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An Empirical Fit for Viscoelastic Simulations of Tertiary Tides

Tertiary tides (TTs), or the continuous tidal distortion of the tertiary in a hierarchical triple system, can extract energy from the inner binary, inducing within it a proclivity to merge. Despite previous work on the subject, which established that it is significant for certain close triple systems, it is still not a well-understood process. A portion of our ignorance in this regard stems from our inability to integrate a simulation of this phenomenon into conventional stellar evolution codes, since full calculations of these tidal interactions are computationally expensive on stellar evolution timescales. Thus, to attain a better understanding of how these TTs act on longer timescales, an empirical expression of its effects as a function of parameters of the triple system involved is required. In our work, we evaluate the rate at which TTs extract energy from the inner binary within a series of constructed hierarchical triple systems under varying parameters, and study the rate at which the inner binary orbital separation shrinks as a function of those parameters. We find that this rate varies little with the absolute values of the masses of the three component objects, but is very sensitive to the mass ratio of the inner binary $q$, the tertiary radius $R_{\rm 3}$, the inner binary orbital separation $a_{\rm 1}$, the outer orbital separation $a_{\rm 2}$, and the viscoelastic relaxation time of the tertiary $τ$. More specifically, we find that the percentage by which $a_{\rm 1}$ shrinks per unit time can be reasonably approximated by (1/$a_{\rm 1}$)(d$a_{\rm 1}$/d$t$)=$\left(2.22{\times}10^{-8}{\rm yrs}^{-1}\right)4q\left(1+q\right)^{-2}(R_{\rm 3}/100{\rm R}_{\odot})^{5.2}(a_{\rm 1}/0.2{\rm AU})^{4.8}(a_{\rm 2}/2{\rm AU})^{-10.2}$ $(τ/0.534{\rm yrs})^{-1.0}$. We also provide tests of how precise this fitting function is.