11institutetext:  1 Max Planck Institute for Astronomy, D-69117 Heidelberg,
Germany
2 Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität
Heidelberg, D-69120 Heidelberg, Germany
3 Sternberg Astronomical Institute, M.V. Lomonosov Moscow State University,
Moscow, 119234 Russia

(GVS)

# Measurements of the Expansion Velocities of Ionized-Gas Superbubbles in
Nearby Galaxies Based on Integral Field Spectroscopy Data

Grigorii V. Smirnov-Pinchukov1 Oleg V. Egorov2,3<EMAIL_ADDRESS>
(April 30, 2021/Revised: September 6, 2021/Accepted: September 6, 2021)

###### Abstract

The study of dynamic properties of bubbles in the interstellar medium is
important for understanding the feedback mechanisms from star-formation
processes in galaxies. The ongoing integral field spectroscopy of nearby star-
forming galaxies reveals many expanding bubbles and superbubbles identified by
the local increase in gas velocity dispersion. The limited angular resolution
often prevents bona fide measures of the expansion velocities in galaxies
outside the Local Group, even despite sufficient spectroscopic resolution. We
present a method that makes it possible to measure the expansion velocity of
bubbles surrounding massive stars and clusters based on the data about local
variations in gas velocity dispersion. We adapt the method for the Fabry–Perot
interferometers used with the 6-m telescope of the Special Astrophysical
Observatory of the Russian Academy of Sciences, as well as for any integral
field spectrograph with a Gaussian line spread function. We apply the method
described to analyze the kinematics of ionized superbubbles gas and the only
known supernova remnant in the IC 1613 galaxy. The estimate of the kinematic
age of the supernova remnant (on the order of 3100 years) agrees well with the
previously obtained independent estimate based on X-ray data.

###### keywords:

ISM: bubbles; methods: data analysis; ISM: kinematics and dynamics; ISM:
supernova remnants; galaxies: individual: IC 1613

##  1 INTRODUCTION

Stars with masses $M>8~{}M_{\odot}$ play an important role in the evolution of
the interstellar medium in galaxies. Their wind, ionizing radiation, and
subsequent supernova explosions are the source of energy for the formation of
shell-like structures surrounding clusters and OB-associations. The sizes of
the observed bubbles vary over a wide range—from several parsecs to several
hundred parsecs in the case of supershells/superbubbles (Oey and Clarke, 1997;
Nath et al., 2020). Sustained energy inflow from several generations of stars
results in further growth of superbubbles to sizes of 1–3 kpc (Weisz et al.,
2009; Warren et al., 2011). In some dwarf galaxies such giant superbubbles are
the dominant structure of the interstellar medium (Egorov et al., 2018). Such
superbubbles that are larger than a few hundred parsecs are mostly observed in
H I at 21 cm (Bagetakos et al., 2011), although deep images in emission lines
allow detection of similar structures in ionized gas (Egorov et al., 2014,
2017).

The classical model of the evolution of bubbles driven by the winds of massive
stars (Weaver et al., 1977) and the combined wind of multiple stars and
supernova explosions (Mac Low and McCray, 1988) describes the observed
structures in the interstellar medium of galaxies quite well at qualitative
level. According to this model, the radius of a bubble $R$ depends on time $t$
and the inflow of mechanical energy $L$ as $R\sim L^{1/5}t^{3/5}$.
Accordingly, at each moment of time its expansion velocity is $v_{\rm exp}\sim
L^{1/5}t^{-2/5}$. Thus the age of the bubble and the energy required for its
formation can be estimated based on the observed quantities—the size and
expansion velocity. In particular, the age of the shell can be estimated in
terms of the Weaver et al. (1977) model as $t=0.6R/v_{\rm exp}$. Because of
this rather simple analytical dependence the observed parameters of
superbubbles in nearby galaxies can serve as indicators of the properties of
massive stars responsible for the formation of these structures. However, a
numerical comparison of observed superbubbles properties with their estimates
predicted by the (Weaver et al., 1977) model showed a severalfold-factor
discrepancy (Oey and García-Segura, 2004). Radiative energy losses, which can
range from 60% to almost 100% (Sharma et al., 2014; Vasiliev et al., 2015a;
Yadav et al., 2017), are viewed as one of the main causes of this discrepancy.
The new analytical model of superbubble evolution proposed by El-Badry et al.
(2019) takes this factor into account as a parameter. However, it remains an
open question how the efficiency of energy transfer from star clusters into
the mechanical energy of expanding superbubbles changes under different
conditions (e.g., in the case of different metallicities and gas densities).

The development of integral field spectroscopy techniques has made possible a
detailed comparison of observed properties of massive stars and of the
surrounding regions of ionized gas in several nearby galaxies, allowing the
contribution of various feedback processes to the overall process of energy
and momentum exchange with the interstellar medium to be estimated (Egorov et
al., 2014, 2017; McLeod et al., 2019; Ramachandran et al., 2019; McLeod et
al., 2020). However, such studies are significantly limited by the spatial and
spectroscopic resolution of observational data. Thus with the characteristic
sizes of the shells surrounding individual massive stars or small OB-
associations on the order of several tens of parsecs, even at a distance of
4–5 Mpc (e.g., the M 81 group (Karachentsev et al., 2013)) the angular sizes
of such objects often become comparable to the angular resolution of
observational data (on the order of $2\arcsec$–$3\arcsec$), making the
determination of superbubbles sizes and expansion velocities quite a
challenging task. Furthermore, the characteristic observed superbubbles
expansion velocities 20-50 km s-1 make it impossible for most modern integral
field spectrographs to resolve the emission line profiles into components to
reliably estimate their expansion velocities. Observations of a number of
nearby dwarf galaxies made with the Fabry–Perot interferometer (FPI) in the
H$\alpha$ line demonstrate the presence of a substantial number of regions of
increased ionized gas velocity dispersion whose position on the
“intensity–velocity dispersion” diagram ($I$–$\sigma$) allows us to interpret
them as spectroscopically unresolved expanding superbubbles (Moiseev and
Lozinskaya, 2012). Numerical simulations of the interaction of multiple
supernova remnants yield a close to the observed pattern in the ($I$–$\sigma$)
parameter space, which confirms the above interpretation (Vasiliev et al.,
2015b). As shown by Guerrero et al. (1998), such superbubbles should exhibit
radial gradient of the emission line profile width, which can be used to
estimate the expansion velocity, however, such measurements require a rather
high angular resolution. On the other hand, Tenorio-Tagle et al. (1996) showed
that the presence of a spatially unresolved shell also broadenes the observed
emission line profile. In particular, Moiseev and Lozinskaya (2012) proposed
to use this effect to search for unique emission objects with high stellar
wind speeds (e.g., Wolf-Rayet stars, luminous blue variables) on the
($I$–$\sigma$) dependence for the galaxies studied. Thus, even in cases of
insufficient spatial and spectroscopic resolution, expanding bubbles around
stars and clusters can be identified by changes in the width of the spectral
line profile. However, the physical parameters of such bubble (in particular,
its expansion velocity) often cannot be estimated directly without invoking
model assumptions about its geometry.

