  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.
