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Tidal radii of main sequence stars -- I. Physical tidal radius, semi-analytic model and their implications

A star is tidally disrupted by a supermassive black hole when their separation is shorter than the "tidal radius". This quantity is often estimated on an order-of-magnitude basis without reference to the star's internal structure. Using MESA models for main sequence stars and fully general relativistic dynamics, we find the physical tidal radius for complete disruption $\cal{R}_t$ for a $10^6M_\odot$ black hole (BH). We find that across a factor $\sim20$ in stellar mass $M_*$, i.e., $0.15M_{\odot}\leq M_*\leq3M_\odot$, $\cal{R}_t\sim27\times$(BH's gravitational radius). When comparing $\cal{R}_t$ with the commonly used order-of-magnitude estimate $r_t$, we find that $\cal{R}_t\sim1.05-1.45r_t$ for $0.15M_\odot\leq M_*\leq0.5M_\odot$, but between $0.5 M_\odot$ and $1 M_\odot$, $\cal{R}_t$ drops to $\sim 0.45r_t$, and it remains at this value up to $10 M_\odot$. The near-constancy of $\cal{R}_t$ implies a weaker dependence of the full disruption rate on $M_*$ than when predicted with $r_t$. The characteristic energy width of the debris $ΔE$ ranges from $\sim1.2Δ\cal{E}$ for low-mass stars to $\sim 0.35Δ\cal{E}$ for higher-mass stars, where $Δ\cal{E}=GM_{\rm BH}R_*/\cal{R}_t^{2}$. We present analytic fits for the $M_*$ dependence of $\cal{R}_t$ and $ΔE$; these fits lead to analytic expressions for the time of peak mass fallback rate and the maximal mass fallback rate. Our results also bear on the fraction of events leading to fast or slow circularization, as well as on the character of the tidal event occurring when the remnant of a partial disruption returns to the black hole. Using a semi-analytic model, we show that $\cal{R}_t$ is primarily determined by the star's central density rather than its mean density. For high-mass stars, the full disruption rate is roughly 1/4 the partial disruption rate, while this ratio is close to unity for low-mass stars.