  A polytropic quark star model is suggested in order to establish a general
framework in which theoretical quark star models could be tested by
observations. The key difference between polytropic quark stars and the
polytropic model studied previously for normal (i.e., non-quarkian) stars is
related to two issues: (i) a constant term representing the contribution of
vacuum energy may be added in the energy density and the pressure for a quark
star, but not for a normal star; (ii) the quark star models with non-vanishing
density at the stellar surface are not avoidable due to the strong interaction
between quarks. The first one implies that the vacuum inside a quark star is
different from that outside, while the second one is relevant to the effect of
color confinement. The polytropic equations of state are stiffer than that
derived in conventional realistic models (e.g., the bag model) for quark
matter, and pulsar-like stars calculated with a polytropic equation of state
could then have high maximum masses (> 2 M_sun). Quark stars can also be very
low massive, and be still gravitationally stable even if the polytropic index,
n, is greater than 3. All these would result in different mass-radius
relations, which could be tested by observations. In addition, substantial
strain energy would develop in a solid quark star during its accretion/spindown
phase, and could be high enough to take a star-quake. The energy released
during star-quakes could be as high as ~ 10^{47} ergs if the tangential
pressure is ~ 10^{-6} higher than the radial one.