In this paper, we propose a method that allows us to estimate the expansion
velocity of the bubbles and superbubbles in the interstellar medium from
measurements of the emission line profile width in their integrated spectra
and from the average velocity dispersion of the interstellar medium in the
galaxy. The method is based on the analytical model of a uniform spherically
symmetric expanding shell in the turbulent interstellar medium, however we
show that the estimates are also valid for the case of non-uniform shells, and
agree well with “direct” measurements of the expansion velocities and ages of
observed superbubbles and supernova remnants (based on the example of the IC
1613 galaxy). The method described above is applicable, in particular, to
spatially unresolved (or poorly resolved) shells, as well as in the cases
where spectroscopic resolution prevents decomposition of the line profile into
kinematically isolated components. In this paper we consider two types of
spectrographs: with an instrumental profile (line spread function, LSF) in the
form of a Lorentz function (as in FPI) and with a Gaussian instrumental
profile (applicable to the case of many classical integral field
spectrographs). We have considered Lorentzian LSF widths corresponding to the
FPIs used111Actual list:
https://www.sao.ru/hq/lsfvo/devices/scorpio-2/ifp_eng.html in SCORPIO-2
(Afanasiev and Moiseev, 2011) focal reducer of the 6-m telescope of the
Special Astrophysical Observatory of the Russian Academy of Sciences (SAO RAS)
(see review of the method in Moiseev et al., 2021).

This paper has the following layout. Section 2 describes the analytical model
of a uniform expanding bubble in a turbulent medium and the factors that
affect the shape of its integrated emission line profile (turbulence of the
interstellar medium, thermal and instrumental broadening), and provides a
qualitative comparison of the constructed model with the H$\alpha$ line
profiles of observed objects. Section 3 describes the proposed method for
measuring the expansion velocity of a bubble from the observed velocity
dispersion, and the results of its Monte Carlo testing. In Section 4, we use
the method described above to estimate the expansion velocity of the expanding
superbubble of ionized gas in IC 1613 and determine the kinematic age of the
only supernova remnant in this galaxy. We summarize the main conclusions of
this work in Section 5.

##  2 MODELING THE PROFILE OF AN EXPANDING BUBBLE

###  2.1 A Uniform Expanding Bubble in Unperturbed Interstellar Medium

Figure 1: Scheme of a uniform expanding bubble model. The velocity of an
emitting atom from spherical layer $dV$ with polar angle $\theta$ relative to
the center of the shell is equal to the sum of the bubble expansion velocity
$v_{\rm exp}$ (red) and the random velocity of thermal and turbulent motions
$v_{I}$ (blue) that causes the emission line broadening.

According to the classical model of the evolution of a bubble whose expansion
is driven by the energy inflow from the wind of massive stars and supernovae
(Weaver et al., 1977; Lozinskaya, 1992)), propagation of shocks in the
interstellar medium results in the formation of a rather thin layer of swept-
up matter with the same gas pressure inside this layer. According to formula
(73) from Weaver et al. (1977), in the case of characteristic bubble expansion
velocities $v_{\rm exp}\sim 20$–$50$ km s-1 and the sound speed in ionized gas
equal to $c_{s}\sim 10$ km s-1 the thickness of the newly formed bubble equals
to 2–7% of its radius. The actually observed shells are usually somewhat
thicker (e.g., 7–25% of the radius in the LMC (Oey, 1996), on average about
25% for infra-red bubbles in the Galaxy (Churchwell et al., 2006)) but their
thickness is still small compared to the diameter, which allows us to use the
uniform thin sphere model to describe observed gas kinematics toward it (see
Fig. 1).

Consider a segment of the spherical layer $(\theta,\theta+d\theta)$ of gas
with volume $dV$, whose radial velocity is concentrated in the interval
$(v,v+dv)$, where $v$ is determined exclusively by the expansion velocity of
the bubble as $v=v_{\rm sys}+v_{\rm exp}\ \theta$. Here $v_{\rm sys}$ is the
radial velocity of the bubble center and $\theta$ is the angle between the
direction to the observer and the normal vector to the surface element (polar
angle). The radial velocity differential for such an element is $dv=-v_{\rm
exp}\sin{\theta}d\theta$. We assume that the thickness of the shell is
constant and equal to $h$ to obtain

$\begin{array}[]{rcl}dV&=&\int\limits_{0}^{2\pi}\left(hr^{2}\sin{\theta}d\theta\right)d\varphi\\\
&=&2\pi hr^{2}\sin{\theta}d\theta\\\ \\\
&=&-2\pi\dfrac{hr^{2}}{v_{exp}}dv,\end{array}$ (1)

where $r$ is the radius of the shell and $\varphi$ is the azimuthal angle in
the plane perpendicular to the sky plane (see Fig. 1). It follows from this
that

$\frac{dV}{dv}=-2\pi\frac{hr^{2}}{v_{\rm exp}}=const.$ (2)

The radiation flux toward the observer from each element $dV$ is proportional
to the number of radiating atoms in it (Osterbrock and Ferland, 2006).
Assuming the density of matter is the same throughout the shell, we obtain

$F_{v}(v)dv\propto dV=const\times dv.$ (3)

Figure 2: Schematic representation of the expanding shell (top) and examples
of the emission-line profiles in its different parts (bottom). Bottom left:
integrated emission-line profile of the shell, bottom right: line profile
towards the center of the shell. The blue rectangle shows the model spectrum
excluding thermal, turbulent, and instrumental broadening. The blue curve
shows the line profile including thermal, turbulent, and instrumental
broadening, the red line shows the one- or two-component Voigt profile fit.

