 Solution

Limb darkening makes the central stellar surface brighter than the disk mean. A small planet at projected position $\mathbf{x}$ blocks fractional flux approximately $\delta (\mathbf{x})=(R_p/R_{\star }{)}^{2}I(\mathbf{x})/⟨I⟩$. A central transit is therefore deepest near mid-transit and rounded rather than flat-bottomed; a near-limb chord can be shallower. The geometrical contact times do not change for fixed radii and orbit, although a threshold-based measured duration can change because ingress and egress are fainter.