The sectional category of spherical fibrations

We give homological conditions which determine sectional category, secat, for rational spherical fibrations. In the odd dimensional case the secat is the least power of the Euler class which is trivial. In the even dimensional case secat is one when a certain homology class in twice the dimension of the sphere is $-1$ times a square. Otherwise secat is two. We apply our results to construct a fibration $p$ such that $\mathrm{secat}(p)=2$ and genus $(p)=\infty$. We also observe that secat, unlike cat, can decrease in a field extension of $\mathbb{Q}$.