11institutetext: Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117
Heidelberg, Germany 22institutetext: Indian Institute of Astrophysics,
Koramangala II Block, Bengaluru 560034, India 33institutetext: Department of
Physics, Indian Institute of Science, Bangalore 560012, India

# Dynamical interplay of disc thickness and the interstellar gas: implication
for the longevity of spiral density waves

Soumavo Ghosh E-mail<EMAIL_ADDRESS>Chanda J. Jog E-mail:
<EMAIL_ADDRESS>
(Received XXX; accepted YYY)

A typical galactic disc has a finite thickness and in addition to stars it
also contains a finite amount of interstellar gas. Here, we investigate the
physical impact of the finite thickness of a galactic disc on the disc
stability against the non-axisymmetric perturbations and on the longevity of
the spiral density waves, with and without the presence of gas. The longevity
is quantified via group velocity of density wavepackets. The galactic disc is
first modelled as a collisionless stellar disc with finite height and then
more realistically as a gravitationally-coupled stars plus gas system (with
different thickness for stars and gas). For each case, we derive the
appropriate dispersion relation in the WKB approximation, and study the
dynamical effect of the disc thickness on the life-time of spiral density
waves via a parametric approach. We find the generic trend that the effective
reduction in disc self-gravity due to disc thickness makes it more stable
against the non-axisymmetric perturbations, and shortens the life-span of the
spiral density waves. Further, the interstellar gas and the disc thickness are
shown to have a mutually opposite dynamical effect on the disc stability as
well as the longevity of the spiral density waves. While the gas supports the
non-axisymmetric features for a longer time, the disc thickness has an
opposite, quenching effect. Consequently, the net change is set by the
relative dominance of the opposite effects of the interstellar gas and the
disc thickness.

###### Key Words.:

galaxies: evolution - galaxies: kinematics and dynamics - galaxies: spiral -
galaxies: structure - instabilities - hydrodynamics

##  1 Introduction

Spiral features are one of the most common non-axisymmetric structures (apart
from a bar) in disc galaxies in the local Universe (e.g Elmegreen et al.,
2011; Yu et al., 2018; Savchenko et al., 2020). The occurrence of spiral
structure in high-redshift (up to $z\sim$ 1.8) disc galaxies is also known
observationally (e.g. Elmegreen & Elmegreen, 2014; Willett et al., 2017; Hodge
et al., 2019). Simultaneous occurrence of spiral structure with an $m=2$ bar
or an $m=1$ lopsidedness in disc galaxies is commonplace (e.g., see Rix &
Zaritsky, 1995; Bournaud et al., 2005; Buta et al., 2010; Zaritsky et al.,
2013; Kruk et al., 2018; Ghosh et al., 2021). The Milky Way is a barred spiral
galaxy (Weinberg, 1992; Gerhard, 2002) as well. In the past, a plethora of
physical mechanisms ranging from bar-induced spirals (e.g., Salo et al.,
2010), or due to tidal encounters (e.g., Toomre & Toomre, 1972; Dobbs et al.,
2010), or swing amplification of noise (Goldreich & Lynden-Bell, 1965; Julian
& Toomre, 1966; Toomre, 1981), due to disc response to giant molecular clouds
(D’Onghia et al., 2013), due to interactions with other spirals (Masset &
Tagger, 1997), and due to recurrent groove modes (Sellwood & Lin, 1989;
Sellwood, 2012; Sellwood & Carlberg, 2019) to manifold-driven spirals
(Athanassoula, 2012), have been proposed for exciting spirals in disc
galaxies.

Regardless of the physical mechanism(s) triggering the spiral instability in
the disc, it is vital to address the nature and the longevity of spirals. Past
studies of Goldreich & Lynden-Bell (1965); Julian & Toomre (1966); Toomre
(1981) proposed the spiral arms as transient material arms arising due to
joint effects of epicyclic motion of stars, shear due to differential
rotation, and the disc self-gravity. On the other hand, another set of studies
by Lin & Shu (1964, 1966) envisaged the spirals as quasi-stationary density
waves which rotate with respect to the disc with a well-defined pattern speed
(for a detailed recent review, see Shu, 2016). Implicit to the formalism of
swing amplification, the material arms do not last for a long time (beyond a
few dynamical timescales or a few $\times 10^{8}$ yr), and get wound up due to
the disc differential rotation, whereas by definition, the spiral density
waves are assumed to be stationary or they last forever. However, Toomre
(1969) showed that, due to the radial group transport, a wavepacket made of
such density waves eventually gets destroyed within $\sim 1\mbox{$\>{\rm
Gyr}$}$ time-scale, thereby posing a challenge to the stationary picture of
the density wave theory. Spirals, generated in $N$-body simulations, are
almost always found to be a transient phenomenon (e. g., see Sellwood, 2011;
Grand et al., 2012; Baba et al., 2013, for a review the reader is referred to
Dobbs & Baba (2014)). However, a high-resolution $N$-body simulation by
D’Onghia et al. (2013) showed that the spirals can sustain (at least in the
statistical sense) for much longer time-scale due to the non-linear disc
response to the perturbation caused by giant molecular cloud-like mass
concentration. Another recent work by Saha & Elmegreen (2016) showed that
bulges play a pivotal role in the sustenance of spirals in $N$-body
simulations, and intermediate-sized bulges can help the spiral density wave to
last longer ($\sim 5\mbox{$\>{\rm Gyr}$}$) by providing a high Toomre
$Q$-barrier in the inner region. Interstellar gas is shown to play a pivotal
role too in the longevity of spiral density waves in disc galaxies. Earlier
work by Sellwood & Carlberg (1984) demonstrated the role of a dissipative
component in cooling the stellar disc and facilitating the generation of fresh
spiral waves. Jog (1992) included gas along with stars in the framework of
swing amplification mechanism for a more realistic treatment which yielded
broad stellar arms as observed. Further, Ghosh & Jog (2015, 2016) showed that
the interstellar gas helps spiral density waves to survive for a longer time
(several billion years).

Most of the past analytical studies of spiral structure in disc galaxies
treated the galactic disc as being infinitesimally-thin, for simplicity (but
see Goldreich & Lynden-Bell, 1965). This assumption is valid when the height
of the disc is small as compared to the wavelength of the perturbation (see
discussions in Toomre, 1964; Binney & Tremaine, 2008). However, in reality, a
galactic disc has a finite height. López-Corredoira & Molgó (2014) showed that
the Galactic disc flares substantially between Galactocentric radii
$8\mbox{$\>{\rm kpc}$}$ to $25\mbox{$\>{\rm kpc}$}$ (also see Li et al., 2019)
with the thin-disc component having a scale-height of $\sim 300\>{\rm pc}$ in
the Solar neighbourhood (e. g., see Jurić et al., 2008). This flaring of
galactic disc (i.e., increment of scale-height) is a generic phenomenon (e.g.,
see de Grijs & Peletier, 1997; Narayan & Jog, 2002a; Sarkar & Jog, 2019;
García de la Cruz et al., 2021). In addition, the existence of a thick-disc
component is now well-established observationally in both, external galaxies,
as well as the Milky Way (e.g., see Tsikoudi, 1979; Burstein, 1979; Yoachim &
Dalcanton, 2006; Comerón et al., 2011a, b, 2018). Spirals play a pivotal
dynamical role in disc dynamics by transporting angular momentum (Lynden-Bell
& Kalnajs, 1972); by causing radial migration of stars without heating
(Sellwood & Binney, 2002; Roškar et al., 2008; Schönrich & Binney, 2009), and
by exciting vertical breathing motions (Debattista, 2014; Faure et al., 2014;
Ghosh et al., 2020). Thus, a more realistic study of the spiral structure in
disc galaxies should take into account the finite thickness of the disc.

Past theoretical studies have shown that the introduction of a finite
thickness of a galactic disc results in a net reduction in the radial force in
the mid-plane (e.g., see Toomre, 1964; Jog & Solomon, 1984; Jog, 2014). This,
in turn, facilitates the galactic disc to become more stable against the
axisymmetric perturbations (Toomre, 1964; Jog & Solomon, 1984). Julian &
Toomre (1966) demonstrated that the finite thickness of the stellar disc
decreases the amplitude of the density transforms, in a local patch of the
disc. Further, Ghosh & Jog (2018) showed the generic trend of suppressing the
growth of the swing-amplified spirals due to the introduction of a finite
thickness for the disc. However, little is known about the plausible role of
the disc finite thickness on the radial group transport and the longevity of
spiral density wave.

In this paper, we investigate the physical effect of the finite thickness of a
galactic disc on the persistence of spiral density wave, both in absence and
in presence of the interstellar gas. To achieve that, we first derive the
appropriate dispersion relation in the WKB (Wentzel-Kramers-Brillouin) limit,
for a collisionless stellar disc with a finite thickness and for a
gravitationally-coupled two-component (stars plus gas) system with different
thickness for the stellar and gas discs. Then, we systematically vary
different input parameters, namely, the Toomre $Q$ parameter, thickness of the
disc, and the gas fraction, to check the dependence of the longevity of the
spiral density wave, if any, on the finite thickness of the disc. The rest of
the paper is organised as follows: Sect. 2 provides the derivation of the
relevant dispersion relations in the WKB limit whereas Sect. 3 presents the
dimensionless form of the corresponding dispersion relations. Sect. 4 gives
our results covering the effect of finite thickness on the persistence of the
spiral density waves, with and without the interstellar gas. Sects. 5 and 6
contain discussion and the main findings of this work, respectively.

##  2 WKB dispersion relation for galactic disc with finite thickness

Here, we first derive the dispersion relation for a one-fluid disc, and then
extend to a collisionless stellar disc with finite height (Sects. 2.1–2.2).
Finally, we consider a more realistic gravitationally-coupled two-component
(stars plus gas) system where the stellar and the gaseous discs can have
different scale-heights (Sect. 2.3), as observed in real galaxies. The
underlying disc is taken to be axisymmetric and the spiral structure is
treated as only a small perturbation on the steady-state axisymmetric disc, so
that linear perturbation approach will be valid (for details see Binney &
Tremaine, 2008). A cylindrical coordinate system $(R,\phi,z)$ is used
throughout the formulation.

###  2.1 One-component fluid disc

Following the treatment given in Binney & Tremaine (2008), we start with an
infinitesimally thin fluid disc, and then we modify the formulation by
introducing the effect due to the finite thickness of the fluid disc. The
fluid disc is characterised by the disc surface density $\Sigma_{\rm d}$ and
the sound speed $c$. We assume that the pressure acts only in the disc plane.
Now, for such a system, the Euler’s equations of motion in the cylindrical
coordinates become

$\frac{\partial v_{R}}{\partial t}+v_{R}\frac{\partial v_{R}}{\partial
R}+\frac{v_{\phi}}{R}\frac{\partial
v_{R}}{\partial\phi}-\frac{v^{2}_{\phi}}{R}=-\left(\frac{\partial(\Phi+{\mathcal{H}})}{\partial
R}\right)_{z=0}\,,$ (1)

and,

$\frac{\partial v_{\phi}}{\partial t}+v_{R}\frac{\partial v_{\phi}}{\partial
R}+\frac{v_{\phi}}{R}\frac{\partial
v_{\phi}}{\partial\phi}+\frac{v_{R}v_{\phi}}{R}=-\frac{1}{R}\left(\frac{\partial(\Phi+{\mathcal{H}})}{\partial\phi}\right)_{z=0}\,.$
(2)

Here, ${\mathcal{H}}$ is the specific enthalpy of a polytropic fluid with an
equation of state $p=K\Sigma_{d}^{\gamma}$ where $\gamma=(n+1)/n$, $n$ being
the polytropic index, and $K$ being a proportionality constant, and the form
of ${\mathcal{H}}$ is given by (Binney & Tremaine, 2008)

${\mathcal{H}}=\frac{\gamma}{\gamma-1}K\Sigma_{d}^{\gamma-1}\,.$ (3)

Now, assuming the spiral density wave to be a small perturbation, we write
$v_{R}=v_{R_{1}};v_{\phi}=v_{\phi_{0}}+v_{\phi_{1}}$, where $v_{R_{1}}$ and
$v_{\phi_{1}}$ are small perturbations. Further assuming the random motion to
be small compared to the rotation, from the Euler equation, it can be seen
that the unperturbed motion gives rise to

$v_{\phi_{0}}=\sqrt{R\frac{d\Phi_{0}}{dR}}=R\Omega(R)\,,$ (4)

where $\Phi_{0}$ is the unperturbed potential and $\Omega(R)$ is the circular
frequency. The linear perturbed equations of motion become

$\frac{\partial v_{R_{1}}}{\partial t}+\Omega\frac{\partial
v_{R_{1}}}{\partial\phi}-2\Omega(R)v_{\phi_{1}}=-\left(\frac{\partial(\Phi_{1}+{\mathcal{H}}_{1})}{\partial
R}\right)_{z=0}\,,$ (5)

and,

$\frac{\partial v_{\phi_{1}}}{\partial t}+\left[\frac{d(\Omega
R)}{dR}+\Omega\right]v_{R_{1}}+\Omega\frac{\partial
v_{\phi_{1}}}{\partial\phi}=-\frac{1}{R}\left(\frac{\partial(\Phi_{1}+{\mathcal{H}}_{1})}{\partial\phi}\right)_{z=0}\,,$
(6)

where ${\mathcal{H}}_{1}$ is the perturbed specific enthalpy.

