Superbubble evolution in disk galaxies
I. Study of blowout by analytical models
^{1} Institut für Astrophysik, Universität Wien, Türkenschanzstr. 17, 1180 Vienna, Austria
email: verena.baumgartner@univie.ac.at
^{2} Zentrum für Astronomie und Astrophysik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
Received: 8 February 2013
Accepted: 1 August 2013
Context. Galactic winds are a common phenomenon in starburst galaxies in the local universe as well as at higher redshifts. Their sources are superbubbles driven by sequential supernova explosions in star forming regions, which carve out large holes in the interstellar medium and eject hot, metal enriched gas into the halo and to the galactic neighborhood.
Aims. We investigate the evolution of superbubbles in exponentially stratified disks. We present advanced analytical models for the expansion of such bubbles and calculate their evolution in space and time. With these models one can derive the energy input that is needed for blowout of superbubbles into the halo and derive the breakup of the shell, since RayleighTaylor instabilities develop soon after a bubble starts to accelerate into the halo.
Methods. The approximation of Kompaneets is modified in order to calculate velocity and acceleration of a bubble analytically. Our new model differs from earlier ones, because it presents for the first time an analytical calculation for the expansion of superbubbles in an exponential density distribution driven by a timedependent energy input rate. The timesequence of supernova explosions of OBstars is modeled using their main sequence lifetime and an initial mass function.
Results. We calculate the morphology and kinematics of superbubbles powered by three different kinds of energy input and we derive the energy input required for blowout as a function of the density and the scale height of the ambient interstellar medium. The RayleighTaylor instability timescale in the shell is calculated in order to estimate when the shell starts to fragment and finally breaks up. Analytical models are a very efficient tool for comparison to observations, like e.g. the Local Bubble and the W4 bubble discussed in this paper, and also give insight into the dynamics of superbubble evolution.
Key words: ISM: bubbles / ISM: supernova remnants / galaxies: ISM / galaxies: halos
© ESO, 2013
1. Introduction
Most massive stars are born in OBassociations in coeval starbursts on timescales of less than 1−2 Myr (Massey 1999). These associations can contain a few up to many thousand OBstars, socalled super star clusters, but typically have 20−40 members (McCray & Kafatos 1987). Energy and mass are injected through strong stellar winds and subsequent supernova (SN) explosions of stars with masses above 8 M_{⊙}. The emerging shock fronts sweepup the ambient interstellar medium (ISM) and, as the energy input in form of SNexplosions continues, superbubbles (SBs) are produced, which may reach dimensions of kiloparsecsize (e.g. TenorioTagle et al. 2003). The sweptup ISM collapses early in the evolution of the SB into a cool, thin, and dense shell (Castor et al. 1975) present in HI and Hα observations. The bubble interior contains hot (>10^{6} K), rarefied material, usually associated with extended diffuse Xray emission (Silich et al. 2005). Due to the stratification of the ISM in disk galaxies, the superbubbles can accelerate along the density and pressure gradient and blow out into the halo, appearing as elongated structures. Examples of huge bubbles and supergiant shells are the Cygnus SB with a diameter of 450 pc in the Milky Way (Cash et al. 1980) and the Aquila supershell extending at least 550 pc into the Galactic halo (Maciejewski et al. 1996). Our solar system itself is embedded in an HI cavity with a size of a few hundred parsecs called the Local Bubble (Lallement et al.2003), and is most likely generated by stellar explosions in a nearby moving group (Berghöfer & Breitschwerdt2002). Also in external systems like in the LMC (Chu & Mac Low 1990), in NGC 253 (Sakamoto et al. 2006) and M101 (Kamphuis et al. 1991) such bubbles, holes and shells are observed.
The acceleration of the shell promotes RayleighTaylor instabilities and after it is fully fragmented, only the walls of the SB are observed. Through such a chimney the hot pressurized SNejecta can escape into the halo. The walls may be subject to the gravitational instability and interstellar clouds can form again, which triggers star formation (e.g. McCray & Kafatos 1987). This is observed, for example, on the border of the OrionEridanus SB (Lee & Chen 2009).
The knowledge of SB evolution is crucial for understanding the socalled diskhalo connection and it also gives us information about the chemical evolution of the galaxies, the enrichment of the intergalactic as well as the intracluster medium. The thick extraplanar layer of ionized hydrogen seen in many galaxies has probably been blown out of the disk into the halo by photoionization of OBstars and correlated SNe (Tüllmann et al. 2006). With the high star formation rate of starburst galaxies, the energy released by massive bursts of star formation can even push the gas out of the galactic potential, forming a galactic wind. Outflow rates of 0.1 − 10 M_{⊙}/yr are common for starburst driven outflows (BlandHawthorn et al. 2007). If the hot and metal enriched material is brought to the surrounding intergalactic medium, it will mix after some time, increasing its metallicity. Galactic winds are observed in nearby galaxies (e.g. Dahlem et al. 1998), as well as up to redshifts of z ~ 5 (e.g. Swinbank 2007; Dawson et al. 2002), thus being an ubiquitous phenomenon in star forming galaxies. If the energy input is not high enough, the gas will fall back onto the disk due to gravity, after loss of pressure support, forming a galactic fountain (Shapiro & Field 1976; de Avillez 2000). In this case, the heavy elements released by SNexplosions are returned back to the ISM in the disk, presumably spread over a wider area, and future generations of stars will incorporate them.
Spitoni et al (2008), also using the Kompaneets approximation, have investigated the expansion of a superbubble, its subsequent fragmentation and also the ballistic motion of the fragments including a drag term, which describes the interaction between a cloud and the halo gas. In addition, these authors have analyzed the chemical enrichment of superbubble shells, and their subsequent fragmentation by RayleighTaylor instabilities, in order to compare the [O/Fe]ratios of these blobs to high velocity clouds (HVCs). They find that HVCs are not part of the galactic fountain, and even for intermediate velocity clouds (IVCs), which are in the observed range of velocities and heights from the galactic plane, the observed [O/Fe]ratios can only be reproduced by unrealistically low initial disk abundances. The chemical enrichment of the intergalactic medium will be the subject of a forthcoming paper.
This paper is structured as follows: Sect. 2 shows how the evolution of superbubbles can be described. In Sect. 3 we present the results of this work, which is mainly the expansion of a bubble in space and time and also the onset of RayleighTaylor instabilities in the shell. In Sect. 4 the models are used to analyze two MilkyWay superbubbles. A discussion follows in Sect. 5 and summary and conclusions are presented in Sect. 6.
2. Superbubble evolution
2.1. ISM stratification
For a homogeneous ISM, the propagation of a shock front originating from an instantaneous release of energy was described by Sedov (1946) and Taylor (1950), while the expansion of an interstellar bubble with continuous wind energy injection was studied by Castor et al. (1975) and Weaver et al. (1977). Yet, the description of SB evolution in such a uniform ambient medium is only valid for early stages of evolution. The vertical gas density distribution has a major effect on the larger superbubbles which have sizes exceeding the thickness of a galactic disk. On these scales, the ISM structure is far from homogeneous. In a MilkyWaytype galaxy, the warm neutral HI is planestratified and can be described by a symmetric exponential atmosphere with respect to the midplane of a galaxy (Lockman 1984). The scale height of this layer is ~500 pc. The Reynolds layer of warm ionized gas has a scale height of H = 1.5 kpc (Reynolds 1989) and the highly ionized gas of the hot (10^{6}−10^{7} K) halo surrounds the galaxy extending to ~20 kpc (Bregman & LloydDavies 2007) with a scale height of ~4.4 kpc (Savage et al. 1997). In the disk, the cold neutral and molecular gas are found to have H ~ 100 pc.
The expansion of superbubbles in a stratified medium was studied since many decades, both analytically (e.g. Maciejewski & Cox 1999) and numerically (Chevalier & Gardner 1974; Tomisaka & Ikeuchi 1986; Mac Low & McCray 1988, hereafter MM88). For an analytic description of superbubbles, Kompaneets’ approximation (1960) is a very good choice. Although it involves several simplifications (e.g. no magnetic field), it represents the physical processes involved very well (Pidopryhora et al. 2007).
2.2. Blowout phenomenon and fragmentation of the shell
A focus of this paper is to analyze the blowout phenomenon: in an exponential as well in an homogeneous ISM, a bubble decelerates first. But due to the negative density gradient of the exponentially stratified medium and the resulting encounter with very rarefied gas, the bubble starts to accelerate into the halo and even beyond, if the energy input is high enough, at a certain time in its evolution. MM88 call this process blowout, which is also used by Schiano (1985) and Ferrara & Tolstoy (2000), but there, blowout involves complete escape of the gas from the galaxy. From a more phenomenological point of view, Heiles (1990) distinguishes between breakthrough bubbles, which break out of the dense disk and are observed as shells and holes in the ISM, whereas blowout bubbles break through all gas layers and inject mass and metals into the halo. In our definition (see Sect. 2.3), a SB will blow out of a specific gas layer at the time, when the outer shock accelerates, and if the shock stays strong all the time. In particular, we want to determine the energy input required for blowout into the halo, and its dependence on ISM parameters (see Sect. 3.2.).
Shortly after the acceleration has started, RayleighTaylor instabilities (RTIs) appear at the interface between bubble shell and hot shocked bubble interior. As the amplitudes of the perturbations grow, fingerlike structures develop at the interface and vorticity of the flow increases due to shear stresses. Finally, in the fully nonlinear phase of the instability, turbulent mixing of the two layers starts, and the shell will eventually breakup and fragment. An azimuthal magnetic field in the shell will limit the growth rate of the instability due to magnetic tension forces, but not for all wave modes of the instability. In essence, only modes above a critical wave number will become unstable, giving a lower limit to the size of blobs (Breitschwerdt et al. 2000). This happens first at the top of the expanding bubble, where the acceleration is highest. The clumps generated this way and the hot gas inside the bubble – including the highly enriched material – are expelled into the halo of the galaxy or even into intergalactic or intracluster space, contributing to the chemical enrichment of galactic halo or intracluster medium.
After deriving the acceleration of the outer shock in Sect. 3.2, we analyze the timescales for the development of RTIs in the shell in Sect. 3.3.
2.3. Modeling superbubbles
Originally developed to describe the propagation of a blastwave due to a strong nuclear explosion in the Earth’s atmosphere, the approximation found by Kompaneets (1960) is also applicable to investigate the evolution of a superbubble in a disk galaxy analytically. We modify Kompaneets’ approximation (KA) in order to describe not only a bubble driven by the energy deposited in a single explosion or as a continuous wind (e.g. Schiano 1985), but to produce analytical models for the expansion using a timedependent energy input rate due to sequential SNexplosions of massive stars according to a galactic initial mass function (IMF). The axially symmetric problem is described in cylindrical coordinates (r, z). In order to use the KA for the investigation of SB evolution the following assumptions have to be made: (i) the pressure of the shocked gas is spatially uniform; (ii) almost all of the sweptup gas behind the shock front is located in a thin shell, and (iii) the outer shock has to be strong all over the evolution of the bubble. Using the third assumption we get our blowout condition: if the outer shock has a Mach number M ≥ 3, i.e. the upstream velocity of the gas in the shock frame is at least 3 times the sound speed of the warm neutral ISM at the transition of deceleration to acceleration, then the bubble will blowout into the halo and reach regions of higher galactic latitudes. Mac Low et al. (1989, hereafter MMN89) find via comparisons to their numerical simulations that the KA can be even used after the instabilities in the shell set in, because the pressure inside the bubble is not released very quickly. If the condition is not fulfilled (i.e. no strong shock), the system is obviously not energetic enough, hence cannot be described by the KA and will finally slow down and merge with the ISM like in the case of disk supernova remnants. Blowout usually occurs when the shell reaches between one (Veilleux et al. 2005) and three scale heights (Ferrara & Tolstoy 2000). In the next section, this is confirmed and exact values for the height of the bubble are given, using different descriptions of the ambient ISM and different ways of energy input (see Table 2). Compared to other groups using a dimensionless dynamical parameter introduced by MM88 to decide if blowout will happen, our criterion’s advantage is that the result is given in explicit numbers, and hence easier to use when compared to observations. Additionally, we investigate if the acceleration of the bubble at the time where fragmentation occurs will exceed the gravitational acceleration near the galactic plane. Using this simple comparison we can ensure that fragments of the bubble and the hot bubble interior will be expelled into the halo instead of falling back onto the disk.
A pure exponential atmosphere was used in the KA, which was already modified for a radially stratified medium (Korycansky 1992) and an inversesquare decreasing density (Kontorovich & Pimenov 1998). We adopted the original calculations to model the expansion of superbubbles in a more realistic fashion and investigate two cases of density distribution in our paper: the first one corresponds to a bubble evolving in an exponentially stratified medium symmetric to the midplane (1)where ρ_{0} is the density in the midplane and H is the scale height of the ISM. The cluster is located at z = 0. Since it is an idealization that OBassociations are only found in the galactic midplane, but are rather offset in zdirection, we examine in our second case an offplane explosion, where the density law is given by (2)with z_{0} being the center of the explosion. The density at z = z_{0} is either derived from the relation ρ_{1} = ρ_{0}·exp [ − z_{0}/H] or can be a known value. Any real case encountered will be described by one of the two or lie in between, i.e. the offset is small enough for the midplane to be pierced by further explosions.
To get a spatial solution the RankineHugoniot conditions for a strong shock for every point of the azimuthally symmetric shock surface have to be solved (e.g. BisnovatyiKogan & Silich 1995). This gives the normal component of the expansion speed (3)where γ is the ratio of specific heats (γ = 5/3 for a monatomic ideal gas). We need to know the pressure P(t) in the bubble, (4)with Ω(t) being the volume confined by the shock and E_{th} the thermal energy in the hot shocked gas region. The volume is given by the integral (5)Introducing a transformed time variable (in units of a length) (6)makes it easier to solve the PDE which is obtained after rearranging and equating Eq. (3) and the time derivative of Eq. (6) and by assuming that the shock front is a timedependent surface, defined as f(r,z,t) = 0 (7)By separation of variables one gets the solution r = r(y, z), which describes the evolution of the half widthextension of the bubble parallel to the galactic plane. For the symmetric density law this is (8)and for an offplane explosion we find (9)Due to the explicit relation between time t and y, the parameter y is used to represent the evolution of the bubble. At this point we introduce a dimensionless parameter . Figures 1 and 2 show the position of the outer shock at different values of for the symmetric and offplane model, respectively.
Fig. 1 Position of the shock front in the symmetric model at certain values of the dimensionless time variable , 1.0, 1.4, 1.7, 1.9, 1.98, and 2.0 with the energy source in the galactic midplane. 

