Winds in Star Clusters Drive Kolmogorov Turbulence

Draft version June 29, 2020 Typeset using LATEX twocolumn style in AASTeX61

WINDS IN STAR CLUSTERS DRIVE KOLMOGOROV TURBULENCE

Monica Gallegos-Garcia,1, 2 Blakesley Burkhart,3, 4 Anna Rosen,5, 6, 7 Jill P. Naiman,8 and Enrico Ramirez-Ruiz9, 10

1Department of Physics and Astronomy, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA 2Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA),1800 Sherman, Evanston, IL 60201, USA 3Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA 4Department of Physics and Astronomy, Rutgers, The State University of New Jersey, 136 Frelinghuysen Rd, Piscataway, NJ 08854, USA 5Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA 6Einstein Fellow 7ITC Fellow 8School of Information Sciences, University of Illinois, Urbana-Champaign, IL, 61820 9Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA 10Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark

ABSTRACT Intermediate and massive stars drive fast and powerful isotropic winds that interact with the winds of nearby stars in star clusters and the surrounding interstellar medium (ISM). Wind-ISM collisions generate astrospheres around these stars that contain hot T 107 K gas that adiabatically expands. As individual bubbles expand and collide they ∼ become unstable, potentially driving turbulence in star clusters. In this paper we use hydrodynamic simulations to model a densely populated young star cluster within a homogeneous cloud to study stellar wind collisions with the surrounding ISM. We model a mass-segregated cluster of 20 B-type young main sequence stars with masses ranging from 3–17 M . We evolve the winds for 11 kyrs and show that wind-ISM collisions and over-lapping wind-blown ∼ bubbles around B-stars mixes the hot gas and ISM material generating Kolmogorov-like turbulence on small scales early in its evolution. We discuss how turbulence driven by stellar winds may impact the subsequent generation of star formation in the cluster.

Keywords: ISM: Turbulence – stars: winds arXiv:2006.14626v1 [astro-ph.GA] 25 Jun 2020