Thus in the uniform bubble model considered the observed integrated line
profile has a rectangular shape with the same radiation flux density
$F_{v}(v)$ over the velocity interval $[v_{\rm sys}-v_{\rm exp};v_{\rm
sys}+v_{\rm exp}]$ also in the absence of radiation outside this interval (see
the left-hand panel in Fig. 2). In contrast, if only the central part of such
an idealized model is observed, the emission line-profile has the form of two
narrow rectangular peaks with velocities corresponding to the ray velocities
of the approaching and receding sides of a bubble, and their difference is
equal to $2\times v_{\rm exp}$ (see the right-hand panel in Fig. 2).

In the above scheme only the contribution of the bubble expansion to the line
profile is taken into account, ignoring the thermal and natural broadening and
the contribution of the turbulence of the interstellar medium. Let us consider
these effects in the next section.

###  2.2 Estimating Thermal and Turbulent Line Broadening

The width of emission lines in the interstellar medium is influenced by three
main factors: natural line broadening (due to quantum uncertainty of the level
energy with a finite lifetime), thermal line broadening and gas turbulence.
The contribution of natural line broadening in this case is negligible
compared to other factors, so we will consider only the effect of the latter
two factors.

Under the assumption of local thermodynamic equilibrium, the velocity of
thermal motion of electrons in H II-regions is described by the Maxwell
distribution, according to which the dispersion of the velocity projection is
equal to

$\sigma_{\rm{th}}=\sqrt{\dfrac{k_{B}T_{e}}{m}},$ (4)

which corresponds to $\sigma_{\rm th}\sim 9.1$ km s-1 for HII for the adopted
characteristic electron temperature $T_{e}=10^{4}$ K (Osterbrock and Ferland,
2006)

The situation with estimating the velocity dispersion of turbulent motions in
the interstellar medium is more complicated. Observations indicate supersonic
velocity dispersion for turbulent motions of ionized gas in galaxies (Moiseev
et al., 2015; Varidel et al., 2020), which may be due to gravitational
instability combined with feedback from massive stars (Krumholz et al., 2018).
The observed dispersion of ionized gas velocities in this case correlates with
the star-formation rate in the galaxy, and the characteristic velocity
dispersions in nearby dwarf galaxies are $\sigma_{\rm turb}\sim 12$–$35$ km
s-1 (Moiseev et al., 2015), although they can be several times greater for
galaxies with high star-formation rates. With integral field spectroscopy data
available this quantity can be estimated for each individual object under
study, for example, as the flux-weighted average of the velocity dispersion of
bright H II regions (Moiseev and Lozinskaya, 2012) in the H$\alpha$ line.

Thus because of thermal broadening and turbulent motions in the interstellar
medium the observed emission lines originating from outside the shell have a
Gaussian profile with velocity dispersion

$\sigma_{\rm ISM}=\sqrt{\sigma_{\rm th}^{2}+\sigma_{\rm turb}^{2}}\sim
15\hbox{--}36~{}\text{km\,s${}^{-1}$}$ (5)

Assuming that the above mechanisms produce the comparable broadening of the
emission lines from the walls of a bubble and from the surrounding
interstellar medium, the final line profile from the shell can be obtained by
convolving the rectangular distribution obtained in the previous section with
a Gaussian function with a width of $FWHM=2\sqrt{2\ln{2}}\,\sigma_{\rm ISM}$.

###  2.3 Comparison of Model and Observed Line Profiles

The observed shape and width of a spectral line depend on the instrumental
profile of the spectrograph. The instrumental profile of the FPI in SCORPIO-2
is well described by the Lorentz function, which means that the final observed
emission-line profile can be approximated by the Voigt function, which is a
convolution of the Gaussian and Lorentz functions (Moiseev and Egorov, 2008).
Fig. 2 shows that after thermal, turbulent, and instrumental broadening are
taken into account, the line profile from the model considered is indeed
similar to those observed with the FPI (see Section 4). The integrated line
profile can be well described by a single-component Voigt function, and the
line profile in the direction of the center of a bubble, by a two-component
Voigt function with the components centered on the velocities of the
approaching and the receding sides.

We now use the model constructed to illustrate how insufficient angular
resolution can distort the estimate of the bubble expansion velocity
determined by decomposing the observed profile into components. Fig. 3 shows
examples of simulated shell line profiles (expanding with a velocity of
$v_{\rm exp}=35$ km s-1 in a medium with $\sigma_{\rm ISM}=15$ km s-1 and
observed with an IFP751 FPI having LSF width $FWHM_{\rm LOR}=18$ km s-1)
integrated over different apertures. The smallest aperture corresponds to the
central part of the shell, and the largest, to the integrated spectrum from
the entire region. While in the former case the line profile clearly breaks
into two components, there is no such separation in the case of the integrated
spectrum corresponding to the case of a completely unresolved shell. The
separation between the individual components can be seen to decrease with
increasing relative area of the aperture compared to the total area occupied
by a bubble in the sky plane. In the case considered, the shell expansion
velocity is underestimated by a factor of 1.5 for an aperture with 50% of the
total bubble area. Thus even if spectroscopic resolution is sufficient to
decompose the profile into kinematically distinct components, insufficient
angular resolution may result in a significantly underestimated shell
expansion velocity and hence overestimated kinematic age and underestimated
energy inflow required for the formation of these components. To correctly
determine these parameters the angular resolution of the data has to be 0.5 of
the shell size or better. This imposes a significant limitation on the use of
estimates obtained by line-profile decomposition for bubbles smaller than 100
pc in the case of observations of galaxies beyond the Local Group.