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Fig. 2 Same as Fig. 1, but for the offplane model. The energy source is located at z_{0} = 0.7H above the plane (indicated by the dashed line). 

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A few points of the bubble’s surface are special, because explicit equations are available for them from the solution (Eqs. (8) and (9)). When , the top z_{u} and bottom z_{d} of the bubble can be evaluated. In the case of the symmetrically decreasing density, the top of the bubble as a function of is given by (10)and the bottom of the bubble is simply . In the other case, the coordinate z_{u} shows an offset of z_{0} and we derive (11)and for the bottom of the bubble (12)Furthermore, the halfwidth maximum extension of the bubble, r_{max}, parallel to the galactic disk is found, where , (13)which is valid for both density laws.
Obviously, the top of the bubble reaches infinity by the time where approaches the value 2. This happens, because the newly sweptup mass asymptotically goes to zero, due to the exponential decrease in density. The solution then breaks down, because numerically the shock reaches infinity in a finite time. Also the remnant volume goes to infinity (see Figs. 1 and 2). In reality, a nonzero density at infinity leads to restricted values of the shock front velocity (Kontorovich & Pimenov 1998). Actually, blowout occurs earlier in the evolution of a bubble with values of well below 2.
3. Results
We use different energy input schemes to calculate blowout timescales, the energy input that is needed for blowout, and the instability timescales. In the first and simplest case, a single, huge explosion forms the superbubble like in the original version of Kompaneets (1960). A time dependent energy input rate is implemented in the next model, where the number and sequence of SN explosions in a star cluster is given by an initial mass function. Finally, the wind model uses a constant energy input rate to drive the expansion of the bubble, which is just a special case of the IMFmodel. In all calculations, a cluster is defined to have at least two member stars.
We apply these models to two cases of ISM density distribution. First, we study the evolution of a bubble in an ISM representing the Lockman layer of a MilkyWaytype galaxy (number density n_{0} = 0.5 cm^{3}, scale height H = 500 pc) with the center of the explosion placed in the midplane and exponentially, symmetrically decreasing density above and below the disk (Eq. (1)). In the second case, the star cluster is displaced by z_{0} from the galactic midplane and the SB is expanding into a rather dense (n_{0} = 10 cm^{3}), lowscale height (H = 100 pc) medium described by a simple exponential law (cf. Eq. (2)). In this offplane configuration the bubble shall not expand below the midplane before blowout (i.e. roughly the time when RTIs start to appear in the shell), thus it is not affected by the increasing density below the plane until that time, which results in a onesided blowout. In other words, the value of z_{0} has to correspond to the absolute value of the coordinate (Eq. (12)) in the case of an unshifted bubble (z_{0} = 0) at the time when the acceleration sets in. Values of at range between 0.45H and 1.0H, thus we take an average value and put the explosion at z_{0} = 0.7H^{1} in order to be able to compare the models. This corresponds very well to MM88’s criterion for ’onesided superbubbles’: bubbles for which the association is found above 0.6H blow out on one side of the disk only, and the bottom of the bubble should be decelerating more strongly than a spherical one would do.
3.1. Thermal energy
Basu et al. (1999) show that the thermal energy in the hot interior of a SB expanding into an exponentially stratified medium is very close to the value in the case of a homogeneous ISM until about four times the dimensionless timescale t/t_{D}, which is rather late in the evolution of the superbubble. We find that at the time of fragmentation ~2 − 3 t/t_{D} are reached (see Table 3). Thus, we can estimate the thermal energy in the region of hot shocked gas following the calculations of Weaver et al. (1977) for a windblown bubble in a uniform ISM. These take into account the equations of energy and momentum conservation, as well as the radius of a spherical bubble. The inner shock, where the energy conversion takes place is always roughly spherical, since it is close to the energy source. However, with increasing time, the dynamics of a bubble in an exponentially stratified medium will differ from that in a homogeneous medium, where no blowout will happen.
3.1.1. SNmodel
All SNe explode at the same time. We find that the fraction of the total SNenergy converted into thermal energy at the inner shock is E_{th, SN}(t) = 2/3·E_{SN}·N_{SN} (see Appendix A). N_{SN} is the total number of SNe, with each explosion releasing a constant energy of E_{SN} = 10^{51} erg.
3.1.2. IMF and windmodel
A timedependent energy input rate L_{SB}(t) = L_{IMF}·t^{ δ} is used, where the energy input rate coefficient L_{IMF} and the exponent δ are characterized by the slope of the IMF and the main sequence lifetime of massive stars. The calculations of the timedependent energy input rate follow Berghöfer & Breitschwerdt (2002) and Fuchs et al. (2006) and shall be presented here briefly. The IMF describes the differential number of stars in a mass interval (m,m + dm) by a power law (14)where Γ is the slope of the IMF and m is always given in solar mass units. Integration from a lower mass limit m_{l} to an upper mass limit m_{u} gives the cumulative number of stars or – having m_{l} ≥ 8M_{⊙} – the number of OB stars to explode as SNe in this stellar mass range (15)In our general treatment of the IMF either N_{0} or m_{u} will be given and in order to get N_{OB} from the equation above, a correlation between N_{0} and m_{u} is needed. We use integer mass bins and simply fix the number of stars in the last mass bin N(m_{u} − 1,m_{u}) = N_{OB} = 1, i.e. there is exactly one star in the mass bin of the most massive star. Hence the normalization constant is . When dealing with a real association, the number of stars N_{OB} in a certain mass range (m_{l}, m_{u}) can be deduced from observations and thus the normalization constant will be estimated. Moreover, using this information is statistically more relevant than using THE most massive star of the cluster because the distribution of stars in a real clusters may not follow integer mass bins.
Since the distribution of the stars by their mass is given by the IMF, and L_{SB} follows the timesequence of massive stars exploding as SNe, we get the energy input rate L_{SB}(t) = d/dt[(N(m)·E_{SN})]. We just need to express the function N(m) as a timesequence, thus we treat the number of stars between (m,m + dm) as a function of mass and use the main sequence lifetime (t,t − dt) of massive stars (16)The main sequence lifetime and the mass of a star are connected through t(m) = κ·m^{− α} or m(t) = (t/κ)^{− 1/α}. The values of Fuchs et al. (2006) are used throughout this paper, where κ = 1.6 × 10^{8} yr and α = 0.932. The timederivative of m(t) together with dN/dm from Eq. (14) are inserted into Eq. (16) (17)Since L_{SB}(t) ∝ t^{ δ}, the exponent must be δ = −^{(}Γ/α + 1^{)}. After summarizing the constants one obtains the energy input rate coefficient (18)The full equation of the thermal energy as a function of time for the IMFmodel is (see Appendix A for details) (19)Using an IMF yields a more realistic framework for galactic SB expansion compared to a simplified point explosion. Therefore, we investigate the effect of changing the slope of the IMF in our calculations. We compare three different IMFslopes for massive stars: Γ_{1} = −1.15 (Baldry & Glazebrook 2003), Γ_{2} = −1.35 (Salpeter 1955), and Γ_{3} = −1.7 (Brown et al. 1994; Scalo 1986) resulting in δ_{1} = 0.23, δ_{2} = 0.45, and δ_{3} = 0.82.
With an exponent δ = 0, Eq. (19) respresents the thermal energy in case of a constant energy input rate, where the number of all SNe is averaged over the whole lifetime of the association, thus this model simply has a slope of Γ_{0} = −0.932 (windmodel). Now the larger time intervals between SNexplosions of stars with higher main sequence lifetime compensate the growing number of SNe per mass interval going to lower mass stars as it would be the case in the IMFmodel. With a constant energy input rate L_{w}, our result (Eq. (19)) checks with Weaver et al’s relation of E_{th,w}(t) = 5/11·L_{w}·t.
3.2. Evolution of a superbubble until blowout
The transformed time variable (Eq. (6)) contains not only the energy input and the ambient density, but also the volume of a bubble at a given time. Since this equation gives us an explicit correlation between time t and the time variable it is possible to describe the volume of the bubble as a function of instead of being dependent on time. To proceed with our analytical description, we need to find simple expressions of the volume for the two cases of density distribution.
Symmetric model
The bubble contour on the + zside of the midplane can be approximated by being part of an ellipse with a semimajor axis of (20)and corresponding to the bottom of an unshifted bubble in a pure exponential atmosphere. The semiminor axis is equal to (see Eq. (13)). The center of the ellipse is located on the zaxis (21)such that the ellipse is given by (22)Rotating this curve around the zaxis and multiplying by two results in the total volume of the superbubble (23)The approximation works very well and we find a deviation of Eq. (23) from the numerical integration of the volume according to Eq. (5) of only ~2.2% at very late stages of evolution ().
Fig. 3 Blowout timescales at the coordinate z_{u} for the SNmodel. Left: double exponential layer with H = 500 pc and n_{0} = 0.5 cm^{3} (symmetric model); right: exponentially stratified medium with H = 100 pc and n_{0} = 10 cm^{3} (offplane model). The dashed lines indicate the number of SNe required for blowout and the corresponding blowout time. 

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Offplane model
The volume will be approximated by treating the 3D shape of the bubble as a prolate ellipsoid (see also Maciejewski & Cox 1999). It is not dependent on the offset z_{0}, thus the bubble volume as a function of reads (24)At , the analytical result differs from the numerical value according to Eq. (5) by only ~1.8 %.
Obviously, the parameter is important for estimating the evolutionary status of a bubble, since it relates the size of the bubble to time. We can now calculate the age t of a bubble at any value of . Replacing and using the nondimensional volume instead of Ω (t) helps to obtain a useful relation of the time derivative of Eq. (6) (25)When performing the integration, the thermal energy for each model, i.e. for different kinds of energy input, has to be replaced by the corresponding formula. Substituting and integrating yields the general expression (26)which will be solved separately for all three models in the remainder of this subsection. In order to simplify the integration on the right hand side, and especially to avoid a double integral in the case of in the calculations to follow, we use a series expansion of the bubble’s volume: In both cases of density distribution we expand until 43rd order to make sure the simplified integral has a deviation of ≪1 % from the numerical value at a time of .
For further investigation of the blowout phenomenon we need to know the velocity and acceleration of the shock front. It is possible to derive these properties analytically at certain points of the bubble’s surface, where explicit equations exist. The velocity can be calculated for top, bottom and maximum radial extension of the bubble, and also at for the symmetric model as well as at for the offplane model. We will concentrate on calculating the velocity at the top of the SB (which is equal to the absolute value at the bottom for the symmetric model) in this paper, since this is crucial in determining if blowout happens or not. For all models, the velocity at the top of the bubble is given by (29)with the derivative of z_{u} with respect to (30)Whereas the velocity given by Eq. (3) depends on time t and the coordinate z, the velocity in Eq. (30) is only dependent on . This makes it easier to find general results in terms of the yparameter at the time of blowout for each model specification. The second derivative of z_{u} gives us the acceleration at this coordinate (31)The calculation of (Eq. (25)) and thus need to be done separately for each model, which is shown in Appendix B.
Once velocity and acceleration are obtained, we get the value of , where the velocity of the top of the bubble has its minimum, i.e. the transition from deceleration to acceleration along the density gradient. The value of the dimensionless variable is the same for all SBs of each model, but corresponds to a different time in the evolution of a bubble depending on the ISM parameters and the energy input. The time interval until is called blowout timescale (see Figs. 3−5 for SN, IMF, and windmodel, respectively).
Fig. 4 Same as Fig. 3, but for the IMFmodel (left: symmetric model; right: offplane model). Thin lines indicate the number of OBstars needed for blowout. Γ is the IMF exponent (for details, see text). 

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Fig. 5 Same as Fig. 3, but as a function of a constant energy input rate (left: symmetric model; right: offplane model). The number of SNe is obtained by converting the wind luminosity to an energy input averaged over a time interval of 20 Myr, from the explosion of the most massive star until the last SN. 

