As the parent of teenage boys who have to be dragged out of bed on school days, I had been looking forward to earlier sunrises once the winter solstice was past. But early January mornings seemed darker than ever while at the same time, the sky was clearly lighter around 5 p.m.

It turned out that what I suspected was actually true — by Jan. 2, there were 12 more minutes of sunlight in the afternoons, but 3 fewer minutes in the morning. It also turned out that the reasons for it were complicated, as I discovered in a series of phone and e-mail conversations with Jay M. Pasachoff, a professor of astronomy at Williams College, and a former student of his, Joseph Gangestad, who received his Ph.D. in orbital mechanics from Purdue.

They pointed me to the Equation of Time, a grandly named formula relating to the fact that not all days are 24 hours, if you track noon by the position of the Sun instead of on a clock.

We’ve all seen a readout of the Equation of Time, Dr. Pasachoff said. It’s that uneven figure 8 that can be found on globes in a deserted part of the Pacific, a shape known as an analemma.