Why tilt, not distance, makes northern summer

The puzzle

It is early July, and Earth is as far from the Sun as it ever gets. Yet this northern town is in the middle of summer. How can the sunlight up north be strongest now, when the tilt only leans Earth a little?

Zoom in on one band of sunlight landing straight on the town's ground. It makes a short, bright patch. If strength depends only on distance, then the same distance must give the same strength. Now tilt the ground, so the same band arrives at a slant, with the Sun no farther away. Make a guess.

Slant and strength

The band is just as wide, but the lit patch stretches long. The same light is spread thinner. Slide the Sun from overhead down toward the horizon, and watch the patch grow. At thirty degrees above the horizon, the patch is twice as long. Each bit of ground gets half the light, and the distance never changed. That is what matters: light per bit of ground, which is what warms it.

The tilt sets the angle

Back to the whole Earth in July. The axis leans toward the Sun, so at forty degrees north the noon Sun is high, about seventy-three degrees. The patch is short. Six months later the axis leans away. The noon Sun is only about twenty-seven degrees up, and the same band covers a patch about twice as long. The tilt is small, but it changes the angle a lot. The distance changes only a few percent. So northern summer comes from the angle, even with the Sun farthest away.