The Role of Tides and Differential Rotation in the Angular Momentum Evolution of Low Mass Stars

Abstract

Stellar rotation is a fundamental property that governs magnetic activity, angular momentum evolution, and the dynamical interactions of stars with their companions. Despite the transformative datasets provided by \textit{Kepler} and \textit{TESS}, extracting physical information about stellar surfaces and tidal interactions from photometric time series remains challenging due to model degeneracies, computational expense, and the lack of physically interpretable statistical frameworks. This thesis develops novel statistical methods to address these challenges and applies them to advance our understanding of stellar rotation, tidal dissipation, and starspot variability in low-mass stars and binary systems. In the second chapter, I present \texttt{alabi}, an open-source framework for accelerating Bayesian inference with computationally expensive forward models. By training Gaussian process surrogates via active learning, \texttt{alabi} reduces the number of expensive model evaluations by orders of magnitude while preserving posterior accuracy, achieving speedups of 10--1000$\times$ for problems with up to 64 dimensions. In the third chapter, I apply \texttt{alabi} to investigate the prospects for constraining equilibrium tidal dissipation in low-mass binary stars. Using global sensitivity analysis and simulated Bayesian inference with coupled tidal, stellar evolution, and magnetic braking models, I show that constraining the tidal quality factor $\mathcal{Q}$ for individual systems is fundamentally limited by degeneracies with unknown initial conditions, even with ideal observational precision. Moreover, I identify population-level approaches, particularly overdensities in the observed distribution of orbital periods, rotation periods, and eccentricities, as a more promising path forward. In the fourth chapter, I derive a physically motivated Gaussian process kernel for stellar photometric variability from first principles. Starting from an analytic starspot model, I obtain a closed-form autocovariance function with hyperparameters (the equatorial rotation period, differential rotation shear, stellar inclination, spot lifetime, and spot emergence/decay timescale) that map directly onto measurable physical properties. I validate the kernel against numerical simulations, demonstrate that it captures harmonic structure absent in standard phenomenological kernels, and show that a banded Cholesky solver reduces the computational scaling from $\mathcal{O}(N^3)$ to $\mathcal{O}(Nb^2)$, where $b$ is the bandwidth of the covariance matrix. Applied to Kepler-63, the kernel recovers rotation, differential rotation, and inclination consistent with independent transit and spectroscopic measurements. This result constitutes the first demonstration that differential rotation can be inferred from one-dimensional photometric lightcurves using a physically interpretable Gaussian process.