin to the
calculated bar extent, plus two extra bins to encompass the bar end regions.
While this indicates that the physical size of bins is different for each
result, this difference is sufficiently negligible to the effect of scale on
SFE results in this case. As in Figure 10, the star formation time is $\Delta
t_{\rm SF}=10$ Myr and the time evolution is mapped by coloured bars
corresponding to time steps of 20 Myr evolving from dark to light. Similarly,
the bin coinciding with the bar end is shaded in grey.

Along the bar, we are looking to assess two key features: symmetry about $x=0$
to consider whether star formation is symmetrical along the bar; and whether
there is any noticeable trend in the SFE($x$) profile such as significant
peaks or troughs in SFE at key regions (i.e. at the bar ends). Interestingly,
in both cases the SFE profile along the bar does not seem particularly
symmetrical at any of the time periods shown in Figure 12. However, neither
does it seem consistently asymmetric or skewed to one particular direction at
all times for either case.

Within each bin, there is also no standard trend of SFE changing with time,
similar to the radial dependence. There is, however, noticeable evolution in
the shape of the bar-wide trends in both cases. In the case of IsoB, at
earlier times the SFE seems to decrease steeply toward the bar ends
($t^{\prime}<100$ Myr). In the period after $t^{\prime}=100$ Myr, the SFE
values for each bin appear variable along the bar axis in the small scale but
are generally constant on average. Comparatively, the TideB result also seems
to predominantly show SFE decreasing steeply towards the bar end for the
earliest times. After $t^{\prime}\sim 60$ Myr, however, the SFE appears to
drop in the middle bar ($2\delta_{\mathrm{b}}\lesssim|x|\lesssim
5\delta_{\mathrm{b}}$) before rising again towards the bar edges. This is
consistent with the previous section where a significant change in the SFE
gradient with radius of the TideB case was observed at almost exactly this
time period, showing lower SFE in the inner region of the disc compared to
outer radii.

To further quantify this variation of SFE along the bar, we re-orient Figure
12 as a heat-map of the SFE along the bar axis in Figure 13. In this figure,
the SFE values for each time are smoothed by fitting a spline interpolant and
then plotted along a strip of bar axis values in kpc for each case. Time
evolution is defined as downward along the $y$-axis and the calculated result
for each 10 Myr. While no obvious trend appears outside the peak of the
centre-most nucleus in the IsoB case, the bar in the TideB case quickly
evolves to show strong SFE at the bar ends (delineated by white vertical
dashed lines) and low SFE at approximately half the bar radius, alongside the
expected high peak in the central region. The IsoB case does show a similar
SFE deficiency at some time periods but this appears more serendipitous as a
part of random variation which fluctuates along the entire length of the bar.
From this figure, it is also evident that there is some variation in the
intensity of SFE in the central region with time. This trend is similar
between the two cases, appearing suddenly and showing brightest in the early
period of the bar before decreasing slightly but steadily as the bars evolve.

##  5 Discussion

Studies of the interacting target (NGC 3627) specifically note observations
show higher SFE at the bar ends and centre compared to other parts of the bar
(Watanabe et al., 2011; Law et al., 2018). In comparison, studies of the
isolated target (NGC4303) more commonly report more average values for SFE
along the bar (Momose et al., 2010), although Yajima et al. (2019) also
specifically investigate the bar end to also find higher results than the
central bar or arm regions studied. In Figure 13 it can be seen clearly that
while there is certainly some variation along the axis of a bar formed in
isolation, it is much more likely to appear as featureless on average. In this
case, resolution and the specific locations and shapes used to classify
features could significantly impact the overall results observed.
Comparatively, there must be some active effect in the TideB case which is
obviously causing star formation to be consistently strongest at the bar edges
and centre but severely suppressed in the regions between these areas.

Variation in SFE within the bar region has previously been attributed to
possible non-circular motions generating strong shocks or shear motions along
the bar and disrupting many bar-located molecular clouds (Schinnerer et al.,
2002; Dobbs et al., 2014). It is true that the TideB case certainly shows
significantly higher non-axisymmetric motion (see Figure 4), however these
stronger motions are also evident across the entire disc. Law et al. (2018)
draw a possible correlation between kinetic temperature and SFE in NGC 3627
(TideB target), which may support this.