Corresponding author: Monica Gallegos-Garcia [email protected] 2

1. INTRODUCTION spectra of density and momentum but do impact the Feedback from stellar winds play an important role in Fourier velocity spectrum. They conclude that stel- shaping the structure of the interstellar medium (ISM, lar winds with high mass-loss rates can contribute to Krumholz 2014). Massive stars produce powerful winds turbulence in molecular clouds. since the mass loss rates and wind velocities are de- A natural extension in studying how wind-blown bub- termined by the star’s radiation output (Castor et al. bles interact with the ISM and contribute to large-scale 1975b). Intermediate- and low-mass stars also con- turbulence in molecular clouds is to study how these tribute to producing ionized bubbles in the ISM, i.e. bubbles interact with one another in clustered environ- so-called astrospheres (Wood 2004; Mackey et al. 2016), ments. Expanding shells have been observed around which are a potential source of local ISM turbulence small star clusters like the ρ-Oph cluster, which con- (Burkhart & Loeb 2017), cosmic rays (del Valle et al. tains five B-stars located in the Ophiuchus molecular 2015), dust processing (Katushkina et al. 2017), and can cloud (Lada & Lada 2003; Chen et al. 2020, in prep). In be used to identify runaway stars (Peri et al. 2012). this scenario, fast winds ejected from stars collide with In regards to massive stars, early theoretical models winds from neighboring stars causing the bubbles to by Castor et al.(1975a) and Weaver et al.(1977) demon- overlap and form a collective “cluster wind” (Cantóet al. strated that the interaction between fast, isotropic stel- 2000). The resulting “super-bubble,” which is filled with lar winds and the surrounding ISM produces a large cav- hot and diffuse gas, eventually expands beyond the star ity or “bubble” surrounded by a thin shell of dense, cold cluster itself (Bruhweiler et al. 1980; Stevens & Hartwell material. In agreement with these models, parsec-scale 2003; Rodr´ıguez-Gonzálezet al. 2007, 2008). Similar to circular cavities are regularly found in regions of high- the single wind-blown shell, where Rayleigh-Taylor and mass star formation (Churchwell et al. 2006, 2007; Beau- Kelvin-Helmholtz instabilities lead to turbulent mixing mont & Williams 2010; Deharveng et al. 2010). Such (McKee et al. 1984; Nakamura et al. 2006), wind-wind features likely contribute to parsec-scale turbulence in collisions in a multiple star system may also lead to in- these environments and drive density fluctuations that stabilities within the cluster wind and produce small- influence subsequent generations of stars (Offner & Arce scale turbulence within the ISM. This turbulent motion 2015; Burkhart 2018). may act in the same way as the single star case, intro- Although it was previously thought that only winds ducing energy and turbulence into its environment as from O or early B-type stars could drive bubbles in the super-bubble grows. molecular clouds, numerous shells have been found in Motivated by this, in this Letter we perform hydrody- low- and intermediate-mass star forming regions (Arce namic simulations to model the collective cluster wind et al. 2011; Li et al. 2015). These studies concluded from a dense star cluster of young B-type stars em- that these bubbles are likely driven by stellar winds bedded in a uniform molecular cloud to determine how from intermediate-mass stars and the energetics of these wind-wind collisions and overlapping bubbles can drive bubbles may help sustain turbulence in the Perseus and turbulence in star clusters. This is in contrast to Cantó Taurus star-forming regions, which may explain the ob- et al.(2000) in which only a single mass of star was served density and velocity power spectrum in Perseus used in the cluster simulations. Offner & Arce(2015) (Pingel et al. 2018; Padoan et al. 2006). use an isothermal equation of state and therefore only To study the development and expansion of wind- follow the momentum injection by winds of young inter- blown bubbles around intermediate-mass stars and mediate mass stars. Here we use an adiabatic equation their contribution to sustaining turbulence in molecular of state and calculate the energy losses using a realistic clouds, Offner & Arce(2015) performed isothermal mag- cooling function, which allows us to fully capture the netohydrodynamic (MHD) simulations that modeled kinetic energy and momentum injection from the fast stellar wind momentum feedback from intermediate- stellar winds and to follow the expansion of the result- mass main sequence stars embedded in a turbulent ing super bubble. We investigate on what time scales molecular cloud. Similar to Arce et al.(2011), they turbulence can be effectively generated within a cluster find that for a random distribution of stars whose by these intermediate- and high-mass stars. individual winds do not interact, a mass-loss rate of This Letter is organized as follows: in Section2 we 7 1 1 describe the stellar wind properties, the initial condi- 10− M yr− and a wind velocity of 200 km s− is ≥ required to drive the shells observed in a Perseus-like tions, and the corresponding physics of our simulation. molecular cloud. Their study also showed that the In Section 3.1, we describe the bulk properties of our stellar winds that produce and drive the expansion of simulations and show how overlapping wind bubbles can these shells do not produce clear features in the Fourier drive turbulence in young star clusters. In Section 3.2 we 3 show the evolution of the density-weighted power spec- from Dalgarno & McCray(1972) for 10 K T 104 K. ≤ ≤ trum and PDFs of physical properties of interest such Metallicity is fixed at solar. as the temperature, density and Mach numbers. In Sec- The star cluster modeled is embedded in a non- tion 3.3 we show the cooling efficiently of the collective turbulent background so that we can self-consistently cluster wind. Finally, in Section4 we summarize our follow the driving of turbulence generated only by winds. findings and discuss their implications. While our initial condition of a uniform background density is certainly idealized, it is likely that turbulence is significantly damped on the scales of a few tenths of a 2. METHODS parsec due to various viscous and MHD damping mech- anisms (Li et al. 2008; Burkhart et al. 2015a; Xu et al. We assume a star cluster mass of 400 M with indi- 2016; Qian et al. 2018). As we are interested to study vidual star masses chosen by stochastically sampling the the direct impact of turbulence produced by the star Kroupa initial mass function (Kroupa 2001). We only cluster we restrict ourselves to a case in which the am- model the 20 most massive stars in the cluster (masses bient medium is uniform. The ambient medium has a ranging between 3.2 – 17 M ) because the energy and 3 3 density of namb = 10 cm− and a cloud temperature momentum injected by their winds dominate over the of 10 K. The box size is (1.24 pc)3 with a finest spatial total momentum and energy of the entire stellar popu- resolution of 120AU. For reference, the shell radii in lation in the cluster (Rosen et al. 2014). The 20 stars in ∼ Perseus identified by Arce et al.(2011) range within 0.14 our cluster are mass-segregated, with a cluster radius of 4 3 – 2.79 pc. They also use a cloud density 10 cm− to r = 0.14 pc and a stellar density profile resembling the ∼ calculate mass-loss rates of the stars embedded within Orion Nebula Cluster (Da Rio et al. 2014). We simulate the shells. Chen et al.(2020, in prep) find an average the wind-wind interactions in the star cluster for 11 kyr, radius of 1.36 pc for the shell in Ophiuchus, which is up to the point where the cluster wind bubble expands ∼ likely being driven by 5 B-type stars. to a radius of 0.22 pc. ∼ To model wind feedback on the ISM by the stars, For the mass range chosen, the stellar winds are ra- we inject the stellar wind over a spherically-symmetric diatively driven (Vink et al. 2001). The values of sphere surrounding each star with a diameter of 16 cells the isotropic wind mass-loss (M˙ ), and wind tempera- (corresponding to 1000 AU in radius). This follows the ture (T ) are taken from the Modules for Experiments ∼ w results of Ruffert(1994) that suggests >8 cells are re- in Stellar Astrophysics (MESA) Isochrones and Stel- quired for sink or source grid sources in hydrodynamical lar Tracks (MIST, Choi et al. 2016). Wind velocity simulations. Within this sphere, the wind density de- (v ) is taken to be the escape velocity of the stars, w creases proportional to the inverse square distance from an adequate approximation for the stars simulated here the center, while the wind velocity magnitude and tem- (Naiman et al. 2018). The mass-loss values range from 12 8 1 perature remain constant. 2 10− – 2 10− M yr− , the wind temperature ∼ × × 3 range from 13 – 29 10 K, and the wind velocities are 3. RESULTS × 1 between 790 – 940 km s− . These values correspond ∼ to a young star cluster with an age between 4 – 8 Myr. 3.1. Evolution of the Wind-blown Bubbles in Star We perform our simulations with FLASH, a 3D adap- Clusters tive mesh refinement grid-based hydrodynamics code, Figure1 shows a time series of slices through the sim- which allows us to include self-gravity and cooling (Fryx- ulation for gas density (top row), gas temperature (mid- ell et al. 2000). We use the default refinement criteria in dle row), and the velocity magnitude of the gas (bottom FLASH with density as the refinement variable and ad- row). We denote star mass and star locations projected ditionally enforce maximum refinement for cells near the on the slice plane with colored stars. stars. We assume an ideal equation of state where the The density panels in Figure1 show the result of the gas pressure is given by P = (γ 1)eT , where ρ is the gas stellar winds from the most massive stars near the cen- − density, eT is the thermal energy density per unit mass, ter quickly colliding and merging into one large bubble, and γ is the adiabatic index, which we take to be 5/3. similar to those observed in molecular clouds (Lada & We assume the energy equation is modified by a cool- Lada 2003; Rosen et al. 2014) and predicted by Cantó ing rate of the form Q(~r, t) = ni(~r, t)ne(~r, t)Λ(T,Z). et al.(2000). The first panel at 1.7 kyrs shows three This is derived from the electron and ion number den- bubbles formed by the central most massive stars. The sities, ne(~r, t) and ni(~r, t) and cooling function for gas larger bubble deforms the spherical shape of the other of temperature T and metallicity Z, where Λ(T,Z) is two bubbles as it expands. At t 2 kyr the largest shell ∼ taken from Gnat & Sternberg(2007) for T > 104 K and bursts along this plane when it reaches the position of 4

