jmc

Since $9=3^2$, we can convert directly to base 3 by expanding each base 9 digit into two base 3 digits: $$\begin{array}{rl}{8}_9& =22_3\\ 1_9& =01_3\\ 3_9& =10_3\end{array}$$ Putting the base 3 digit pairs together, we get $813_9=\boxed{220110_3}$.