The complexity of Scott sentences of scattered linear orders

Given a countable scattered linear order $L$ of Hausdorff rank $\alpha < \omega_1$weshowthatithasa$d$\text{-}$\Sigma_{2\alpha+1}$ Scott sentence. Ash calculated the back and forth relations for all countable well-orders. From this result we obtain that this upper bound is tight, i.e., for every $\alpha < \omega_1$ there is a linear order whose optimal Scott sentence has this complexity. We further show that for all countable $\alpha$ the class of Hausdorff rank $\alpha$ linear orders is ${\mathrm{\Sigma }}_{2\alpha +2}$ complete.