16.5

15.0 -21 10 13.5 ] ] 12.0 -22 3 ¯ 10 − M

10.5 m [ c

9.0 -23 g 10 [ M

7.5 ρ 6.0 10-24 t = 1.7 kyr t = 3.7 kyr t = 10.5 kyr 4.5

108 16.5

15.0 107 ] K

13.5 [ 6 e ] 10 12.0 r ¯ u

5 t M 10.5

[ 10 a r

9.0 e 4 10 p M 7.5 m 6.0 103 e T 4.5 t = 1.7 kyr t = 3.7 kyr t = 10.5 kyr 102

16.5 103 15.0 13.5 ] ] 12.0 2 1 ¯ 10 − s

M 10.5 [ m

9.0 k [

M 1

7.5 10 v 6.0 t = 1.7 kyr t = 3.7 kyr t = 10.5 kyr 4.5 100

Figure 1. Time series of slices along the y-z plane of the simulation. We show gas density (top row), gas temperature (middle row), and the velocity magnitude of the gas (bottom row). Star markers indicate the projected stellar locations onto the slice plane and denote star mass with color. The first column shows the wind-blown bubbles before the single cluster wind is formed. The second column shows a snapshot when areas of significant mixing of material appear. The third column shows one of the final snapshot of our simulation. another star. The three bubbles begin to coalesce with at the shell-bubble boundary (Rosen et al. 2014). The each other to form the resulting cluster wind and engulf last panel at 10.5 kyr shows the cluster wind expanded the lower-mass stars at larger radii in the same man- to a radius of 0.21 pc. During the entire evolution ∼ ner. The second density panel at 3.7 kyr shows mixing the gas motion is dominated by the wind feedback from of low-density stellar wind material with high-density the central stars. This can be seen by the fact that material that was initially between the separate bub- the central region of the cluster bubble is kept at lower bles (see first panel). These mixing features are most densities. This is due to the high velocities of the stel- prominent when the bubbles merge near the edges of lar winds pushing material in this vicinity away at all the cluster wind since that is where more high-density times. Although self-gravity of the gas is calculated in swept-up material resides. We associate the mixing fea- these simulations, we find that self-gravity is dynami- tures to be turbulent instabilities that are likely a result cally unimportant within the shell since the total mass of Kelvin-Helmholtz instabilities caused by the wind- within the bubble, MB 0.01 M , is low. ∼ wind collisions. In agreement with our results, Krause The second row in Figure1 shows the temperature et al.(2013) also finds these instabilities develop at the evolution of the star cluster. As the winds collide with locations of intersecting wind bubbles. Throughout the the surrounding ISM and other wind material, the ki- simulation mixing features are also prominent along the netic energy is thermalized resulting in high tempera- inner edge of the shell as high-temperature and low- tures of T = 107–108 K within the wind bubble. This density gas is pushed onto it leading to turbulent mixing hot gas adiabatically expands, which we properly ac- 5 ] 3 ] ] 3 g cm [M 21 ? [g cm M [10 ⇢ ⇢ t = 1.7 kyr t = 3.7 kyr t = 7.4 kyr t = 10.5 kyr

0.25 15.0 0.00

12.5 0.25 ]

0.50 M [M 10.0 10 ?

0.75 log M 7.5 1.00 1.25 1.7kyr 3.7kyr 7.4kyr 10.5kyr 5.0 1.50

Figure 2. Top row: density-weighted projections of density through the x-y plane of the simulation. Bottom row: Mach number M slices along the x-y plane of the simulation. Green is supersonic material and pink is subsonic. Most of the wind material is subsonic except for the outer shell. This is because the high densities in this region allow the gas to cool to lower temperatures, decreasing the speed of sound in this region. count for since we use an adiabatic, rather than isother- the turbulent material inside the dense shell is subsonic mal, equation of state. It is this adiabatic expansion at all times. It is important to point out here that the that dominates the over-all expansion of the combined realistic adiabatic equation of state used in these simu- wind bubble (Cantóet al. 2000). Similar to the den- lations is critical for the correct calculation of the sonic sity panels, we can see a mixing of hot wind material Mach number. In our simulations we are able to follow with cooler shell material. Again this mixing is most thermodynamics of the gas and hence the temperature prominent when the bubbles merge and at the bubble and Mach number fluctuations. edges. The cluster gas does not cool significantly over In the time series of the mass-weighted density pro- timescales shown here. This cooling inefficiency is likely jections we ignore the ambient material by only includ- due to low cooling rates that are achieved by the lowing gas with T > 10 K. These density projections show density and high-temperature gas at solar metallicity gas configurations similar to the density slice plots. In (Rosen et al. 2014). In contrast, the high-density and the first panel at t = 1.7 kyr we can see the locations low-temperature shell cools slightly over the timescale of high-density shells that have not completely merged. shown here, which we discuss in more detail in Sec- These correspond to the dark, higher-density curves on tion 3.3. the top right. As the simulation progresses the high- The last row of Figure1 shows the velocity magni- density shells of individual bubbles merge and the gas tude in the simulation. We see that the high-velocity becomes more homogeneous within the cluster shell. material dominates the inner regions of the cluster wind In Figure3 we show the PDFs for density, temperature where the density is the lowest. As the shell expands and and Mach number for different snapshots. The narrow 21 3 sweeps up material from the ambient medium, it slows peaks at ρ 10− g cm− ,T 10 K, and the peaks be- ∼ 4 2 ∼ and cools (cooling of the shell shown in the middle panel tween 10− –10− correspond to the ambient mate- M ∼ of Figure3). rial. The density PDFs (left panel) show non-lognormal Figure2 shows a time series of mass-weighted density behavior throughout the evolution of the bubble. The projections (top row) and slices through the simulation density PDF of subsonic and supersonic turbulence has of Mach number (bottom row), the ratio of the gas been extensively studied and, for isothermal turbulence, M velocity magnitude to the sound speed, v/cs. Pink cor- takes on a lognormal form (Vazquez-Semadeni 1994; responds to subsonic, < 1, and green corresponds to Federrath et al. 2008; Burkhart et al. 2009; Kainulainen M supersonic, > 1. In the slices we see that most of & Tan 2013; Burkhart & Lazarian 2012; Burkhart et al. M M 6 PDF PDF PDF