Contrary to the above, Beuther et al. (2018) assert that the surface densities
of NGC 3627 are too high for shear to be sufficient to effectively dampen star
formation activity at the bar ends, and that differing pattern speeds between
the bar and arms may be the most favourable condition to promote intense star
formation in these regions. The arms in the TideB model do frequently decouple
from the bar ends and reconnect, significantly more so than the IsoB case.
Figure 14 shows a magnified view of the TideB gas in the central region for
two time periods either side of the $t^{\prime}=100$ Myr result shown in
previous sections. The arch-like arm features bracket the bar ends,
highlighting their decoupling from the bar; a time-dependent process which
appears to occur frequently over orbital time-scales. Calculating the pattern
speed for both the bar and arms it is possible to determine that there is a
difference in rotation speed at radii about the bar end. The bar pattern speed
was determined by tracking the angular displacement of the defined bar axis at
the given radius $R_{\rm{bar}}$ over the duration of interest $t^{\prime}>0$
which is in line with common methods considered appropriate for resolving
transient features (Grand et al., 2013; Pettitt et al., 2017). Throughout the
relatively short period of interest the bar pattern speed does not
significantly change from a value of $\Omega_{\rm{bar}}=25.9$ km s-1 kpc-1 for
the TideB case. Considering the arm pattern speed to be dynamic, transient and
tidally driven, we assume an arm pattern speed of
$\Omega_{\rm{sp}}=\Omega({\rm disc})-\kappa/2$ in line with Pettitt et al.
(2016). At the bar end $R_{\rm{bar}}$, this radially dependent pattern speed
is found to be $\Omega_{\rm{sp}}=41.9$ km s-1 kpc-1 which is significantly
greater than the bar speed and thus, evidence of such decoupling between the
two features.

Figure 14: The TideB central zoom-in to highlight the bar-spiral overlap
regions. Two time steps equally spaced either side of $t^{\prime}=100$ Myr.

Considering the continued influence of the companion, it is also possible that
the sustained torques from the interaction may impact the development of such
an obviously high SFE at the bar ends. It is possible that the initial
starburst effect triggered by the interaction may be responsible. Under this
kind of driven bar formation, the interaction generates a large inflow toward
the centre and causing increased star formation in the entire central region.
However, over time the available matter may become insufficient to support the
same amount of star formation after the initial in-flow amount is converted.
This may be in line with observations of NGC 3627 where Casasola et al. (2011)
postulate that the bar end may be aligned with co-rotation and that the
absence of an inner Lindblad resonance (ILR) means the torques are negative
between the bar and the nucleus. In this state, NGC 3627 can be considered a
so-called ‘smoking gun’ of inner gas inflow, wherein the dynamical resonances
and kinematically decoupled inner bar serve to directly supply gas to the
central region and AGN (Casasola et al., 2011). Although Casasola et al.
(2011) note that this process should be unsustainable after the bar has had
time to slow under secular evolution, the bar pattern speed of TideB does not
significantly slow over the relatively short duration of this analysis period
and so discernible traces should still be evident. However, calculating the
co-rotation radius for the TideB disc gives a radius $R_{\rm{CR}}=4.97$ kpc
which is not especially close to the determined bar extent
($R_{\rm{bar}}=3.12$ kpc). So, dynamical resonances directly supplying the
inner region with gas in this way is unlikely. Additionally, such an
explanation does not necessarily account for why there should remain only
preferentially strong star formation at the bar ends, in addition to the
centre.

These are all possible factors which may impact the formation of such a
consistent feature in tidally driven bars which does not appear to persist for
any length of time in the more isolated evolutionary scenario. This appears to
indicate there is a need for further, more directed studies that will
facilitate a determination of the root cause and relative probability of such
a feature occurring in different galaxy populations, as well as how such
features may propagate into the later stages of bar and disc evolution. This
is particularly pertinent as these clear fissures in the SFE profile are both
prominent and enduring in the early stages of the tidal bar, so would make a
fitting metric by which observational surveys may determine the formation
mechanism of a given galactic bar.

##  6 Conclusions

The primary aim of this work was to consider the effects that the formation
mechanism driving a spiral galaxy to form a bar may have on the resulting
properties of the host galaxy, in particular on its star forming potential. To
achieve this, two hydrodynamical $N$-body simulations were performed: IsoB
which would form a bar by isolated, independent evolution; and TideB which
required the impetus of a passing companion to trigger bar formation through
tidal interaction. These two scenarios were designed to mimic the nearby
spirals NGC 4303 (IsoB) and NGC 3627 (TideB), popular barred-spiral targets
for observational surveys. These simulations produced two clearly barred-
spiral galaxies with similar morphologies to the real target galaxies. Both
cases show the evolution of similar exponential type bars with a two-arm,
grand design spiral-like structure in the stellar and gas components. Velocity
profiles in the gas disc of both cases are also relatively similar in terms of
the shape of features across the disc. The degree of non-axisymmetric motion
in the tidally driven TideB case is, however, significantly greater than that
of the isolated IsoB case at all evolutionary time periods.