In addition to the decomposition of the observed profile into kinematically
isolated components, there are a number of other methods for estimating the
expansion velocities of nebulae from published observational data. Guerrero et
al. (1998) suggested using radial gradient of the line profile width to
compute the expansion velocities of nebulae. However, this method is not
applicable in the case of insufficient angular resolution or nonuniform
nebulae. Dewey (2010) described yet another method in his paper on the
interpretation of the kinematics of unresolved young supernova remnants. Like
in this paper (cf. Section 2.1), the above author assumes that the integrated
spectrum of an expanding supernova remnant has a uniform distribution with
velocities from $-v_{\rm exp}$ to $v_{\rm exp}$, but he ignores the effects
due to turbulence in the surrounding interstellar medium (which is true in
first approximation given that characteristic shock velocities in young
supernova remnants are one to two orders of magnitude higher than those
discussed in our paper) or the non Gaussian shape of the spectrograph
instrument profile. Dewey computes the dispersion of the uniform distribution
from $a$ to $b$ (from $-v_{\rm exp}$ to $v_{\rm exp}$) as

$\sigma=\sqrt{\dfrac{(b-a)^{2}}{12}}=\sqrt{\dfrac{v_{exp}^{2}}{3}}\approx
0.577\ v_{\rm exp},$ (6)

to determine the expansion velocity of supernova remnants from the observed
line width. Note that this is not entirely correct, because the velocity
dispersion estimated by fitting the distribution to a Gaussian function
($\sigma\approx 0.715\ v_{\rm exp}$) differs from that given above for the
uniform distribution. With the above fact taken into account, the solution
proposed by Dewey (2010) coincides with a special case of the method we
consider below (see Section 3) in the case of zero turbulent motions in the
interstellar medium.

Thus, simple approaches can yield significantly (by tens of percent)
underestimated or overestimated results. Below we propose a method for
estimating shell expansion velocities from the measured width of the
integrated line profile based on modeling that takes into account the main
effects described above that affect the observed shape of the profile.

Figure 3: Model emission-line profile of a bubble expanding with velocity
$v_{\rm exp}=35$ km s-1 (marked with vertical lines) in the medium with
$\sigma_{\rm ISM}=15$ km s-1 and “observed” with spectrograph with Lorentzian
LSF $FWHM_{\rm LOR}=18$ km s-1. Different colors correspond to different
apertures in the integration: 1% of the total area in the sky plane
corresponds to the center of the bubble and 100%, to the entire shell.

##  3 MEASURING A BUBBLE EXPANSION VELOCITY BASED ON THE OBSERVED VELOCITY
DISPERSION

As noted above, when the angular and/or spectral resolution of the employed is
insufficient, the expansion of a bubble shows up as an increase in the
measured velocity dispersion. This effect is often apparent in observations.
Thus one can efficiently search for expanding superbubbles in nearby galaxies
by analyzing ($I$–$\sigma$) diagrams in the H$\alpha$ line (Muñoz-Tuñón et
al., 1996; Moiseev and Lozinskaya, 2012; Egorov et al., 2017, 2018, 2021),
where such objects can be distinguished by their increased velocity
dispersion. Compact nebulae around individual energetic objects (WR, LBV
stars) and unresolved supernova remnants also stand out on the
($I$–$\sigma$)-diagram as regions with high intensity and high velocity
dispersion (Moiseev and Lozinskaya, 2012). Below we investigate how to
estimate the expansion velocities of such bubbles based on the data about the
spatial distribution of velocity dispersion in their host galaxy.

To reveal the relationship pattern between the measured spectral line width
and the actual shell expansion velocity, we constructed and analyzed a series
of model spectra computed for the case of a uniform bubble expanding in
turbulent interstellar medium (see Section 2). In our analysis we varied the
shell expansion velocity and the velocity dispersion in the interstellar
medium (which includes turbulent and thermal line broadening) in the intervals
$v_{\rm exp}=5$–$95$ km s-1 and $\sigma_{\rm ISM}=3$–$70$ km s-1,
respectively. As mentioned above (see Section 2.3), the instrumental profile
of the FPI can be well described by the Lorentz function. At the same time,
the LSF shape of classical integral field spectrographs have an LSF is close
to Gaussian (e.g., MaNGA (Law et al., 2021), MUSE (Bacon et al., 2010), IFU
within SCORPIO-2 (Afanasiev et al., 2018)). We considered four types of LSF in
our analysis:

  * 1) Gaussian LSF with the profile width $FWHM_{\rm LSF}$. In this case the effect of instrumental broadening on the integrated line profile is similar to that due to turbulent line broadening, so we consider the quantity $\sqrt{\sigma_{\rm ISM}^{2}+\sigma_{\rm LSF}^{2}}$, where $\sigma_{\rm LSF}=FWHM_{\rm LSF}/(2\sqrt{2\ln 2})$, as a parameter for further analysis. Here we estimate the “observable” velocity dispersion $\sigma_{\rm obs}$ measured from the integrated spectrum of the shell by fitting the line profile to a Gaussian function.

  * 2) The case of the LSF in the form of a Lorentz function. In that case we set $\sigma_{\rm LSF}=0$ in all further computations. Note that here the measured velocity dispersion $\sigma_{\rm obs}$ inferred by fitting the observed profile to a Voigt function is free from the contribution of instrumental broadening, because it refers to the Gaussian component (Moiseev and Egorov, 2008). We considered three values of the Lorentzian profile width corresponding to the three FPIs used in SCORPIO-2 on the 6-m telescope of the SAO RAS:

    * $\bullet$

$FWHM_{\rm LOR}=18$ km s-1 (for IFP751);

    * $\bullet$

$FWHM_{\rm LOR}=36$ km s-1 (for IFP501);

    * $\bullet$

$FWHM_{\rm LOR}=78$ km s-1 (for IFP186).

Figure 4: Relationship between the expansion velocity $v_{\rm exp}$, observed
velocity dispersion $\sigma_{\rm obs}$, unperturbed interstellar medium
velocity dispersion $\sigma_{\rm ISM}$, and the shape and width of the
instrumental profile ($FWHM_{\rm LOR}$, $\sigma_{\rm LSF}$) as inferred in
terms of the uniform expanding bubble model. The solid lines show the result
of the best approximation by a relation of the form (7).