####  2.1.1 Introduction of finite thickness of the fluid disc

For simplicity, we assume that the disc has a constant density that does not
vary with $z$, and the disc has a total thickness of $2h$. For an
infinitesimally thin, axisymmetric disc, and an axisymmetric perturbation, the
solution of the Poisson equation is given by (Toomre, 1964)

$\Phi_{1}=-(2\pi G/|k|)\Sigma_{1}exp(-k|z|)\,,$ (7)

where $k$ is the wavenumber of perturbation and $\Phi_{1}$ and $\Sigma_{1}$
are the perturbation (or imposed) potential and the corresponding surface
density, respectively. The perturbation surface density is taken to have the
form exp$[i(kr-\omega t)]$. For such a disc, the radial force at the mid-plane
($z=0$) due to a vertical layer between $z$ and $z+dz$ is proportional to
exp$[-k|z|dz]$. Hence, the net radial force at $z=0$ is obtained by
integrating over $z$ to be ( see Toomre, 1964)

$\left(\frac{\partial\Phi_{1}}{\partial R}\Big{|}_{z=0}\right)_{\rm
net}=-i2\pi G{\Sigma_{1}}\delta\,,$ (8)

where $\delta$ is the reduction factor which denotes the reduction in the
radial force at the mid-plane due to the finite height for a constant density
disc. This can also be thought of as a reduction in the disc surface density,
and its form is given as (Toomre, 1964)

$\delta=[1-exp(-kh)]/kh\,.$ (9)

In an analogous fashion, by integrating the contribution of force due to
layers at different $z$, we can show that the azimuthal force in the mid-plane
is also reduced by an identical reduction factor.

We point out that, the perturbation potential and the surface density in the
framework of WKB approximation, follow the same relation (Eq. (7)) as above
(see Eqs. (6)-(18) in Binney & Tremaine, 2008). Hence, a similar analysis (as
mentioned above) will yield a same reduction factor $\delta$ (Eq. (9)) in the
radial, as well as the azimuthal force at mid-plane, for the WKB approximation
as well. Therefore, in case of a fluid disc with a finite thickness, Eqs. (5)
and (6) become

$\frac{\partial v_{R_{1}}}{\partial t}+\Omega\frac{\partial
v_{R_{1}}}{\partial\phi}-2\Omega(R)v_{\phi_{1}}=-\left(\frac{\partial\Phi_{1}}{\partial
R}\delta+\frac{\partial{\mathcal{H}}_{1}}{\partial R}\right)_{z=0}\,,$ (10)

and,

$\frac{\partial v_{\phi_{1}}}{\partial t}+\left[\frac{d(\Omega
R)}{dR}+\Omega\right]v_{R_{1}}+\Omega\frac{\partial
v_{\phi_{1}}}{\partial\phi}=-\frac{1}{R}\left(\frac{\partial\Phi_{1}}{\partial\phi}\delta+\frac{\partial{\mathcal{H}}_{1}}{\partial\phi}\right)_{z=0}\,.$
(11)

We assume the trial solutions are of the form

$\begin{split}v_{R_{1}}=Re\left[v_{R_{a}}(R)e^{i(m\phi+kR-\omega t)}\right]\\\
v_{\phi_{1}}=Re\left[v_{\phi_{a}}(R)e^{i(m\phi+kR-\omega t)}\right]\\\
\Phi_{1}=Re\left[\Phi{{}_{a}}(R)e^{i(m\phi+kR-\omega t)-k|z|}\right]\\\
\Sigma_{d_{1}}=Re\left[\Sigma_{d_{a}}(R)e^{i(m\phi+kR-\omega t)}\right]\\\
{\mathcal{H}}_{1}=Re\left[{\mathcal{H}}_{a}(R)e^{i(m\phi+kR-\omega
t)}\right]\,.\end{split}$ (12)

Putting these in Eqs. (10) and (11), and after some algebraic simplification
we get,

$v_{R_{a}}=\frac{i}{\Delta}\left[(\omega-m\Omega)\left(\delta\frac{d\Phi_{a}}{dR}+\frac{d{\mathcal{H}}_{a}}{dR}\right)-\frac{2m\Omega}{R}(\Phi_{a}\delta+{\mathcal{H}}_{a})\right]\,,$
(13)

and,

$v_{\phi_{a}}=-\frac{1}{\Delta}\left[2B\left(\delta\frac{d\Phi_{a}}{dR}+\frac{d{\mathcal{H}}_{a}}{dR}\right)+\frac{m(\omega-m\Omega)}{R}\left(\Phi_{a}\delta+{\mathcal{H}}_{a}\right)\right]\,,$
(14)

where $B$ is the Oort constant and
$\Delta\equiv\kappa^{2}-(\omega-m\Omega)^{2}$, $\kappa$ being the epicyclic
frequency. Also, the perturbed equation of state gives
${\mathcal{H}}_{a}=c^{2}\frac{\Sigma_{d_{a}}}{\Sigma_{0}}$, where $\Sigma_{0}$
is the unperturbed disc surface density (for details see Binney & Tremaine,
2008).

Similarly, the perturbed continuity equation in cylindrical coordinates is

$\frac{\partial\Sigma_{d_{1}}}{\partial
t}+\Omega\frac{\partial\Sigma_{d_{1}}}{\partial\phi}+\frac{1}{R}\frac{\partial}{\partial
R}\left(Rv_{R_{1}}\Sigma_{0}\right)+\frac{\Sigma_{0}}{R}\frac{\partial
v_{\phi_{1}}}{\partial\phi}=0\,,$ (15)

which, after substituting the trial solution (Eq. (12)) becomes

$-i(\omega-m\Omega)\Sigma_{d_{a}}e^{ikR}+\frac{1}{R}\frac{d}{dR}\left(Rv_{R_{a}}e^{ikR}\Sigma_{0}\right)+\frac{im\Sigma_{0}}{R}v_{\phi_{a}}e^{ikR}=0\,.$
(16)

####  2.1.2 Dispersion relation

Now, we invoke the WKB approximation to derive an analytical dispersion
relation. In this limit, Eqs. (13)-(14) reduce to (for details, see Binney &
Tremaine, 2008)

$\begin{split}v_{R_{a}}=-\frac{(\omega-m\Omega)}{\Delta}k(\Phi_{a}\delta+{\mathcal{H}}_{a})\hskip
5.69046pt\mbox{and}\hskip
5.69046ptv_{\phi_{a}}=-\frac{2iB}{\Delta}k(\Phi_{a}\delta+{\mathcal{H}}_{a})\,,\end{split}$
(17)

Also, the continuity equation (Eq. (16)) reduces to

$-(\omega-m\Omega)\Sigma_{d_{a}}+k\Sigma_{0}v_{R_{a}}=0\,,$ (18)

as in the WKB approximation, the last term in Eq. (16) involving
$v_{\phi_{a}}$ is smaller than the first two terms and can be dropped (Binney
& Tremaine, 2008). Now, on substituting the values for $v_{R_{a}}$,
${\mathcal{H}}_{a}$ as obtained above and $\Phi_{a}=-2\pi G\Sigma_{a}/|k|$, as
obtained in the WKB limit using Eqs. (7) and (12), where $\Phi_{a}$ and
$\Sigma_{a}$ are the imposed perturbation potential and the corresponding
imposed perturbation surface density; and after some algebraic manipulation,
Eq. (18) reduces to

$\Sigma_{d_{a}}=\frac{2\pi
G|k|\Sigma_{0}\delta}{\kappa^{2}-(\omega-m\Omega)^{2}+c^{2}k^{2}}\Sigma_{a}\,,$
(19)

where $\Sigma_{d_{a}}$ is the disc response density. Now, for a self-sustained
density wave, the quantities $\Sigma_{a}$ and $\Sigma_{d_{a}}$ should be
equal, and hence the dispersion relation becomes (Binney & Tremaine, 2008)

$(\omega-m\Omega)^{2}=\kappa^{2}-2\pi G|k|\Sigma_{0}\delta+c^{2}k^{2}\,,$ (20)

where $m$ is a positive integer, denoting the $m$-fold rotational symmetry of
the perturbation. We note that, in the limit of $h\rightarrow 0$, so that
$\delta\rightarrow 1$, the dispersion relation (Eq. (20)) reduces to the
corresponding dispersion relation for an infinitesimally-thin fluid disc (see
Binney & Tremaine, 2008), as expected. In other words, Eq. (20) is a
generalisation of the dispersion relation for an infinitesimally thin fluid
disc. Here, the additional factor $\delta$ comes in from the inclusion of the
effect of finite height in the calculation.

###  2.2 One-component stellar disc

A cold stellar disc is dynamically equivalent to a fluid disc with zero
pressure or zero enthalpy, and hence, the perturbation $\bar{v}_{R_{1}}$ in
the mean radial velocity of the stars can be obtained from Eqs. (5) and (6)
(assuming zero pressure) as (Binney & Tremaine, 2008)

$\bar{v}_{R_{a}}=-\frac{\omega-m\Omega}{\Delta}k\Phi_{a}\,.$ (21)

where $\Delta$ is as defined earlier. If we assume the disc to have a finite
thickness of $2h$ and a constant density $\rho$ (as in Sect. 2.1.1), then in
an analogous fashion, the radial force in the mid-plane is reduced by a
reduction factor $\delta$ (defined in Eq. (9)). Then following the same
procedure as done in Sect. 2.1.1, namely solving the Euler equation (Eq. 10)
with zero enthalpy, the solution for $\bar{v}_{R_{a}}$ is obtained in the WKB
limit to be

$\bar{v}_{R_{a}}=-\frac{\omega-m\Omega}{\Delta}k(\Phi_{a}\,\delta)\,.$ (22)

As expected, this can be obtained from Eq. (17) by setting the enthalpy to be
zero.

Now, if the stellar disc is not sufficiently cold, i.e., if the typical
epicyclic amplitude is not small enough as compared to the wavelength
($2\pi/k$) of perturbation, then the net response velocity measured at a given
location is due to stars with large epicyclic amplitudes and hence would have
sampled different values of the spiral potential. This results in a partial
cancellation in the mean velocity perturbation response to the imposed
potential (for details see Binney & Tremaine, 2008). Therefore, the resulting
expression for the mean velocity perturbation for a thin disc is given by

$\bar{v}_{R_{a}}=-\frac{\omega-m\Omega}{\Delta}k\Phi_{a}\mathcal{F}\,,$ (23)

where $\mathcal{F}$ is the factor by which the response of the disc to a given
spiral perturbation is diminished below the value for a cold disc. The form of
$\mathcal{F}$ is discussed later in Sect. 3.2.