The star formation history and evolution of features in a SFR projection onto
the disc plane shows that both cases have relatively similar levels for
average SFR once the bar is formed, including a prolonged burst of bar-driven
star formation. However, in the TideB case there is also an additional burst-
like period of star formation which is likely caused by the interaction
strengthening the arms before the bar forms. This boost is conincident with a
period where significant star formation is apparent in the arm regions of
TideB that is not seen in the history of the IsoB case. Both cases also show
clearly distinguishable star forming structures within the bar radius which
differ independently with time, as well as between each formation mechanism.
The general shape of the $\Sigma_{\rm gas}$ vs. $\Sigma_{\rm SFR}$
distribution clearly varies depending on the bar formation mechanism. The
TideB case shows a much higher distribution of inefficient but high gas
density regions, though these seem to be confined to tidal debris in the outer
disc. However, both cases appear to conform to the literature Kennicutt-
Schmidt relation averaged over all galaxies.

While many general properties of each case appear similar, there are in fact,
significant variations in star formation properties within each of the bar,
arm or even inter-arm regions. Considering the Kennicutt-Schmidt relation in
terms of different morphological components (bar, arm and inter-arm) shows
that there is little visually distinguishable difference between the arm and
inter-arm regions except for the arms being further along the KS relation than
the inter-arm on average, and the bar further along than both. The bar region
in all cases follows a significantly steeper profile than any other component
and the disc as a whole. The arm region is comparatively similar to the full
disc average for both the isolated and tidal bars. The inter-arm region
represents the shallowest profile in all cases, comparable only to the disc
average slope if tidal remnants in the far outer radii are included in the
TideB case.

The overall radial dependence of SFE in the early stages of the bar evolution
in both cases shows a strongly decreasing profile with increasing radius. For
the IsoB case, this profile quickly flattens once outside of the very central
region ($R>1$ kpc). However, the TideB case eventually reverses the gradient
of SFE across the disc, showing a steady positive gradient with SFE,
increasing towards outer radii (excluding the centre-most region) at later
times. This is attributed to the apparent development of a region of
suppressed star formation in the middle extent of the bar in the TideB case.
These features are likely related to the presence of the companion in some
way, possibly due to shocks or shearing effects from the strong non-
axisymmetric motions present, tidal torques or the result of the initial burst
quickly absorbing available resources in that region. However, it remains that
the tidally driven case shows a particularly consistent and identifiable SFE
profile along the bar axis with peaks at the bar ends and in the centre, not
seen in the isolated bar. Further investigation into the presence, propagation
and drivers of this SFE deficiency is identified as worthwhile as it may allow
for a consistent identification method for bar formation mechanism in
observable galaxies, since it is only evident in the tidally driven bar.

## Acknowledgements

We thank the anonymous referee whose comments helped improved the quality of
this manuscript. EJI acknowledges the support of Japanese Government MEXT
Scholarship for Foreign Students. ARP acknowledges the support of The Japanese
Society for the Promotion of Science (JSPS) KAKENHI grant for Early Career
Scientists (20K14456). TO acknowledges the support of MEXT KAKENHI grants
18H04333, 19H01931, and 20H05861. Images and partial analysis were made using
the pynbody python package (https://github.com/pynbody/pynbody, (Pontzen &
Governato, 2012) and the MathWorks matlab programming and numeric computing
platform. Observational data was acessed from the PHANGS-HST survey (Lee et
al., 2021). Numerical computations were carried out on the Cray XC50 at Center
for Computational Astrophysics, National Astronomical Observatory of Japan. We
thank K. Sorai, Y. Yajima and S. Benincasa for helpful discussions realated to
this work.

## Data Availability

The data underlying this article can be shared upon reasonable request to the
corresponding author.

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##  Appendix A Rotation Curve Evolution

In Figure 15 we show the rotation curve in both models (IsoB top, TideB
bottom) at different evolutionary stages. There is clearly a significant
variance in the shape of the rotation curve over the time period investigated.
Both curves tend to rise up in the inner disc as the bar grows, with outer
undulations decaying on 100 Myr time-scales. Of particular note is the spike
around 4 kpc in the TideB case that lines up particularly well with the NGC
3627 profile at $t^{\prime}=0$ Myr, which is also coincidentally when the
model shows the closest morphological match.

Figure 15: Rotation curves for each case at the key intervals
$t^{\prime}=[-100,0,100,200]$ Myr as well as the initial condition and
observation results from the PHANGS survey data (Lang et al., 2020).