Fig. 4 shows the dependence of the model-estimated expansion velocity of a
bubble on the velocity dispersion $\sigma_{\rm obs}$ measured from its
integrated spectrum, for different values of $\sigma_{\rm ISM}$. For each of
these, the resulting dependencies are well approximated by a relation of the
form

$v_{\rm exp}=k(\sigma_{\rm obs}^{2}-(\sigma_{\rm ISM}^{2}+\sigma_{\rm
LSF}^{2}))^{a}+v_{0}.$ (7)

The coefficients $k$, $v_{0}$ and exponent $a$ in the formula depend on the
average dispersion of gas velocities in the galaxy $\sigma_{\rm ISM}$ as well
as on the form and width of the instrumental profile of the spectrograph
(parameters $\sigma_{\rm LSF}$ and $FWHM_{\rm LOR}$). Figure 5 shows the
behavior of the dependence of each parameter on these quantities. Note that
the parameter $v_{0}$ is relevant only in the case of high velocity dispersion
$\sigma_{\rm ISM}$ of unperturbed gas and is most important in the case of the
use of the low spectral resolution FPI, which is poorly suited to study gas
velocity dispersion variations in galaxies. We can assume $v_{0}\simeq 0$ in
most cases of typical velocity dispersions $\sigma_{\rm ISM}\sim 12$–$35$ km
s-1 in nearby galaxies (Moiseev, 2015).

Figure 5: Dependence of parameters $k$, $a$ and $c$ in equation (7) on the
shape and width of the instrumental spectrograph contour $\sigma_{\rm LSF}$,
$FWHM_{\rm LOR}$ and on the average gas velocity dispersion $\sigma_{\rm ISM}$
in the galaxy. The dashed line shows the result of the best polynomial
approximation according to the equations (8)–(10) and Table 1.

The variation of each of the parameters $k$, $a$, $v_{0}$ in formula (7) as a
function of $\sqrt{\sigma_{\rm ISM}^{2}+\sigma_{\rm LSF}^{2}}$ can be
described by the following polynomial relations

$\displaystyle k(x)$ $\displaystyle=$ $\displaystyle
c_{k4}x^{4}+c_{k3}x^{3}+c_{k2}x^{2}+c_{k1}x+c_{k0},$ (8) $\displaystyle a(x)$
$\displaystyle=$ $\displaystyle c_{a2}x^{2}+c_{a1}x+c_{a0},$ (9)
$\displaystyle v_{0}(x)$ $\displaystyle=$ $\displaystyle
c_{v3}x^{3}+c_{v2}x^{2}+c_{v1}x+c_{v0},$ (10)

, where $x=\sqrt{\sigma_{\rm ISM}^{2}+\sigma_{\rm LSF}^{2}}$ ($\sigma_{\rm
LSF}=0$ in the case of the Lorentz instrumental profile). The coefficients in
equations (8)–(10) are listed in Table 1 for all the four LSF types
considered.

Table 1: Polynomial expansion coefficients of parameters $k$, $a$ and $v_{0}$ according to equations (8)–(10) for different instrumental contour widths $FWHM_{\rm LOR}$ | $FWHM_{\rm LOR}$
---|---
Parameter | 0 km s-1 | 18 km s-1 | 36 km s-1 | 78 km s-1
| (Gaussian LSF) | (IFP751) | (IFP501) | (IFP186)
$c_{k0}$ | $1.74$ | $2.0$ | $2.49$ | $2.45$
$c_{k1}$ | $0.0257$ | $0.0741$ | $0.0433$ | $0.102$
$c_{k2}$ | $4.60e-03$ | $4.29e-04$ | $5.99e-04$ | $-5.57e-03$
$c_{k3}$ | $-1.42e-04$ | $-7.34e-05$ | $-7.62e-05$ | $6.37e-05$
$c_{k4}$ | $1.06e-06$ | $6.75e-07$ | $7.48e-07$ | $-1.65e-07$
$c_{a0}$ | $0.471$ | $0.472$ | $0.452$ | $0.438$
$c_{a1}$ | $-4.27e-03$ | $-6.30e-03$ | $-5.26e-03$ | $-3.48e-03$
$c_{a2}$ | $5.54e-05$ | $1.33e-04$ | $1.38e-04$ | $1.50e-04$
$c_{v0}$ | $0.139$ | $-2.0$ | $-2.17$ | $-2.18$
$c_{v1}$ | $-0.23$ | $0.0922$ | $0.0215$ | $-0.091$
$c_{v2}$ | $2.81e-03$ | $-0.0128$ | $-9.37e-03$ | $-0.0011$
$c_{v3}$ | $-3.18e-07$ | $2.73e-04$ | $2.78e-04$ | $2.61e-04$
Figure 6: Dependence of uniform shell expansion velocity measured by the
equations (7)–(10) on the input velocity given in the simulation. For each LSF
type and width, model line profiles are generated for 4000 bubbles with
$S/N=10$ (the orange symbols) and $S/N=100$ (the blue symbols), expansion
velocity $v_{\rm exp}$ (5 to 110 km s-1), ambient gas velocity dispersion
$\sigma_{\rm ISM}$ (10 to 70 km s-1, it includes the instrumental profile
width in the case of Gaussian LSF). Panels for different LSF types and widths
are presented. The histogram in each panel shows the distribution of the
relative measurement error $(v_{\rm exp}[\mathrm{measured}]-v_{\rm
exp}[\mathrm{model}])/v_{\rm exp}[\mathrm{model}]$. The bottom of each panel
shows the distribution of the relative error on input $v_{\rm exp}$.

