  In this paper we consider the unfolding of saddle-node \[ X=
\frac{1}{xU_a(x,y)}\Big(x(x^\mu-\varepsilon)\partial_x-V_a(x)y\partial_y\Big),
\] parametrized by $(\varepsilon,a)$ with $\varepsilon\approx 0$ and $a$ in an
open subset $A$ of $\mathbb R^{\alpha},$ and we study the Dulac time $\mathcal
T(s;\varepsilon,a)$ of one of its hyperbolic sectors. We prove (Theorem A) that
the derivative $\partial_s\mathcal T(s;\varepsilon,a)$ tends to $-\infty$ as
$(s,\varepsilon)\to (0^+,0)$ uniformly on compact subsets of $A.$ This result
is addressed to study the bifurcation of critical periods in the Loud's family
of quadratic centers. In this regard we show (Theorem B) that no bifurcation
occurs from certain semi-hyperbolic polycycles.
