Interstellar bubbles. II - Structure and evolution

“Thermal conduction and mass loading both introduce cooler and denser material into the interior of superbubbles, which dominate the X-ray emission as the luminosity is proportional to the square of the density (c.f. Weaver et al. 1977; Hartquist et al. 1986).”

“This cone, in turn, is enclosed in a thin shell of even denser shocked ISM (e.g., Weaver et al. 1977).”

“The Weaver et al. (1977) description of a bubble blown by a high-mass star predicts a shell-like H II region surrounding a region of shocked stellar wind.”

“…and R W R respectively) for all these stars based on the standoff distance relationship R o = 1.74×10 19 ˙ m * 1/2 v 1/2 w v −1 rel * µ 1/2 H n −1/2 H cm (Weaver et al. 1977), where R o is the standoff distance, ˙ m * is the stellar mass loss rate – 9 – in units of 10 −6 M ⊙ yr −1 , v w is the…”

“With an average expansion velocity of 6.5 km s−1 this results in a dynamic age of 3.2× 106 yr for a pure stellar wind bubble produced by a single star (Weaver et al. 1977).”

“The poor spatial correlation statistic of OB associations and Hi expanding shells is contrary to the popular theory of the formation of stellar wind-driven Hi expanding shells (Weaver et al, 1977).”

“Following Weaver et al. (1977) and Madau et al. (2001), we define the bubble radius as 𝑅bubble (𝑡) = ( 125 154𝜋 )1/5 ( 𝐿SN 𝑛mH )1/5 𝑡3/5 cm, (13) where 𝐿 = const is the average SN luminosity associated to the stellar population in the halo, 𝑡 the time since the burst of SF, and 𝑛 ≡…”

“This hot star originates a substantial stellar wind, and the formation of a internal reverse shock where the gas flows enter a subsonic regime is a natural outcome in the early adiabatic phase of expansion (Castor et al. 1975; Weaver et al. 1977).”

“This phase is called the snowplow phase, and the radius depends on the amount of injected energy, the density of the medium, and how much of the energy is radiated away (Weaver et al. 1977; McKee & Ostriker 1977; Cooper et al. 2008; ?”

“(30) For adiabatic and PDS blasts, α2 = 1.61 and 1.18 (Weaver et al. 1977; Ostriker & McKee 1988).”

“Most of this energy is thermalized and indirectly can be traced by X-ray emission emerging from hot, coronal gas (Weaver et al. 1977; Heckman 2002; Strickland et al. 2002).”

“The ionisation of the outer shell dependsmostly on the radiation of the central star (Weaver et al. 1977).”

“The main sequence wind creates a large, spherical bubble of hot, shocked wind material, which expands into the ambient medium, sweeping up a shell, following the pattern predicted by the analytical model (Weaver et al. 1977).”

“The energy injection rate (dE/dt) in a wind-blown bubble of a radius R expanding at velocity v into an infinite homogeneous medium with density n0 can be expressed as (Weaver et al 1977), dE dt ∼ 3.3× 1035R2kpcv 3 km/sn0,cm−3 erg s −1.”

“…density seen radially outward through the shell is Ns(H) = ns∆R = nR 3 = 2.75× 1016L1/538 n4/5−5 t3/57 cm−2 (14) The gas density ns in the shell is given by (Weaver et al 1977): ns = n Ṙ2 + c2 c2s , (15) where c is the sound speed in the undisturbed IGM and cs is the sound speed in the shell.”

“It has been suggested that these structures are created by feedback from stellar processes (e.g., stellar winds and supernovae (SNe) in OB associations; Weaver et al. 1977; Cash et al. 1980; McCray & Kafatos 1987; Tenorio-Tagle & Bodenheimer 1988).”

“According to the numerical simulations that studied the interaction of the ISM with the strong stellar winds from an O-type star (Weaver et al. 1977), the gas density inside the evacuated wind bubble is ∼0.01 cm−3, which applies to the interior of the cavity.”

“The analogy between the dynamics of superbubbles and the one of winds with constant energy input rate (Weaver et al. 1977), which has been made in the literature, is inadequate in several respects: the X-ray luminosity is strongly linked to the time variability of the energy input rate;…”

“The underlying physical processes are collectively termed stellar feedback, i.e. stellar winds (Castor et al. 1975; Weaver et al. 1977; Wünsch et al. 2011), ionizing radiation (Spitzer 1978; Hosokawa & Inutsuka 2006; Dale et al. 2012; Walch ?”

“Using equation (67) in Weaver et al. (1977) and making the simplifying assumption that this is a spherical shell, we can estimate the hydrogen number density of the shell ns = n0(vs/c0)2 (13) where c0 is the sound speed in the neutral gas outside, and which we approximate to be 1 km s−1.”

“One way to arrive at such a distribution is with a low-density wind medium followed by a thin, dense shell, as is often seen around massive stars (Weaver et al. 1977; Cappa et al. 2003; Chu 2003, 2008), and in the nebula around SN 1987A (Blondin & Lundqvist 1993).”