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By fixing the velocity of the outer shock at the time , the energy input required for blowout of the disk can be derived. We assume that for M ≥ 3 at z_{u} the shock is sufficiently strong and the blowout condition is fulfilled. Thus, the velocity of the shock has to be at least 3·c_{s} with respect to an ambient ISM at rest. Using a temperature of the surrounding medium of T ~ 6000 K typical for the Lockman layer of the Galaxy (Crawford et al. 2002), the minimum velocity corresponds to cm/s, where k_{B} is Boltzmann’s constant; the mean atomic mass is in a gas with mass density where the mean molecular weight of the neutral ISM of μ = 1.3 and the hydrogen mass of m_{H} = 1.7 × 10^{24} g are used. This criterion is valid for the lowscale height, highdensity ISM as well, because onesided SBs blow out of the dense disk and start to accelerate into the halo at ~2H = 200 pc (Table 2). At these distances from the midplane the presence of the warm gas layer already influences the evolution of the bubble. The expressions for the minimum energy input are derived in the following subsections.
Polynomial fits of three typical IMFrelations.
3.2.1. SNmodel
Since the evolution of the bubble depends on the scale height H, on the density ρ_{0,1} of the ambient medium near the energy source, and on the energy E_{th,SN}, a timescale (in units of seconds) can be constructed from these quantities (32)Solving Eq. (26) yields the time t elapsed since the explosion for any chosen value of . For a constant value of energy released by a number of supernovae, the integration on the left hand side is simple. Expressing the time t as a function of , rearranging the equation and making use of the timescale (Eq. (32)) yields the following equation (33)For the calculation of the velocity, we first express Eq. (25) in terms of the timescale t_{SN}(34)Now, using the expressions from above, the velocity at the top of the bubble can be written as (35)The calculation of the acceleration at the top of the bubble is found in Appendix B. The critical value , where the acceleration of the outer shock starts, the corresponding dimensionless timescale and the coordinate of the top of the bubble at this time are summarized in Table 2 for both cases of density distribution. Fig. 3 shows the age of the bubble at the transition from deceleration to acceleration as a function of the number of SNexplosions in the range of 2 − 500 SNe. The dashed lines indicate the number of SNe that are required for blowout for each kind of density distribution and the corresponding timescale for blowout. In the case of the symmetric density law with H = 500 pc and n_{0} = 0.5 cm^{3} about 56 SNe have to explode at once and the acceleration starts ~40 Myr after the initial explosion. Only three SNe are sufficient for an offplane explosion in a pure exponential atmosphere at z_{0} = 0.7H with H = 100 pc and n_{0} = 10 cm^{3}. In this case, blowout happens after ~7.2 Myr.
In order to obtain the general dependence of the minimum number of SNe on the properties of the surrounding medium at the coordinate z_{u}, the thermal energy E_{th, SN} appearing in t_{SN} (Eq. (32)) is replaced by 2/3·E_{SN}·N_{SN} as derived in Appendix A and the velocity at z_{u} (Eq. (35)) is solved for N_{SN}: (36)The velocity is fixed to be cm/s at the time of blowout and the constants k = 1 for the symmetric model and k = exp( − z_{0}/H) for the offplane model, respectively, are used.
3.2.2. IMF and windmodel
Next, we want to calculate the age, velocity and acceleration of a bubble driven by a timedependent energy input rate, where different slopes of the IMF will be used. As a special case of this model, we can describe the evolution of a bubble powered by a constant energy input rate. Using a procedure similar as above, the characteristic timescale for the IMFmodel in terms of the energy input rate coefficient L_{IMF}, the density and the scale height of the ISM is found to be (37)After inserting the thermal energy derived for the IMFmodel (Eq. (19)) into Eq. (26) and solving the integral, one gets the time as a function of (38)Moreover, inserting the thermal energy for this model given by Eq. (19) into Eq. (25) and using the time t and the timescale t_{IMF} gives (39)where ϵ = 1/(δ + 3). Using further substitutions (40)and (41)yields a simplified expression of Eq. (39) (42)As it was done in the previous model, multiplying this equation by (Eq. (30)) results in the velocity at the top of the expanding superbubble (43)For the calculation of the acceleration , see Appendix B. Now that general expressions for velocity and acceleration of the bubble are known, we can estimate when a bubble starts to accelerate into the halo. The results for the IMFmodel using different slopes and for the windmodel are listed in Table 2.
Characteristic values of superbubble blowout.
Characteristic values of superbubble fragmentation.
In order to present the blowout timescales for superbubbles driven by a timedependent energy input rate as a function of the number of OBstars, N_{OB}, instead of the normalization constant N_{0}, which is contained in t_{IMF} (Eq. (38)), we have to make use of the fit N_{0}(N_{OB}) from Table 1. The coefficients of the powerlaw function found with a nonlinear leastsquares fit provide an excellent fit to the relation with errors less than 1% for 2 ≤ N_{OB} ≤ 500 for all IMFmodels. The timescales in Fig. 4 are shown for 2−500 association members, where the dashed lines represent the minimum number of OBstars and the corresponding time until blowout. Using IMFs with Γ_{1} = −1.15, Γ_{2} = −1.35, and Γ_{3} = −1.7, respectively, 27, 25, and 24 OBstars are needed with an upper mass limit of 19, 18 and 17 M_{⊙}. The resulting blowout timescales are 22.8, 20.1, and 15.2 Myr for a symmetric density distribution with H = 500 pc and n_{0} = 0.5 cm^{3} (Table 2 and Fig. 4, left). In the case of an explosion at 70 pc above the plane in an ISM with H = 100 pc and n_{0} = 10 cm^{3} (Table 2 and Fig. 4, right), the association needs to have at least ~8, 13, and 29 massive stars or an upper mass limit of 13, 15 and 18 M_{⊙} (same order of IMFslopes). Approximately 4.0, 3.5, and 2.6 Myr pass from the first SNexplosion until blowout. We are interested again in obtaining an analytical expression for the minimum number of OBstars to get blowout. Equation (43) needs to be solved for the normalization constant, after L_{IMF} was replaced by Eq. (18), which yields (44)for any set of the parameters scale height and ISMdensity. This has to be converted to a number of stars N_{OB, blow} by using the fit presented in Table 1, which is obtained with the same fitting procedure as before. We find that our fits are very good for all IMFslopes with an average deviation of ~1.4% in the range of 2 − 500 OBstars. In Fig. 11 we compare the number of stars required for blowout as a function of the scale height using certain values of the density for the SN and the IMFmodels. The discussion of the results is found in the next section.
Using a constant energy input rate (δ = 0), we obtain a dimensionless wind coefficient (45)instead of the normalization constant. Following Eq. (17), we can derive the minimum wind luminosity that fulfills the blowoutcriterion (46)in units of erg/s. We assume that the wind luminosity is the energy input of all SNe averaged over the period of Δt ≈ 20 Myr, the lifetime of the association, which is in general the time between the first and the last SNexplosion^{2}. Hence, the minimum wind luminosity can be converted into a total number of stars N_{w,blow} = L_{w,blow}·Δt/E_{SN}. Figure 5 shows the time elapsed since until the point of acceleration, for wind luminosities between 3 × 10^{36} and 8 × 10^{38} erg/s, corresponding to ~2−500 SNe calculated over a timespan of 20 Myr. The values for the minimum wind luminosity and corresponding ages of the SB are given by the dashed lines. For a symmetric expansion into the Lockman layer, an energy input rate of ~7.4 × 10^{37} erg/s is necessary for blowout which corresponds to ~47 SNe. Such a bubble needs 26.1 Myr until acceleration begins. Expansion of a SB into the lowscale height, highdensity medium produced by an offplane explosion at z_{0} = 0.7H requires ~1.9 × 10^{37} erg/s or about 12 SNe and takes ~4.7 Myr.
3.3. RayleighTaylor instabilities in the shell
Infinitesimal perturbations at the interface between a denser fluid supported by a lighter fluid in a gravitational field generate waves with amplitudes growing exponentially with time in the initial phase. For an incompressible, inviscid, nonmagnetic fluid, the timescale τ_{rti}, characterizing the growth of the instability, results from a linear stability analysis combined with the conservation equations (47)where λ is the perturbation wavelength, g is the gravitational acceleration and ρ_{1} and ρ_{2} are the densities of the light and heavy fluid, respectively. In the case of a superbubble, where the dense shell is accelerated by the hot, tenuous gas, the most important perturbation wavelengths out of the Fourier spectrum are the ones, which are comparable to the thickness of the shell (which is itself a function of time), because these distort the shell so strongly that breakup can occur. Furthermore, identifying the gravitational acceleration with the acceleration at the top of the bubble results in an instability timescale at this coordinate (48)The density in the shell at the coordinate z_{u} is always given by for an adiabatic strong shock^{3}. The density of the bubble interior as a function of time is with being the ejecta mass and being the volume of the bubble confined by the inner boundary of the shell. The new semimajor and semiminor axes simply reduced by the thickness of the shell are and . To calculate the thickness of the shell we use the fact, that the mass of the gas in a volume of undisturbed ISM is the same as that in the shell of sweptup ISM (if we neglect effects of evaporation, heat conduction or mass loading). In a symmetrically decreasing density distribution the bubble contour has an hourglassshape, thus, due to symmetry, we need to include half of the bubble only in the calculation (49)Half of the shell mass is , which we obtain by integration over the density gradient. The bubble’s volume, confined by the surface A_{ell} of an ellipsoid, is included between the coordinates z = 0 and z_{u}. Using the new semimajor and semiminor axes – each itself a function of – in the equation of the ellipse means solving a double integral in the calculation of the volume (Eq. (23)). However, we can approximate by using an average thickness of the shell over the time in this case. The new semimajor axis is as large as ~91% of the regular value and the new semiminor axis ~90% (averaged over the time span ).
In the case of an offplane explosion the thickness of the shell is given by (50)with the mass inside the shell with the surface A_{ell} of the complete ellipsoid.
Fig. 6 Fragmentation timescales for the SNmodel; the dashed line marks the minimum number of SNe for a blowout superbubble and the corresponding time until full fragmentation of the shell at the top of the bubble (for details see text and Tables 2 and 3). Left: double exponential layer with H = 500 pc and n_{0} = 0.5 cm^{3} (symmetric model); right: exponentially stratified medium with H = 100 pc and n_{0} = 10 cm^{3} (offplane model). 

