To investigate the missing compact star of Supernova 1987A, we analyzed both the cooling and the heating processes of a possible compact star based on the upper limit of observational X-ray luminosity. From the cooling process we found that a solid quark-cluster star, which has a stiffer equation of state than that of conventional liquid quark star, has a heat capacity much smaller than a neutron star. It can cool down quickly, which can naturally explain the non-detection of a point source (neutron star or quark star) in X-ray band. On the other hand, we consider the heating process from magnetospheric activity and possible accretion, and obtain some constraints to the parameters of a possible pulsar. We conclude that a solid quark-cluster star can be fine with the observational limit in a large and acceptable parameter space. A pulsar with a short period and a strong magnetic field (or with a long period and a weak field) would has luminosity higher than the luminosity limit if the optical depth is not large enough to hide the compact star. The constraints of the pulsar parameters can be tested if the central compact object in 1987A is discovered by advanced facilities in the future.
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