# Simulation of a binary white dwarf merger

Hideyuki Saio & Simon Jeffery are exploring models for the evolution of stars formed by the merger of two white dwarfs. The attached movie provides a simulation of the major events involved in the merger process. The simulation has been constructed using ray-tracing software from POV-Ray. This page presents the simulation and provides details of how it was constructed. Details of the evolutionary models are presented elsewhere.

### Initial Parameters

An initial binary system consists of two white dwarfs of masses M1 and M2 (given) with an orbital period (P). In this simulation, we have used M1 = 0.6 solar masses, M2 = 0.3 solar masses and P = 6 hours. All orbits are assumed to be circular. We adopt a simple expression for the white dwarf radius as a function of mass (in solar units):
R = 0.01 / (sqrt(M/0.6))

### Orbital Decay

The merger theory postulates that the binary orbit will shrink due to the loss of orbital angular momentum, either through magnetic interaction between the stars and their atmospheres or by the generation of gravitational waves. There is currently no satisfactory theory which will predict reliable decay times, but they are clearly long. To facilitate the simulation, we define an orbital decay time (td) of a few orbital periods, 10 in the current case.

The orbital period and the separation of the white dwarfs are governed by Kepler's laws. In order to simplify the maths, we define the orbital decay in terms of the initial separation and an elapsed time t:

a = a0 ( 1 - (t/td)² )2/3
whence Kepler's laws give
P = P0 ( 1 - (t/td)² )
and phase is given analytically by
φ= (2π/P0) x (td/2) x ln | (1+t/td) / (1-t/td) |
Assuming no distortion, contact between the white dwarfs occurs at
tc = td x sqrt ( 1 - ((R1+R2)/a0)3/2 )

The simulation starts with the binary observed from an inclination of π/2 and a distance of 1 solar radius. The camera pans around to i = π/9 and then zooms in to 0.1 solar radius to watch the final spiral in. The captions show (in red) the nominal elapsed time (t) and (in yellow) the current orbital period (P/hours) and separation (a/solar radii). From time to time a red dot appears at the system center of mass.

### Merger

At tc, previous calculations show that the less massive white dwarf will break up on a dynamical timescale, approximately a few orbital periods. The process is a complex problem in radiation hydrodynamics involving a transition from a highly assymetric sytem to a cylindrically symmetric system. Simple physical arguments suggest that the initial product will be a Keplerian disk of near-degenerate plasma surrounding the more massive white dwarf.

In our simulation, the breakup phase is represented by a group of toroidal segments in Keplerian orbits. These are distributed randomly in phase, and extent and with an inverse square radial distribution. This phase lasts for a limited interval (not well defined at present), after which the toroidal segments are combined to form an oblate spheroidal disk (aspect ratio 1:5). The surviving white dwarf may be seen within this disk in our simulation, although in nature the disk and fragments will be optically thick.

The camera position remains fixed throughout this sequence. The red caption shows real time (tm=t-tc) from the merger event (nominal hours).

### Expansion

As colliding disk particles lose angular momentum, they will be accreted by the white dwarf and form a non-degenerate envelope around the core, causing its overall radius to increase. When sufficient mass has been accreted, nuclear burning may be ignited at the core-envelope boundary, causing further expansion.

The merged star evolution is simulated by a quadratic expansion
R = 0.01 * tm²
where tm is the time since merger in hours and R is in solar units. Expansion is halted (arbitrarily) when R = 10 solar radii. The accretion disk is allowed to decay during the expansion until all material has been transferred from the disk to the merged star.

The camera aspect remains the same, but moves back to keep the star in shot as it expands. After the star reaches 0.1 solar radii, an octal is cut away to reveal the surviving disk and white dwarf core. The red caption (x) is a nominal time counter since merger. A rod of length initially 0.1 and later 1 solar radius is shown just in front of the star.

### Outstanding Problems

Physics. It is emphasized that this simulation is only a cartoon of something that is believed to occur in nature. Some of the details of the process have been worked out with real physical models, but there are many that are still to be resolved.

Artwork. Although ray-tracing allows a lot of physics to be incorporated in setting up a scene, it has limitations for visualising stellar sources. Some of these are discussed on the separate page: POV-Ray stars.