Now, it is straightforward to show that, for a stellar disc with finite
thickness, where we start with $\bar{v}_{R_{a}}$ as for a finite hight case
above (Eq. (22)), and then taking account of the reduction in the
collisionless disc response when the disc is not cold, then these two effects
manifest simultaneously and the expression for the mean velocity perturbation
becomes

$\bar{v}_{R_{a}}=-\frac{\omega-m\Omega}{\Delta}k(\Phi_{a}\delta)\mathcal{F}\,.$
(24)

Also, the perturbed continuity equation in the WKB limit becomes (see e.g.,
Binney & Tremaine, 2008)

$-(\omega-m\Omega)\Sigma_{d_{a}}+k\Sigma_{0}\bar{v}_{R_{a}}=0\,.$ (25)

Note that this is identical to Eq. (18) since the continuity equation has the
same form for fluid and collisionless cases. On substituting $\bar{v}_{R_{a}}$
from Eq. (24), the above reduces to

$-(\omega-m\Omega)\Sigma_{d_{a}}+k\Sigma_{0}\left[-\frac{\omega-m\Omega}{\Delta}k(\Phi_{a}\delta)\mathcal{F}\right]=0\,.$
(26)

Finally, invoking the WKB approximation as before (Sect. 2.1.1), we substitute
$\Phi_{a}=-2\pi G\Sigma_{a}/|k|$ (Sect. 2.2), and obtain

$\Sigma_{d_{a}}=\frac{2\pi
G\Sigma_{0}|k|}{\Delta}\delta{\mathcal{F}}\Sigma_{a}\,.$ (27)

Again, for a self-sustained density wave, the disc response surface density
($\Sigma_{d_{a}}$) should be equal to the imposed surface density
($\Sigma_{a}$). This, in turn, yields the dispersion relation for the
collisionless disc in the WKB limit as

$(\omega-m\Omega)^{2}=\kappa^{2}-2\pi G|k|\Sigma_{0}\delta\mathcal{F}\,.$ (28)

Here also we note that in the limit of $h\rightarrow 0$, so that
$\delta\rightarrow 1$, the dispersion relation (Eq. (28)) reduces to the
corresponding dispersion relation for an infinitesimally-thin collisionless
stellar disc (see Binney & Tremaine, 2008), as expected. In other words, Eq.
(28) is a generalisation of the dispersion relation for an infinitesimally-
thin stellar disc. Here, the additional factor $\delta$ comes in from the
inclusion of the effect of finite height in the calculation.

###  2.3 Two-component star-gas system with different finite thickness

Here, we treat a galactic disc as a gravitationally coupled two-component
(stars plus gas) system 111 The main difference between this work with that
presented in Ghosh & Jog (2015) is that here we take into account the finite
thickness of both the stellar and the gaseous discs.. The stellar disc is
taken to be collisionless in nature and is characterised by a surface density
$\Sigma_{\rm 0s}$, a one-dimensional velocity dispersion, $\sigma_{\rm s}$,
and a total thickness of $2h_{s}$, while the gas disc is treated as a fluid
and is characterised by the surface density $\Sigma_{\rm 0g}$, a one-
dimensional velocity dispersion or the sound speed, $c_{\rm g}$, and a total
thickness of $2h_{g}$. Since the stars and the gas are gravitationally-
coupled, therefore, their motion will be governed by the joint potential
($\Phi_{\rm tot}$) which is set by both the stellar and the gas discs, i.e.,
$\Phi_{\rm tot}=\Phi_{\rm s}+\Phi_{\rm g}$. The right hand side of Eqs. (1)
and (2) will now contain a derivative of the total potential. Thus, the
steady-state unperturbed motion will now be given by

$v_{\phi_{0i}}=\sqrt{R\frac{d\Phi_{\rm tot,0}}{dR}}=R\Omega(R)\,.$ (29)

Now, for such a system, the reduction due to the finite height affects the
corresponding radial force of each component separately, and each component is
affected by the net force due to both components (see e.g. Jog & Solomon,
1984). Hence, the R.H.S. of the perturbed Euler equation (Eq. (10)) becomes
$(\partial\Phi{s_{1}}/\partial R)\delta_{s}+(\partial\Phi_{g_{1}}/\partial
R)\delta_{g}+(\partial{\mathcal{H}}_{1_{g}}/\partial R$) for $i=g$ (gas),
while for the collisionless stellar disc ($i=s$), only the first two terms
above (without the enthalpy term) are kept, as discussed in Sect. 2.2. The
same correction applies to the azimuthal perturbed equation of motion as well.
Now assuming trial solutions similar to those as shown in Eq. (12), and
invoking the WKB approximation, the solutions of the Euler equations give the
amplitudes of perturbed velocity along the radial and the azimuthal directions
for the stellar and the gas disc respectively as

$\begin{split}v_{R_{a_{s}}}=-\frac{(\omega-m\Omega)}{\Delta}k(\Phi_{a_{s}}\delta_{1}{\mathcal{F}}+\Phi_{a_{g}}\delta_{g})\\\
v_{\phi_{a_{s}}}=-\frac{2iB}{\Delta}k(\Phi_{a_{s}}\delta_{1}{\mathcal{F}}+\Phi_{a_{g}}\delta_{g})\,,\end{split}$
(30)

and,

$\begin{split}v_{R_{a_{g}}}=-\frac{(\omega-m\Omega)}{\Delta}k(\Phi_{a_{s}}\delta_{s}+\Phi_{a_{g}}\delta_{g}+{\mathcal{H}}_{a_{g}})\\\
v_{\phi_{a_{g}}}=-\frac{2iB}{\Delta}k(\Phi_{a_{s}}\delta_{s}+\Phi_{a_{g}}\delta_{g}+{\mathcal{H}}_{a_{g}})\,.\end{split}$
(31)

Substituting the expression for $v_{R_{a_{s}}}$ from Eq. (30) in the perturbed
continuity equation (analog of Eq. (25)), we get

$\Sigma_{d_{a_{s}}}\left[{\kappa^{2}-(\omega-m\Omega)^{2}}\right]=k\Sigma_{0_{s}}\left[k(\Phi_{a_{s}}\delta_{1}{\mathcal{F}}+\Phi_{a_{g}}\delta_{g})\right]\,.$
(32)

On substituting the expression for the perturbation potentials in terms of the
surface densities in the WKB limit, namely $\Phi_{a_{s}}=-2\pi
G\Sigma_{a_{s}}/|k|$ and similarly for $\Phi_{a_{g}}$, and setting
$\Sigma_{a_{s}}$, the imposed or perturbation stellar surface density equal to
$\Sigma_{d_{a_{s}}}$, the stellar response surface density for a self-
consistent solution, the above reduces to

$\frac{\Sigma_{d_{a_{s}}}}{\Sigma_{a_{g}}}=\frac{2\pi
Gk\Sigma_{0_{s}}\delta_{g}}{\left(\kappa^{2}-(\omega-m\Omega)^{2}-2\pi
G|k|\Sigma_{0_{s}}\delta_{s}{\mathcal{F}}\right)}\,.$ (33)

Similarly for the gas disc, the continuity equation (Eq. (18)) combined with
the perturbed velocity component in the WKB limit (Eq. (31)) gives:

$\Sigma_{d_{a_{g}}}\left[{\kappa^{2}-(\omega-m\Omega)^{2}}\right]=k\Sigma_{0_{g}}\left[k(\Phi_{a_{s}}\delta_{s}+\Phi_{a_{g}}\delta_{g}+{\mathcal{H}}_{a_{g}})\right]\,.$
(34)

On substituting the expression for the perturbation potentials in terms of the
surface densities in the WKB limit, namely $\Phi_{a_{s}}=-2\pi
G\Sigma_{a_{s}}/|k|$ and similarly for $\Phi_{a_{g}}$; and further for self-
consistency setting $\Sigma_{a_{g}}=\Sigma_{d_{a_{g}}}$, the above equation
reduces to

$\frac{\Sigma_{a_{s}}}{\Sigma_{d_{a_{g}}}}=\frac{\left(\kappa^{2}-(\omega-m\Omega)^{2}+k^{2}c_{g}^{2}-2\pi
G|k|\Sigma_{0_{g}}\delta_{g}\right)}{2\pi Gk\Sigma_{0_{g}}\delta_{s}}\,.$ (35)

Combining Eqs. (33) and (35) and setting the condition of self-consistency
($\Sigma_{a_{s}}=\Sigma_{d_{a_{s}}}$ and similarly for the gas disc), we get
the dispersion relation for this joint star-gas system as

$\frac{2\pi G\Sigma_{\rm
0s}|k|\delta_{s}{\mathcal{F}}\Big{(}\frac{\omega-m\Omega}{\kappa},\frac{k^{2}\sigma^{2}_{\rm
s}}{\kappa^{2}}\Big{)}}{\kappa^{2}-(\omega-m\Omega)^{2}}+\frac{2\pi
G\Sigma_{\rm 0g}\delta_{g}|k|}{\kappa^{2}-(\omega-m\Omega)^{2}+c^{2}_{\rm
g}k^{2}}=1\,.$ (36)

We checked that for an infinitesimally thin disc (where the reduction factors
$\delta_{i}\rightarrow 1\ (i=s,g)$, see Eq. (9)), the above reduces to the
dispersion relation for a star-gas case obtained in Ghosh & Jog (2015), as
expected.

We next define

$\begin{split}\alpha_{\rm s}=\kappa^{2}-2\pi G\Sigma_{\rm
0s}\delta_{s}|k|{\mathcal{F}}\Big{(}\frac{\omega-m\Omega}{\kappa},\frac{k^{2}\sigma^{2}_{\rm
s}}{\kappa^{2}}\Big{)}\\\ \alpha_{\rm g}=\kappa^{2}-2\pi G\Sigma_{\rm
0g}\delta_{g}|k|+k^{2}{c}^{2}_{\rm g}\\\ \beta_{\rm s}=2\pi G\Sigma_{\rm
0s}\delta_{s}|k|{\mathcal{F}}\Big{(}\frac{\omega-m\Omega}{\kappa},\frac{k^{2}\sigma^{2}_{\rm
s}}{\kappa^{2}}\Big{)}\\\ \beta_{\rm g}=2\pi G\Sigma_{\rm
0g}\delta_{g}|k|\,.\end{split}$ (37)

Upon substitution in Eq. (36) and after some algebraic simplification, we get

$(\omega-m\Omega)^{4}-(\alpha_{\rm s}+\alpha_{\rm
g})(\omega-m\Omega)^{2}+(\alpha_{\rm s}\alpha_{\rm g}-\beta_{\rm s}\beta_{\rm
g})=0\,.$ (38)

This is a quadratic equation in $(\omega-m\Omega)^{2}$. Solving it we get

$(\omega-m\Omega)^{2}=\frac{1}{2}\left[(\alpha_{\rm s}+\alpha_{\rm
g})\pm\left\\{(\alpha_{\rm s}+\alpha_{\rm g})^{2}-4(\alpha_{\rm s}\alpha_{\rm
g}-\beta_{\rm s}\beta_{\rm g})\right\\}^{1/2}\right]\,.$ (39)

The additive root for $(\omega-m\Omega)^{2}$ always leads to a positive
quantity, hence it indicates always oscillatory perturbations under all
conditions (same as for axisymmetric case; see Jog & Solomon, 1984). In order
to study the stability of the system and its further consequences, we
therefore consider only the negative root which is

$(\omega-m\Omega)^{2}=\frac{1}{2}\left[(\alpha_{\rm s}+\alpha_{\rm
g})-\left\\{(\alpha_{\rm s}+\alpha_{\rm g})^{2}-4(\alpha_{\rm s}\alpha_{\rm
g}-\beta_{\rm s}\beta_{\rm g})\right\\}^{1/2}\right]\,.$ (40)

We mention that, the underlying formalism, presented here, closely follows
that presented in Jog & Solomon (1984). This builds on the treatment for one-
component stellar and gas discs of finite height as in Sects. 2.1 and 2.2. On
solving the coupled equations, the resulting dispersion relation is given by
Eq. (36), which has the solution Eq. (40). Also, we caution the reader that
the form for the solution of the dispersion relation (Eq. (40)) and the
subsidiary variables $\alpha_{\rm s}$, $\alpha_{\rm g}$, $\beta_{\rm s}$ and
$\beta_{\rm g}$ (Eq. (37)) (in terms of which Eq. (40) is written) may appear
similar to that in Jog & Solomon (1984, which treated a two-fluid case), and
Ghosh & Jog (2015, which treated a star-gas case where the effect of stellar
dispersion was included in terms of the reduction factor $\mathcal{F}$). In
fact these were defined to have a similar form by construction because all
three are two-component formulations and have a similar underlying
mathematical symmetry, but with different treatment for stars and gas, and the
current work includes the effect of finite height.

