  We introduce a parameterized high-density equation of state (EOS) in order to
systematize the study of constraints placed by astrophysical observations on
the nature of neutron-star matter. To obtain useful constraints, the number of
parameters should be smaller than the number of neutron-star properties that
have been measured or will have been measured in the next several years. And
the set must be large enough to accurately approximate the large set of
candidate EOSs. We find that a parameterized EOS based on piecewise polytropes
with 3 free parameters matches to about 4% rms error an extensive set of
candidate EOSs at densities below the central density of 1.4 solar mass stars.
Adding observations of more massive stars constrains the higher density part of
the EOS and requires an additional parameter. We obtain constraints on the
allowed parameter space set by causality and by present and near-future
astronomical observations. In particular, we emphasize potentially stringent
constraints on the EOS parameter space associated with two measured properties
of a single star; and we find that a measurement of the moment of inertia of
PSR J0737-3039A can strongly constrain the maximum neutron-star mass. We also
present in an appendix a more efficient algorithm than has previously been used
for finding points of marginal stability and the maximum angular velocity of
stable stars.