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Furthermore, we have to estimate the mass inside the hot bubble interior, which is the mass ejected by the SNexplosions. The total mass of each star belonging to the association goes into ejecta, except ~1.4 M_{⊙} for a neutron star remnant, which is lower than the OppenheimerVolkoff limit, because most SN progenitors are lower mass stars among the massive stars. Thus, we have to consider the mass of per star with in units of solar masses and the total number of stars . In order to obtain the ejecta mass for the IMFmodel as a function of time (i.e. time variable ), we fix the upper mass limit – which is related to the first SNexplosion – and introduce a variable lower mass limit . Again, connecting the mass of a star to its main sequence lifetime gives a time dependent ejecta mass. In order to account for the mass included in the mass interval (m_{u} − 1,m_{u}) for integral mass bins at time , we have to correct for the main sequence lifetime of stars with mass m_{u} − 1 (51)Integrating over the mass range of the association gives the ejecta mass of SNe exploded until some time (52)So we are able to derive the mass ejected until the time , where instabilities start to appear in the shell. Unfortunately, this formula does not hold for small associations, since all SNe may have exploded in a rather small time interval Δτ = τ_{ms}(8) − τ_{ms}(m_{u} − 1), possibly long before . In that case, the ejecta mass is replaced by the total mass of the ejecta after the last SNexplosion, otherwise the mass and thus the RTItimescale would be overestimated. Including the ejecta mass (Eq. (52)) in the density ρ_{in} yields an instability timescale (Eq. (48)) as a function of the most massive star m_{u} in the IMFmodel, but we prefer expressing it as a function of the total number of stars in an association. The relation between these two parameters is fitted with an approximation of the form (see Table 1) and has average errors of 1.2% over the range of 2−500 OBassociation members.
In the case of the SNmodel, all the mass of exploding stars (except of 1.4 M_{⊙} per star) is released in the initial explosion. We find a useful powerlaw approximation for the total ejected mass of a star cluster as a function of the number of SNe (using an IMF with Γ = −1.35) (53)Errors for this approximation are about 3% for small associations (2 OBstars) and ~1% for large clusters (500 stars). It can be used to obtain the ejecta mass for the windmodel as well, but instead of the number of supernovae, N_{SN}, we have to include the energy input rate into the bubble until the point of acceleration in units of the standard SN energy . Calculating fragmentation this way only works for a wind, which is produced by averaging SNe and thus includes the mass of the exploded stars, but not for a true stellar wind.
Instabilities will dominate when the RTItimescale at the top of the bubble becomes smaller than the dynamical timescale τ_{dyn,I} = a/ż_{u} (symmetric bubble) or τ_{dyn,II} = z_{u}/ż_{u} (offplane model) of the system. In terms of the dimensionless time variable this happens at and we have to find the value, where for all models, which is shortly after the acceleration sets in. The exponentially growing instability is usually fully developed and the shell will breakup at when (see Table 2). This means that the exponentially growing amplitude of the perturbation has reached a size of e^{3} ≃ 20 times of the initial one, sufficiently large to assume full fragmentation.
Fig. 7 Same as Fig. 6, but for the IMFmodel and indicating the minimum number of OBstars in an association for different IMF slopes by the corresponding thin line (left: symmetric model; right: offplane model). 

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Fig. 8 Same as Fig. 6, but for the wind model and a minimum constant energy input rate equal to a number of SNe distributed over 20 Myr (left: symmetric model; right: offplane model). 

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Figure 6 shows the fragmentation timescale as a function of the total number of stars for the SNmodel, Fig. 7 shows the same for the IMFmodel for different slopes and finally, the timescale as a function of wind luminosity is presented in Fig. 8.
Due to the large fragmentation timescales for very small associations, the abscissa is chosen to range from 20−500 OBstars for the symmetric model to highlight the behavior of this function. The offplane model is shown again for 2−500 stars. For the windmodel these numbers correspond to 3.2 × 10^{37}−7.9 × 10^{38} erg/s (symmetric model) and 3.2 × 10^{36}−7.9 × 10^{38} erg/s (offplane model).
In both cases of ISM density distribution, fragmentation is easily achieved for clusters with N_{blow} (i.e. those producing blowout superbubbles) within 50 Myr, a reasonable timespan for fragmentation to take place before galactic rotation or turbulences have had a major influence on the bubble structure (see Figs. 6− 8, right panel) and no additional fragmentation criterion is needed. However, we want to check at this point whether the fragmenting SB can escape from the disk and accelerate into the halo. Thus, we compare the acceleration of the top of the bubble^{4} at with the gravitational acceleration near the galactic plane (we do not require the bubble to escape completely from the gravitational potential of the galaxy). We use g_{z}(R,z) = −∂φ/∂z with φ, the disk potential in cylindrical coordinates (Miyamoto & Nagai 1975), given by (54)where R is the galactocentric radius, G is the gravitational constant, a = 7.258 kpc, b = 0.520 kpc, and the disk mass M = 2.547 × 10^{11}M_{⊙} (Breitschwerdt et al. 1991) for a MilkyWaytype galaxy. This yields, e.g. for the galactocentric distance of the Sun at R_{⊙} = 8.5 kpc a value of − g_{z}(R = R_{⊙},z = 1 pc) = 3.47 × 10^{11} cm s^{2}. We find for all models that SBs driven by the minimum blowout energy, N_{blow}·E_{SN}, have an acceleration larger than − g_{z}(R,z = 1 pc) for all galactrocentric radii R. The complete set of results can be found in Table 3.
4. Application to the W4 superbubble
Winds of young massive stars of the cluster OCl 352 are supposed to be the energy source of the W4 superbubble. The cluster is located at a height of ~35 pc (Dennison et al. 1997) above the disk at a distance of ~2.35 kpc (West et al. 2007). In the following, we want to apply our offplane windmodel to this superbubble.
According to West et al. (2007) only the structure above OCl 352 should be termed superbubble or chimney (G134.4+3.85), while the lower part is the W4 loop. The superbubble is in the process of evolving into a chimney, because the ionized shell of G134.4+3.85 already started to fragment at the top of the bubble, where the shell is expected to breakup first due to instabilities. From HI observations they derive a scale height of the ambient ISM of 140 ± 40 pc and they get bubble coordinates of pc (i.e. pc) and pc. With this ratio of z_{u,I}/r_{max} = 2.57 the bubble has reached an evolutionary parameter of in the Kompaneets model. At this time, the bubble’s extension from the star cluster to the top of the bubble is 6.66H and from the cluster to the bottom it is about − 1.35H. Comparing this to the physical dimensions yields a scale height of only ~32 pc in this region of the Milky Way. The scale height of 140 pc cannot be confirmed by the KA, since this would result in a value for z_{d} of almost 190 pc, i.e. the bottom of the shell reaching 150 pc below the midplane. Our model predicts that z_{d} should have reached only ~8 pc below the Galactic plane. The cluster itself is found at z_{0} = 1.09H. We obtain an age of the SB of 1.8 Myr or 2.3 Myr, taking a density of n_{0} = 5 cm^{3} and n_{0} = 10 cm^{3}, respectively, and using the energy input rate of 3 × 10^{37} erg/s (Normandeau et al 1996).
Fig. 9 Left: color image of W4 (West et al. 2007) combining HI, Hα and infrared data. For comparison with our model, due to the faint structures and low contrast, the overlay is shown in a separate figure (right panel). The height of the star cluster above the plane is z_{0} = 35 pc, marked by the black dot. Right: Kompaneets bubble at overlaid on the same image. The upper part of the contour fits quite well (solid line), while the part of the shell below the cluster is not seen in the observations (dotted line). 