##  3 Dimensionless form of dispersion relations in the WKB limit

In the literature, usually the dispersion relations are described in terms of
some dimensionless quantities, for the sake of convenience. Here, we follow
the same procedure for the dispersion relations derived in the previous
sections.

###  3.1 One-component fluid disc with finite thickness

Dividing both sides of Eq. (20) by $\kappa^{2}$, and after some algebraic
simplification we get

$s^{2}=1-x\delta+\frac{1}{4}x^{2}Q^{2}\,,$ (41)

where $s=({\omega-m\Omega})/{\kappa}$, and and $x$ (= $|k|/k_{\rm crit}$) are
the dimensionless frequency and wavenumber of the perturbation, respectively;
and $k_{\rm crit}(=\kappa^{2}/2\pi G\Sigma_{\rm 0})$ is the largest stable
wavenumber for a pressure-less stellar disc. $Q(=\kappa c/\pi G\Sigma_{0})$ is
the usual Toomre $Q$ parameter for a fluid disc (Toomre, 1964). Also, $|k|h$
can be expressed as $|k|h=(|k|/k_{crit})\times(k_{crit}h)=x\beta$. Here,
$\beta$ is defined to be equal to $k_{\rm crit}\times h$. Therefore, the
dispersion relation becomes

$s^{2}=1-x\delta+\frac{1}{4}x^{2}Q^{2}\,,$ (42)

where the form of $\delta$ reduces to

$\delta=\frac{1-exp(-x\beta)}{x\beta}\,.$ (43)

Now, the value of $\beta$ is dependent on the chosen values of $k_{\rm crit}$,
i.e., for the same thickness of a disc, the values of $\beta$ will be
different depending on the values of $k_{\rm crit}$. We note that in the Solar
neighbourhood, a circular velocity ($v_{\rm c}$) of $\sim$ 220 km s-1 and
$\Sigma$ $\sim$ 45 M⊙ pc-2 (e.g. see Mera et al., 1998; Narayan & Jog, 2002b)
will produce $k_{\rm crit}$ $\sim$ 1 kpc-1. However, recent studies have
reported slightly different values for the circular velocity and the Solar
position (e.g., see Gillessen et al., 2009; Schönrich et al., 2010; Schönrich,
2012; McMillan et al., 2018; Schönrich et al., 2019). The Galactocentric
distance of the Sun is $8.27\mbox{$\>{\rm kpc}$}$ (Schönrich, 2012) which is
in agreement with the other measurements (within their error-bars Gillessen et
al., 2009; McMillan et al., 2018). Also, the circular velocity at the Solar
radius is $237.8\mbox{$\>{\rm km\,s^{-1}}$}$ (Schönrich et al., 2010) which,
in turn, gives circular frequency at the Solar radius as $28.8\mbox{$\>{\rm
km\,s^{-1}\,kpc^{-1}}$}$. Assuming a flat rotation curve, the corresponding
epicyclic frequency becomes $40.7\mbox{$\>{\rm km\,s^{-1}\,kpc^{-1}}$}$. We
use these latest values of $\Omega$ and $\kappa$ and consider the group
transport at the solar neighbourhood ($R=8.27$ kpc) in the subsequent
sections, unless stated otherwise. Using these recent values, we estimate
$k_{\rm crit}$ $\sim$ 1.3 kpc-1. Therefore, for the sake of uniformity, we
chose $k_{\rm crit}$ = 1.3 kpc-1 for all cases considered here, unless stated
otherwise. The resulting behaviour of the reduction factor $\delta$ is shown
in Appendix A.

###  3.2 One-component stellar disc with finite thickness

Dividing both sides of Eq. (28) by $\kappa^{2}$ and after some algebraic
simplification we get

$s^{2}=1-x\delta\mathcal{F}(s,\chi)\,,$ (44)

where, $\chi$= ${k^{2}\sigma^{2}_{\rm s}}/{\kappa^{2}}$ = $0.286Q_{\rm
s}^{2}x^{2}$, and $\delta$ is already given in Eq. (43). The form for
${\mathcal{F}}$ for a razor-thin disc whose stellar equilibrium state is
described by the Schwarzschild distribution function, is given by (Binney &
Tremaine, 2008):

${\mathcal{F}}(s,\chi)=\frac{2}{\chi}\exp(-\chi)(1-s^{2})\sum_{n=1}^{\infty}\frac{I_{n}(\chi)}{1-s^{2}/n^{2}}\,,$
(45)

where $I_{n}$ is the modified Bessel function of first kind.

###  3.3 Two-component star-gas system with different finite thickness

Proceeding as before, on dividing both sides of Eq. (40) by $\kappa^{2}$ and
after some algebraic simplification we get

$s^{2}=\frac{1}{2}\left[(\alpha^{\prime}_{\rm s}+\alpha^{\prime}_{\rm
g})-\left\\{(\alpha^{\prime}_{\rm s}+\alpha^{\prime}_{\rm
g})^{2}-4(\alpha^{\prime}_{\rm s}\alpha^{\prime}_{\rm g}-\beta^{\prime}_{\rm
s}\beta^{\prime}_{\rm g})\right\\}^{1/2}\right]\\\ ,$ (46)

where

$\begin{split}\alpha^{\prime}_{\rm
s}=1-(1-\epsilon)x\delta_{s}{\mathcal{F}}(s,\xi)\\\ \alpha^{\prime}_{\rm
g}=1-\epsilon x\delta_{g}+\frac{1}{4}Q^{2}_{\rm g}\epsilon^{2}x^{2}\\\
\beta^{\prime}_{\rm s}=(1-\epsilon)x\delta_{s}{\mathcal{F}}(s,\xi)\\\
\beta^{\prime}_{\rm g}=\epsilon x\delta_{g}\\\ \end{split}$ (47)

where, $\xi$= ${k^{2}\sigma^{2}_{\rm s}}/{\kappa^{2}}$ = $0.286Q_{\rm
s}^{2}(1-\epsilon)^{2}x^{2}$. The three dimensionless parameters $Q_{\rm s}$,
$Q_{\rm g}$ and $\epsilon$ are, respectively the Toomre $Q$ parameters for
stars $Q_{\rm s}$(=$\kappa\sigma_{\rm s}/(3.36G\Sigma_{\rm 0s})$), and for gas
$Q_{\rm g}$ = ($\kappa c_{\rm g}/(\pi G\Sigma_{\rm 0g})$) and, $\epsilon$
=${\Sigma_{\rm 0g}}/({\Sigma_{\rm 0s}+\Sigma_{\rm 0g}})$ the gas mass fraction
in the disc, respectively. Also, the forms of $\delta_{i}$ are given by

$\delta_{i}=\frac{1-exp(-x\beta_{i})}{x\beta_{i}}\,.$ (48)

where $i=s,g$ for stars and gas, respectively. We note that, when the height
$h_{i}\rightarrow 0$, and so $\delta_{i}\rightarrow 1$, Eq. (46) reduces to
the dispersion relation for the gravitationally coupled two-component (stars
plus gas) system which is infinitesimally thin (see Ghosh & Jog, 2015), as
expected.

##  4 Results

Here, we present the results related to the dynamical effect of inclusion of a
finite thickness of the disc, with or without the presence of the interstellar
gas on the disc stability and the longevity of a spiral density wave. For a
bi-symmetric ($m=2$) spiral density wave, the pattern speed $\Omega_{\rm
p}=\omega/2$ and $s=(\omega-2\Omega)/\kappa=2(\Omega_{\rm p}-\Omega)/\kappa$.
Here, $s=0$ corresponds to the corotation (CR) point and $|s|$ = 1 gives the
Lindblad resonances (Ghosh & Jog, 2016).

To quantify the disc stability, we define a quantity $|s|_{\rm cut-off}$ as
the lowest value of the dimensionless frequency ($s$), corresponding to an
appropriate dispersion relation, for which one is able to obtain a real or
stable wave solution at any given $R$ (for details, see e.g., Binney &
Tremaine, 2008; Ghosh & Jog, 2016). For $s$ lying in the forbidden region
namely, between the corotation or $s=0$ and $s_{\rm cut-off}$, the solution is
imaginary and the wave is transient, at any given radius $R$ (for details see
discussions in Binney & Tremaine, 2008; Ghosh & Jog, 2016). In other words,
the value of $|s|_{\rm cut-off}$ denotes the edge of the forbidden region. A
decrease in the value of $|s|_{\rm cut-off}$ signifies the decrease in the
forbidden zone; this results in the disc becoming more prone to being unstable
against the perturbations. Conversely, an increase in the value of $|s|_{\rm
cut-off}$ denotes the increase in the forbidden zone, and the disc becomes
more stable against the perturbations.

As for the longevity of the spiral density wave, following Toomre (1969), we
study the radial group transport of a wavepacket of such a density wave. To
achieve that, we calculate the radial group velocity from the local dispersion
relation, appropriate for a particular system considered here. It is known
that the information from a disturbance, generated at a certain radius $R$,
propagates in the disc with its group velocity $v_{\rm g}$. When the medium is
inhomogeneous, the group velocity, at a given radius $R$, is defined as (e.
g., see Whitham, 1960; Lighthill, 1965)

$v_{g}(R)=\frac{\partial\omega(k,R)}{\partial k}\,.$ (49)

The value of the group velocity ($v_{\rm g}$), at a given $R$, can be
estimated from the slope of the local dispersion relation (when expressed in a
dimensionless form) by using the following equation (e. g., see Toomre, 1969;
Binney & Tremaine, 2008)

$v_{g}(R)=sgn(ks)\left(\frac{\kappa}{k_{\rm crit}}\right)\frac{ds}{dx}\,.$
(50)

Here, $s$ and $x$ are the dimensionless frequency and the wavenumber of the
perturbation, respectively, and $sgn(ks)=\pm 1$ depending on whether $ks>0$ or
$ks<0$. Here, the slope is obtained at a point $x$ where the observed value of
$s$ intersects the dispersion relation curve. Thus, the location (or the $x$
value) where the slope is to be calculated is determined by the pattern speed
($\Omega_{\rm p}$) value as well as the underlying mass distribution (which in
turn sets the values of $\Omega$ and $\kappa$), for more details see Sect.
4.1.

A decrease in the group velocity implies that a wavepacket of such density
wave would take a longer time to reach the centre of the disc, and eventually
get absorbed. In other words, a decrease in the group velocity signifies a
longer persistence of the spiral density waves in the disc (for further
discussions, see, e. g., Toomre, 1969; Ghosh & Jog, 2015). Sect. 4.1 provides
the details of the effect of finite thickness on the disc instability and
longevity of spiral density wave for the one-component stellar disc whereas
Sect. 4.2 gives the same for the gravitationally-coupled two-component (stars
plus gas) disc. We mention that the methodology of investigating the disc
stability against the $m=2$ spiral density wave is similar to that in Ghosh &
Jog (2016) whereas the treatment for calculation of the group velocity is
similar to that in Ghosh & Jog (2015), except here we use the appropriate
dispersion relations for a one-component stellar disc, and a two-component
disc with finite height, as derived in the previous section.

