  This paper deals with the period function of the reversible quadratic centers
\begin{equation*} X_{\np}=-y(1-x)\partial_x+(x+Dx^2+Fy^2)\partial_y,
\end{equation*} where $\np=(D,F)\in\R^2.$ Compactifying the vector field to
$\Sc^2$, the boundary of the period annulus has two connected components, the
center itself and a polycycle. We call them the inner and outer boundary of the
period annulus, respectively. We are interested in the bifurcation of critical
periodic orbits from the polycycle $\out_\np$ at the outer boundary. A critical
period is an isolated critical point of the period function. The criticality of
the period function at the outer boundary is the maximal number of critical
periodic orbits of $X_\np$ that tend to $\out_{\np_0}$ in the Hausdorff sense
as $\np\to\np_0.$ This notion is akin to the cyclicity in Hilbert's 16th
Problem. Our main result (Theorem A) shows that the criticality at the outer
boundary is at most 2 for all $\np=(D,F)\in\R^2$ outside the segments
$\{-1\}\times [0,1]$ and $\{0\}\times [0,2]$. With regard to the bifurcation
from the inner boundary, Chicone and Jacobs proved in their seminal paper on
the issue that the upper bound is 2 for all $\np\in\R^2.$ In this paper the
techniques are different because, while the period function extends
analytically to the center, it has no smooth extension to the polycycle. We
show that the period function has an asymptotic expansion near the polycycle
with the remainder being uniformly flat with respect to~$\np$ and where the
principal part is given in a monomial scale containing a deformation of the
logarithm. More precisely, Theorem~A follows by obtaining the asymptotic
expansion to fourth order and computing its coefficients, which are not
polynomial in~$\np$ but transcendental.
