A variety of descriptions of the conversion of a neutron into a strange star have appeared in the literature over the years. Generally speaking, these works treat the process as a mere phase transition or ignore everything but microscopic kinetics, attempting to pin down the speed of the conversion and its consequences. We revisit in this work the propagation of the hypothetical "combustion" n $\to$ SQM in a dense stellar environment. We address in detail the instabilities affecting the flame and present new results of application to the turbulent regime. The acceleration of the flame, the possible transition to the distributed regime and further deflagration-to-detonation mechanism are addressed. As a general result, we conclude that the burning happens in (at least) either the turbulent Rayleigh-Taylor or the distributed regime. In both cases the velocity of the conversion of the star is several orders of magnitude larger than vlam, making the latter irrelevant in practice for this problem. A transition to a detonation is by no means excluded, actually it seems to be favored by the physical setting, but a definitive answer would need a full numerical simulation.
The interest in hypothetical absolutely stable phases of QCD at high density ( [1]) has been going on for more than two decades. In addition to the celebrated and widely studied SQM, renewed work on pairing in dense matter ( [2]) opened the possibility of an extra gain of energy, possibly enhancing the prospects for a "∆SQM"stability ( [3]).
There are many approaches to this problem dealing with evolved configurations of compact stars made from SQM or ∆SQM (refs) and their observable features. Another important question is how exactly the “normal” (proto) neutron stars become exotic objects. This could happen early in their lives (on ∼ seconds via classical nucleation or later if the conversion is quenched and depends ultimately on accretion, driven by quantum effects (bombaci). Even though the compression of the central regions of the star to ρ ≥ 3ρ 0 and temperatures T ∼ 10MeV favor the conversion n → SQM,it is still possible that the nucleation could be postponed [4]. In the remaining of this work we will assume a “hot” conversion on ∼ s timescale, focusing on the precise form of propagation and related energetic issues.
The fate of a just-nucleated stable drop of SQM (or ∆ SQM) matter has been addressed in several works (for example, [5], [6]). Usually,it was assumed that the propagation of the conversion front proceeds by diffusion of the s-quarks ahead, carrying the seed of flavor conversion that releases energy after relaxation. This “chemical” equilibration depends on weak interactions and occurs in a timescale τ w ∼ 10 -9 s,much faster than dynamical times of the order of a ms. In some works not even the formation of a front is acknowledged, and a macroscopic synchronization of the conversion is assumed, that is, deconfinement followed by a decay of d into s quarks, a process which is unlikely given that the conditions for that should be extreme in a region of the order of ∼ 10km ( [7], [8]).
In general, kinetic descriptions postulating diffusion lead to a well-known slow combustion mode termed deflagration, which is familiar to anyone. Moreover, in these works only the laminar regime is studied,leading to front velocities ≪ c sound ∼ c,neglecting therefore all complications related to the behavior actually observed in flames. Specifically,it has been known for years ( [9]) that at least two instabilities are important for the flame propagation: the Landau-Darrehius and Rayleigh-Taylor modes. While the first desestabilizes all wavelengths at the liner level (but is controlled by non-linear terms),the latter acts on large scales and is always active and important. Both instabilities develop on timescales ∼ τ dynamical ∼ 10 -3 s, and therefore their effects should be considered from the scratch, right after the beginning of the propagation.
After a few ms, the action of the LD instability leads to a wrinkling of the flame and the development of a so-called cellular structure. As a result of this, the fuel consumption rate rises (because it is controlled by the total area, which has become bigger) and the flame accelerates. However, the non-linear effects neglected by Landau play an important role stabilizing the flame, which looks nested Several approaches have been attempted One of these employed concepts of fractal geometry to argue that the velocity of the flame (by itself a fractal) can be described by the expression [10]
Whereas the maximum length l max is not difficult to find, and is ultimately bounded by the largest scale in the flame, the minimum length is more tricky: if l th represents the (microscopic) scale of the dissipation,then a related scale ∼ 100 × l th exists. This is known as the Markstein length. An inspection of eq.( 1) shows that the value of the exponent,however, does not allow a vary large increase of the velocity. Actually the flame speeds up in numerical simulations ( [11]) but not dramatically. Independently of this, the cellular stage is short and more dramatic effects happen along the propagation as described below.
The action of gravity directed against the propagation speed has been observed to disrupt the cellular flame generating a turbulent cascade also on short timescale. Above a length L at which u cell = u ′ (L) (where u ′ is the velocity fluctuating part), disruption of bubbles occur. This is the so-called Gibson scale. The estimation of this length requires the specification of the turbulent spectrum. Assuming a Kolmogorov form, one can check that
since this is a small length, but still ≫ l th because the thermal scale is truly microscopic, we can classify the propagation as belonging to the flamelet regime. In this situation, the flame propagation is still controlled by diffusion, but the total burning rate is determined by turbulence, and termed the flame brush. When the flame brush is developed, turbulent eddies turnover control the transport and fuel consumption rate, thus u turb is now unrelated to diffusion. A rough depiction of the evolution of t
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