3 log ⇢ [g/cm ] log T [K] log 10 10 10 M

Figure 3. Probability density functions for density, temperature, and Mach number at three snapshots in the simulation. The peaks at ρ ∼ 1021 g cm−3,T∼ 10 K, and the peak between M ∼ 10−4–10−2 correspond to the background material. Left: Density PDF. Middle: The pink, dotted-dashed peak at T∼ 104 K corresponds to the cluster shell at t = 1.7 kyr. As the simulation evolves this region cools the most. Right: The cluster bubble is dominated by sub-sonic material. 2015b). This is primarily attributed to the application ture at T 107 K corresponds to the material inside ∼ of the central limit theorem to a hierarchical (e.g. turbu- the cluster bubble. Unlike the shell material this only lent) density field generated by a multiplicative process, cools slightly. This is likely because the material inside such as shocks. However, for non-isothermal turbulence the bubble is kept at low densities and is constantly ex- or for turbulence with self-gravity (e.g. adiabatic equa- periencing wind-wind collisions that shock heat the ma- tion of state) a lognormal is no longer observed (Fed- terial to high temperatures where the cooling function, errath & Klessen 2012; Collins et al. 2012; Nolan et al. Λ(T,Z), is very low. 2015; Mocz et al. 2017; Burkhart 2018). The density The Mach number PDF (right panel in Figure3) PDF is significantly affected by temperature variations shows three features. The only sonic feature log > 0 10 M (left panel of Figure3), as expected for the case of non- corresponds to the bubble shell. The other feature at isothermal gas with heating/cooling (Scalo et al. 1998; log > -2 corresponds to the collective cluster wind. 10 M Mandelker et al. 2020). The fact that the density PDF is These two are the dominating bubble features. As the highly non-lognormal once stellar winds become impor- simulation evolves we see that these features do not tant may indicate that, for second generations of stars change but only increase in magnitude due to the larger forming near wind-blown bubbles, star formation theo- bubble and shell volumes as the simulation evolves. The ries that rely on the lognormal density PDF may not be third feature between log -2 – -4 corresponds to 10 M ∼ applicable. the background material. This peak evolves towards The temperature PDF (middle panel in Figure3) has higher Mach number at later times because, although three main features: a peak at T = 10 K, a peak at the temperature of the background medium remains T 104 K, and a peak at T 107 K. The first peak constant, the material’s velocity increases slightly as it ∼ ∼ corresponds to the ambient material. The middle peak becomes gravitationally attracted to the cluster bubble. corresponds to the material just interior to bubble shell. The material below this peak value, but above 10 K, 3.2. Turbulent Power Spectrum corresponds to the swept up material in the shell. This As the wind blown bubble expands into the ambient shell material begins to cool for two reasons since radia- medium, turbulent density fluctuations develop in the tive cooling (L n2 Λ(T,Z)) depends on the cool- cool ∝ X inner bubble post-shock region. Since these fluctuations ing function Λ(T,Z) and the election number density are largely subsonic (i.e. developed in the postshock of the material nX. First, the material reaches higher gas) we may expect to find Kolmogorov-like turbulence densities when it is accumulated onto the outer shell. inside the bubble. Second, the cooling function has a local maximum of ef- In the limit of incompressible (sub-sonic) turbulence, ficiency for material at T 104 K at solar metallicity. ∼ density fluctuations are not relevant and the density and Because of this we see that the shell is the only mate- kinetic energy power spectrum should evolve in a simi- rial that efficiently cools during the simulation, which lar fashion. Therefore for incompressible turbulence the we discuss in more detail in Section 3.3. The final fea- Fourier power spectrum slope is expected to remain close 7

10 2 10 2 2 1.7 kyr 12 k− 10 5/3 3.8 kyr k− 7.5 kyr

k 10.6 kyr 11 k 4 k 10 4 10 5/3 10

) d k− ) d ) d k k ( ( k 10 v v (

3 10 3 v / / 1 1 ρ ρ 6 E 6 E 10 E 10 109 20483 5123 10243 2563 108 100 101 102 100 101 102 100 101 102 k k k

