The state of super-dense matter is essential for us to understand the nature of pulsars; however, non- perturbative quantum chromodynamics makes it very difficult to make direct calculations of the state of cold matter at realistic baryon number densities inside compact stars. Nevertheless, from an observational point of view, it is conjectured that pulsars could be made up of quark clusters since the strong coupling between quarks might render the quarks to be grouped in clusters. In this paper, we attempt to find an equation of state of condensed quark-cluster matter in a phenomenological way. Supposing that the quark-clusters could be analogized to inert gases, we apply here the corresponding-state approach to derive the equation of state of quark-cluster matter, as was similarly demonstrated for nuclear and neutron-star matter in the 1970s. According to the calculations that we have presented, the quark-cluster stars, which are composed of quark-cluster matter, could have a high maximum mass that is consistent with observations and, in turn, further observations of pulsar mass could also place a constraint on the properties of quark-cluster matter. We will also briefly discuss the melting heat during the solid-liquid phase conversion and its related astrophysical consequences.
Pulsar glitches, i.e. the sudden spin-ups of pulsars, have been detected for most known pulsars.The mechanism giving rise to this kind of phenomenon is uncertain, although a large data set has been built.In the framework of the starquake model, based on Baym ＆ Pines, the glitch sizes（the relative increases of spin-frequencies during glitches） △Ω/Ω depend on the released energies during glitches, with less released energies corresponding to smaller glitch sizes. On the other hand, as one of the dark matter candidates,our Galaxy might be filled with so called strange nuggets（SNs） which are relics from the early Universe.In this case collisions between pulsars and SNs are inevitable, and these collisions would lead to glitches when enough elastic energy has been accumulated during the spin-down process. The SN-triggered glitches could release less energy, because the accumulated elastic energy would be less than that in the scenario of glitches without SNs. Therefore, if a pulsar is hit frequently by SNs, it would tend to have more small glitches, whose values of ？？/？ are smaller than those in the standard starquake model（with larger amounts of released energy）. Based on the assumption that in our Galaxy the distribution of SNs is similar to that of dark matter, as well as on the glitch data in the ATNF Pulsar Catalogue and Jodrell Bank glitch table, we find that in our Galaxy the incidences of small glitches exhibit tendencies consistent with the collision rates between pulsars and SNs. Further testing of this scenario is expected by detecting more small glitches（e.g.,by the Square Kilometre Array）.
Several X-ray-dim isolated neutron stars （XDINSs）, also known as the Magnificent Seven, exhibit a Planck-like soft X-ray spectrum. In the optical/ultraviolet （UV） band, there is an excess of radiation compared to an extrapolation from the X-ray spectrum. However, the majority exhibits ＂spec- tral deviations＂： the fact that there is more flux at longer wavelengths makes spectra deviate from the Rayleigh-Jeans law. A model of bremsstrahlung emission from a nonuniform plasma atmosphere is proposed in the regime of a strangeon star to explain the optical/UV excess and its spectral devi- ation as well as X-ray pulsation. The atmosphere is on the surface of strangeon matter, which has negligible emission, and is formed by the accretion of ISM-fed debris disk matter moving along the magnetic field lines to near the polar caps. These particles may spread out of the polar regions which makes the atmosphere non-uniform. The modeled electron temperatures are ~ 100 - 200 eV with radi- ation radii Ropt ~ 5 - 14km. The spectra of five sources （RX J0720.4-3125, RX J0806.4-4123, RX J1308.6＋2127, RX J1605.3＋3249, RX J1856.5-3754） from optical/UV to X-ray bands can be fitted well by the radiative model, and exhibit Gaussian absorption lines at ~ 100 - 500 eV as would be expected. Furthermore, the surroundings （i.e., fallback disks or dusty belts） of XDINSs could be tested by future infrared/submillimeter observations.
Pulsar-like compact stars usually have strong magnetic fields, with strengths from -10^8 to -10^12 G on the surface. How such strong magnetic fields can be generated and maintained is still an unsolved problem,which is, in principle, related to the interior structure of compact stars, i.e., the equation of state of cold matter at supra-nuclear density. In this paper we are trying to solve the problem in the regime of solid quark-cluster stars.Inside quark-cluster stars, the extremely low ratio of number density of electrons to that of baryons ne /nb and the screening effect from quark-clusters could reduce the long-range Coulomb interaction between electrons to short-range interaction. In this case, Stoner＇s model could apply, and we find that the condition for ferromagnetism is consistent with that for the validity of Stoner＇s model. Under the screened Coulomb repulsion, the electrons inside the stars could be spontaneously magnetized and become ferromagnetic, and hence would contribute non-zero net magnetic momentum to the whole star. We conclude that, for most cases in solid quark-cluster stars, the amount of net magnetic momentum, which is proportional to the amount of unbalanced spins ξ =（n＋- n-）/ne and depends on the number density of electrons ne =n＋ ＋ n-, could be significant with non-zero ξ. The net magnetic moments of electron system in solid quark-cluster stars could be large enough to induce the observed magnetic fields for pulsars with B ~ 10^11 to ~ 10^13 G.
