1. The Christians didn't borrow from a pagan festival. Rather, it was the other way around--the pagans imitated the Christians. Further, the ancient Roman cults didn't even have a winter solstice festival!

2. Here’s how it happened: (1) there was an ancient Jewish belief that the great prophets were to have an “integral age” (where you die on the same day as either your birth or your conception); (2) there arose a consensus that Christ was conceived on March 25 (the Feast of the Annunciation, when the angel appeared to Mary); (3) therefore, it was figured that Christ was born 9 months later on December 25.

Thus, December 25th as the date of the Christ’s birth appears to owe nothing whatsoever to pagan influences upon the practice of the Church during or after Constantine’s time. It is wholly unlikely to have been the actual date of Christ’s birth, but it arose entirely from the efforts of early Latin Christians to determine the historical date of Christ’s death.

And the pagan feast which the Emperor Aurelian instituted on that date in the year 274 was not only an effort to use the winter solstice to make a political statement, but also almost certainly an attempt to give a pagan significance to a date already of importance to Roman Christians.