Figure 4. Left: Time evolution of the density-weighted velocity power spectrum. Color corresponds to a different snapshot in the simulation. We find that it follows a Kolmogorov power spectrum (gray dashed line) Middle: Time evolution of the velocity power spectrum. The colors correspond to the same times as in the left plot. We compare this to -5/3 Kolmogorov power spectrum (gray dashed line) and -2 Burgers slope (black dotted line). Right: Resolution test for the density-weighted velocity power spectrum. The color corresponds to the maximum levels of refinement for the AMR grid. to the Kolmogorov index of -5/3. (Goldreich & Sridhar In particular, for the most advanced time snapshot, we 1995; Chepurnov et al. 2015). If instead, we expected su- find that the density-weighted velocity power spectrum personic turbulence or a strong signature of self-gravity, (left panel) follows a Kolmogorov scaling and the veloc- the density power spectral slope would be significantly ity power spectrum (middle panel) in our simulations flatter than Kolmogorov (Kowal et al. 2007; Burkhart obeys a Kolmogorov/Burgers scaling after 10 kyrs. Our et al. 2010; Collins et al. 2012). For the velocity power results agree with previous studies of turbulence driven spectrum, supersonic flows approach the limit of Burg- by stellar winds, such as Offner & Arce(2015), who find ers turbulence with a slope of -2. As for the power spec- that wind-blown bubbles affect the velocity power spectrum of density, shocks can create small-scale density trum. enhancements (e.g., Beresnyak et al.(2005); Kowal & We also perform a convergence study to determine Lazarian(2007), which in turn induce more power on how the power spectrum depends on the AMR grid res- small scales and significantly flatten the spectral slope olution in the right panel of Figure4. The effective grid as compared to incompressible turbulence. Kritsuk et al. resolution including AMR is 20483 and the grid resolu- (2007) proposed to use the density-weighted velocity tion for the power spectrum calculation is 5123. From power spectrum, u ρ1/3v, in order to restore the Kol- the right panel of Figure4, the spectral slope in the ≡ mogorov scaling in the power spectra and second order range of k 10–20 seems to be converged for resolution ∼ structure function in compressible high Mach number greater than 5123. Kritsuk et al.(2007) suggested that hydrodynamic turbulence. the power spectral scaling based on their time-averaged Following Kritsuk et al.(2007), we show the density- statistics from a 10243 driven turbulence simulation may 1/3 weighted velocity power spectrum u ρ v (left panel) begin to correspond to Re . Similarly, we find our ≡ → ∞ and the velocity power spectrum (middle panel) in Fig- simulations of wind driven turbulence also begin to con- ure4. Different time snapshots are represented with verge at around this resolution. different colors, where red represents the most advance snapshot and yellow shows the earliest snapshot. The 3.3. Shell Properties and Cooling Time straight gray dashed line shows the -5/3 Kolmogorov Our simulations show that after 5 kyr the combined ∼ prediction and the black dotted line shows the predicted wind of the stars leaves the cluster itself, forming a col- slope for Burgers’ turbulence. lective cluster shell. We calculate the shell radius as the The different colored lines show how the power spec- density-weighted average distance from the origin fol- trum evolves in time in our simulations. At early times lowing equation 23 in Rosen et al.(2017). We obtain (i.e., yellow line), the wind-blown bubble is expanding Rshell = 0.2 pc at t = 10.5 kyr corresponding to an from the smallest scales to larger scales and the velocity approximate expansion rate vexp = Rshell/t of the shell 1 power spectrum is not a well defined power law. How- to be 18.7 km s− . For comparison, we also calculate ever, as time increases a power law like feature forms a density-weighted average velocity expansion of 9.05 1 with a slope consistent with a value between -2 and -5/3. km s− , which is a factor of 2 lower than the value de- ∼ 8

1 9.05 km s− from above. In each cell we calculate the 0.2 102 cooling time tcool given by, E (3/2)1.9n kT 100 t = thermal = X . (1) 0.1 cool 2 Lcool 0.9nXΛ(T,Z) -2 p

10 x ) e

c Here E is the thermal energy of the gas parcel,

t thermal p / ( 0.0 l -4 o L is the energy loss rate via cooling, and Λ(T,Z)

o cool z 10 c

t is the cooling function taken from Gnat & Sternberg -6 (2007). Here we have used ni = 0.9nX for a fully ion- 0.1 10 ized plasma of solar composition, where nX and ni are 10-8 the electron and ion number density respectively. To 0.2 t = 10.5 kyr calculate nX in each cell we use ρ = 1.9µmpnX, where 0.2 0.1 0.0 0.1 0.2 µ = 0.62 assumes all helium is ionized at solar metallic- y (pc) ity. In the top panel of Figure5 we show tcool/texp using the characteristic bubble expansion time t 21 kyr exp ∼ in all cells. We ﬁnd a cell-mass weighted average of 107 t 340 kyr and a cell-mass weighted average of cool ∼ 0 t /t 16 for the hot stellar winds inside the bub- 106 10 cool exp ∼ ) ble shell. The majority of the gas within the shell, K r ( e

5 b especially at the center where the density is lowest,

e 10 r m u u has tcool texp. At larger radii closer to the shell t N

r 4 boundary we ﬁnd t t . This is likely due to

h cool exp e 10 & -1 c p 10 a the turbulent mixing between the low-density, high- m M e 3 T 10 temperature shock-heated gas and high-density, low- temperature shell. Since t t the energy loss cool exp 102 via cooling is negligible within the bubble and instead t = 10.5 kyr 10-2 the thermalized wind energy is transferred via adiabatic 10-24 10-23 10-22 10-21 expansion, thereby driving the dense bubble shell. We Density ( g ) note that we find that the regions around the most mas- cm3 sive stars at the center of the simulation are dominated by numerical effects and we have removed these from Figure 5. Top panel: Ratio of the cooling time for the the calculations in Figure5 as indicated by the white bubble material tcool to the characteristic bubble expansion region. Since the majority of the gas has t t time texp at t = 10.5 kyr. For these calculations we remove cool exp the cells that are dominated by numerical effects. Bottom this region does not affect our results. panel: Mach number weighted by cell-mass within the cluster The bottom panel of Figure5 shows the Mach num- wind as a function of temperature and density at t = 10.5 ber as a function of temperature and density at t = 10.5 kyr. We show only gas with T > 10 K so that we do not kyr within the cluster wind. We ignore the background include contributions from the background ISM. medium by only including gas with T > 10 K. The low- temperature (T 4 103 K) and high-density material . × rived previously. Typically, observations infer the bub- in Figure5 corresponds to the outer radius material of ble age from the dynamical timescale, tdyn = Rshell/vexp the bubble shell. Material at T 104 K and mach num- ∼ where both Rshell and vexp are observationally inferred ber > 1 correspond to the bulk shell material. Mov- M values of the shell radius and shell expansion rate, re- ing upwards to higher temperature, T = 104–107 K and 24 22 3 spectively (e.g., Li et al. 2015). Our measured values lower densities, ρ = 10− – 10− g cm− , corresponds stated above for the shell velocity imply that inferring to the collective cluster winds. Most of this material is the bubble age from these quantities will underestimate sub-sonic except at the highest temperature and lowest the dynamical age of the bubble since the shell velocity density. is decreasing in time. To study the stellar wind energy of the cluster we In the top panel of Figure5 we compare the time for calculate the fraction of the total energy injected by the bubble material to cool, tcool, to the characteristic the stellar winds that go into the following: thermal bubble expansion time texp = Rshell/vexp using vexp = energy of the hot gas interior to the shell Ethermal = 9

