  Precision tests of gravity can be used to constrain the properties of
hypothetical very light scalar fields, but these tests depend crucially on how
macroscopic astrophysical objects couple to the new scalar field. We develop
quasi-analytic methods for solving the equations of stellar structure using
scalar-tensor gravity, with the goal of seeing how stellar properties depend on
assumptions made about the scalar coupling at a microscopic level. We
illustrate these methods by applying them to Brans-Dicke scalars, and their
generalization in which the scalar-matter coupling is a weak function of the
scalar field. The four observable parameters that characterize the fields
external to a spherically symmetric star (the stellar radius, R, mass, M,
scalar `charge', Q, and the scalar's asymptotic value, phi_infty) are subject
to two relations because of the matching to the interior solution, generalizing
the usual mass-radius, M(R), relation of General Relativity. We identify how
these relations depend on the microscopic scalar couplings, agreeing with
earlier workers when comparisons are possible. Explicit analytical solutions
are obtained for the instructive toy model of constant-density stars, whose
properties we compare to more realistic equations of state for neutron star
models.
