Dynamical Tides Reexpressed

TL;DR

This paper revises the classical formula for stellar dynamical tides, simplifying it by expressing the torque in terms of boundary parameters and a nearly universal coefficient, thereby reducing errors and implementation difficulties.

Contribution

It introduces a new, simplified expression for the dynamical tide torque using boundary parameters and a nearly universal coefficient, improving accuracy and ease of use.

Findings

The new formula reduces errors caused by previous large prefactors.

The coefficient $eta_{2}$ is approximately unity and nearly independent of stellar properties.

The revised expression is easier to implement in stellar evolution codes.

Abstract

Zahn (1975) first put forward and calculated in detail the torque experienced by stars in a close binary systems due to dynamical tides. His widely used formula for stars with radiative envelopes and convective cores is expressed in terms of the stellar radius, even though the torque is actually being applied to the convective core at the core radius. This results in a large prefactor, which is very sensitive to the global properties of the star, that multiplies the torque. This large factor is compensated by a very small multiplicative factor, $E_{2}$. Although this is mathematically accurate, depending on the application this can lead to significant errors. The problem is even more severe, since the calculation of $E_{2}$ itself is non-trivial, and different authors have obtained inconsistent values of $E_{2}$. Moreover, many codes (e.g. BSE, StarTrack, MESA) interpolate (and sometimes extrapolate) a fit of $E_{2}$ values to the stellar mass, often in regimes where this is not sound practice. We express the torque in an alternate form, cast in terms of parameters at the envelope-core boundary and a dimensionless coefficient, $\beta_{2}$. Previous attempts to express the torque in such a form are either missing an important factor, which depends on the density profile of the star, or are not easy to implement. We show that $\beta_{2}$ is almost independent of the properties of the star and its value is approximately unity. Our formula for the torque is simple to implement and avoids the difficulties associated with the classic expression.