To estimate the relative error of the shell expansion velocity inferred from
equation (7) and the robustness of the equations (7)–(10) with respect to
noise, we performed an additional Monte Carlo simulation. For each of the four
LSF types, we simulated 4000 spectra of bubbles with random parameters
constrained in the intervals $\sqrt{\sigma_{\rm ISM}^{2}+\sigma_{\rm
LSF}^{2}}=10$–$70$ km s-1 and $v_{\rm exp}=5$–$110$ km s-1, and added noise
with final signal-to-noise ratios of $S/N=10$ and $S/N=100$. We applied
equations (7)–(10) to the resulting synthetic profiles and estimated the
bubble expansion velocity in each case. A comparison of the resulting estimate
with its true value set in the model demonstrates the accuracy of the
technique. Fig. 6 shows the dependence of these two quantities on each other
as well as the distribution of the relative measurement error computed as
$(v_{\rm exp}[\mathrm{measured}]-v_{\rm exp}[\mathrm{model}])/v_{\rm
exp}[\mathrm{model}]$. As is evident from the figure, the technique described
above is quite robust with respect to noise and the shell expansion velocity
to be estimated with an accuracy of about 10% over a wide range of velocities.

The method described above is based on the uniform bubble model, but real H II
regions have a more complex morphology. First, the observed superbubbles in
extragalactic star-forming regions form as a result of energy inflow from
multiple massive stars, i.e., in fact, they are complexes of merging local
bubbles at different stages of evolution. As was shown by Tenorio-Tagle et al.
(1996), the integrated spectrum of such unresolved complexes ca still be well
described by the model of a single uniform superbubble, but there may be an
additional broad low-intensity component in the line wings—its contribution
does not affect the analysis considered in this paper. The second important
point is that real H II regions are not uniform but highly clumped with
significant local density and temperature inhomogeneities. Let us check to
what extent this effect affects our estimates.

Figure 7: Same as in Fig. 6, but for the models of bubbles with nonuniform
brightness distribution. The dependences are shown only for the LSF in the
form of a Lorentz function with profile width $FWHM_{\rm LOR}=18$ km s-1.

We now consider a shell in the form of a thin expanding sphere, but this time
the brightness of each element $dV$ in Fig. 1 varies with angles $\theta$ and
$\varphi$. For this purpose, we set the distribution $I(\theta,\varphi)$ in
the form of the sum of spherical harmonics up to and including the eighth
order with random amplitudes and with the dominant contribution of the angle-
independent (zero) component. The final intensity distribution from each
element in the bubble is a set of separate spots of various sizes with higher-
and lower-than average brightness. As in the case of the uniform model, the
integrated spectrum of the resulting shell is convolved with a Gaussian to
account for thermal and turbulent broadening and with a Gaussian or Lorentz
function describing the instrumental profile of the spectrograph. To test the
applicability of equations (7)–(10) to such structures with non-uniform
brightness distributions, we performed Monte Carlo simulations similar to
those described above, but this time with synthetic profile generated in terms
of a nonuniform bubble model. Fig. 7 shows the result of comparing the
measured and modeled bubble expansion velocities (for $FWHM_{\rm LOR}=18$ km
s-1). The number of regions exhibiting a significant deviation of the result
from the model is slightly higher than in the case of a uniform shell, but on
the average the relative error is still at the level of about 10%. Hence the
method described is also applicable in the case of shell structures with
nonuniform brightness distribution.

Figure 8: Integrated model line profile from an expanding spherical bubble
(the blue line) in the form of a hemisphere at different orientations. In the
case where the bright part of the bubble is in the sky plane (the green line),
the resulting integrated profile is significantly asymmetric. When the bright
part of the bubble is perpendicular to the sky plane (the orange line) the
resulting line profile is indistinguishable from the case of spherical
symmetry. The vertical lines correspond to the preset expansion velocity
$v_{\rm exp}=35$ km s-1.

In the cases where star formation occurs at the boundary of a molecular cloud,
significant gas density gradient of gas density around massive stars can cause
the formation of asymmetric shells with azimuthal gradient of brightness and
velocity. Typical examples are blister-type nebulae (see, e.g., Egorov et al.
(2010)). To test the applicability of the above technique to such objects, we
simulated a shell structure with smooth azimuthal variation of brightness and
velocity. Given that the brightness of the emission-line nebula depends on gas
density $I\sim n_{e}^{2}\sim\rho^{2}$ (Osterbrock and Ferland, 2006), and that
relation $v\rho^{2}\sim const$ (McKee and Cowie, 1975) is usually fulfilled
for radiative shocks, we can specify the expansion velocity of our asymmetric
shell at each point by the law $v\sim I^{-1}$. From observational point of
view we can distinguish two extreme cases: the bright (dense) side of the
shell is in the sky plane (face-on) or perpendicular to it (edge-on). In the
first case such a shell appears as an ordinary symmetric H II region, whereas
in the second case there should be clear brightness gradient (in the case of
sufficient angular resolution). Fig. 8 shows the simulated integrated profiles
for each of these cases. As is evident from the figure, the integrated line
profile of a blister-type profile is indistinguishable from the spherically
symmetric case in its edge-on orientation, and hence the method we have
described above is fully applicable to such objects. On the other hand, in the
case of face-on orientation the integrated profile is significantly
asymmetric, resulting in underestimated measured velocity dispersion and,
consequently, underestimated bubble expansion velocity. Thus, in the case
shown in Fig. 8, a formal fit of the asymmetric face-on profile of the shell
by a single-component Voigt function and application of equations (7)–(10)
yields an expansion velocity estimate of $v_{\rm exp}=30$ km s-1 instead of 35
km s-1 value incorporated into the model, and the discrepancy is significantly
greater in the case of higher expansion velocities. Unfortunately, even in the
case of sufficiently high angular resolution distinguishing such face-on
blister-type nebulae from spherically symmetric ones is by no means a trivial
task. In this connection, it is worth noting that the method presented in this
paper may be unreliable in cases where the integrated line profile of a nebula
is significantly asymmetric.

The method described in this section allows us to use equations (7)–(10) to
determine the expansion velocities of spatially unresolved (or poorly
resolved) bubbles in the interstellar medium of galaxies based on the gas
velocity dispersion $\sigma_{\rm obs}$ measured from the integrated spectrum
of a bubble and the average gas velocity dispersion in the galaxy $\sigma_{\rm
ISM}$. The gas velocity dispersion $\sigma_{\rm ISM}$ in this case can be
inferred, for example, from the estimated flux-weighted average gas velocity
dispersion of the bright H II regions in the galaxy, which stand out in the
($I$–$\sigma$) diagrams. The presented method can also be useful in the case
of spatially resolved shell structures when insufficient spectral resolution
prevents the decomposition of the profile into kinematically isolated
components. In the next section we test the method on real observational data.

