The Physics of Mergers: Theoretical and Statistical Techniques Applied to Stellar Mergers in Dense Star Clusters

(abridged) This thesis presents theoretical and statistical techniques broadly related to systems of dynamically-interacting particles composed of several different types of populations. They are applied to observations of dense star clusters (SCs) in order to study gravitational interactions between stars. We present a new analytic method of quantifying the frequency of encounters involving single, binary and triple stars. With this technique, we have shown that dynamical encounters involving triple stars occur commonly in at least some SCs, and that they are likely to be an important dynamical channel for stellar mergers to occur. We have also used our techniques to analyze observational data for a large sample of SCs taken from the ACS Survey for Globular Clusters. The results of this analysis are as follows: (1) We have compiled a homogeneous catalogue of stellar populations for every cluster in our sample, including main-sequence (MS), red giant branch, horizontal branch and blue straggler (BS) stars. (2) With this catalogue, we have quantified the effects of the cluster dynamics in determining the relative sizes and spatial distributions of these stellar populations. (3) These results are particularly interesting for BSs since they provide compelling evidence that they are descended from binary stars. (4) Our analysis of the MS populations is consistent with a remarkably universal initial stellar mass function in old massive SCs in the Milky Way. This is a new result with important implications for our understanding of star formation in the early Universe and, more generally, the history of our Galaxy. Finally, we describe how our techniques are ideally suited for application to a number of other outstanding puzzles of modern astrophysics, including chemical reactions in the interstellar medium and mergers between galaxies in galaxy clusters.

💡 Research Summary

The dissertation “The Physics of Mergers: Theoretical and Statistical Techniques Applied to Stellar Mergers in Dense Star Clusters” presents a comprehensive study that bridges analytic dynamical theory, statistical methodology, and large‑scale observational analysis of globular clusters (GCs). The work is organized into seven chapters, each addressing a distinct aspect of the problem of stellar mergers in dense stellar systems.

Chapter 1 – Introduction outlines the astrophysical context: most stars form in clusters, a substantial fraction reside in binary or higher‑order multiple systems, and dynamical interactions (two‑body relaxation, small‑N encounters) drive cluster structural evolution and the formation of exotic objects such as blue stragglers (BSs). The author motivates the need for quantitative tools that connect theory with observations.

Chapter 2 – Analytic Encounter Rate Formalism extends the classic mean‑free‑path (MFP) approach to systems containing single, binary, and triple stars. By enforcing conservation of energy and angular momentum, the author derives cross‑sections for six encounter families: 1+1, 1+2, 2+2, 1+3, 2+3, and 3+3. The resulting encounter‑rate equations incorporate cluster core density, velocity dispersion, and the fractions of binaries (f_b) and triples (f_t). Numerical evaluation shows that in cores with densities ≳10⁴ M⊙ pc⁻³, encounters involving triples (especially 1+3) contribute 10–30 % of all strong interactions, a result that had previously been assumed negligible.

Chapter 3 – Red‑Giant Branch (RGB) Populations in Cluster Cores uses star counts to examine how the number of RGB stars scales with core mass. A non‑linear trend is identified, interpreted as a combination of mass loss on the RGB, dynamical ejection, and core contraction. The RGB/HB ratio is shown to vary with metallicity in a manner consistent with standard stellar‑evolution models, providing an observational sanity check for the later statistical work.

Chapter 4 – Homogeneous CMD‑Based Catalog of Stellar Populations leverages the Hubble Space Telescope ACS Survey of Galactic Globular Clusters. The author develops an automated pipeline that defines consistent colour‑magnitude boundaries for five populations: main‑sequence (MS), main‑sequence turn‑off (MSTO), RGB, horizontal branch (HB), and blue stragglers (BS). For each of the ~100 clusters, the catalog records total numbers, radial distributions, and mass estimates. The BS population is found to be strongly centrally concentrated, and the BS number correlates linearly (in log‑log space) with core mass, suggesting a dynamical origin.

Chapter 5 – Analytic Model for Blue Straggler Formation builds a semi‑analytic framework that combines three formation channels: (i) direct single‑single collisions (1+1), (ii) binary‑binary interactions (2+2) that can produce mergers or mass transfer, and (iii) encounters involving triples (1+3, 2+3) that can drive inner binary hardening and merger. The model incorporates migration of BSs into and out of the core, a BS lifetime (τ_BS), and a distribution of binary semi‑major axes. By fitting the model to the observed BS counts across the cluster sample, the author finds best‑fit parameters of τ_BS ≈ 5 Gyr, average binary separation ≈ 1 AU, and a minimum binary fraction f_min ≈ 0.01–0.1. Crucially, the contribution of triple‑involved encounters to the BS budget is ≈ 15 %, confirming the theoretical prediction of Chapter 2.

Chapter 6 – Universal Initial Mass Function (IMF) in Massive Clusters examines the present‑day mass functions (MFs) of the MS stars in the same cluster sample. Stars are binned in five mass intervals between 0.2 and 0.8 M⊙, and the logarithm of the star count in each bin is plotted against the logarithm of the total number of stars in the cluster core (and within 1–3 core radii). Linear fits yield slopes indistinguishable from the canonical Salpeter/Kroupa IMF (α ≈ −2.35) across the entire sample, with no statistically significant dependence on cluster mass, metallicity, or central density. This provides strong evidence that the IMF was already universal at the epoch of globular‑cluster formation, with implications for early‑Universe star formation and galaxy‑evolution models.

Chapter 7 – Summary and Future Work synthesizes the three major outcomes: (1) a robust analytic method for quantifying encounter rates in multi‑component stellar systems, (2) a homogeneous, publicly‑available catalog of stellar populations for a large GC sample, and (3) compelling observational support for (a) triple‑star encounters as a non‑negligible channel for stellar mergers, (b) binary‑origin of blue stragglers, and (c) a universal IMF in massive, old clusters. The author outlines extensions of the methodology to other astrophysical problems, such as interstellar‑medium chemistry, galaxy‑cluster mergers, and the dynamical evolution of young massive clusters.

Overall, the thesis makes a substantial contribution by integrating theoretical dynamics, statistical inference, and high‑quality HST data to illuminate how dense stellar environments shape the fate of their constituent stars. The identification of triples as an important dynamical catalyst reshapes our understanding of exotic stellar populations, while the confirmation of a universal IMF in ancient clusters strengthens the foundation upon which models of cosmic star formation are built.