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According to West et al. (2007), Basu et al.’s (1999) way of treating the W4 SB and W4 loop as one entity (and fitting a Kompaneets model to that) seems inappropriate. However, we wanted to see if our model could improve their findings. Since Basu et al. (1999) do not use an offplane model, we first checked their results with our model for an explosion in the midplane. We infer an age of ~2.3 Myr () when inserting all the values they use, which corresponds quite well to their derived age of 2.5 Myr. In this case, the distance of the top of the bubble to the star cluster of z_{1} = 246 pc was used, which corresponds to the coordinate in the offplane model including the offset z_{0}. In fact, the coordinate z_{u,I} = 211 pc should be used as the distance of the top of the bubble to the cluster when determining the aspect ratio, which is z_{u,I}/r_{max} = 211/74 = 2.85. With a slightly larger elongation than in the previous calculation, the bubble has reached the time parameter . With a height of 7.81H from z_{0} to z_{u,II}, this yields a scale height of H ≅ 27 pc close to Basu et al.’s value of 25 pc. But, the fact that the bubble is shifted to a lower density environment changes the age of the bubble significantly. With an offset of z_{0} = 1.30H, the age of the bubble is only ~1.7 Myr using n_{0} = 10 cm^{3}. This age is found within the previous estimates for the cluster of 1.3 − 2.5 Myr (cf. Dennison et al. 1997 and references therein) and is therefore consistent with the assumption that the bubble is blown by the wind of the Ostars in the cluster.
As the W4 loop and the W4 bubble are not dynamically connected, we concentrate in our approach on fitting only the SB shell above the cluster with our offplane model to Fig. 10 of West et al. (2007). If we use and the coordinates given in West et al. (2007) we find that the contour of the model does not match the shell in the observations very well. The model would fit almost perfectly, if it was shifted upwards by about one scale height, but then the position of the star cluster would be located outside the contour like in Fig. 11 of West et al. (2007). The somewhat more elongated bubble at respresents nicely the shell above OCl 352 (Fig. 9, right) with the cluster – although not exactly in the center – matching the offset z_{0} in the observations. We suggest that the part of the shell below OCl 352 (Fig. 9, right, dashed line) was decelerated due to the presence of the W4 loop and appears now flattened or has even merged with the upwards expanding part of the W4 loop.
Applying our criterion from Sect. 3, we find that blowout of a bubble with the association at around one scale height in an ISM with H = 27 pc and n_{0} = 10 cm^{3} is guaranteed for a wind luminosity as low as 8 × 10^{35} erg/s. We thus support Basu et al.’s statement that the bubble is already on its way of blowing out into the Galactic halo. Even if the bubble was not shifted above the plane, around 4 × 10^{36} erg/s would be sufficient for blowout, which is well below the 1.67 × 10^{38} erg/s found by MMN89 (as cited in West et al. 2007). The acceleration of the bubble, i.e. blowout, has started already at , more than 1 Myr ago.
Fig. 10 Symmetric Kompaneets bubble at overlaid on the absorption map of neutral interstellar NaI of Lallement et al. (2003). Isodensity contours with an equivalent width of 20 m (inner contour) and 50 m (outer contour) represent the rarefied cavity. 

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4.1. Local Bubble
The Local Bubble (LB) was most likely produced by 14 − 20 SNe, which exploded 10 − 15 Myr ago as the Pleiades subgroup B1 was moving through our Galactic neighborhood (Berghöfer & Breitschwerdt 2002; Fuchs et al. 2006). Numerical simulations suggest that 19 SNe are responsible and the bubble is ~13.6 Myr old with the last SN having exploded 0.5 Myr ago (Breitschwerdt & de Avillez 2006; de Avillez & Breitschwerdt 2012). Observations (Welsh et al. 1999; Sfeir et al. 1999) show that the bubble is not confined at higher galactic latitudes and thus, should be termed “Local Chimney”, but an elongated structure (the chimney walls) still exists, extending to ~250 − 400 pc above and below the midplane (Lallement et al. 2003). Although the Local Bubble is tilted about 20° to the midplane and expands perpendicularly to the plane of Gould’s Belt, we apply here our symmetric model with a timedependent energy input rate.
The walls of the Local Bubble (Fig. 5 of Lallement et al. 2003) can be fit very well with a Kompaneets model using an evolutionary parameter of . With the coordinate corresponding to ~370 pc, this yields a scale height of H = 80 pc. The radius of the bubble in the plane is almost 150 pc and r_{max} has an extension of 180 pc at in our model, corresponding to the observations (see Fig. 10). Since it was suggested that 19 SNe exploded in a time interval of 13.1 Myr and with a given lower mass limit of m_{l} = 8.2 M_{⊙} (Fuchs et al. 2006), we can infer – using main sequence lifetimes of the stars – that the upper mass boundary should be m_{u} = 20.9 M_{⊙} (independent of the IMF). So far, we assumed that there is exactly one star in the mass bin (m_{u} − 1,m_{u}) for general modeling. But this way of mass binning has to be modified and the mass interval of the most massive star needs to be adopted, because we know both m_{u} and N_{OB} in the case of the LB. Using an IMF with Γ_{2} = −1.35, the mass bin containing the most massive star is N(m_{u} − 1.9,m_{u}) = 1 and a normalization constant of N_{0} = 613 is found. This simply means, that the remaining 18 stars are distributed within the mass interval from 8.2 − 19 M_{⊙}. A density of the undisturbed ISM of n_{0} ≈ 7 cm^{3} is obtained to infer the presumable age of the LB. Acceleration of the shell started already at in this configuration, which was 3.3 Myr after the first SN exploded. The velocity of the bubble at this time was about 20 km s^{1}; thus the bubble fulfills the Kompaneets criterion of blowout into the halo. RayleighTaylor instabilities started to appear at the top of the bubble at , about 3.4 Myr after the initial SNexplosion and full fragmentation took place at a time of 5.2 Myr. With an acceleration of ~2.5 × 10^{9} cm s^{2} at exceeding the gravitational acceleration near the galactic plane by two orders of magnitude, the fragmenting shell will not fall back onto the disk. Actually, it is even higher than the local vertical component of the gravitational acceleration − g_{z}(R = 8.5 kpc, z = 114 pc) = 2.37 × 10^{9} cm s^{2} such that blowout of the Galaxy’s gravitational potential could be achieved. But since the LB is already disrupted on its poles before having reached one scale height of the Lockman layer, the bubble will not be able to expand to highz regions, but the ejected material will probably mix with the lower halo gas, leaving behind a “Local Chimney”.
5. Discussion
5.1. Comparisons of the models: blowout
Fig. 11 Numbers of SNe/OBstars needed for blowout are compared for SN, IMF and windmodel (energy input rate over 20 Myr gives the number of SNe for the windmodel). a) Symmetric case, midplane number density: 0.5 cm^{3}, scale height range: 220−500 pc; b) symmetric case, midplane number density: 5 cm^{3}, scale height range: 80−500 pc; c) offplane explosion at z_{0} = 0.7H, midplane number density: 1 cm^{3}, scale height range: 270−500 pc; d) offplane explosion at z_{0} = 0.7H, midplane number density: 10 cm^{3}, scale height range: 90 − 500 pc. 

