Investigating Convective Overshoot in White Dwarfs

Figure 5: Spread (full width at half maximum) variation of tracer profiles which were added at seven discrete depths within the overshoot region of the same simulation shown in Figure 2. Depths correspond to those on the x-axis of Figure 2. A line of t1/2 is drawn (dashed) to show the expected behaviour of a diffusion process.

Figure 1: Characteristic velocities in the envelope of a 11 500 K and log g = 8.0 DA white dwarf as a function of stellar mass. The 1D MLT/α=0.8 velocity (solid black) is shown along with diffusion velocities of Mg (green), Ca (magenta), and Fe (cyan) and a crude macroscopic diffusion velocity (red dashed) extrapolated from 3D simulations (Tremblay et al. 2015). Metals are removed from the mixed regions when convective velocity equals diffusion velocity. The change in the mass within which the metals are mixed can be read from the distance between the 1D (black) and 3D (red) circles on the x-axis.

Tim Cunningham, Pier-Emmanuel TremblayDepartment of Physics, University of Warwick, Coventry CV4 7AL

Figure 2: Evolution of a normalised tracer particle distribution over 155 seconds of stellar time. From left to right each blue line represents a lapse of 100 ms whilst each purple line demarcates 10s intervals. Simulation of a pure-hydrogen (DA) white dwarf with effective temperature Teff = 12 000 K and surface gravity log g = 8.00 using CO5BOLD. The blue-filled rectangle indicates the location of the 1D convection zone according to the Schwarschild criterion for instability.

Why Convective Overshoot is Important For decades, understanding the diffusion of heavy elements through stellar plasmas has been central to research in solar and stellar evolution and accretion onto white dwarfs (Paquette et al. 1986). In modelling convection, most commonly an approximate approach has been taken with static 1D models defining a discrete boundary between convective and diffusive regions using the Schwarschild stability criterion.

There is, however, mounting evidence to suggest that the 1D treatment of convection underestimates the amount of material which is convectively mixed.

Results of the 2D simulations of Freytag et al. (1996) suggest that, for a DA white dwarf, 100 times more mass is ∼convectively mixed than is contained within the 1D convection zone. A further result was that beneath the convection zone macroscopic mixing was found to scale approximately as the square RMS velocities.

AcknowledgementsThis project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements No 677706 – WD3D).

Characterising Overshoot with Tracer ParticlesBuilding on the 2D work of Freytag et al. (1996) we are

adding tracer particles to deep 3D simulations of DA white dwarfs to probe the extent of convective mixing. Figure 5 (below) shows the spread of tracer particles as a function of time from which we will extract a depth dependent overshoot coefficient as

ConclusionsFigure 2 highlights the disparity between the 1D convection

zone (blue rectangle) and the extent over which material is mixed in 3D. As expected, the tracer particle spread decreases with depth as macroscopic diffusion becomes less efficient.

Our immediate aim is to create a figure akin to Figure 1, via this methodology, and for the other stable simulations we currently have in the range Teff = 11 300 – 15 500 K.

ReferencesFreytag, B., Ludwig, H.-G., & Steffen, M. 1996, A&A, 313, 497

Paquette, C., Pelletier, C., Fontaine, G., & Michaud, G. 1986, ApJS, 61, 197

Tremblay, P.-E., Ludwig, H.-G., Freytag, B., et al. 2015, ApJ, 799, 142

A basic adaptation of this model by Tremblay et al. (2015) is presented above (Figure 1) as a function of the enclosed stellar mass. In this crude, exponentially decaying, overshoot model, the amount of mass (x-axis) in which the accreted material is mixed is increased by 4 orders of magnitude. The remainder of the research shown in this presentation pertains to improving on this first attempt by finding a more physically robust model of overshoot.

Figure 3: Schematic representation of the discernible layers of physical processes adapted from Freytag et al. (1996). The convection zone proper comprises the top layer and the three layers immediately beneath make up the overshoot region.

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convectively unstable,bubbles accelerated down

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Figure 4: For the simulation shown in Figure 2 (DA, log g = 8.00, Teff = 12 000 K) entropy (solid) and radiative flux (dot-dash) profiles at t = 152 s after tracer profile added. The left-most and palest region is the same as that shown in Figure 2; namely, the 1D convection zone. The subsequent regions indicate the changing physical processes described in Figure 3.