Approximate solutions for general Riemann-Cartan-typeR+R2theories of gravitation

In the case of a weak, spherically symmetric and static field, approximate solutions for the nine-parameter $R+R^2$ gravitational theories with torsion are obtained. Corresponding to some solutions with torsion, metrics ($g_{\mu \nu }$) differ from the Schwarzschild metric to first order in $M$. One can define a distance-dependent gravitational "constant" $G\left(r\right)\mathrm{}$. For example, if $\frac{c\lambda }{(c-\lambda )(p-r+2s)\mathrm{}}>0\mathrm{}$ and $-\frac{a\lambda }{(a+\lambda )(2p-2r+s)\mathrm{}}<0\mathrm{}$, a distance-dependent gravitational constant $G\left(r\right)=G_0[1+\alpha (1+\mu r\left)e^{-\mu r}\right]$ is obtained.