##  4 THE GALAXY IC 1613: TESTING THE METHOD ON OBSERVATIONAL DATA

Nearby star-forming dwarf galaxies are an excellent laboratory for studying
the interaction processes of massive stars and the interstellar medium.
Because of the lack of spiral density waves and the thick gaseous disk
extended supershell complexes are often observed in the interstellar medium of
such galaxies. Such a pattern can be clearly seen in IC 1613—a nearby Local
Group galaxy (at a distance of $D\sim 760$ kpc (Karachentsev et al., 2013). A
complex of expanding superbubbles of ionized gas extending over about 1 kpc is
observed in its eastern part (see Fig. 9). This complex of superbubbles has
been studied in detail previously by Lozinskaya et al. (2003) and is well
suited for testing the method described in Section 3 on real data.

Figure 9: Complex of superbubbles in the galaxy IC 1613 according to
H$\alpha$-line SCORPIO observations on the 6-m telescope of the SAO RAS. Top
left panel: H$\alpha$-line image of the region, the position of the cut (the
black line) for the “position-velocity” diagram (shown in the bottom left
panel). The right panel shows the observed H$\alpha$-line profiles by
integration in the apertures shown by the green circles in the image. The
profiles #1, #3 are obtained for the central part of the considered regions,
and profiles #2, #4—for the entire region. Profile #5 corresponds to the
bubble surrounding the young SNR S8 supernova remnant. The dashed green line
shows the individual components, and the red line, the final fit of the
observed profile by a one- or two-component Voigt function.

We use H$\alpha$-line data obtained on September 13, 2001 with the FPI
(IFP501) of the SCORPIO focal reducer (Afanasiev and Moiseev, 2005) mounted on
the BTA 6-m telescope of the SAO RAS. For a detailed description of the
observations and data processing see Lozinskaya et al. (2003).

As is evident from Fig. 9, the two regions labeled Reg. 1 and Reg. 2 exhibit a
clear shell-like morphology and reveal a characteristic velocity ellipse in
the “position-velocity” diagrams, indicating the observed superbubble
expansion. The angular size of these superbubbles is on the order of
$80\arcsec$ and $55\arcsec$ (290 and 200 pc), respectively, which
significantly exceeds the angular resolution of the observational data (about
$1.5\arcsec$). The observed H$\alpha$ profile in the central part of the
shells (#1 and #3 in Fig. 9) is clearly separated into components
corresponding to the approaching and the receding sides. The expansion
velocities of the Reg. 1 and Reg. 2 shells computed from the maximum
separation of the H$\alpha$ profile components are equal to $v_{\rm exp}=51$
km s-1 and 37 km s-1 respectively.

Superbubbles similar to Reg. 1 and Reg. 2 observed in more distant galaxies
will have smaller angular sizes, and they should be practically unresolvable
at $D\sim 10$ Mpc. In such a case their integrated spectrum will be observed,
as in the case of profiles #2 and #4 in Fig. 9. No separation into individual
components can be observed, but the H$\alpha$ profile is broadened. The
measured velocity dispersion (free of instrumental broadening) is $\sigma_{\rm
obs}=39$ km s-1 and 26 km s-1 for Reg. 1 and Reg. 2, respectively. We can
estimate the velocity dispersion of the surrounding interstellar medium as the
flux-weighted average H$\alpha$-line velocity dispersion in regions with
intensities higher than the median I (H$\alpha$) in the galaxy (to exclude the
contribution of diffuse ionized gas). The inferred velocity dispersion
$\sigma_{\rm ISM}=15$ km s-1 is consistent with the estimate of the velocity
dispersion of individual components on the profile #3, as well as with an
estimate for separate bright H II regions, which exhibit no signs of
expansion. We can adopt these values $\sigma_{\rm ISM}$ and $\sigma_{\rm obs}$
and $FWHM_{\rm LOR}=36$ km s-1 for IFP501 and use equations (7)–(10) to
estimate expansion velocity by the method proposed in this paper. The inferred
expansion velocity estimates $v_{\rm exp}=53$ km s-1 and 33 km s-1 for Reg. 1
and Reg. 2, respectively, agree well with those found above, which are based
on the H$\alpha$-line profile decomposition within the previously quoted
relative error of the method (on the order of 10%).

The only known supernova remnant in the IC 1613 galaxy—SNR S8—is also observed
toward the complex considered, but, unlike Reg. 1 and Reg. 2 regions, it is
practically unresolved, making it impossible to directly measure directly its
expansion velocity. Observational data obtained with MPFS spectrograph
attached at the 6-m telescope of the SAO RAS Lozinskaya et al. (1998) revealed
no velocity dispersion gradient that could clearly indicate SNR expansion, and
yielded a rough estimate $v_{\rm exp}\sim 50$–$100$ km s-1 based on the
variation of the observed radial velocity toward the SNR ($v_{\rm exp}<300$ km
s-1 if weak emission structures are also considered). A detailed study of SNR
S8 is also given in Schlegel et al. (2019). Direct measurements of the
expansion velocity of the supernova remnant are still lacking, but new high-
resolution X-ray data have allowed the authors to estimate SNR S8 properties,
including its age, independent of kinematic parameters.