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In order to compare the efficiencies of the three different models, we have to take a look at the number of SNe needed for blowout and at the corresponding timescales.
For an ISM with properties of the Lockman layer of the Galaxy, the SNmodel requires the largest number of SNe for blowout. If taking into account an OBassociaton’s lifetime of 20 Myr, the wind model’s efficiency is between the IMF and the SNmodel. The IMFmodel is the most efficient one and needs only half the number of SNe to explode over the whole lifetime of the SNmodel. In fact, the IMF with a slope of Γ_{3} = −1.7 needs the lowest number of OBstars to produce a blowout SB, followed by the one with Γ_{2} = −1.35 and Γ_{1} = −1.15, but the numbers are approximately the same (see Table 2). Additionally, the blowouttimescales for the IMFmodels are shortest with a steeper slope yielding a shorter timescale.
For the low scale heighthigh density ISM used in combination with an offplane explosion, the numbers of N_{blow} are lower in general. The IMFmodel with Γ_{3} = −1.7 is an exception, since the number of OBstars N_{blow} is now higher than in the symmetric case. The SNmodel is the most efficient one with only three SNe needed for blowout, and also the behavior of the IMFslopes is the opposite as above. The minimum energy input for the windmodel is somewhere in between, similar to that of the IMFmodel with Γ_{2}. But the timescales for blowout are unaffected by this change and are still longest for the SNmodel and shortest for the Γ_{3}model. This is simply because the acceleration starts later in terms of the time variable for the SNmodel, thus a larger volume has to be carved out by the SNexplosions.
We find the mathematical explanation for the blowout efficiency of the models when looking at Eqs. (36), (44), and (46), which are used to describe N_{SN, min}, N_{OB, min} and L_{w, min}, respectively. The dependence on the density is linear for the SN and windmodel and slightly increasing with an exponent b_{2} from Table 1 for the IMFmodels. The SNmodel is strongly influenced by the scale height, since N_{SN, min} ∝ H^{3}. For the windmodel, L_{w, min} ∝ H^{2} is found. The different IMFslopes give us the following relations: N_{OB, min} ∝ H^{1.9} for Γ_{1}, H^{1.6} for Γ_{2}, and finally H^{1.2} for Γ_{3}.
In Fig. 11 we illustrate these facts by comparing the symmetric and the offplane model for a lowdensity and highdensity ISM each. The values were chosen such that the Lockmanlayer density of n_{0} = 0.5 cm^{3}, (a), and a highdensity layer (d) with n_{0} = 10 cm^{3}, were used as reference (see Sect. 3). According to that, we wanted to investigate what happens to N_{min} with a density 10 times higher, i.e. n_{0} = 5 cm^{3}, in the symmetric case (b) or a 10 times lower density, i.e. n_{0} = 1 cm^{3}, in the offplane scenario (c). The plots start with a scale height on the xaxis, where at least N_{min} ≥ 2 are needed, which is the smallest association we use in this paper, and end at a scale height of 500 pc. We find that the models behave similarly throughout this scale height range irrespective of the density distribution, but the order of the efficiency of all models changes at H ~ 300−450 pc, which marks the transition of a lowscale height to a highscale height medium. In the lowscale height regime (until ~350 pc) the windmodel is least efficient, followed by the Γ_{3}model and the SNmodel. The Γ_{2}IMF is slightly better and an IMF with Γ_{1} needs the lowest number of SNe. For a scale height above 400 pc, the sequence of the models is the same as the one calculated for the Lockman layer (described above), except for the offplane model with n_{0} = 1 cm^{3}, where the IMFmodels do not intersect before ~500 pc according to Fig. 11c.
5.2. Comparison of the models: fragmentation
Next, we want to compare the timescales until fragmentation and explore the efficiencies of the different models. The energy input needed for a bubble to achieve fragmentation is given by the minimum blowout energy, i.e. the energy required for reaching a certain velocity v_{acc} at (see Table 2). The trend until the beginning of the fragmentation process (i.e. at ) is the same as that for blowout for both ISM cases, in terms of and in absolute timescale: the SNmodel is the least efficient one and an IMF with a steep slope yields the best results (shown in Table 3). Also, it takes the SNmodel the longest total time ) until full fragmentation occurs for bubbles driven by N_{blow}. But the IMFmodel with Γ_{2} = −1.35 now shows the lowest total timescale followed by Γ_{1} = −1.15 with no obvious correlation among the models. We find that the SNmodel is the fastest model in completing the fragmentation process – which means from until – and a steep IMF (Γ_{3}) yields the highest values of this timespan. Since this is the opposite to the efficiency sequence until , no general trend for the fragmentation timescales can be seen.
Concerning the interpretation of these facts one has to look at the RTinstability timescale itself. It is decreasing more rapidly for the SNmodel and a flat IMFmodel, because the acceleration is increasing in a slightly shallower way than for steep IMFs. Thus, a value of 1/3 τ_{rti} is reached easier for a flat IMF than for a steep one.
6. Summary and conclusions
We have developed analytical models based on the Kompaneets approximation (KA) in order to derive in a fairly simple and straightforward manner the physical parameters of observed superbubbles and their ambient medium and to gain physical insight into the blowout phenomenon associated with star forming regions.
In this paper we have deliberately refrained from building more complex models, which e.g. include stellar wind and WolfRayet wind phases, because we have put our focus on the important dynamical phenomena of blowout and fragmentation of the outer shell, and their dependence on the energy input source over time. A more detailed description of superbubble evolution models including further stellar evolutionary phases will be the subject of a forthcoming paper.
In our work the key aspect was to work out analytically the dynamics of (unfragmented) superbubbles for different energy input modes. We modified the KA to implement a more realistic way of energy input, i.e. modeling the time sequence of exploding stars in an OB association by including the main sequence lifetime of the massive stars and describing the numbers per mass interval by an initial mass function. We tested three different IMF slopes and also compared the IMFmodel to a simple SNmodel with an instantaneous release of energy and to a wind model with a constant energy input rate. Two different density distributions of the ISM were applied, a symmetric medium with parameters of the Lockman layer and a highdensity, lowscale height pure exponential atmosphere with the star cluster dislocated from the galactic plane. Velocity and acceleration of the shock front can be calculated analytically and the question how many SNe are needed for blowout into a galactic halo can be answered. The exact position of the outer shock in scale height units when the acceleration starts can be given. Furthermore, the timescale for the development of RayleighTaylor instabilities in a SB shell is calculated, and thus, a fragmentation timescale can be derived. The overall pattern shows that at larger scale heights (H > 400 pc), independent of the ISM density, the SNmodel needs the highest energy input, followed by the windmodel, whereas an IMF with a steep slope is the most efficient one. The same ranking applies to the blowout timescales of these models (Figs. 3−5 and Table 2). At low scale heights (H ~ 100 pc) and moderate or high densities (n_{0} ≥ 5 cm^{3}), the picture changes completely, i.e. the SNmodel requires the lowest energy input and the IMFmodel with Γ_{3} = −1.7 is the least efficient one. The explanation is that a single release of energy is more powerful in sweeping up a thin layer of ISM, whereas it is easier for the IMF model with an increasing energy input with time (L_{SB} ∝ t^{δ}, 0 < δ < 1, Eq. (17)) to sustain the supply of energy over a larger distance. When comparing fragmentation timescales (Figs. 6−8 and Table 3), the IMFmodels exhibit the lowest values, favouring flatter IMFs with Γ_{1} and Γ_{2}. In terms of and absolute time, fragmentation happens first for the Γ_{2}model for SBs driven by the minimum number of SNe for blowout.
Still, the KA is a rather simple model. It does not account for magnetic fields, ambient pressure, inertia and evaporation of the shell. Also a galactic gravitational field and cooling of the shocked gas inside the cavity are neglected.
In our model, we include galactic gravity in a rudimentary way: if a bubble fulfills the blowout criterion and additionally the shell’s acceleration at the top of the bubble at the time of fragmentation exceeds the vertical component of the gravitational acceleration in the disk, the SNejecta and fragmenting shell are expelled into the halo. We find that this is true for all bubbles created by an energy input of N_{blow}·E_{SN}.
