Globular Cluster formation in a collapsing supershell

Abstract

Primordial clouds are supposed to host the so-called population III stars. These stars are very massive and completely metal-free. The final stage of the life of population III stars with masses between 130 and 260 solar masses is a very energetic hypernova explosion. A hypernova drives a shock, behind which a spherically symmetric very dense supershell forms, which might become gravitationally unstable, fragment, and form stars. In this paper we study under what conditions can an expanding supershell become gravitationally unstable and how the feedback of these supershell stars (SSSs) affects its surroundings. We simulate, by means of a 1-D Eulerian hydrocode, the early evolution of the primordial cloud after the hypernova explosion, the formation of SSSs, and the following evolution, once the SSSs start to release energy and heavy elements into the interstellar medium. Our results indicate that a shell, enriched with nucleosynthetic products from SSSs, propagates inwards, towards the center of the primordial cloud. In a time span of a few Myr, this inward-propagating shell reaches a distance of only a few parsec away from the center of the primordial cloud. Its density is extremely high and its temperature very low, thus the conditions for a new episode of star formation are achieved. We study what fraction of these two distinct populations of stars can remain bound and survive until the present day. We study also under what conditions can this process repeat and form multiple stellar populations. We extensively discuss whether the proposed scenario can help to explain some open questions of the formation mechanism of globular clusters.

Appendix A: The role of the external pressure on supershell fragmentation

Dinnbier et al. (2017) suggest that the relative importance of the pressure $P_{\mathit{ext}}$ confining a shell influences the fragmentation not only quantitatively, but also qualitatively. If the confining pressure dominates the self-gravity, the shell breaks into many gravitationally stable objects which subsequently coalesce until they form enough mass to collapse gravitationally. If the self-gravity dominates the confining pressure, the condensing fragments collapse directly into gravitationally bound objects. While the dispersion relation proposed by Elmegreen and Elmegreen (1978) provides a reasonable estimate for fragment properties for self-gravity dominated shells, no linearised estimate holds for pressure dominated shells. For pressure dominated shells, Dinnbier et al. (2017) indicate that the Jeans mass is a proxy for the fragment mass. This determines our choice of fragment radius $d$ and the approximations made in Eq. (2) in these two cases. For pressure-dominated shells, we assume that the most unstable fragment contains one Jeans mass while for self-gravity dominated shells, we assume that the radius corresponds to the most unstable wavelength in the dispersion relation proposed by Elmegreen and Elmegreen (1978). Therefore,

where parameter

$$ A = \frac{1}{\sqrt{1 + 2P_{\mathit{ext}}/(\pi G \varSigma^{2})}} $$

(A.2)

indicates the relative importance of the external pressure versus self-gravity (Elmegreen and Elmegreen 1978). Parameter $A$ lies in the interval $(0, 1)$, where pressure dominated shells have $A$ near zero and self-gravity dominated ones $A$ near unity.

Since fragmentation occurs at substantially different times for pressure dominated and self-gravity dominated shells, we investigate which case is more appropriate to our supershell model and use the approximation for the fragmenting time accordingly. Let we assume that the supershell fragments in the pressure dominated case. For this reason, we omit the second term on the right hand side of Eq. (2) because unstable fragments are delivered by coalescence with no role of the pressure gradient. The sound speed of the supershell $a_{s}$ is insensitive to the position within the supershell wall (see Fig. 4). The sound speed also changes very little during the expansion. This enables us to express the fragmenting time in the closed form.

Let us assume a supernova explosion at the density peak of a medium with a general power–law density distribution $n(r) = n_{c} (r/r_{c})^{- \omega }$. The exponent of the expanding law $\alpha$ (Eq. (3)) depends on the value of $\omega$ and whether the shell is in the energy conserving phase or the momentum conserving phase. While for expansion in the energy conserving phase $\alpha = 2/(5 - \omega )$, for expansion in the momentum conserving phase $\alpha = 1/(4 - \omega )$ (Ostriker and McKee 1988).

The surface density of a supershell of radius $R$ is

$$ \varSigma (R) = \frac{M(R)}{4 \pi R^{2}} = \frac{n_{c} \bar{m} r_{c}^{2} K^{1-\omega } t^{\alpha (1-\omega )}}{3 - \omega }, $$

(A.3)

where $\bar{m}$ denotes the mean gas particle mass. The supershell is confined by two pressures: thermal pressure from the cavity acting on its inner surface, and ram pressure from the accreting medium acting on its outer surface. In the momentum conserving phase, the latter of the pressures is significantly higher than the former. For the adopted model of shell fragmentation, it is not a priori clear, which of these pressures characterises fragmentation better. We choose the ram pressure, which is higher, to represent $P_{\mathit{ext}}$, and we shall see that the shell fragments in self-gravity dominated case, so the choice of the particular pressure is not important. Thus,

$$ P_{\mathit{ext}} = n(r) \bar{m} ( {\mathrm{d}R}/{\mathrm{d}t} ) ^{2} = n_{c} \bar{m} r_{c}^{2} K^{2 - \omega } \alpha^{2} t^{2\alpha - \alpha \omega -2}. $$

(A.4)

Substituting Eq. (A.3), Eq. (A.4) and the parameter $A$ for pressure dominated shells (small $A$; $P_{\mathit{ext}} \gg \pi G \varSigma^{2}$) where Eq. (A.2) becomes

$$ A \simeq \biggl\{ \frac{\pi G \varSigma^{2}}{2 P_{\mathit{ext}}} \biggr\} ^{1/2} = \sqrt{ \frac{\pi G n_{c} \bar{m} r_{c}^{2}}{2}} \frac{K^{-\omega /2} t^{-(\alpha \omega + 2)/2}}{\alpha (3 - \omega )}, $$

(A.5)

into Eq. (A.1a) and Eq. (2), one obtains the fragmenting time ($t_{\mathit{frg}} = t_{\mathit{col}}$),

$$ t_{\mathit{frg}} = \biggl\{ \frac{9 \pi a_{s}^{4} (3 - \omega )^{6} (1 + \alpha ^{2})^{4} K^{7 \omega }}{8 \alpha^{2} (G n_{c} \bar{m} r_{c}^{2})^{7} K^{8}} \biggr\} ^{1/(6 + 8\alpha - 7\alpha \omega )}. $$

(A.6)

We use $\alpha = 1/2$ and $\omega = 2$ for the presented model. Substituting these values to Eq. (A.6) and using Eq. (A.2) with $t = t_{\mathit{frg}}$, we realise that the supershell fragmented when $A \simeq 0.8$, i.e. in the self-gravity dominated case. This finding contradicts the original assumption that the supershell fragments in the pressure dominated case.

Therefore, to estimate the fragmenting time, we use the estimate of fragment radius in the self-gravity dominated case and also take into account the term in Eq. (2) representing the pressure gradient. Substituting Eq. (A.1b) into Eq. (2), one finds that the collapse time-scale is given by

$$ t_{\mathit{col}} = \biggl( \frac{G^{2} \varSigma^{2}}{4 a_{s}^{2}}-\frac{\alpha ^{2}}{t^{2}} \biggr) ^{-1/2}. $$

(A.7)

Note that for shells expanding in the momentum conserving phase, the second term on the right hand side of Eq. (A.7) decreases faster than the first term if $\omega < 5/2$. Thus, for the choice $\omega = 2$, there always exists an expansion time $t$ which is larger than the fragment collapse time $t_{\mathit{col}}$, and the shell fragments.