Anisotropic hydrodynamics, bulk viscosities, and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>r</mml:mi></mml:math>-modes of strange quark stars with strong magnetic fields

Abstract

In strong magnetic fields the transport coefficients of strange quark matter become anisotropic. We determine the general form of the complete set of transport coefficients in the presence of a strong magnetic field. By using a local linear response method, we calculate explicitly the bulk viscosities ${\ensuremath{\zeta}}_{\ensuremath{\perp}}$ and ${\ensuremath{\zeta}}_{\ensuremath{\parallel}}$ transverse and parallel to the $B$ field, respectively, which arise due to the nonleptonic weak processes $u+s\ensuremath{\leftrightarrow}u+d$. We find that for magnetic fields $B&lt;{10}^{17}\text{ }\text{ }\mathrm{G}$, the dependence of ${\ensuremath{\zeta}}_{\ensuremath{\perp}}$ and ${\ensuremath{\zeta}}_{\ensuremath{\parallel}}$ on the field is weak, and they can be approximated by the bulk viscosity for the zero magnetic field. For fields $B&gt;{10}^{18}\text{ }\text{ }\mathrm{G}$, the dependence of both ${\ensuremath{\zeta}}_{\ensuremath{\perp}}$ and ${\ensuremath{\zeta}}_{\ensuremath{\parallel}}$ on the field is strong, and they exhibit de Haas--van Alphen--type oscillations. With increasing magnetic field, the amplitude of these oscillations increases, which eventually leads to negative ${\ensuremath{\zeta}}_{\ensuremath{\perp}}$ in some regions of parameter space. We show that the change of sign of ${\ensuremath{\zeta}}_{\ensuremath{\perp}}$ signals a hydrodynamic instability. As an application, we discuss the effects of the new bulk viscosities on the $r$-mode instability in rotating strange quark stars. We find that the instability region in strange quark stars is affected when the magnetic fields exceed the value $B={10}^{17}\text{ }\text{ }\mathrm{G}$. For fields which are larger by an order of magnitude, the instability region is significantly enlarged, making magnetized strange stars more susceptible to $r$-mode instability than their unmagnetized counterparts.