“Stellar wind from HD 20336 We use the interstellar windblown bubble model by Weaver et al. (1977) to estimate the time needed to blow a bubble of radius 6 pc with a stellar wind from a B2V star.”

“When spherical stellar winds interact with typical interstellar medium (ISM) gas (nISM ≈ 1 cm−3, P ISM ≈ 10−12 dyn cm−2), they are well described by the model of Weaver et al. (1977).”

“The radius of a superbubble can be estimated from the standard wind bubble theory (Weaver et al. 1977): RSB ≃ (22 pc)t 3/5 MyrN 1/5 ∗,30n −1/5 H,100 , (1) where tMyr is the time in units of Myr after the onset of massive star formation (assumed to be coeval for all stars), N∗,30 = N∗/30 where N∗ is…”

“Following Mac Low & McCray (1988) in treating a superbubble as as a large stellar wind bubble, we use the expansion law obtained by Weaver et al. (1977), r = ( 125 154π )1/5( Lt3 ρ )1/5 , where r is the superbubble radius, L = SN η νSNESN is the mechanical luminosity provided by the supernovae…”

“Weaver et al. (1977) predict that the radius of a wind-blown bubble should increase much faster than the thickness of the swept up shell when the bubbles are in their first expansion stage.”

“Castor et al. (1975) and Weaver et al. (1977) extended this analysis to include the effects of strong stellar winds from O and early B stars.”

“Weaver et al. (1977) modeled a conductive interface of a different kind, at the shell of a wind-blown bubble around a massive star.”

“The fast wind of a massive star can sweep the ambient medium into a shell and form a wind-blown bubble (c.f. Weaver et al. 1977).”

“…Rosen et al. (2014), and others, there are large discrepancies between observations of the hot gas pressure produced by winds and the classical Weaver et al. (1977) model, and in Lancaster et al. (2021a, 2021b), we proposed that energy losses due to turbulently mediated radiative cooling…”

“At later times t? tfree, the ejecta shell begins to appreciably decelerate as its swept-up mass exceeds that injected by the wind, causing the shell radius to grow more gradually in time, R∝ t3/5 (Weaver et al. 1977).”

“This collective wind terminates at the wind termination shock (WTS), whose radius can be estimated theoretically as (Weaver et al. 1977): 𝑅WTS ≈ 𝑁3/10∗ ( 𝑛ext 100 cm−3 )−3/10 ( 𝑡c 1 Myr )2/5 pc , (1) where 𝑁∗ ≈ 10 − 1000 is the total number of massive (𝑀 & 8𝑀 ) stars in the cluster, 𝑛ext ≈…”

“…the ambient medium; however, the precise degree to which this mechanism is operational and effective for GRB progenitors remains an open question (Weaver et al. 1977; Marston 1997; Moore et al. 2000; Ramirez-Ruiz et al. 2001; Chevalier et al. 2004; Krause et al. 2013; Smith 2014; Margutti et al.…”

“This method shows excellent agreement with theoretical predictions for wind bubbles given by Castor et al. (1975) and Weaver et al. (1977).”

“Press) Cioffi,D. F., McKee, C. F., & Bertschinger, E. 1988, ApJ, 334, 252 Crosthwaite, L. P., Turner, J. L., &Ho, P. T. P. 2000, AJ, 119, 1720 de Avillez,M. A. 2000,MNRAS, 315, 479 de Avillez,M. A., & Berry, D. L. 2001,MNRAS, 328, 708 de Avillez, M. A., & Breitschwerdt, D. 2003a, Rev. Mexicana Astron.”

“We run these models with the same underlying assumptions as in Paper I, especially considering that the stellar radiation field is not trapped by the bow shock (Weaver et al. 1977).”

“At the interface between the shocked fast wind and the dense nebular shell, heat conduction lowers the temperature and mass evaporation raises the density of the shocked fast wind, producing optimal conditions for X-ray emission (Weaver et al., 1977).”

“…that the leakage of the shocked wind gas through the holes in the expanding shell may solve a long time standing problem of the shell growth descipancy what implies that Weaver et al. (1977) model often overpredicts the sizes and expansion velocities of the wind-driven shells (see Oey 1996).”

“We estimated the kinematic ages of the identified local H ii shells in terms of Weaver et al. (1977) model by the following relation: t = 0.6R/Vexp, where R is the radius in pc, Vexp – the expansion velocity in km s−1 and t – the age in Myr.”

“Oey & Clarke (1997, 1998) proposed that H II regions undergo energy-conserving expansion powered by stellar winds and supernovae (see also Weaver et al. 1977) until the internal pressure is on the order of the ambient pressure.”

