Outer crust of a cold non-accreting magnetar

The outer-crust structure and composition of a cold, non-accreting magnetar are studied. We model the outer crust to be made of fully equilibrated matter where ionized nuclei form a Coulomb crystal embedded in an electron gas. The main effects of the strong magnetic field are those of quantizing the electron motion in Landau levels and of modifying the nuclear single-particle levels producing, on average, an increased binding of nucleons in nuclei present in the Coulomb lattice. The effect of a homogeneous and constant magnetic field on nuclear masses has been predicted by using a covariant density functional in which induced currents and axial deformation due to the presence of a magnetic field that breaks time-reversal symmetry have been included self-consistently in the nucleon and meson equations of motion. Although not yet observed, for $B\gtrsim {10}^{16}$ G both effects contribute to produce different compositions—odd-mass nuclei are frequently predicted—and to increase the neutron-drip pressure as compared to a typical neutron star. Specifically, in such a regime, the magnetic-field effects on nuclei favor the appearance of heavier nuclei at low pressures. As $B$ increases, such heavier nuclei are also preferred up to larger pressures. For the most extreme magnetic field considered, $B={10}^{18}$ G, and for the models studied, almost the whole outer crust is made of ${}_{40}^{92}{\mathrm{Zr}}_{52}$.

Figure 1

Binding energy $BE$ predicted by DD-ME2 as a function of the strength of the magnetic field $B$ for several typical nuclei appearing in the outer crust. A quadratic fit is also shown to indicate the average growing dependence of $BE$.

Figure 2

(a) Proton and neutron trends and (b) proton fraction $y\left(P\right)\equiv Z\left(P\right)/\left[N\right(P)+Z(P\left)\right]$ trends predicted by Duflo–Zuker model (blue solid line), by the DD-ME2 model (red dashed line), and by the NL3 model (green dash-dotted lines), in the absence of a magnetic field.

Figure 3

Compositions, i.e., $Z\left(P\right)$ and $N\left(P\right)$ trends, of the outer crust of a magnetar obtained by employing the DD-ME2 model for three external magnetic-field $B$ values: (a) $B={10}^{14}\mathrm{G}$, (b) $B={10}^{16}\mathrm{G}$, (c) $B={10}^{18}\mathrm{G}$. Solid (blue) lines include effects of magnetic field on nuclei and electrons, i.e., $B_{\mathrm{nucl}}=B$. Dashed (red) lines consider only magnetic effects on electron gas only, i.e., $B_{\mathrm{nucl}}=0\mathrm{G}$. Note that upper lines correspond to $N\left(P\right)$ and lower lines to $Z\left(P\right)$.

Figure 4

Compositions, i.e., $Z\left(P\right)$ and $N\left(P\right)$ trends, of the outer crust of a magnetar obtained by employing the NL3 model for three external magnetic-field $B$ values: (a) $B={10}^{14}\mathrm{G}$, (b) $B={10}^{16}\mathrm{G}$, (c) $B={10}^{18}\mathrm{G}$. Solid (blue) lines include effects of magnetic field on nuclei and electrons, i.e., $B_{\mathrm{nucl}}=B$. Dashed (red) lines consider only magnetic effects on electron gas, i.e., $B_{\mathrm{nucl}}=0\mathrm{G}$. Note that the upper lines correspond to $N\left(P\right)$ and the lower lines to $Z\left(P\right)$.

Figure 5

Neutron-drip transition pressures $P_{\mathrm{drip}}$ as a function of five external magnetic fields $B$: $B\approx 0\mathrm{G}, B={10}^{14}\mathrm{G}, B={10}^{15}\mathrm{G}, B={10}^{16}\mathrm{G}, B={10}^{17}\mathrm{G}$, and $B={10}^{18}\mathrm{G}$. Circles (blue) refer to the results considering the effects of the magnetic field on both electrons and nuclei, i.e., $B_{\mathrm{nucl}}=B$, while triangles (red) refer to calculations where the effects on nuclear binding energy have been neglected, i.e., $B_{\mathrm{nucl}}=0$.

Figure 6

Equations of state ($n$ versus $P$) predicted by the DD-ME2 model, for six different external magnetic-field values ($B=0\mathrm{G}, B={10}^{14}\mathrm{G}, B={10}^{15}\mathrm{G}, B={10}^{16}\mathrm{G}, B={10}^{17}\mathrm{G}$, and $B={10}^{18}\mathrm{G}$), including ($B_{\mathrm{nucl}}=B$) and neglecting ($B_{\mathrm{nucl}}=0\mathrm{G}$) the effects of magnetic fields on nuclei [panels (a) and (b), respect-ively]. Curves are plotted up to the neutron-drip transition point.