Stability of strange star crusts and strangelets

We construct strangelets, taking into account electrostatic effects, including Debye screening, and arbitrary surface tension $\sigma$ of the interface between vacuum and quark matter. We find that there is a critical surface tension ${\sigma }_{\mathrm{crit}}$ below which large strangelets are unstable to fragmentation and below which quark star surfaces will fragment into a crystalline crust made of charged strangelets immersed in an electron gas. We derive a model-independent relationship between ${\sigma }_{\mathrm{crit}}$ and two parameters that characterize any quark matter equation of state. For reasonable model equations of state, we find ${\sigma }_{\mathrm{crit}}$ typically of order a few $\mathrm{MeV}/{\mathrm{fm}}^2$. If $\sigma ⩽{\sigma }_{\mathrm{crit}}$, the size-distribution of strangelets in cosmic rays could feature a peak corresponding to the stable strangelets that we construct.

Figure 1

A strangelet profile with radius $R=1.6{\lambda }_D$, and $\sigma =0.1325n_Q^2{\lambda }_D/{\chi }_Q$. (This corresponds to the barely-stable strangelet at the critical surface tension—see Sec. 3.) The horizontal axis is $x=r/{\lambda }_D$. The charge density $q$ is plotted in units of $n_Q$, ${\mu }_e$ is plotted in units of $n_Q/{\chi }_Q$, the electric field is plotted in units of $n_Q/\sqrt{{\chi }_Q}$ and the pressure $p_{\mathrm{QM}}$ is plotted in units of $n_Q^2/{\chi }_Q$. Because of the electric field, the charge density in the strangelet is pushed towards the outer edge. The pressure gradient within the strangelet is balanced by the electric force on the charged matter.

Figure 2

The Gibbs free energy (i.e. energy per quark) relative to neutral uniform quark matter with $p=0$ for strangelets of various sizes and surface tensions. We plot $\overline{\Delta g}(x)$, which is $\Delta g$ in units of $n_Q^2/(n{\chi }_Q)$, as a function of the radius $x=R/{\lambda }_D$, for various values of the surface tension ($\overline{\sigma }$ is $\sigma$ in units of $n_Q^2{\lambda }_D/{\chi }_Q$). The solid lines (red online) are obtained from (24); the dashed lines (blue online) are obtained from a numerical solution of the Poisson equation, including the electrons and not making any approximations in the evaluation of $E(R)$. The numerical results confirm the validity of the approximations made in deriving (24, 25). (The numerical analysis is for a bag model of unpaired quark matter with ${\mu }_{\mathrm{crit}}=305\mathrm{MeV}$ and $m_s=200\mathrm{MeV}$. The $n_Q$ and ${\chi }_Q$ for this model are given in Sec. 4.) At the critical surface tension ${\overline{\sigma }}_{\mathrm{crit}}=0.1325$ one can just barely construct a strangelet that is favored over uniform quark matter. The radius of this critical strangelet is $R_{*}=1.606{\lambda }_D$. For $\overline{\sigma }<{\overline{\sigma }}_{\mathrm{crit}}$, large strangelets and bulk neutral quark matter are unstable to fragmentation. For $\overline{\sigma }>{\overline{\sigma }}_{\mathrm{no}\mathrm{\text{−}}\mathrm{barrier}}=0.1699$, there is no fusion barrier: strangelets of any size can lower their free energy by fusing should they encounter each other. For ${\overline{\sigma }}_{\mathrm{crit}}<\overline{\sigma }<{\overline{\sigma }}_{\mathrm{no}\mathrm{\text{−}}\mathrm{barrier}}$, there are “metastable” droplets, whose free energy is greater than that of bulk quark matter but which must overcome a barrier in order to fuse.

Figure 3

The critical surface tension for strangelets made of unpaired quark matter, as a function of the strange quark mass, at ${\mu }_{\mathrm{crit}}=305\mathrm{MeV}$. Other allowed values of ${\mu }_{\mathrm{crit}}$ give almost indistinguishable curves. The solid curve is given by (27) whereas the dashed curve was obtained by solving the Poisson equation numerically.