“Thus, the separation between the two stars should at least be as large as the characteristic scale of the wind-bubble created by WR114, i.e. (Weaver et al. 1977): Rb = 27 pc ( Lw 1036 erg s−1 )1/5 ( n0 1 cm−3 )−1/5 ( t 1Myr )3/5 , where Lw = Ṁv 2 ∞/2 is the mechanical luminosity of the star and t…”

“In this figure we can observe the classical twoshock configuration (Weaver et al. 1977): a forward shock is propagating through the ISM.”

“3+0.2 could result from relative motion between an ionizing star and its surrounding molecular material, analogous to elongated structures predicted for such stars moving rapidly though the more diffuse interstellar medium (Weaver et al. 1977).”

“Our description of the inner bubble structure is based on the Weaver et al. (1977) similarity solution, and assumes that the density n and temperature T profiles can be approximated by n = nc(1 − x) −λ, (12) T = Tc(1 − x) λ, (13) where nc and Tc are the central hot gas number density and…”

“…of Weaveret al. (1977) Rbub = κ ( 1 2 Ṁwv 2 w )1/5 ρ −1/5 0 t 3/5, (30) where κ is a numerical coefficient which, in the simple case considered by Weaver et al. (1977) wasκ = [125/(154π)]1/5, while in a realistic case it can be found from numerical modeling based on a stellar evolution…”

“The hot, shocked gas flows away from the cluster as a superwind, colliding with the ambient ISM and forming a bubble that grows in an energy-conserving mode (Weaver et al. 1977; Mac Low & McCray 1988).”

“Following the treatment in the previous literature (Weaver et al. 1977; Faucher-Giguère & Quataert 2012; Wang & Loeb 2015), we consider the so-called thin-shell approximation for regionc, which assumes negligible thickness of the shocked ambient gases (i.e., Rcd;Rs) and that all the shocked gases…”

“…occur in a low density environment, subsequent explosion add mass to the shell and heat the interior of the bubble which can lead to the formation of ’super-bubbles’ (Castor et al. 1975; Weaver et al. 1977; McCray & Kafatos 1987; Tomisaka & Ikeuchi 1986; Mac Low & McCray 1988; Koo & McKee 1992).”

“For r ≥ Rts, there is a shocked wind region, which is separated from the ambient medium via a contact discontinuity (for details, see Weaver et al. 1977).”

“The radii of the shells are supposed to be 30–150 pc and the velocity of expansion Vsh must be less than Vsh < 15 km s−1 (Castor, McCray & Weaver 1975; Weaver et al. 1977; Elmegreen & Lada 1977; Lozinskaya & Sitnik 1988; Lozinskaya 1999, and others).”

“The process was studied theoretically by, for example, Weaver et al. (1977).”

“The density inside the bubble is nb = 0.01(L636n190 t−22Myr )1/35 cm−3 (Weaver et al. 1977).”

“The analytical, self-similar description of the expansion of an adiabatic, spherical wind in a homogeneous environment gives a power-law dependence of the radius on time, R(t) ∝ t3/5 (Weaver et al. 1977).”

“Weaver et al. (1977) derives (Equation 51) the radius of an interstellar bubble as a function of time t6 in Myr, ambient density n0, and mechanical energy injection L36, in units of 10 36 erg s−1 .”

“We derive rotation velocity Vrot(Rij) at the radius of Rij by using equation (Weaver et al. 1977): Vrot(Rij) = VMagellan cosθijsinI , (B3) where Rij is the radius from the center on the plane of the galaxy given as Rij= √ x2 + (y/cosI)2, I is the inclination angle., The minor axis and major axis…”

“It has long been known that the solution for multiple SN events is qualitatively different from a single explosion (e.g., Castor et al. 1975; Weaver et al. 1977; Mac Low & McCray 1988; Keller et al. 2014).”

“As pointed out and discussed by Weaver et al. (1977), the WDS can be considered as a thin shell after the cooling in the WDS is significant.”

“…50% higher than the one derived for an expanding superbubble with the age and size given above and density of 12 cm~3 (Martin & Kennicutt 1995), using the self-similar solution of Weaver et al. (1977), and is more than sufficient to produce the observed X-ray luminosity (Strickland & Stevens 1999).”

“The SBs expand according to the solution of Weaver et al (1977); they propagate into the hot pervasive phase, their (modest) interaction with the cold clouds is neglected.”

“In the ‘standard’ picture, H I shells result from the combined effect of hot stellar winds from O and B stars and of supernova (SN) shocks (e.g. Weaver et al. 1977; McCray & Kafatos 1987).”

“Previously, observed dynamics of bubbles blown by single stars or superbubbles have been compared with pressuredriven bubble models by Weaver et al. (1977) or variations of such models, and it has been commonly concluded that the observed stellar wind strength is too high for the observed bubble…”

“This is the case for many standard situationsof interstellar bubbles (e.g. Weaver et al. 1977).”

“The 105 K gas comes from the evaporation of the disc gas and adiabatically expanded wind material, and, the hot gas ( T ∼ 5 × 106 K) comes from the shocked ISM and wind material.”

“The analytical solution (Weaver et al. 1977, lines in Fig.”