(3/2)NkT , the kinetic energy of the shell Eshell = spectrum in Figure4) but are likely not to be the domi- 2 (1/2)Mshellvexp, and turbulent energy of the hot gas nate driver on larger scales in GMCs (Cunningham et al. 2 Erms = (1/2)MBvrms. Throughout the simulation the 2006; Wang et al. 2010; Offner & Arce 2015; Offner & ˙ 2 energy injected by stellar winds, Lw = (1/2)Mwvw, is Liu 2018). For sub-parsec length-scales in the vicinity of 34 1 L = 1.36 10 erg s− . At t = 3.75 kyr the fraction of intermediate mass star clusters and, for the timescales w × the total energy in thermal energy is f 0.63 and we consider ( 10 kyr), stellar winds may be the domi- thermal ≈ ≈ the fraction in kinetic energy of the shell is f 0.17. nate source of local turbulent motions. This is because shell ≈ We find that the fraction in turbulent energy is compa- turbulence from a larger scale cascade will be damped to 1 rable to fshell. At t = 10.5 kyr the results are sim- velocities below 1 km s− (Larson 1981; Burkhart et al. ilar except the thermal energy is slightly lower with 2015a; Qian et al. 2018) while the expansion speed of 1 f 0.58. The missing energy is likely associated our bubble is in excess of 9 km s− after 10 kyr and be- thermal ≈ with mixing of hot and cool gas and a small fraction of cause timescales we consider are too short for the stars energy lost by cooling. to go supernova. The winds deposit energy and momentum into the shell but the shell wake exhibits significant 4. DISCUSSION AND CONCLUSIONS non-local perturbations caused by instabilities that develop within the collective cluster wind and mixing be- We use hydrodynamic simulations to investigate the tween the cool shell gas and hot gas within the bubble. impact of stellar winds on their environment in the first This contributes to an evolving turbulent cascade, as is few thousand years of the wind expansion. In particular, evident from the velocity power spectrum. we perform simulations that employ a realistic adiabatic Our results imply that wind driven turbulence around equation of state in order to properly follow the adia- intermediate and high-mass stars may trigger subse- batic expansion of the thermalized shocked wind mate- quent generations of ongoing star formation in the few rial that is produced by winds colliding with the winds parsec vicinity of the cluster that impinge on the shell. of nearby stars and the ISM. We also include radiative This second generation of stars might form in a very dif- cooling and self-gravity. We sample a range of mass- ferent manner than the first generation that is produced loss rates corresponding to intermediate and high-mass by cold collapsing gas from the natal molecular environ- stars for a dense mass-segregated star cluster following ment. The density distribution is largely a power-law a stochastically sampled Kroupa IMF. With these sim- formed during the initial phases of star formation (Kain- ulations, we study how stellar wind-wind collisions can ulainen & Tan 2013; Burkhart 2018). In this work, we drive turbulence within the expanding wind-blown bub- find that the density distribution within the bubble is ble. We show that the turbulence driven by stellar winds not lognormal due to the non-isothermal nature of the is primarily subsonic likely because the shock heated gas and a powerlaw is likely not present due to the fact material cools via adiabatic expansion and has tempera- 6 7 2 3 1 that self-gravity is not important in the expanding re- tures of 10 10 K and velocities of 10 10 km s− . ∼ − − gion of the bubble, at least for the short simulation time This material continues to cool via adiabatic expansion presented here. The bubble gas is too hot to collapse and and expands at high velocities until it interacts with form stars in the vicinity of the cluster and the next gen- the surrounding bubble shell. We find that the cluster eration of stars likely would form further away from the wind material exhibits a velocity power spectrum scal- cluster, triggered by the compression of the shell if the ing between Kolmogorov and Burgers turbulence. The shell is able to sweep up enough gas and cool efficiently power spectral scaling we observe is similar to previous (Li et al. 2014). numerical studies of wind driven turbulence (Offner & Our results should also provide an understanding of Liu 2018). the types of environments likely to be relevant for as- This is in contrast to molecular gas on larger scales in sessing supernova feedback in star clusters. For instance, GMCs (i.e. 1-10 pc) that is predominately isothermal the dynamics of supernova in turbulent medium (Mar- and cold (T 10 K), and has a much higher sonic Mach ∼ tizzi et al. 2015; Kim & Ostriker 2015; Zhang & Cheva- number. We conclude that stellar feedback can drive lier 2019) can be significantly different than in a ho- small-scale turbulence from colliding stellar winds in the mogeneous medium (McKee & Ostriker 1977). What immediate vicinity of high-mass stars and may be able is more, clustering of supernova might occur before the to offset dissipation of turbulent energy cascading down star cluster dissolves, which could potentially enhanced from larger scales. supernova feedback (e.g., Kim et al. 2017; Fielding et al. Our study is in agreement with previous investiga- 2018; Gentry et al. 2019; Karpov et al. 2020). tions that find stellar winds can drive turbulence out to about a 1 pc scale (e.g., the low k bump in the power 10