We now try to estimate the expansion velocity of SNR S8 and its kinematic age
from available observational data obtained with the FPI using equations
(7)–(10). The integrated profile in the H$\alpha$ line (#5 in Fig. 9) is
significantly broadened and slightly asymmetric. The measured velocity
dispersion $\sigma_{\rm obs}=64$ km s-1 corresponds to the optical shell
expansion velocity $v_{\rm exp}=84$ km s-1, which agrees with the approximate
estimate obtained in Lozinskaya et al. (1998). The velocity thus measured
corresponds to the velocity of the radiative shock propagating in regions of
increased supernova remnant density, while the shock velocity outside dense
clouds, which determines the remnant expansion velocity, is related to the gas
density as (McKee and Cowie, 1975): $\beta n_{i}v_{s}^{2}=n_{c}v_{c}^{2}$,
where $\beta=3.15$ for fast shocks, $n_{i}$ and $n_{c}$ are equal to the gas
density in unperturbed interstellar medium and in a dense cloud, respectively;
$v_{s}$ is the SNR shock front velocity in unperturbed medium,
$v_{c}$—radiative shock velocity in a dense cloud. When applied to SNR S8
$v_{c}=v_{\rm exp}\simeq 84$ km s-1, $n_{i}\simeq 1.6$ cm-3 (according to
Schlegel et al. (2019)), $n_{c}=n_{e}/1.2\simeq 1080$ (according to electron
density estimate $n_{e}$ obtained by Lozinskaya et al. (1998) by the [S II]
6717/6731 Å lines ratio). From this, we find the velocity of the shock front
SNR S8 $v_{s}\simeq 1230$ km s-1. Assuming that SNR S8 is at the adiabatic
stage of expansion (similar to Schlegel et al. (2019)), we can describe its
evolution as $R_{s}\sim t^{0.4}$ (Sedov, 1959), which means that
$t=0.4R_{s}/v_{s}$. Taking the estimate $R_{s}\sim 9.5$ pc obtained in
Schlegel et al. (2019) from X-ray images, we determine the kinematic age value
of SNR S8 $t\simeq 3100$ years, which agrees well with the independent
estimate $t\simeq 3380$–$5650$ years obtained in Schlegel et al. (2019) who
did not use an information on the gas kinematics.

We thus used the IC 1613 galaxy as an example to demonstrate that the method
for measuring a bubble/superbubble expansion velocity described in this paper
is applicable to real observational data and yields the expansion velocity and
age estimates that are consistent with other independent measurements.

##  5 SUMMARY

Integral field spectroscopy reveals variations of the observed gas velocity
dispersion in galaxies that are driven by presence of the expanding bubbles
and superbubbles surrounding massive stars and clusters in star-forming
regions (see, e. g., Moiseev and Lozinskaya, 2012). With high enough spectral
resolution the observed emission line profile in such regions can be
decomposed into kinematically isolated components allowing the bubble
expansion velocity to be measured (see, e.g., Lozinskaya et al. (2003); Egorov
et al. (2014, 2017); Oparin et al. (2020)), but this is rarely possible for
galaxies located beyond the Local Group because of limited spatial resolution.
In this paper, we propose a method allowing one to overcome this limitation
and estimate the shell expansion velocity from the dispersion of gas
velocities measured from integrated spectrum.

We considered an analytical model of a uniform thin shell expanding in a
turbulized interstellar medium and found how thermal, turbulent, and
instrumental broadening and bubble expansion affect the shape of the observed
emission line profile. We used this model to demonstrate that the shell
expansion velocity obtained by decomposing the profile into kinematically
separated components can be significantly underestimated if the angular
resolution is insufficient. This results in overestimated kinematic age and
underestimated energy inflow required for the formation of the shell, and
these biases can affect the estimated efficiency of the transfer of energy
from stars to kinetic energy of superbubbles when comparing observations and
theoretical models.

The models we have built showed that the bubble expansion velocity can be
determined by equation (7), which includes the gas velocity dispersion of the
bubble (measured from the integrated spectrum) and in the surrounding
unperturbed interstellar medium (which can be estimated for each galaxy as a
flux-weighted average value for bright H II regions). We showed that the
parameters used in equation (7) depend on the average gas velocity dispersion
in the galaxy and can be approximated by polynomials (8)–(10). The
coefficients in these polynomials depend on the type of instrumental profile
of the spectrograph. We considered the cases of LSF in the form of Gaussian
functions (applicable to most classical integral field spectrographs) with
arbitrary spectral resolution and Lorentz functions (for FPI) with the
resolution corresponding to IFP751, IFP501, and IFP186, which are used in
SCORPIO-2 instrument at the 6-m telescope of the SAO RAS. Table 1 lists the
coefficients inferred for these cases.

We applied the method described to synthetic spectra of bubbles having random
parameters and different $S/N$ ratios in their spectra and showed that the
relative error of the measurement of the bubble expansion velocity is on
average about 10%. To further test the applicability of the method, we also
analyzed a bubble model with a non-uniform brightness distribution and found
the result to be the same. We showed that the method is also applicable to
bubbles with large-scale inhomogeneities (for example, those having the form
of a hemisphere), but in this case, the spatial orientation of bright regions
plays an important role.

We used the method described to estimate the expansion velocities of well-
resolved shells of ionized gas and of the only known supernova remnant (SNR
S8) in the nearby galaxy IC 1613. The estimates obtained for the superbubbles
agree well with measurements made by decomposing the line profile into
kinematically separated components. For the first time we estimated the
kinematic age of SNR S8 ($t\simeq 3100$ years) and our result agrees well with
the age obtained earlier from X-ray data (Schlegel et al., 2019).

The method proposed in this paper allows measuring the expansion velocities of
spatially unresolved (or poorly resolved) superbubbles in galaxies outside the
Local Group. In particular, it is intended to be used to measure the expansion
velocities, kinematic ages, and energy of superbubbles identified by increased
gas velocity dispersion from observations of nearby dwarf galaxies with a FPI
at the 6-m telescope of the SAO RAS within the framework of SIGMA-FPI
catalog222http://sigma.sai.msu.ru. (Egorov et al., in prep.).

###### Acknowledgements.

The authors are grateful to A. V. Moiseev for the discussion on the
preliminary results of the work and to the anonymous referee for valuable
comments.

## FUNDING

This work was supported by the Russian Science Foundation (Grant No.
19-72-00149). Observations on the 6-m telescope of the Special Astrophysical
Observatory of the Russian Academy of Sciences are supported by the Ministry
of Science and Higher Education of the Russian Federation (contract No.
05.619.21.0016, project identifier RFMEFI61919X0016). The renovation of
telescope equipment is currently provided within the national project
”Science.”

## CONFLICT OF INTEREST

The authors declare that there is no conflict of interest.

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Translated by A. Dambis