Compact stars containing quark matter may masquerade as neutron stars in the range of measured mass and radius, making it difficult to draw firm conclusions on the phase of matter inside the star. The sensitivity of core g-mode oscillations to the presence of a mixed phase may alleviate this difficulty. In hybrid stars that admit quark matter in a mixed phase, the g-mode frequency rises sharply due to a marked decrease in the equilibrium sound speed. Resonant excitation of g-modes can leave an imprint in the waveform of coalescing binary compact stars. We present analytic and numeric results to assess the sensitivity displayed by g-mode oscillations to quark matter in a homogeneous or mixed phase and also compute relevant damping times in quark matter due to viscosity.

We calculate the mass-radius relationship of quark stars with the magnetized density-dependent quark mass model in this work, considering two magnetic field geometries: a statistically isotropic, tangled field and a force-free configuration. In both cases, magnetic field production decreases in the case of maximum quark star mass. Furthermore, a tangled, isotropic magnetic field has a relatively smaller impact on the mass and radius, compared to the force-free configuration, which implies that the geometry of the interior magnetic field is at least as important as the field strength itself when the influence of the strong magnetic field on the mass and radius is assessed.

The main purpose of this study is to interpret the possibilities of hybrid star configurations under different phase transition paths and provide a general description of the conditions and features of the different configurations. We assume that there are two possible phase transition paths, i.e., from a nuclear phase to a 2flavor(2f)/3flavor(3f) quark phase directly, or first from a nuclear phase to a 2f quark phase, and then from that phase to a 3f quark phase sequentially. In addition, we consider Maxwell and Gibbs constructions based on the assumption of a first-order transition, which yields multiple configurations of hybrid stars: N-2f, N-3f, and N-2f-3f for a Maxwell construction, and N-2fmix-2f, N-3fmix-3f, N-2f3fmix, and N-2fmix-3f for a Gibbs construction. From the radii analysis of different hybrid star configurations with the same mass of , the appearance of the quark matter (from nuclear to 2f or 3f quark matter) causes a radius difference of 0.5km~2km and provides the possibility of detection by NICER in the future. However, the sequential transition from 2f to 3f quark matter is difficult to detect because the transition does not lead to too high of a change in radius (far smaller than ). The dependence solely on the measurements of the stellar radii to probe the equation of state of dense matter in neutron stars causes difficulties. Multi-messenger observations can help us to infer the interior of a neutron star in the future.