###  4.1 Effect of finite thickness on one-component stellar disc

To study the dynamical effect of the disc finite thickness, first we calculate
the dispersion relation (Eq. (44)) for Toomre $Q=1.1$ while varying the disc
thickness, or equivalently $\beta$ from 0.1 to 0.3. Fig. 1 shows the
corresponding dispersion relations. For comparison, we also show the
corresponding dispersion relation for an infinitesimally-thin stellar disc
with $Q=1.1$. Toomre (1969) assumed an $Q=1$ which denotes the neutral
stability of the stellar disc. Here, instead, we assume a slightly higher
value of $Q=1.1$ such that stellar disc is stable against axisymmetric
perturbation, but the self-gravity is still important (for details, see
Toomre, 1964; Jog & Solomon, 1984).

Figure 1: Dispersion relation for the one-component stellar disc (Eq (44)) is
shown for $Q=1.1$ and for $\beta$ = 0.1–0.3. For comparison, the corresponding
dispersion relation for the zero-thickness (or infinitesimally-thin) stellar
disc is also shown (black dashed line). The horizontal dashed line (in black)
denotes the $|s|_{\rm obs}$ value (calculated at $R=8.27\mbox{$\>{\rm kpc}$}$)
corresponding to an assumed pattern speed of $12.5\mbox{$\>{\rm
km\,s^{-1}\,kpc^{-1}}$}$.

####  4.1.1 Disc stability against spiral density waves

A visual inspection of Fig. 1 reveals that with increasing thickness of the
disc, the $|s|_{\rm cut-off}$ values increase steadily. This implies that the
stellar disc becomes more and more stable against the non-axisymmetric
perturbations. In order to study this more quantitatively, we calculate the
$|s|_{\rm cut-off}$ values from the dispersion relation while systematically
varying the Toomre $Q$ values from $1.1$ to $2$, and $\beta$ = 0.1, 0.3, and
0.5. The resulting variation of the $|s|_{\rm cut-off}$ values for different
$\beta$ and Toomre $Q$ parameter values are shown in Fig. 2. For reference, we
also show the corresponding $|s|_{\rm cut-off}$ values for a one-component
stellar disc with zero thickness (i. e., $\beta=0$). As seen clearly from Fig.
2, for a fixed Toomre $Q$ parameter, the $|s|_{\rm cut-off}$ value increases
monotonically with increasing $\beta$ values. Although this trend remains true
for the whole range of Toomre $Q$ values considered here, the effect of
increase in the $|s|_{\rm cut-off}$ value due to disc thickness is more
prominent for lower Toomre $Q$ values considered here. For example, for
$Q=1.1$, the $|s|_{\rm cut-off}$ values increase almost by a factor of $2$ on
changing the $\beta$ from 0 (zero-thickness) to 0.5 ($\sim 385$ pc); whereas
the corresponding change is much smaller, by $\sim 7\%$ for $Q=2$. This trend
is not surprising, since for lower values of $Q$, the disc self-gravity is
more relevant, and consequently the reduction in the self-gravity (due to
finite-thickness) will be more pronounced as compared to the case with a
higher Toomre $Q$ value. The physical implications of the increasing $|s|_{\rm
cut-off}$ value with thickness is discussed below.

Figure 2: One-component stellar disc: Variation of the $|s|_{\rm cut-off}$
values as a function of the Toomre $Q$ parameter are shown for different disc
thickness ($\beta$). The black dashed line shows the corresponding $|s|_{\rm
cut-off}$ values for a one-component stellar disc which is infinitesimally-
thin. For a fixed Toomre $Q$, the $|s|_{\rm cut-off}$ values increase
monotonically with $\beta$.

####  4.1.2 Effect of finite thickness on the allowed range of pattern speed
values

As the $|s|_{\rm cut-off}$ value increases with increasing thickness, this
implies that the forbidden region (i. e., the region between the CR and
$|s|_{\rm cut-off}$) also increases. In other words, introduction of a finite
thickness of the disc helps to stabilise it against the non-axisymmetric
perturbations. The increase in the $|s|_{\rm cut-off}$ value with increasing
thickness has far-reaching impact on the pattern speed ($\Omega_{\rm p}$)
value of a stable, spiral density wave. In a collisionless stellar disc, a
spiral density wave exists only in those regions where

$\Omega-\kappa/2\leq\Omega_{\rm p}\leq\Omega+\kappa/2\,,$ (51)

is satisfied, and the equality holds only at the Lindblad resonance points
(for details see, e. g., Binney & Tremaine, 2008). Also, at a certain radius
$R$, and for a given pattern speed ($\Omega_{\rm p}$), the density wave will
be stable if the corresponding $|s|(=m|\Omega_{\rm p}-\Omega|/\kappa)$ is
greater than the $|s|_{\rm cut-off}$ value, say $\alpha$, of the corresponding
dispersion relation at that radius $R$ (for detailed discussion see Binney &
Tremaine, 2008; Ghosh & Jog, 2016). In other words, for obtaining a stable
spiral density wave, with pattern speed $\Omega_{\rm p}$, the following
condition needs to be satisfied

$\Omega_{\rm p}\geq\Omega+\alpha\kappa/2\hskip 5.69046pt\mbox{or, }\hskip
5.69046pt\Omega_{\rm p}\leq\Omega-\alpha\kappa/2\,,$ (52)

at radius $R$, depending on whether the radius $R$ falls outside the CR or
inside the CR. Therefore, by combining Eqs. (51) and (52), the allowed range
of pattern speed ($\Omega_{\rm p}$) values for a stable, spiral density wave
is

$\Omega_{\rm p}\in[\Omega-\kappa/2,\Omega-\alpha\kappa/2]\hskip
5.69046pt\mbox{or, }\hskip 5.69046pt\Omega_{\rm
p}\in[\Omega+\alpha\kappa/2,\Omega+\kappa/2]\,,$ (53)

depending on whether the radius $R$ is inside the corotation or outside the
corotation. Now, as the value of $\alpha$ (or $|s|_{\rm cut-off}$) increases
with thickness, the allowed range of pattern speed values for a stable spiral
density wave become progressively narrower. This is another finding of this
work. Note that the inclusion of gas allows a larger range of allowed pattern
speed values (Ghosh & Jog, 2016). Thus, the effects of the gas and the disc
thickness have an opposite effect on the range of pattern speed values
corresponding to a stationary (non-evanescent) spiral density wave. This is
further discussed in Sect. 4.2.

####  4.1.3 Radial group transport and effect on longevity of spiral density
waves

Next, we study whether the radial group transport, and hence the longevity of
a spiral density wave gets altered with the inclusion of finite thickness of
the collisionless stellar disc. In classical density wave theory, the pattern
speed ($\Omega_{\rm p}$) of the spiral arms is a free parameter (e. g., see
Lin & Shu, 1964, 1966). Observationally, the pattern speed of spiral density
waves has been measured only for a few external galaxies (e. g, Fathi et al.,
2007, 2009) apart from the Milky Way. Therefore, driven by purely theoretical
interest, we treat the pattern speed ($\Omega_{\rm p}$) as a free parameter.
For investigating the effect of disc thickness on the group velocity, we first
assume the pattern speed to be $\Omega_{\rm p}=12.5\mbox{$\>{\rm
km\,s^{-1}}$}$, the same value as used in Toomre (1969). Also, we choose
$Q=1.1$ here, and vary $\beta$ from 0 to 0.3. Now, using the definition of the
dimensional quantity $s$ as given earlier in this section (i.e.,
$|s|=2(|\Omega_{p}-\Omega|)/\kappa$) and using $\Omega$ and $\kappa$ values
for a particular galaxy at a given radius, and a given value of $\Omega_{\rm
p}$ gives $|s|_{\rm obs}$, the observed value of $|s|$. This is shown by the
horizontal line in Fig. 1. Then, we compute the group velocity ($v_{g}$) at
$R=8.27\mbox{$\>{\rm kpc}$}$, from the slopes of the corresponding dispersion
relations (as shown in Fig. 1) with varying $\beta$ values, the slope is
obtained where the line $|s|_{\rm obs}$ intersects the dispersion relation.
The resulting values of the group velocity ($v_{g}$) and the time that one
such wavepacket (of density wave) would take to travel a distance of
$10\mbox{$\>{\rm kpc}$}$ are listed in Table 1. Before proceeding to
interpreting dynamical effect of thickness on the group velocity, the location
where the slopes are being calculated, merits a discussion. For an assumed
$\Omega_{\rm p}$ value and the values of $\Omega$, and $\kappa$ (set by the
underlying mass distribution), the corresponding $|s|_{\rm obs}$ would
intersect the dispersion relations at two points : one at the long wavelength
branch (lower $x$ value), and another at a short wavelength branch (higher $x$
value). For a given $\Omega_{\rm p}$ value, we always calculate the group
velocity in the short wavelength branch regime, as the WKB approximation works
better there (for details, see Binney & Tremaine, 2008).

Table 1: Group velocity for one-component stellar disc (with $Q=1.1$) for various thickness values, calculated at $R=8.27\mbox{$\>{\rm kpc}$}$. $\Omega_{\rm p}$ | $|s|_{\rm obs}$ | $\beta$ | $ds/dx$ | $v_{g}$ | Time to travel
---|---|---|---|---|---
($\>{\rm km\,s^{-1}\,kpc^{-1}}$) |  |  |  | $(\mbox{$\>{\rm km\,s^{-1}}$})$ |  $10\mbox{$\>{\rm kpc}$}$ (Gyr)
10 | 0.92 | 0 | 0.032 | 1.3 | 7.5
|  | 0.1 | 0.04 | 1.62 | 6.03
|  | 0.2 | 0.045 | 1.83 | 5.3
|  | 0.25 | 0.047 | 1.91 | 5.1
|  | 0.3 | 0.05 | 2.03 | 4.8
12.5 | 0.8 | 0 | 0.107 | 4.34 | 2.25
|  | 0.1 | 0.123 | 5 | 1.95
|  | 0.2 | 0.137 | 5.56 | 1.75
|  | 0.25 | 0.142 | 5.77 | 1.69
|  | 0.3 | 0.143 | 5.81 | 1.68
15 | 0.68 | 0 | 0.181 | 7.35 | 1.32
|  | 0.1 | 0.199 | 8.1 | 1.2
|  | 0.2 | 0.203 | 8.25 | 1.18
|  | 0.25 | 0.204 | 8.29 | 1.17
|  | 0.3 | 0.205 | 8.3 | 1.16

From Table 1, it is clearly seen that for $\Omega_{\rm p}=12.5\mbox{$\>{\rm
km\,s^{-1}\,kpc^{-1}}$}$ and $Q=1.1$, the value of the group velocity
increases steadily with increasing disk thickness ($\beta$). As the $\beta$
value changes from 0 to 0.3, the group velocity ($v_{g}$) increases by $\sim
35$ per cent. This implies that a wavepacket would take less time to reach the
centre of the disc (and get absorbed eventually) for a disc with finite
thickness when compared to that of an infinitesimally-thin disc. In other
words, a spiral density wave would survive for a lesser time for a stellar
disc with finite thickness when compared to an infinitesimally-thin stellar
disc. Next, we choose $\Omega_{\rm p}=10\mbox{$\>{\rm
km\,s^{-1}\,kpc^{-1}}$}$, and $15\mbox{$\>{\rm km\,s^{-1}\,kpc^{-1}}$}$, and
recalculate the variation of the group velocity with changing disc thickness.
Here also we choose $Q=1.1$. The corresponding results are also given in Table
1. We find that, for these pattern speed values, the group velocity increases
with the disc thickness, similar to the case of $\Omega_{\rm
p}=12.5\mbox{$\>{\rm km\,s^{-1}\,kpc^{-1}}$}$. However, the amount by which
the group velocity changes, varies with the assumed pattern speed values. To
elaborate, the group velocity increases by $\sim 56$ percent for $\Omega_{\rm
p}=10\mbox{$\>{\rm km\,s^{-1}\,kpc^{-1}}$}$ whereas the the group velocity
increases by $\sim 13$ percent for $\Omega_{\rm p}=15\mbox{$\>{\rm
km\,s^{-1}\,kpc^{-1}}$}$, once the thickness ($\beta$) is increased from 0 to
0.3. We also consider a higher Toomre $Q$ value, namely, $Q=1.5$ and study the
variation of the group velocity with disc thickness. For the sake of brevity,
the detailed variations in the group velocity values are not shown here.
However, we find that, for an $Q=1.5$, and $\Omega_{\rm p}=12.5\mbox{$\>{\rm
km\,s^{-1}\,kpc^{-1}}$}$, the corresponding group velocity ($v_{g}$) increases
by $\sim 16$ per cent for an increase in thickness from $\beta=0$ to
$\beta=0.3$, as compared to $\sim 35$ per cent for $Q=1.1$ obtained earlier.

