Chern-Simons (CS) modified gravity is an extension to general relativity (GR) in which the metric is coupled to a scalar field, resulting in modified Einstein field equations. In the dynamical theory, the scalar field is itself sourced by the Pontryagin density of the space-time. In this paper, the coupled system of equations for the metric and the scalar field is solved numerically for slowly rotating neutron stars described with realistic equations of state and for slowly rotating black holes. An analytic solution for a constant-density nonrelativistic object is also presented. It is shown that the black hole solution cannot be used to describe the exterior space-time of a star as was previously assumed. In addition, whereas previous analysis were limited to the small-coupling regime, this paper considers arbitrarily large coupling strengths. It is found that the CS modification leads to two effects on the gravitomagnetic sector of the metric: (i) Near the surface of a star or the horizon of a black hole, the magnitude of the gravitomagnetic potential is decreased and frame-dragging effects are reduced in comparison to GR. (ii) In the case of a star, the angular momentum J, as measured by distant observers, is enhanced in CS gravity as compared to standard GR. For a large coupling strength, the near-zone frame-dragging effects become significantly screened, whereas the far-zone enhancements saturate at a maximum value Delta J(max) similar to (M/R)J(GR). Using measurements of frame-dragging effects around the Earth by Gravity Probe B and the LAGEOS satellites, a weak but robust constraint is set to the characteristic CS length scale, xi(1/4) less than or similar to 10(8) km.
