ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: show that after ∼5 kyr the combined wind of the stars leaves the cluster itself, forming a collective cluster shell. We calculate the shell radius as the density-weighted average distance from the origin following Equation (23) in Rosen et al. ( 2017 ). We obtain at corresponding to an approximate expansion rate of the shell to be 18.7 . For comparison, we also calculate a density-weighted average velocity expansion of 9.05 , which is a factor of ∼2 lower than the value derived previously. Typically, observations infer the bubble age from the dynamical timescale, where both and are observationally inferred values of the shell radius and shell expansion rate, respectively (e.g., Li et al. 2015 ). Our measured values stated above for the shell velocity imply that inferring the bubble age from these quantities will underestimate the dynamical age of the bubble because the shell velocity is decreasing in time. In the top panel of Figure 5 we compare the time for the bubble material to cool, , to the characteristic bubble expansion time using from above. In each cell we calculate the cooling time given byHere is the thermal energy of the gas parcel, is the energy loss rate via cooling, and is the cooling function taken from Gnat Sternberg ( 2007 ). Here we have used for a fully ionized plasma of solar composition, where and are the electron and ion number density, respectively. To calculate in each cell we use , where assumes all helium is ionized at solar metallicity. Figure 5. Top panel: ratio of the cooling time for the bubble material to the characteristic bubble expansion time at t = 10.5 kyr. For these calculations we remove the cells that are dominated by numerical effects. Bottom panel: Mach number weighted by cell-mass within the cluster wind as a function of temperature and density at t = 10.5 kyr. We show only gas with K so that we do not include contributions from the background ISM. In the top panel of Figure 5 we show using the characteristic bubble expansion time kyr in all cells. As the cooling timescale has a strong dependence, density enhancements due to clumping will lead to faster cooling times. To take this into account we calculate on a cell-by-cell basis. We find a cell-mass−weighted average of kyr and a cell-mass weighted average of for the hot stellar winds inside the bubble shell. Figure 5 also shows that the majority of the gas within the shell, especially at the center where the density is lowest, has . At larger radii closer to the shell boundary we find . This is likely due to the turbulent mixing between the low-density, high-temperature shock-heated gas and high-density, low-temperature shell. Because the energy loss via cooling is negligible within the bubble, and instead the thermalized wind energy is transferred via adiabatic expansion, thereby driving the dense bubble shell. We note that we find that the regions around the most massive stars at the center of the simulation are dominated by numerical effects and