Young stellar clusters across nearly five orders of magnitude in mass appear to follow a power-law mass–radius relationship (MRR), $R_{\star }\propto M_{\star }^{\alpha }$, with α ≈ 0.2–0.33. We develop a simple analytic model for the cluster mass–radius relation. We consider a galaxy disc in hydrostatic equilibrium, which hosts a population of molecular clouds that fragment into clumps undergoing cluster formation and feedback-driven expansion. The model predicts a mass–radius relation of $R_{\star }\propto M_{\star }^{1/2}$ and a dependence on the kpc-scale gas surface density $R_{\star }\propto \Sigma _{\rm g}^{-1/2}$, which results from the formation of more compact clouds (and cluster-forming clumps within) at higher gas surface densities. This environmental dependence implies that the high-pressure environments in which the most massive clusters can form also induce the formation of clusters with the smallest radii, thereby shallowing the observed MRR at high-masses towards the observed $R_{\star }\propto M_{\star }^{1/3}$. At low cluster masses, relaxation-driven expansion induces a similar shallowing of the MRR. We combine our predicted MRR with a simple population synthesis model and apply it to a variety of star-forming environments, finding good agreement. Our model predicts that the high-pressure formation environments of globular clusters at high redshift naturally led to the formation of clusters that are considerably more compact than those in the local Universe, thereby increasing their resilience to tidal shock-driven disruption and contributing to their survival until the present day.

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关于星团的初始质量-半径关系

质量接近五个数量级的年轻星团似乎遵循幂律质量半径关系 (MRR)，$R_{\star }\propto M_{\star }^{\alpha }$，α ≈ 0.2 –0.33。我们为簇质量-半径关系开发了一个简单的分析模型。我们考虑一个处于流体静力平衡状态的星系盘，它拥有一群分子云，这些分子云分裂成团块，经历星团形成和反馈驱动的膨胀。该模型预测了 $R_{\star }\propto M_{\star }^{1/2}$ 的质量半径关系以及对 kpc 尺度气体表面密度 $R_{\star }\propto \Sigma 的依赖性_{\rm g}^{-1/2}$，这是由于在更高的气体表面密度下形成更致密的云（以及内部形成簇的团块）。这种环境依赖性意味着，可以形成最大质量星团的高压环境也会诱导形成具有最小半径的星团，从而使观测到的大质量 MRR 向观测到的 $R_{\star }\propto M_ 变浅。 {\star }^{1/3}$。在低簇质量下，松弛驱动的膨胀会导致类似的 MRR 变浅。我们将预测的 MRR 与简单的种群综合模型相结合，并将其应用于各种恒星形成环境，发现了良好的一致性。我们的模型预测，高红移球状星团的高压形成环境自然导致形成的星团比本地宇宙中的星团紧凑得多，