It seems that the difference between the solstice and the Natalis Solis Invicti has not to do with the calendar reformations, but as the emperor Julianus said:
"They fixed that feast not just in the exact day of the solar conversion, but in the one in which the return of the sun, from south to north, does appear before the eyes of everybody" ( from http://www.centrostudilaruna.it/saturnalia.html )
The same thing happens with the crescent of the new moon that becoms visible only 3 day about after the astronomical phenomenon.
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From: Tony Ciapo
Well, Julius Caesar must have had the same lapse, when he established December 25 as the date of the winter solstice in 46 BCE.
Unfortunately, the date would have gradually drifted due to the inaccuracy in the Julian Calendar. When Pope Gregory reformed the calendar, he did not start at 46 BCE, but a few hundred years later. Thus the 4 day difference now between Christmas and the winter solstice .
From: Tony Ciapo
Well, Julius Caesar must have had the same lapse, when he established December 25 as the date of the winter solstice in 46 BCE.
Unfortunately, the date would have gradually drifted due to the inaccuracy in the Julian Calendar. When Pope Gregory reformed the calendar, he did not start at 46 BCE, but a few hundred years later. Thus the 4 day difference now between Christmas and the winter solstice .