Finally, our results have interesting implications for 4. We find that the majority of the injected wind en- studies of clustered astrospheres in the ISM. In partic- ergy is in the form of thermal energy of the hot, ular, wind blown bubbles from clusters of intermediate low-density gas within the bubble and that the mass stars may alter the dynamics of cosmic ray prop- shell and turbulent kinentic energy within the bub- agation and diffusion as compared to stars in isolation. ble are similar throughout the simulation. As the Galactic cosmic rays passing through large cavities will simulation evolves, the thermal energy within the have their spectra efficiently cooled and thus bubbles bubble decreases slightly due to adiabatic expan- can give rise to small-scale anisotropies in the direction sion, radiative cooling, and mixing of cold and hot to the observer (Scherer et al. 2015). material near the bubble shell. Future numerical studies of star cluster winds should consider including magnetic fields, in addition to the adiabatic equation of state, in order to quantitatively M.G-G. is grateful for the support from the Ford Foun- connect to cosmic ray observables. dation Predoctoral Fellowship. B.B. is grateful for dis- Our main results from this study are as follows: cussions with the SMAUG collaboration. B.B. gratefully acknowledges generous support from the Simons Foun- 1. Wind-wind collisions from the winds of intermedi- dation Center for Computational Astrophysics (CCA) ate and high-mass stars drives primarily subsonic and the Harvard-Smithsonian Center for Astrophysics turbulence. The turbulence exhibits a power spec- (CfA) Institute for Theory and Computation (ITC) trum between Kolmogorov and Burgers turbulence postdoctoral fellowship. A.L.R. acknowledges support developed within the bubble on the bubble expan- from NASA through Einstein Postdoctoral Fellowship sion time. grant number PF7-180166 awarded by the Chandra X- ray Center, which is operated by the Smithsonian Astro- physical Observatory for NASA under contract NAS8- 2. An adiabatic equation of state, heating, and ra- 03060. J.P.N. acknowledges support from the Institute diative cooling are all important effects to include for Theory and Computation (ITC) postdoctoral fellow- when treating the physics of wind blown bubbles. ship and the National Science Foundation Astronomy An adiabatic equation of state enables a more real- and Astrophysics postdoctoral fellowship award num- istic treatment of the kinetic energy injection from ber 1402480. E.R.-R. thanks the Heising-Simons Foun- the fast stellar winds and the adiabatic expansion dation and the Danish National Research Foundation of the bubble. (DNRF132) for support. A.L.R. would also like to thank her “coworker,” Nova Rosen, for “insightful conversa- tions” and unwavering support at home while this paper 3. Dense and low-temperature shells can be a poten- was being written during the Coronavirus pandemic of tial site for star formation if the mass accumula- 2020. Her contributions were not sufficient to warrant tion is an on-going process. co-authorship due to excessive napping.1

REFERENCES Arce, H. G., Borkin, M. A., Goodman, A. A., Pineda, J. E., Burkhart, B., Lazarian, A., Balsara, D., Meyer, C., & Cho, & Beaumont, C. N. 2011, ApJ, 742, 105 J. 2015a, ApJ, 805, 118 Beaumont, C. N., & Williams, J. P. 2010, ApJ, 709, 791 Burkhart, B., Lee, M. Y., Murray, C., & Stanimirovic, S. Beresnyak, A., Lazarian, A., & Cho, J. 2005, ApJ, 624, L93 2015b, ApJ, 811, 28 Bruhweiler, F. C., Gull, T. R., Kafatos, M., & Sofia, S. Burkhart, B., & Loeb, A. 2017, ApJL, 849, L10 1980, ApJ, 238, L27 Burkhart, B., Stanimirović,S., Lazarian, A., & Kowal, G. Burkhart, B. 2018, ApJ, 863, 118 2010, ApJ, 708, 1204 Burkhart, B., Falceta-Gon¸calves, D., Kowal, G., & Cantó,J., Raga, A. C., & Rodr´ıguez,L. F. 2000, ApJ, 536, 896 Lazarian, A. 2009, ApJ, 693, 250 Castor, J., McCray, R., & Weaver, R. 1975a, ApJ, 200, Burkhart, B., & Lazarian, A. 2012, ApJL, 755, L19 L107 Castor, J. I., Abbott, D. C., & Klein, R. I. 1975b, ApJ, 195, 1 Because she is a cat. 157 11