We note that the measured group velocity is critically dependent on the
location where the slope is being measured along the dispersion relation.
Nevertheless, an increment in the group velocity value with the increasing
disc thickness is seen to be a generic phenomenon, as shown here. Lastly, we
point out that for the one-component stellar disc case, we could not explore a
higher Toomre $Q$ value and/or a higher pattern speed value, because for these
cases, the measured $|s|_{\rm obs}$ value is found to be lower that the
$|s|_{\rm cut-off}$ value. Consequently, the $|s|_{\rm obs}$ does not
intersect the corresponding dispersion relation. In other words, for higher
Toomre $Q$ values, and a higher pattern speed value, e.g., $\Omega_{\rm
p}=18\mbox{$\>{\rm km\,s^{-1}\,kpc^{-1}}$}$ does not admit a real solution in
$k$ (or equivalently, $x$), and so we could not calculate the group velocity
using Eq. (49). We will explore this parameter regime in the next section
where the interstellar gas is taken into account.

####  4.1.4 Radial variation of the group transport

So far, we have calculated the group velocity of a typical wavepacket, made of
density wave, at a certain radius, say $R$, to study the effect of the finite
thickness of a stellar disc on the longevity of the spiral density wave. We
have so far considered the group transport at the solar neighbourhood $R=8.27$
kpc. However, in reality, any spiral arm in a disc galaxy has a finite radial
extent. Here, we study how the group velocity of a wavepacket changes at
different radii for different thickness, and consequently how this affects the
longevity of the spiral density wave.

We mention that, for an assumed flat rotation curve in the outer disc region,
as done here, the values of $\kappa$ and $\Omega$ will change at different
radial locations. Therefore, for an assumed value of $\Omega_{\rm p}$ which
remains constant with respect to radius, the corresponding $|s|_{\rm obs}$
value would change at different radial location, so would the $x$ values where
the $|s|_{\rm obs}$ cuts the local dispersion relation. To evaluate this, we
first take $\Omega_{\rm p}=12.5\mbox{$\>{\rm km\,s^{-1}\,kpc^{-1}}$}$ and
$Q=1.1$ (same as Table 1 which was done for $R=8.27$ kpc), and redo the group
velocity calculation at $R=6.27\mbox{$\>{\rm kpc}$}$, $R=7.27\mbox{$\>{\rm
kpc}$}$, and $R=9.27\mbox{$\>{\rm kpc}$}$. The results are given in Table 2.
From Table 2, it is clearly seen that at three different radial locations we
considered here, the variation of the finite thickness from $\beta=0$ to
$\beta=0.3$, leads to a monotonic increase of the group velocity of the
density wavepacket. To express it more quantitatively, at $R=6.27\mbox{$\>{\rm
kpc}$}$, the group velocity increases by $\sim 66.2$ percent when $\beta$ is
varied from $0$ to $0.3$ whereas at $R=7.27\mbox{$\>{\rm kpc}$}$, and
$R=9.27\mbox{$\>{\rm kpc}$}$, the corresponding group velocity increases by
$\sim 49.8$ and $\sim 23.1$ percent, respectively (for the same $\beta$
variation). This trend is in concordance with that seen at
$R=8.27\mbox{$\>{\rm kpc}$}$ (see Table 1). In other words, the finite
thickness has a similar (qualitative) effect on the group velocity as well as
the longevity of spiral density waves. We checked this trend for the other
considered pattern values, namely, $\Omega_{\rm p}=15\mbox{$\>{\rm
km\,s^{-1}\,kpc^{-1}}$}$, and $\Omega_{\rm p}=10\mbox{$\>{\rm
km\,s^{-1}\,kpc^{-1}}$}$ as well. We found a qualitative trend in the results
similar to what is seen for $\Omega_{\rm p}=12.5\mbox{$\>{\rm
km\,s^{-1}\,kpc^{-1}}$}$, as long as the the dispersion relation admits a real
solution in $x$ in the short wavelength brach for the corresponding $|s|_{\rm
obs}$ value (for details, see previous section). For brevity they are not
shown here. To conclude, the finite thickness of the stellar disc has a
similar quenching effect on the longevity of spiral density waves at different
radial locations (covering the radial extent of spirals).

Table 2: Radial variation of group velocity for one-component stellar disc (with $Q=1.1,\Omega_{\rm p}=12.5\mbox{$\>{\rm km\,s^{-1}\,kpc^{-1}}$}$), calculated at three radial locations. $R$ | $|s|_{\rm obs}$ | $\beta$ | $ds/dx$ | $v_{g}$ | Time to travel
---|---|---|---|---|---
($\>{\rm kpc}$) |  |  |  | $(\mbox{$\>{\rm km\,s^{-1}}$})$ |  $10\mbox{$\>{\rm kpc}$}$ (Gyr)
6.27 | 0.94 | 0 | 0.021 | 1.14 | 8.5
|  | 0.1 | 0.027 | 1.46 | 6.7
|  | 0.2 | 0.032 | 1.71 | 5.7
|  | 0.25 | 0.034 | 1.82 | 5.35
|  | 0.3 | 0.035 | 1.9 | 5.1
7.27 | 0.87 | 0 | 0.06 | 2.8 | 3.5
|  | 0.1 | 0.074 | 3.42 | 2.85
|  | 0.2 | 0.083 | 3.8 | 2.56
|  | 0.25 | 0.086 | 3.9 | 2.45
|  | 0.3 | 0.09 | 4.2 | 2.32
9.27 | 0.72 | 0 | 0.156 | 5.6 | 1.73
|  | 0.1 | 0.178 | 6.4 | 1.51
|  | 0.2 | 0.184 | 6.7 | 1.45
|  | 0.25 | 0.188 | 6.8 | 1.43
|  | 0.3 | 0.192 | 6.9 | 1.41

###  4.2 Effect of finite height on two-component star-gas system

In the earlier section, we show that the inclusion of finite thickness of the
stellar disc makes it more stable against the non-axisymmetric perturbations,
and increases the group velocity of a wavepacket, thereby decreasing the
longevity of the spiral density wave. Further, Ghosh & Jog (2015) showed that
for a gravitationally-coupled star-gas system where both the stellar and the
gas discs are infinitesimally-thin, the inclusion of gas helps the spiral
density waves to sustain for a longer time. Therefore, it is natural to
investigate how the longevity of the spiral density wave is affected in a
stars plus gas system with finite disc thickness.

Figure 3: Dispersion relations for the two-component stars plus gas system
with finite thickness (Eq. (46)) are shown for $Q_{\rm s}=1.5$, $Q_{\rm
g}=1.3$, and for different gas-fractions : $\epsilon=0.15$ (top panel),
$\epsilon=0.2$ (bottom panel). The thickness of the stellar disc ($\beta_{\rm
s}$) is varied from 0.1 to 0.3, whereas the thickness of the gas disc
($\beta_{\rm g}$) is kept fixed at 0.1 throughout all cases shown here. For
comparison, the corresponding dispersion relation for the zero-thickness stars
plus gas system is also shown (black dashed line). Figure 4: Two-component
stars plus gas system : Variation of the $|s|_{\rm cut-off}$ values as a
function of Toomre $Q$ parameter for the stellar disc ($Q_{\rm s}$) are shown
for different stellar disc thickness ($\beta_{\rm s}$), and for various Toomre
$Q$ parameter for the gas disc ($Q_{\rm g}$), and gas-fraction ($\epsilon$).
The thickness of the gas disc ($\beta_{\rm g}$) is kept fixed at 0.1
throughout all the cases shown here.

To achieve that, first we calculate the dispersion relation (Eq. (46)) for the
gravitationally-coupled star-gas system for different thickness of stellar
disc ($\beta_{\rm s}$) while assuming $Q_{\rm s}=1.5$, $Q_{\rm g}=1.3$, and
taking two values of $\epsilon$, namely, $0.15$, and $0.2$. We mention that,
the thickness of the gas disc ($\beta_{\rm g}$) is kept fixed at 0.1
throughout this paper, unless stated otherwise. The resulting dispersion
relations are shown in Fig. 3. A visual inspection reveals two broad trends,
namely, for a fixed set of three values $(Q_{\rm s},Q_{\rm g},\epsilon)$, the
value of $|s|_{\rm cut-off}$ increases monotonically with the increase of
thickness of the stellar disc. This trend is consistent with the findings for
the one-component stellar disc. However, we notice that, with increasing gas
fraction ($\epsilon$), the increment in the $|s|_{\rm cut-off}$ value with
thickness, is less (compare top and bottom panels of Fig. 3). Secondly, the
dispersion relation in the short-wavelength branch become increasingly flat
with the increase of disc thickness, when compared with the same for the
infinitesimally-thin star-gas system. This holds true for both the gas-
fraction values considered here.

####  4.2.1 Disc stability against spiral waves, and allowed pattern speed
values

In order to probe the joint effect of the disc thickness and the inclusion of
gas on the variation of $|s|_{\rm cut-off}$ values, we systematically
calculate the relevant dispersion relations (using Eq. (46)) for a wide range
of $Q_{\rm s}$, $Q_{\rm g}$, $\beta_{\rm s}$, and $\epsilon$. Then we compute
the corresponding $|s|_{\rm cut-off}$ values. The resulting variation of
$|s|_{\rm cut-off}$ values are shown in Fig. 4. Fig. 4 clearly demonstrates
the generic trend that, for a fixed value of $(Q_{\rm s},Q_{\rm g},\epsilon)$,
the $|s|_{\rm cut-off}$ increases monotonically while the disc thickness
($\beta_{s}$) is varied from 0 to 0.5. However, the variation in the $|s|_{\rm
cut-off}$ values is more for a lower $Q_{\rm s}$ value (say 1.2) when compared
with a higher $Q_{\rm s}$ value (say, $\sim 2$). These findings are in
compliance with the earlier finding for the one-component stellar disc. Next,
we study how the variation in the $|s|_{\rm cut-off}$ values with thickness,
gets affected by the inclusion of the interstellar gas. We find that, for a
fixed value of ($Q_{\rm g}$, $\beta_{\rm s}$), the $|s|_{\rm cut-off}$ value
for a higher gas-fraction ($\epsilon$) is always lower when compared with the
same for a lower gas-fraction (compare top and bottom panels of Fig. 4). This
trend holds true for the whole range of $Q_{\rm s}$ and $Q_{\rm g}$values
considered here, although the trend is more prominent for the smaller values
of $Q_{\rm s}$. The physical reason behind this trend is when more gas present
in the system (a lower $Q_{\rm g}$ value and a higher $\epsilon$ value), the
destabilising effect of the interstellar gas wins over the stabilising effect
of the finite thickness. This is particularly true when the $Q_{\rm s}$ and
$Q_{\rm g}$ values are smaller (say close to 1) as the self-gravity of the
stars-gas system would be important to determine the dynamical state of the
system. Thus, the net stability of the gravitationally-coupled stars plus gas
system is determined by the joint (mutually opposite) effects of the
interstellar gas and the finite disc thickness. Nevertheless, the
destabilising effect of the interstellar gas gets reduced (at least,
partially) by the stabilising effect of the finite disc thickness in all cases
shown here.Thus, the net range of allowed pattern speed values is higher for a
two-component, finite height disc than a stars-alone, thin disk; but less than
the range allowed for a two-component, thin disc.

