Shock formation in stellar perturbations and tidal shock waves in binaries

TL;DR

This paper models how tidal forces can generate sound waves that steepen into shocks on stellar surfaces, using a hybrid linear-nonlinear approach, and finds shocks are unlikely before Roche lobe overflow in binary systems.

Contribution

It introduces a novel hybrid linear-nonlinear modeling method for stellar surface shocks caused by tidal forcing in binary systems.

Findings

Shocks form at the same point as Roche lobe overflow.

The hybrid model effectively captures shock formation conditions.

Shocks are unlikely to occur before mass transfer begins.

Abstract

We investigate whether tidal forcing can result in sound waves steepening into shocks at the surface of a star. To model the sound waves and shocks, we consider adiabatic non-spherical perturbations of a Newtonian perfect fluid star. Because tidal forcing of sounds waves is naturally treated with linear theory, but the formation of shocks is necessarily nonlinear, we consider the perturbations in two regimes. In most of the interior, where tidal forcing dominates, we treat the perturbations as linear, while in a thin layer near the surface we treat them in full nonlinearity but in the approximation of plane symmetry, fixed gravitational field and a barotropic equation of state. Using a hodograph transformation, this nonlinear regime is also described by a linear equation. We show that the two regimes can be matched to give rise to a single mode equation which is linear but models nonlinearity in the outer layers. This can then be used to obtain an estimate for the critical mode amplitude at which a shock forms near the surface. As an application, we consider the tidal waves raised by the companion in an irrotational binary system in circular orbit. We find that shocks form at the same orbital separation where Roche lobe overflow occurs, and so shock formation is unlikely to occur.