Chen, H., Rosen, A., Goodman, A., & Offner, S. 2020, in Lada, C. J., & Lada, E. A. 2003, Annual Review of prep Astronomy and Astrophysics, 41, 57 Chepurnov, A., Burkhart, B., Lazarian, A., & Stanimirovic, Larson, R. B. 1981, MNRAS, 194, 809 S. 2015, ApJ, 810, 33 Li, H., Li, D., Qian, L., et al. 2015, The Astrophysical Choi, J., Dotter, A., Conroy, C., et al. 2016, ApJ, 823, 102 Journal Supplement Series, 219, 20 Churchwell, E., Povich, M. S., Allen, D., et al. 2006, ApJ, Li, P. S., McKee, C. F., Klein, R. I., & Fisher, R. T. 2008, 649, 759 ApJ, 684, 380 Churchwell, E., Watson, D. F., Povich, M. S., et al. 2007, Li, S., Frank, A., & Blackman, E. G. 2014, Monthly Notices ApJ, 670, 428 of the Royal Astronomical Society, 444, 2884. Collins, D. C., Kritsuk, A. G., Padoan, P., et al. 2012, ApJ, https://doi.org/10.1093/mnras/stu1571 750, 13 Mackey, J., Haworth, T. J., Gvaramadze, V. V., et al. 2016, Cunningham, A. J., Frank, A., Quillen, A. C., & Blackman, A&A, 586, A114 E. G. 2006, ApJ, 653, 416 Mandelker, N., Nagai, D., Aung, H., et al. 2020, MNRAS, Da Rio, N., Tan, J. C., & Jaehnig, K. 2014, ApJ, 795, 55 arXiv:1910.05344 Dalgarno, A., & McCray, R. A. 1972, Annual Review of Martizzi, D., Faucher-Giguère,C.-A., & Quataert, E. 2015, Astronomy and Astrophysics, 10, 375 MNRAS, 450, 504 Deharveng, L., Schuller, F., Anderson, L. D., et al. 2010, McKee, C. F., & Ostriker, J. P. 1977, ApJ, 218, 148 A&A, 523, A6 McKee, C. F., van Buren, D., & Lazareff, B. 1984, ApJ, del Valle, M. V., Romero, G. E., & Santos-Lima, R. 2015, 278, L115 MNRAS, 448, 207 Mocz, P., Burkhart, B., Hernquist, L., McKee, C. F., & Federrath, C., & Klessen, R. S. 2012, ApJ, 761, 156 Springel, V. 2017, ApJ, 838, 40 Naiman, J. P., Ramirez-Ruiz, E., & Lin, D. N. C. 2018, Federrath, C., Klessen, R. S., & Schmidt, W. 2008, ApJL, MNRAS, 478, 2794 688, L79 Nakamura, F., McKee, C. F., Klein, R. I., & Fisher, R. T. Fielding, D., Quataert, E., & Martizzi, D. 2018, MNRAS, 2006, The Astrophysical Journal Supplement Series, 164, 481, 3325 477 Fryxell, B., Olson, K., Ricker, P., et al. 2000, The Nolan, C. A., Federrath, C., & Sutherland, R. S. 2015, Astrophysical Journal Supplement Series, 131, 273 MNRAS, 451, 1380 Gentry, E. S., Krumholz, M. R., Madau, P., & Lupi, A. Offner, S. S. R., & Arce, H. G. 2015, ApJ, 811, 146 2019, MNRAS, 483, 3647 Offner, S. S. R., & Liu, Y. 2018, Nature Astronomy, 2, 896 Gnat, O., & Sternberg, A. 2007, The Astrophysical Journal Padoan, P., Juvela, M., Kritsuk, A., & Norman, M. L. Supplement Series, 168, 213 2006, ApJL, 653, L125 Goldreich, P., & Sridhar, S. 1995, ApJ, 438, 763 Peri, C. S., Benaglia, P., Brookes, D. P., Stevens, I. R., & Kainulainen, J., & Tan, J. C. 2013, A&A, 549, A53 Isequilla, N. L. 2012, Astronomy & Astrophysics, 538, Karpov, P. I., Martizzi, D., Macias, P., et al. 2020, ApJ, A108. 896, 66 http://dx.doi.org/10.1051/0004-6361/201118116 Katushkina, O. A., Alexashov, D. B., Izmodenov, V. V., & Pingel, N. M., Lee, M.-Y., Burkhart, B., & Stanimirović,S. Gvaramadze, V. V. 2017, MNRAS, 465, 1573 2018, ApJ, 856, 136 Kim, C.-G., & Ostriker, E. C. 2015, ApJ, 802, 99 Qian, L., Li, D., Gao, Y., Xu, H., & Pan, Z. 2018, The Kim, C.-G., Ostriker, E. C., & Raileanu, R. 2017, ApJ, 834, Astrophysical Journal, 864, 116. 25 https://doi.org/10.3847%2F1538-4357%2Faad780 Kowal, G., & Lazarian, A. 2007, ApJ, 666, L69 Qian, L., Li, D., Gao, Y., Xu, H., & Pan, Z. 2018, ApJ, Kowal, G., Lazarian, A., & Beresnyak, A. 2007, ApJ, 658, 864, 116 423 Rodr´ıguez-González, A., Cantó,J., Esquivel, A., Raga, Krause, M., Fierlinger, K., Diehl, R., et al. 2013, A&A, 550, A. C., & Velázquez,P. F. 2007, MNRAS, 380, 1198 A49 Rodr´ıguez-González, A., Esquivel, A., Raga, A. C., & Kritsuk, A. G., Norman, M. L., Padoan, P., & Wagner, R. Cantó,J. 2008, ApJ, 684, 1384 2007, ApJ, 665, 416 Rosen, A. L., Krumholz, M. R., Oishi, J. S., Lee, A. T., & Kroupa, P. 2001, MNRAS, 322, 231 Klein, R. I. 2017, Journal of Computational Physics, 330, Krumholz, M. R. 2014, PhR, 539, 49 924 12

Rosen, A. L., Lopez, L. A., Krumholz, M. R., & Vazquez-Semadeni, E. 1994, ApJ, 423, 681. Ramirez-Ruiz, E. 2014, MNRAS, 442, 2701 http://adsabs.harvard.edu/doi/10.1086/173847 Vink, J. S., de Koter, A., & Lamers, H. J. G. L. M. 2001, Ruffert, M. 1994, ApJ, 427, 342 A&A, 369, 574 Scalo, J., Vázquez-Semadeni,E., Chappell, D., & Passot, Wang, P., Li, Z.-Y., Abel, T., & Nakamura, F. 2010, ApJ, T. 1998, ApJ, 504, 835 709, 27 Weaver, R., McCray, R., Castor, J., Shapiro, P., & Moore, Scherer, K., van der Schyff, A., Bomans, D. J., et al. 2015, R. 1977, ApJ, 218, 377 Astronomy & Astrophysics, 576, A97. Wood, B. E. 2004, Living Reviews in Solar Physics, 1, 2 http://dx.doi.org/10.1051/0004-6361/201425091 Xu, S., Yan, H., & Lazarian, A. 2016, ApJ, 826, 166 Stevens, I. R., & Hartwell, J. M. 2003, MNRAS, 339, 280 Zhang, D., & Chevalier, R. A. 2019, MNRAS, 482, 1602