####  4.2.2 Radial Group transport and effect on longevity of spiral waves

Finally, we turn to probe the joint effects of the disc finite thickness and
the interstellar gas on the group velocity of the spiral density wave in a
gravitationally-coupled stars-gas system. For the sake of theoretical
interest, we choose a smaller value of $Q_{\rm s}$, say 1.3. Also, for the
moment, we treat $Q_{\rm s}$ and $\epsilon$ as free parameters, and $Q_{\rm
g}$ is set by these two values via the relation $Q_{\rm g}$ = $(0.306Q_{\rm
s})(1-\epsilon)/\epsilon$ (for details see Ghosh & Jog, 2015). Later, we will
relax this constraint and will treat all three parameters, namely, $Q_{\rm
s}$, $Q_{\rm g}$, and $\epsilon$ as free parameters while studying the group
velocity.

To study how the group velocity changes, first we assume a $\Omega_{\rm
p}=12.5\mbox{$\>{\rm km\,s^{-1}\,kpc^{-1}}$}$ which in turn, gives $|s|_{\rm
obs}=0.8$ at $R=8.27\mbox{$\>{\rm kpc}$}$. Then we compute the group velocity
from the slopes of the dispersion relations (Eq. (46)) for $Q_{\rm s}=1.3$ and
$\beta_{\rm g}=0.1$ while varying the gas-fraction ($\epsilon$) from 0.1 to
0.25, and disc thickness ($\beta_{\rm s}$) from 0 to 0.3. The resulting
variations of the slopes and the group velocity values are listed in Table 3.
As evident from Table 3, the value of the group velocity increases with the
increase of disc thickness, and this trend holds true for all values of gas-
fraction considered here. However, there is subtle change with the increasing
gas-fraction. To elaborate, when the gas-fraction is assumed to be 0.1, the
group velocity increases by $\sim 35$ percent for a variation of $0-0.3$ in
the $\beta$ values. However, for a higher gas-fraction value (say
$\epsilon=0.2$), the group velocity increases only by $\sim 12$ percent for
the same variation of $\beta$ values from 0 to 0.3. Further, when we choose an
even higher value of gas-fraction (say, $\epsilon=0.25$), we find a negligible
increase in the group velocity ($\sim 2$ percent) when we vary $\beta$ from 0
to 0.3. The physical reason is as we choose progressively lower values of
$Q_{\rm g}$ and higher values of $\epsilon$, the overwhelming effect of the
interstellar gas in decreasing the group velocity wins over the opposite
effect of disc thickness on the group velocity values. In other words, when
the self-gravity, and the low dispersion of interstellar gas dominates over
the reduction in the self-gravity due to the finite thickness, the group
velocity of a wavepacket, made of spiral density waves, is predominantly set
by the effect of the interstellar gas. Next, we choose a higher pattern speed
value, namely, $\Omega_{\rm p}=15\mbox{$\>{\rm km\,s^{-1}\,kpc^{-1}}$}$ which
in turn, produces $|s|_{\rm obs}=0.68$ $R=8.27\mbox{$\>{\rm kpc}$}$. Then, we
study how the group velocity changes as we vary simultaneously the gas-
fraction, and the disc finite thickness. We find that, when $\epsilon=0.1$,
the group velocity increases by $\sim 20$ percent when the disc thickness is
increased from 0 to 0.3. However, when the gas-fraction is changed to a higher
value (say $\epsilon=0.2$), the group velocity increases only by $\sim 14$
percent when $\beta_{\rm s}$ is varied from 0 to 0.3. This trend is similar to
what we found for the $\Omega_{\rm p}=12.5\mbox{$\>{\rm
km\,s^{-1}\,kpc^{-1}}$}$. Thus, the change in group velocity in presence of
the disc thickness and the interstellar gas is a complex process, as these two
physical factors have opposite effect on the group velocity. The net change in
the group velocity is set by the relative dominance of the effects of the
interstellar gas and the disc thickness.

Table 3: Group velocity for two-component star-gas system (with $Q_{\rm
s}=1.3$) for various thickness values, calculated at $R=8.27\mbox{$\>{\rm
kpc}$}$.

$\Omega_{\rm p}$ | $|s|_{\rm obs}$ | $\epsilon$ | $\beta$ | $ds/dx$ | $v_{g}$ | Time to
---|---|---|---|---|---|---
($\>{\rm km\,s^{-1}}$ |  |  |  |  | $(\mbox{$\>{\rm km\,s^{-1}}$})$ | travel $10\mbox{$\>{\rm kpc}$}$
kpc-1) |  |  |  |  |  | (Gyr)
12.5 | 0.8 | 0.1 | 0 | 0.081 | 3.3 | 2.95
|  |  | 0.1 | 0.1 | 4.1 | 2.37
|  |  | 0.2 | 0.104 | 4.2 | 2.32
|  |  | 0.3 | 0.11 | 4.5 | 2.16
|  | 0.15 | 0 | 0.067 | 2.72 | 3.58
|  |  | 0.1 | 0.082 | 3.35 | 2.9
|  |  | 0.2 | 0.083 | 3.4 | 2.86
|  |  | 0.3 | 0.084 | 3.41 | 2.85
|  | 0.2 | 0 | 0.074 | 3 | 3.24
|  |  | 0.1 | 0.083 | 3.37 | 2.89
|  |  | 0.2 | 0.0832 | 3.38 | 2.88
|  |  | 0.3 | 0.0833 | 3.39 | 2.87
|  | 0.25 | 0 | - | - | -
|  |  | 0.1 | 0.093 | 3.4 | 2.86
|  |  | 0.2 | 0.095 | 3.42 | 2.84
|  |  | 0.3 | 0.095 | 3.42 | 2.84

* The Toomre $Q$ for gas disc ($Q_{\rm g}$) is set by the values of $Q_{\rm s}$ and $\epsilon$, for details see text. The thickness of the gas disc $\beta_{\rm g}$ is fixed at 0.1 for all cases.

To study further the net effect of the interstellar gas and the disc thickness
where the parameters $Q_{\rm s}$, $Q_{\rm g}$, and $\epsilon$ are treated as
free parameters (unlike the previous case), we choose a case where $Q_{\rm
s}=1.3$, and $Q_{\rm g}=1.3$. For these assumed parameters, and for
$\epsilon=0.2$ and $\Omega_{\rm p}=15\mbox{$\>{\rm km\,s^{-1}\,kpc^{-1}}$}$,
the group velocity increases by $\sim 25$ percent when $\beta_{\rm s}$ is
varied from 0 to 0.3. However, for a higher $\epsilon=0.25$, the change in the
group velocity is negligible when $\beta_{\rm s}$ is varied from 0 to 0.3.
This trend further demonstrate the mutual interplay of the (opposite) effects
induced by the interstellar gas and the disc finite thickness. Lastly, we
assume an even higher value of the pattern speed, namely, $\Omega_{\rm
p}=18\mbox{$\>{\rm km\,s^{-1}\,kpc^{-1}}$}$ which in turn, yields an $|s|_{\rm
obs}=0.53$. For the current assumed parameter space ($Q_{\rm s}$, $Q_{\rm g}$,
$\epsilon$, $\beta_{\rm s}$), it is possible to obtain a real solution in $k$
(or alternatively, in $x$) for the $\Omega_{\rm p}=18\mbox{$\>{\rm
km\,s^{-1}\,kpc^{-1}}$}$; thereby allowing us to probe the mutual effect of
the gas and the thickness for a realistic value (or close to observationally
reported values) of the $\Omega_{\rm p}$. Here also, we find the same trend in
the change of the group velocity value as we increase the value of $\beta_{\rm
s}$. More quantitatively, when $\epsilon=0.2$, the group velocity increases by
$\sim 11$ per cent as we vary $\beta_{\rm s}$ from 0 to 0.3. However, the
change in the group velocity is found to be negligible when $\epsilon$ is set
to $0.25$ while $\beta_{\rm s}$ is varied by a same amount as previous case.
This further accentuates the complex mutual interplay of the opposite effects
of the disc thickness and the interstellar gas on the resulting group velocity
values.

Lastly, we explore the mutual effect of the disc thickness and the
interstellar gas for some higher values of $Q_{\rm s}$ while varying the
thickness of the stellar disc up to $\sim 400$ pc (equivalently, $\beta_{\rm
s}\sim 0.5$). This is close to the observed values of the disc thickness in
the external galaxies (e. g., see de Grijs & Peletier, 1997). For that, we
first choose $Q_{\rm s}=1.6$, $Q_{\rm g}=1.5$, and $\epsilon=0.2$, and then
vary the disc thickness ($\beta_{\rm s}$) from 0 to 0.5. We find that, for
this chosen set of values for ($Q_{\rm s}$, $Q_{\rm g}$, $\epsilon$), and for
an assumed $\Omega_{\rm p}$ of $15\mbox{$\>{\rm km\,s^{-1}\,kpc^{-1}}$}$, the
group velocity increases monotonically with increasing disc thickness. We
estimate that, for a variation of the disc thickness from 0 to 0.5, the group
velocity increases by $\sim 37$ per cent. Further, we choose another set of
values for ($Q_{\rm s}$, $Q_{\rm g}$, $\epsilon$), namely, $Q_{\rm s}=2$,
$Q_{\rm g}=2$, and $\epsilon=0.2$. This set of chosen values is typical for
the outer regions of the discs of Magellanic-type irregular galaxies where the
gas fractions are high (e. g., see Gallagher & Hunter, 1984; Jog, 1992). We
then choose a $\Omega_{\rm p}=12.5\mbox{$\>{\rm km\,s^{-1}\,kpc^{-1}}$}$, and
vary the disc thickness from 0 to 0.5. We find that, for this case, the group
velocity increases by $\sim 13$ percent as the disc thickness is varied from 0
to 0.5. Note that, for this case, we could not use a higher pattern speed
value, say $\Omega_{\rm p}=15\mbox{$\>{\rm km\,s^{-1}\,kpc^{-1}}$}$ or higher
as the group velocity approach cannot be applied here. The reason is, the
$|s|_{\rm obs}$ values corresponding to these higher pattern speed values are
almost always less than the $|s|_{\rm cut-off}$ values obtained from the
dispersion relations corresponding to $Q_{\rm s}=2$, $Q_{\rm g}=2$, and
$\epsilon=0.2$, and $\beta_{\rm s}=0.2-0.5$.

####  4.2.3 Coverage on the parameter space

Thus, to conclude, we study the joint effect of the disc thickness and the
interstellar gas on the stability of the disc against non-axisymmetric
perturbations as well as on the longevity of the $m=2$ spiral density waves.
By exploring a range of parameter space, we demonstrate the complex nature of
the mutually opposite effect of the disc thickness and the interstellar gas.
We mention that, while studying this mutually opposite effect on the longevity
of the spiral density wave, we could not carry out a systematic search in the
parameter space, unlike the case of disc stability (or equivalently, the
variation of $|s|_{\rm cut-off}$). The reasons are as follows: first, the
group velocity is obtained using the slope of the local dispersion relation
graphically (as in Toomre, 1969). Although this approach allows us to
calculate the group velocity conveniently, this is a non-robust procedure (for
details, see Toomre, 1969; Ghosh & Jog, 2015). The slope depends critically on
the exact location of $x$ where the $|s|_{\rm obs}$ (corresponding to a
$\Omega_{\rm p}$ value) intersects the dispersion relation. In other words, it
is a local value. Secondly, for a two-component system, the dispersion
relation is a fourth-order polynomial (e.g. Jog & Solomon, 1984). Thus for
certain choice of parameters, if the $|s|_{\rm obs}$ intersects the dispersion
relation in the region of high $x$ that falls in the region of second minimum,
the slope would tend to flatten regardless of the fact whether or not the
solution in $x$, corresponding to that $|s|_{\rm obs}$ value, is real or
imaginary. Hence, while normally a higher $\beta$ or thickness leads to a
steeper slope, in such cases the flattening effect due to the $k^{4}$
behaviour may dominate and hence one may get a smaller slope for higher
$\beta$, and hence leading to a smaller value of the group velocity. This is
opposite to the typical dependence on disc thickness as shown above where a
higher $\beta$ was shown to result in a higher group velocity. In such cases,
the effect of disc thickness would not oppose, instead it would show a similar
trend to the effect of gas. This complex, mixed behaviour is more likely to be
seen at high gas fraction (see Jog & Solomon, 1984) or high $|s|_{\rm obs}$.
This could contribute to a diverse and complex dynamical behaviour, and hence
the results obtained from varying the $\beta$ parameter for a fixed set of
($Q_{\rm s}$, $Q_{\rm g}$, $\epsilon$) values have to be interpreted with
caution.

