What is Burn-UD

The theory of strong interactions (Quantum Chromodynamics) predicts that hadronic (nucleated) matter subjected to high densities and/or temperatures will de-confine into a quark-gluon plasma (QGP). This transition has already been realized in the low-density/high-temperature regime both in heavy-ion collision experiments and in Nature -during the first few moments after the Big-Bang, where the reverse transition is expected to have occurred. The high-density/low-temperature regime is also expected to occur in Nature at the centers of compact stars.

Although asymptotic freedom would suggest otherwise, even at ultra-high densities - higher than that of a nucleus - hadrons do not spontaneously dissolve into their constituent u & d-quarks because overall the energy per baryon would be higher. But if a significant number of s-quarks are included then the extra quantum number (flavor) allows more quarks to share the same energy state, thus lowering the overall energy per baryon. This phase of matter is referred to as strange-quark matter (SQM). Since the probability of converting many u- & d-quarks into s-quarks at the same time is effectively zero, a conversion from hadronic to SQM can only proceed readily if a substantial number of s-quarks are already present.

The situation considered here is inside a compact star where the central density has reached that of nuclear de-confinement. Subsequently, a seed of SQM is instantiated at the star’s center. It has been speculated that such a situation could occur during the core-collapse phase of a supernova, or, if the nucleation timescale is slower, in an older neutron star whose central density has increased due to spin-down. The details of the seeding mechanism are not considered here, but it is assumed to proceed via either clustering of lambdas, higher-order neutrino “sparking” reactions, or seeding from the outside.

In the boundary region surrounding the SQM seed hadrons will then overlap with s-quarks, allowing the hadrons to dissolve into their constituent u- & d-quarks. Such a region will attempt to equilibrate chemically by producing more s-quarks, which will diffuse into the hadronic regions, forming even more SQM. The situation is then an interface where hadronic matter is on one side and SQM on the other, with the interplay between reactions and diffusion governing the speed at which the interface spreads. This scenario is of a type similar to that of non-premixed combustion, with the important difference being that the reactions in question are not activated above a certain temperature, as is often the case for combustion processes. Instead, the presence of s-quarks activates the burning and the process is governed by particle diffusion rather than heat diffusion.

During the combustion an interface is created with cold (unburnt) fuel on one side and hot (burnt) ash on the other. Subsequent cooling of the ash will result in pressure gradients that induce fluid-like motion of the matter in and around the interface. To conserve energy, fluid velocities across the interface would then differ, causing compression or rarefaction in the burning region which can enhance or quench the combustion. Thus, a fluid-dynamical treatment of the problem is essential to determine the interface speed. There is large uncertainty in the typical speed of the interface - values proposed in the literature range from 103 to 109 cm/s. This variation stems from the strong dependence of the reaction rate on temperature and density, both of which vary throughout the interface. Thus, one cannot define a typical speed from analytic considerations alone; the one considered in Burn-UD therefore uses a numerical approach to this problem.

Burn-UD solves the equations of hydrodynamical combustion. It solves the equations explicitly in time using a fourth-order Runge-Kutta scheme; spatial derivatives are treated separately, as per the Method of Lines. The spatial derivatives include advection – which is treated with a third-order upwinded, flux-limited, finite-volume scheme - and diffusion and pressure terms, which are second-order, not upwinded, and treated separately from the advection terms (ie. not flux-limited). While it may seem that Flux-Corrected Transport would be well suited to this problem, it provides only a negligible increase in accuracy at the cost of excessive overhead.