Further analysis of a fragmenting superbubble would also involve to calculate the motion of the shell fragments, and ultimately the dynamics of a galactic fountain like e.g. in Spitoni et al. (2008). A simple ballistic treatment of the motion of blob fragments in a gravitational potential would, however, be too simplistic, as they would for some time experience a drag force due to the outflowing hot bubble interior. The drag would be proportional to the ram pressure of the hot gas and the cross section of a blob, which would, to lowest order, be proportional to the thickness of the fragmented shell; and finally it would depend on the geometry of the blob, which might be taken as spherical. However, due to compressibility, bowshocks and headtail structures might subsequently be formed, so that for a realistic treatment numerical simulations would be the best choice.
According to MM88 cooling of the bubble interior can be neglected for Milky Way type parameters of the ISM, but should be taken into account for smaller OBassociations or a dense and cool ISM. In general, cooling is not important as long as the timescale for radiative cooling is large compared to the characteristic dynamical timescale of the superbubble. Including a magnetic field would be quite important, but this goes beyond the scope of our analytical model and is left to numerical simulations. Ferriere et al. (1991) find that the presence of a magnetic field could slow down the expansion of a superbubble. Stil et al. (2009) argue that the scale height and age of a bubble are underestimated by ~50% when using a Kompaneets model without magnetic fields. However, they cannot produce such narrow superbubbles like W4 with their MHD simulations. Moreover, 3D high resolution numerical simulations (de Avillez & Breitschwerdt 2005) show that magnetic tension forces are much less efficient in 3D than in 2D in holding back the expanding bubble.
A slightly slower growth of a bubble would be also achieved by taking into account the inertia of the cold massive shell in the calculations. MM88 find a difference of ~10% in radius after comparison with the models of Schiano (1985), which neglect inertia. Also due to the ambient pressure of the ISM superbubbles should expand more slowly as it was suggested by Oey & GarcíaSegura (2004). In our analytical calculations we find that these effects have to be compensated when we make comparisons to observed bubbles by including a rather high ISM density to prevent SBs from expanding too fast. Furthermore, a clumpy ambient medium cannot be considered by the KA.
We applied our models to the W4 superbubble and the Local Bubble, both in the Milky Way. It is certainly not easy to compare a simple model with an observed SB, which is not isolated. In the region of the W3/W4/W5 bubbles, several complexes and clouds are found and multiple epochs of star formation make it difficult to distinguish, which cluster has formed which bubble at what time. However, it is most likely that the cluster OCl 352 is responsible for driving the evolution of the bubble (West et al. 2007). Oey et al. (2005) suggest that winds or SNe of previous stellar generations are responsible for earlier clearing of this region, which could explain the low scale height of around 30 pc. Our calculations suggest that the bubble is younger than found by other authors, which is due to the offset of the association more than one scale height above the Galactic plane. Shifting to lower densities makes it easier to produce a blowout superbubble in a shorter timescale. This is an important result and should be included in the models.
The Local Bubble is one of the rare cases for a doublesided bubble, which can be tested with our symmetric superbubble model. From geometrical properties, we estimate an ISM scale height of ~80 pc. In order to reproduce size and age of the bubble correctly, we deduce from our models that it was an intermediate density region (n_{0} ~ 7 cm^{3}) before the first SNexplosion around 14 Myrs ago. Also this place in the Milky Way is very complex (neighboring Loop I superbubble), which can’t be included in our modeling of superbubbles.
We conclude that blowout energies derived in this paper are lower thresholds and might be higher if e.g. magnetic fields play a role. Accordingly, fitting the models to observed bubbles gives an upper limit for densities of the ambient ISM prior to the first SNexplosion. Observers are encouraged to use the model presented here for deriving important physical parameters of e.g. the energy input sources (number of OB stars, richness of cluster etc.), scale heights, dynamical timescales among other quantities. The solutions of the equations derived in detail here are easy to obtain by simple mathematical programs. Theorists may find it useful to compare our analytic results to high resolution numerical simulations in order to separate more complex effects, such as turbulence, mass loading, magnetic fields etc. from basic physical effects, incorporated in our model.
Acknowledgments
V.B. acknowledges support from the Austrian Academy of Sciences, the University of Vienna, and the Austrian FWF, and the Zentrum für Astronomie und Astrophysik (TU Berlin) for financial help during several short term visits.
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Appendix A: Thermal energy in the bubble
The fraction E_{th} of the total energy, which is converted into thermal energy at the inner shock, is derived. The equation for the thermal energy inside the hot bubble is (A.1)where is the volume of a spherical remnant and R(t) is the radius of the outer shock. Since Basu et al. (1999) have shown that the thermal energy in the hot bubble is comparable to that of a homogeneous one until late evolutionary stages, the simplification of a spherical volume is used in the calculations. The pressure P(t) is obtained by taking into account momentum conservation of the bubble shell (see, for example, Castor et al 1975; Weaver et al 1977) (A.2)with homogeneous ambient density ρ_{0}. This equation is rearranged which gives (A.3)Combining this equation with Eq. (A.1) for the thermal energy results in (A.4)
Appendix A.1: SNmodel
In the case of a bubble created by a single explosion, the radius of the outer shock is (e.g. Clarke & Carswell 2007) (A.5)Inserting R_{SN}(t) and its derivatives Ṙ and to Eq. (A.4) results in E_{th, SN} = 2/3·E_{SN}·N_{SN}.
Appendix A.2: IMFmodel
In order to describe the evolution of a superbubble driven by a timedependent energy input rate – in addition to the equations above – energy conservation of the hot wind gas has to be considered for this problem (A.6)with the total thermal energy in this region given by Eq. (A.1). L_{SB} = L_{IMF}·t^{ δ} is the energy input rate delivered by sequential SNexplosions according to an Initial Mass Function. Additionally it is assumed that the radius of the bubble in a selfsimilar flow scales with time like R_{IMF}(t) = A_{s}·t^{ μ}. The constant is (A.7)and the exponent (e.g. Berghöfer & Breitschwerdt 2002). Now, the fraction E_{th}(t) of the total energy, which is converted into thermal energy at the inner shock, can be calculated. Inserting R_{IMF}(t) and its derivatives into Eq. (A.4) yields
Appendix B: Acceleration of the top of the bubble
The derivative of the bubble volume with respect to is needed in the following calculations, which is the same for all models presented below.
We use the derivative of the series expansion of the volume V_{I} in the symmetric case. For the offcenter model we find (B.1)
Appendix B.1: SNmodel
In order to obtain the acceleration at the top of the bubble, we need to calculate the first term on the RHS of Eq. (31) which is the derivative of Eq. (35) with respect to (B.2)Multiplying the above equation by (Eq. (34)) yields the acceleration (in units of cm/s^{2}) (B.3)
Appendix B.2: IMF and windmodel
Next, we want to determine the acceleration at the top of the bubble for the IMFmodel. Using δ = 0 gives the acceleration for the windmodel. Already, is known (Eq. (42)), therefore only has to be calculated. Determining the derivative of Eq. (43) with respect to yields (B.4)For the complete expression of the acceleration in the IMFmodel, the equation above just has to be multiplied by (Eq. (42)).
Appendix C: List of variables and parameters
γ  ...  ratio of specific heats (γ = 5/3) 
Γ  ...  slope of the IMF (Γ_{0} = −0.932, Γ_{1} = −1.15, 
Γ_{2} = −1.35, Γ_{3} = −1.7)  
H  ...  scale height 
L _{IMF}  ...  IMF energy input rate coefficient 
n _{0}  ...  number density at galactic midplane 
N _{0}  ...  IMF normalization constant 
N _{blow}  ...  minimum number of stars for blowout 
τ _{dyn}  ...  dynamical timescale 
τ _{rti}  ...  RayleighTaylor instability timescale 
...  bubble volume (symmetric model)  
...  bubble volume (offplane model)  
v _{acc}  ...  velocity at time of blowout 
y  ...  transformed time variable 
...  transformed time variable in scale height units  
...  time of blowout  
...  time of fragmentation of the shell  
...  time of onset of RayleighTaylor instabilities  
z _{0}  ...  center of the explosion 
z _{d}  ...  bottom of the bubble 
z _{u}  ...  top of the bubble 
...  velocity of the bubble at z_{u}  
...  acceleration of the bubble at z_{u}  
All Tables
All Figures
Fig. 1 Position of the shock front in the symmetric model at certain values of the dimensionless time variable , 1.0, 1.4, 1.7, 1.9, 1.98, and 2.0 with the energy source in the galactic midplane. 