Also, so far we kept the thickness of the gas disc, $\beta_{\rm g}=0.1$ while
calculating the dispersion relations for a two-component stars-plus-gas
system. Using the $k_{\rm crit}$ value mentioned in section 3.1, this reduces
to a $h_{\rm g}\sim 70$ pc which is typical scale-height for the molecular
hydrogen gas (e.g., Scoville & Sanders, 1987). On the other hand, neutral
hydrogen (H i) shows a typical value for the scale-height as $\sim 150$ pc
(e.g., Lockman, 1984) which corresponds to $\beta_{\rm g}=0.2$. Here, we
briefly state what happens to the findings mentioned above when we set
$\beta_{\rm g}=0.2$ instead of $\beta_{\rm g}=0.1$. We choose $Q_{\rm s}=1.3$,
$Q_{\rm g}=1.3$, and $\epsilon=0.2$, a case already explored in section 4.2.
We then set $\beta_{\rm g}=0.2$, and vary $\beta_{\rm s}$ from 0-0.3, as
before, to see the change in the radial group velocity corresponding to
$\Omega_{\rm p}=15\mbox{$\>{\rm km\,s^{-1}\,kpc^{-1}}$}$, and $\Omega_{\rm
p}=18\mbox{$\>{\rm km\,s^{-1}\,kpc^{-1}}$}$. For a fixed $\beta_{\rm s}$ and
$\Omega_{\rm p}$ values, the group velocity is seen to increase by $\sim
13-55$ per cent for $\beta_{\rm g}=0.2$ when compared with that for
$\beta_{\rm g}=0.1$. This shows that finite thickness of the gas disc also
increases the group velocity for the joint disc (and consequently decreases
the longevity) in a similar fashion as the thickness of the stellar disc. We
also considered a few other set of ($Q_{\rm s}$, $Q_{\rm g}$, $\epsilon$)
values to test this. We find the general trend of an increasing group velocity
with an increase in thickness of the gas disc, for brevity we have not shown
these cases here.

Nevertheless, we stress that, in general, the inclusion of thickness of the
stellar disc and the interstellar gas have opposing effects on the disc
stability against non-axisymmetric perturbations and the radial group
transport, as illustrated in terms of the typical examples in this section
(also see Table 3). In general, for the observed disc thickness and gas
fraction values, as we have discussed in this section, the quenching effect of
the height does not overcome (completely) the supporting role played by the
gas, and hence a disc would still be expected to host non-axisymmetric
features whose longevity will be supported by the gas.

####  4.2.4 Radial variation of the group transport for a two-component disc

As before, we study here the radial variation of the effect of finite
thickness on the group velocity of a wavepacket, made of density wave, for a
gravitationally-coupled stars-gas system. As before, we choose three radial
locations at $R=6.27\mbox{$\>{\rm kpc}$}$, $R=7.27\mbox{$\>{\rm kpc}$}$, and
$R=9.27\mbox{$\>{\rm kpc}$}$. Then we assume a $\Omega_{\rm
p}=12.5\mbox{$\>{\rm km\,s^{-1}\,kpc^{-1}}$}$, and recalculate the group
velocities for different disc thickness and the gas fraction values while
choosing $Q_{\rm s}$ and $Q_{\rm g}$ values identical to what used for Table
3. Here also, we find a similar qualitative trend of mutually-opposing effects
of the finite thickness and the interstellar gas on the group velocity as well
as the longevity of the spiral density waves. To elaborate, at
$R=6.27\mbox{$\>{\rm kpc}$}$, and for $\epsilon=0.1$, and $\Omega_{\rm
p}=12.5\mbox{$\>{\rm km\,s^{-1}\,kpc^{-1}}$}$, the group velocity increases by
$\sim 63.6$ percent when $\beta_{\rm s}$ is varied from $0$ to $0.3$. As the
gas fraction ($\epsilon$) increases, the increment in the group velocity due
to a variation of $\beta$ from 0 to 0.3, starts to diminish monotonically. For
$\epsilon=0.2$, the corresponding increment in the group velocity becomes
$\sim 37.8$ percent, and for $\epsilon=0.25$, the increment in the group
velocity becomes $\sim 33.7$ percent (for the same variation in $\beta$ from 0
to 0.3). In other words, when the gas fraction becomes higher, the dynamical
effect of the interstellar gas in supporting the spiral density wave starts to
dominate over the quenching effect of the finite thickness. This trend is
similar to what was shown for $R=8.27\mbox{$\>{\rm kpc}$}$ (see Table 3). We
further checked for other $\Omega_{\rm p}$ values as well as for other two
radial locations considered here. We found that as long as the dispersion
relation admits a real solution for $x$ in the long wavelength branch for the
corresponding $|s|_{\rm obs}$ value, and the $k^{4}$ behaviour does not
interfere in the locations where the group velocity is being calculated (for
details, see the discussion in previous section), we obtained a similar
qualitative effect of the finite thickness on the group velocity and on the
longevity of the spiral density wave. For brevity, they are not shown here.
Therefore, the finite thickness of the stellar disc has a similar quenching
effect on the longevity of spiral density waves at different radial locations
(covering the radial extent of spirals) for the gravitationally-coupled two-
component disc as well.

##  5 Discussion

Here, we discuss a couple of points relevant for this work.

  * •

At large values of scale height, its effect begins to dominate the effect of
gas, hence the net effect is to increase the group velocity which would lead
to a shorter lifetime of the density wave. A typical galaxy is known to show a
flaring stellar disc as shown observationally by de Grijs & Peletier (1997).
This feature occurs naturally as shown by the theoretical modelling of a
multi-component galactic disc (Narayan & Jog, 2002a). The Milky Way stellar
disc also shows flaring as seen observationally from several surveys, e.g.,
2MASS (Momany et al., 2006), SEGUE (López-Corredoira & Molgó, 2014), LAMOST
(Wang et al., 2018) as well as LAMOST plus Gaia (Yu et al., 2021). Such
flaring was explained by theoretical modelling of a multi-component disc
(Narayan & Jog, 2002b), especially for the outer Galaxy by Sarkar & Jog
(2018). This could explain in a generic way why spiral features are rare in
outer regions even though such regions tend to be gas-rich.

  * •

The formulation presented here employs the linear perturbation theory.
However, the growth of the spirals in a real galaxy can enter the non-linear
regime as well. The study by D’Onghia et al. (2013) demonstrated that the non-
linear response of the disc to the non-axisymmetric perturbations can alter
the life-time of spiral instability in a disc galaxy. While the study
presented here, clearly demonstrates the mutually-opposite effect of the
interstellar gas and the disc thickness on the disc stability and persistence
of spiral density wave, it will be worth pursuing this effect in numerical
simulations of disc galaxies. Also, we mention that, here the spiral structure
is modelled as a two-dimensional structure for simplicity, as is typically
done. However, some of the recent self-consistent simulations of growth of the
spirals in disc galaxies (e. g., see Debattista, 2014; Ghosh et al., 2020)
show that the spirals can well be vertically extended.

##  6 Conclusion

In summary, we investigated the dynamical impact of the disc thickness on the
disc stability, and the longevity of the spiral density wave. For that, we
derived the dispersion relations in the WKB approximation for a collisionless
stellar disc with finite thickness as well as for a gravitationally coupled
stars-plus-gas system with different thickness for the stars and the gas
discs. The main findings are mentioned below.

  * •

The inclusion of finite thickness effectively reduces the self-gravity of the
disc which in turn makes the disc more stable against the non-axisymmetric
perturbations. This result in case of a one-component stellar disc, holds true
for the whole range of Toomre-$Q$ parameter ($1.2-2$) and the disc thickness
($\sim 50-400$ pc) considered in this work. This stabilising effect is more
prominent for lower values of Toomre-$Q$ where the self-gravity of the disc is
more important.

  * •

The stabilising effect of the finite disc thickness has consequences in
setting up the allowed range of pattern speed values for which the system
allows a non-evanescent spiral density wave. With increasing disc thickness,
this allowed range of pattern speed values gets progressively narrower,
especially for lower values of Toomre-$Q$ parameter. For a joint stars-plus-
gas system, the effect of gas is to increase the range of pattern speed values
while the thickness has an opposite effect. Typically, the net range of
allowed pattern speed values is larger for a two-component, finite height disc
than a stars-alone, infinitesimally-thin disc.

  * •

For a one-component stellar disc, the group velocity of a wavepacket increases
monotonically with the increment of disc thickness, thereby implying a
progressively shorter life-time for the spiral density wave. For the same
change in disc thickness by $\sim 250$ pc, the reduction in the life-time can
vary from $\sim 10-60$ per cent when compared to the case of an
infinitesimally-thin stellar disc; and this reduction depends on the assumed
pattern speed values and the Toomre-$Q$ parameter. In a two-component system,
as the gas fraction is increased, the increase in the group velocity caused by
the effect of finite height is reduced. Hence the inclusion of gas opposes the
reduction in lifetime arising due to the finite thickness.

  * •

Even in the presence of the interstellar gas, the disc thickness tends to
stabilise the two-component (stars-gas) system against the non-axisymmetric
perturbation, and this holds true for the whole parameter space ($Q_{\rm
s}=1.2-2$, $Q_{\rm g}=1.2-2$, $\epsilon\sim 0.10-0.25$) considered here.
However, when the dynamical effect of gas is important (e.g., lower values of
$Q_{\rm g}$, and/or higher $\epsilon$ values), the stabilising effect of the
disc thickness becomes minimal.

  * •

The disc thickness tends to diminish the life-time of the spiral density
waves, even when the gas is present. For the same change in disc thickness by
$\sim 400$ pc, the amount of reduction in the life-time can vary from $\sim
5-40$ per cent when compared to an infinitesimally-thin stars-plus-gas system,
depending on the assumed pattern speed and the $Q_{\rm s}$, $Q_{\rm g}$ and
gas-fraction ($\epsilon$) values. However, for gas-rich systems ($\epsilon\geq
0.25$), the effect of finite disc thickness is shown to be negligible.

Thus, to conclude, we demonstrate in this paper that the interstellar gas and
the disc thickness have an opposite dynamical effect on the disc instability
and the persistence of spiral density waves. While the gas makes a galactic
disc more susceptible against non-axisymmetric perturbations and helps the
spiral density wave to survive for a longer time-scale, the disc thickness
influences the system in an opposite manner. Consequently, the net dynamical
effect is set by the relative dominance of these two physical factors. For the
broad range of parameter space we considered, the quenching effect of the
height does not completely suppress the supporting role of the gas, and hence
a disc would still be expected to host non-axisymmetric features whose
longevity will be supported by the gas.

## Acknowledgements

We thank the anonymous referee for useful comments which helped to improve
this paper. C. J. thanks the DST, Government of India for support via a J.C.
Bose fellowship (SB/S2/JCB-31/2014).

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##  Appendix A Behaviour of the Reduction factor

Fig. 5 shows the variation of the reduction factor, $\delta$ (Eq. 43) with the
dimensionless wavenumber $x$, for various disc thickness (or $\beta$ values).
The reduction due to the disc thickness becomes severe for larger values of
disc thickness, as expected. Also, the reduction factor is seen to become
progressively more important in the short-wavelength (large $x$) branch, as
expected from the form of Eq. 43.

Figure 5: Variation of the reduction factor ($\delta$) as a function of
dimensionless wavenumber ($x$) are shown for different disc thickness ($\beta$
values). The values of $\beta$ are colour-coded here.