Superfluid hydrodynamics in the inner crust of neutron stars

The inner crust of neutron stars is supposed to be inhomogeneous and composed of dense structures (clusters) that are immersed in a dilute gas of unbound neutrons. Here we consider spherical clusters forming a body-centered cubic (BCC) crystal and cylindrical rods arranged in a hexagonal lattice. We study the relative motion of these dense structures and the neutron gas using superfluid hydrodynamics. Within this approach, which relies on the assumption that Cooper pairs are small compared to the crystalline structures, we find that the entrainment of neutrons by the clusters is very weak since neutrons of the gas can flow through the clusters. Consequently, we obtain a low effective mass of the clusters and a superfluid density that is even higher than the density of unbound neutrons. Consequences for the constraints from glitch observations are discussed.

Figure 1

Root-mean-square radius $\xi$ of the Cooper pair in the neutron gas (dashed lines) compared with the cell size $L$ of the crystalline lattice (red solid line) and the cluster radius $R$ (blue solid line) as functions of the total baryon density $n_B$ in the inner crust. The neutron gas density $n_{n,1}$ and the shown results for $L$ and $R$ were obtained from calculations detailed in Ref. [17]. The results for $\xi$ as functions of $n_{n,1}$ were obtained respectively by Matsuo [24] and Sun et al. [25] using the Gogny force (purple short dashes), the G3RS force (orange triangles), and the Bonn potential (green long dashes) as pairing interactions.

Figure 2

Schematic illustration of the discretization of a simple cubic cell with a spherical cluster in its center.

Figure 3

Primitive cell of a BCC lattice of spherical clusters.

Figure 4

Cut through a hexagonal lattice of cylindrical rods. The primitive cell is the parallelogram delimited by the white lines.

Figure 5

Streamlines and neutron speed (left) and velocity potentials (right) in a BCC cell of size $L=32.8$ fm, with a cluster of radius $R=7.54$ fm moving with velocity ${\mathbf{u}}_p={\mathbf{e}}_x$. The neutron density inside the cluster is $n_{n,2}=0.0973{\mathrm{fm}}^{-3}$ and outside $n_{n,1}=0.0412{\mathrm{fm}}^{-3}$ (the cluster and cell properties were obtained from calculations described in Ref. [17] and correspond to a baryon density of $n_B=0.0485{\mathrm{fm}}^{-3}$). In the left panels, the streamlines are shown as the white arrows, and the speed of the flow is indicated by the background color from dark purple (slowest) to red (fastest).

Figure 6

Same as Fig. 5, but for a hexagonal cell of size $L=24.7$ fm, containing a cylindrical rod of radius 5.53 fm moving with velocity ${\mathbf{u}}_p={\mathbf{e}}_a$ (upper panels) or ${\mathbf{e}}_b$ (lower panels). The neutron density inside the rod is 0.0942 ${\mathrm{fm}}^{-3}$ and outside 0.0528 ${\mathrm{fm}}^{-3}$ (corresponding to a baryon density of $n_B=0.0624{\mathrm{fm}}^{-3}$).

Figure 7

Schematic illustration of the definition of free and confined neutrons in energy space.

Figure 8

Effective neutron number of the clusters moving through the neutron gas as a function of the baryon density $n_B$. The results of our numerical calculations (blue crosses) are compared with the result of Eq. (18) by Magierski and Bulgac [12, 13, 14] for an isolated cluster in a uniform neutron gas (green dashed line), and with the neutron numbers (19) and (36) of the cluster defined in coordinate (red solid line) and energy (black double-dashed line) space, respectively.

Figure 9

Fraction of superfluid neutrons, $n_n^s/{\overline{n}}_n$ as a function of the baryon density $n_B$. Results of the present superfluid hydrodynamics approach (red solid line) are compared with the result of band-structure calculations by Chamel [5] (black circles). We display also our results for the fraction of (energetically) free neutrons $n_n^f/{\overline{n}}_n$ (green dashes) and those obtained within the band-structure approach [5] (purple squares).

Figure 10

Effective densities of bound neutrons in the 2D (spaghetti) phase for velocities in the directions of the two eigenvectors ${\mathbf{e}}_a$ and ${\mathbf{e}}_b$ as functions of total baryonic density.

Figure 11

Constraints on mass and radius of the Vela pulsar from its observed glitch activity for different superfluid fractions in the crust: hydrodynamic result $I_s/I_{\text{crust}}=0.94$ (red), result from band-structure theory [7] $I_s/I_{\text{crust}}=0.17$ (green), and an intermediate situation $I_s/I_{\text{crust}}=0.64$ (blue) corresponding to hydrodynamics in the gas but no superfluidity in the clusters (see Sec. 6). We also show as an example the mass-radius relation obtained with the SLy4 interaction (dashed line). Note that other equations of state would lead to different mass-radius relations in a band around the shown one (see, e.g., Ref. [45] for an attempt to use observational data to constrain the width of this band).

Figure 12

Superfluid fraction $n_n^s/{\overline{n}}_n$ as a function of the baryon density $n_B$, obtained under the assumption that a fraction $\delta =0$ (green short dashes), 0.5 (blue long dashes), or 1 (red solid line) of the neutrons in the clusters are superfluid. For comparison, the black circles are the result of the band-structure calculations by Chamel [5].