Why is the length of daylight in the northern hemisphere so short on December 21?
This has to do with the earth's orbital inclination.
The earth's rotational axis is tilted in relationship to the plane of its orbit around the sun. From the perspective of an astronaut looking at the earth, they would see that half of the planet is in shadow, and half is in sun, at any given time or position. If the earth had no axial tilt, then an observer standing on the planet, at any point on the planet, would experience exactly 12 hours of night and 12 hours of day, every day. However, let's say we have two observers standing on the planet; one in the Arctic Circle, and one in the Antarctic Circle. Remember that each one experiences the exact same 24 hour day; they have the same angular velocity.
The tilt of the Earth's axis complicates this model by changing the relationship of these observers to the portion of the earth that's in shadow or in light. The observer in the Arctic Circle is tilted away from the sun, so far that his entire circular path around the earth's pole is inside the shadow cast by the sun. The sun never rises for him, or anyone else within 23.5 degrees of the north pole. In contrast, the person in the Antarctic has had their entire circular path pushed into the sunlight, so that the sun is up for the whole day. Anyone in between the two would have a day that is, proportionally, more day-dominant or more night-dominant, depending upon which pole they were closest to. Thus, the reason why daylight is so short on December 21st, the solstice, is that this is the day when the Northern Hemisphere spends the greatest part of its path around the Earth inside the nighttime portion.