The farthest south the vertical rays of the sun ever reach the equator occurs around December 21st or 22nd, which is the Winter Solstice in the Northern Hemisphere.

This is because the Earth is tilted on its axis, and during the Winter Solstice, the Southern Hemisphere is tilted towards the sun, causing the sun's rays to hit the Tropic of Capricorn (the farthest point south where the sun's rays can be directly overhead) at a vertical angle.