Solstice, analemma, pagans and Christmas

According to this wiki "the winter solstice is the day with the shortest period of daylight and longest night of the year, when the Sun is at its lowest daily maximum elevation in the sky," typically Dec. 21. The analemma does not occur on the solstice: "As seen from northern middle latitudes, as the diagram shows, the earliest sunset occurs some time before the December solstice – typically a week or two before it – and the latest sunrise happens a week or two after the solstice. Thus, the darkest evening occurs in early to mid-December, but the mornings keep getting darker until about the New Year."

The pagans come into the picture as Christians appropriated the Roman holiday of Saturnalia.

"Thanks to the Roman Empire’s conquests in Britain and the rest of Europe from the second century B.C. to the fourth century A.D.—and their suppression of older seasonal rites practiced by the Celts and other groups—today’s Western cultures derive many of their traditional celebrations of midwinter from Saturnalia.The Christian holiday of Christmas, especially, owes many of its traditions to the ancient Roman festival, including the time of year Christmas is celebrated. [...] On the Julian calendar, which the Romans used at the time, the winter solstice fell on December 25."