Screening condition in the core of neutron stars

Earlier, the screening condition in neutron star core has been formulated as equality of velocities of superconducting protons and the electrons ${\mathbf{v}}_p={\mathbf{u}}_e$ at wave numbers $q\ll {\lambda }^{-1}$ ($\lambda$ is the London penetration depth) and has been used to derive the force exerted by the electrons on a moving flux tube. By calculating the current-current response, I find that ${\mathbf{v}}_p\ne {\mathbf{u}}_e$ for $l^{-1}<q\ll {\lambda }^{-1}$ ($l$ is the electron mean-free path). I show that at typical realistic parameters the electric field induced by a moving (relative to the electrons) flux tube is not screened by the electron currents. The implication is that the existing picture of the momentum exchange between the electrons and the flux tubes must be reassessed.

Figure 1

The real part of the proportionality coefficient $\zeta$ between the induced current and the test current, Eq. (25), as function of the collision frequency $\nu$ with wave number set as $q=\nu /c$. From left to right, the lines correspond to $\omega =10$, ${10}^2$, ${10}^3$, and ${10}^4$.

Figure 2

The imaginary part of the proportionality coefficient $\zeta$ between the induced current and the test current, Eq. (25), as function of the collision frequency $\nu$ with wave number set as $q=\nu /c$. From left to right, the lines correspond to $\omega =10$, ${10}^2$, ${10}^3$, and ${10}^4$.

Figure 3

The same as Fig. 1 but $q=10\nu /c$.

Figure 4

The same as Fig. 2 but $q=10\nu /c$.

Figure 5

Arrows show the vector pattern of the unscreened (without the electron contribution) microscopic electric field ${\mathbf{E}}_{\mathrm{tube}}\left(\mathbf{r}\right)$ due to a moving array of four vortices. The field ${\mathbf{E}}_{\mathrm{tube}}\left(\mathbf{r}\right)$ is calculated from Eq. (27) with $\lambda /\xi =10$ corresponding to $\mathrm{\Delta }=0.8956$ MeV in Eqs. (2) and (3) and $d_v=\lambda$ corresponding to $H=3.231\times {10}^{15}$ Oe in Eq. (1). The normal cores of the flux tubes are displayed by the dotted circles. The velocity of flux tubes relative to the electron liquid is pointed towards $-{\mathbf{e}}_{\mathbf{x}}$ direction.