PPN Parameter $\gamma$ and Solar System Constraints of Massive Brans-Dicke Theories

Previous solar system constraints of the Brans-Dicke (BD) parameter $\omega$ have either ignored the effects of the scalar field potential (mass terms) or assumed a highly massive scalar field. Here, we interpolate between the above two assumptions and derive the solar system constraints on the BD parameter $\omega$ for {\it any} field mass. We show that for $\omega =O(1)$ the solar system constraints relax for a field mass $m \gsim 20 \times m_{AU}= 20\times 10^{-27}GeV$.