Open with DEXTER  
In the text 
Fig. 2 Same as Fig. 1, but for the offplane model. The energy source is located at z_{0} = 0.7H above the plane (indicated by the dashed line). 

Open with DEXTER  
In the text 
Fig. 3 Blowout timescales at the coordinate z_{u} for the SNmodel. Left: double exponential layer with H = 500 pc and n_{0} = 0.5 cm^{3} (symmetric model); right: exponentially stratified medium with H = 100 pc and n_{0} = 10 cm^{3} (offplane model). The dashed lines indicate the number of SNe required for blowout and the corresponding blowout time. 

Open with DEXTER  
In the text 
Fig. 4 Same as Fig. 3, but for the IMFmodel (left: symmetric model; right: offplane model). Thin lines indicate the number of OBstars needed for blowout. Γ is the IMF exponent (for details, see text). 

Open with DEXTER  
In the text 
Fig. 5 Same as Fig. 3, but as a function of a constant energy input rate (left: symmetric model; right: offplane model). The number of SNe is obtained by converting the wind luminosity to an energy input averaged over a time interval of 20 Myr, from the explosion of the most massive star until the last SN. 

Open with DEXTER  
In the text 
Fig. 6 Fragmentation timescales for the SNmodel; the dashed line marks the minimum number of SNe for a blowout superbubble and the corresponding time until full fragmentation of the shell at the top of the bubble (for details see text and Tables 2 and 3). Left: double exponential layer with H = 500 pc and n_{0} = 0.5 cm^{3} (symmetric model); right: exponentially stratified medium with H = 100 pc and n_{0} = 10 cm^{3} (offplane model). 

Open with DEXTER  
In the text 
Fig. 7 Same as Fig. 6, but for the IMFmodel and indicating the minimum number of OBstars in an association for different IMF slopes by the corresponding thin line (left: symmetric model; right: offplane model). 

Open with DEXTER  
In the text 
Fig. 8 Same as Fig. 6, but for the wind model and a minimum constant energy input rate equal to a number of SNe distributed over 20 Myr (left: symmetric model; right: offplane model). 

Open with DEXTER  
In the text 
Fig. 9 Left: color image of W4 (West et al. 2007) combining HI, Hα and infrared data. For comparison with our model, due to the faint structures and low contrast, the overlay is shown in a separate figure (right panel). The height of the star cluster above the plane is z_{0} = 35 pc, marked by the black dot. Right: Kompaneets bubble at overlaid on the same image. The upper part of the contour fits quite well (solid line), while the part of the shell below the cluster is not seen in the observations (dotted line). 

Open with DEXTER  
In the text 
Fig. 10 Symmetric Kompaneets bubble at overlaid on the absorption map of neutral interstellar NaI of Lallement et al. (2003). Isodensity contours with an equivalent width of 20 m (inner contour) and 50 m (outer contour) represent the rarefied cavity. 

Open with DEXTER  
In the text 
Fig. 11 Numbers of SNe/OBstars needed for blowout are compared for SN, IMF and windmodel (energy input rate over 20 Myr gives the number of SNe for the windmodel). a) Symmetric case, midplane number density: 0.5 cm^{3}, scale height range: 220−500 pc; b) symmetric case, midplane number density: 5 cm^{3}, scale height range: 80−500 pc; c) offplane explosion at z_{0} = 0.7H, midplane number density: 1 cm^{3}, scale height range: 270−500 pc; d) offplane explosion at z_{0} = 0.7H, midplane number density: 10 cm^{3}, scale height range: 90 − 500 pc. 

Open with DEXTER  
In the text 