solar cycle(calendrical)

When used in talking about calendars and not sunspots, the solar cycle is the number of years that must pass before all the days in a year will again fall on the same days of the week. The cycle differs for different calendars. It is the basis of perpetual calendars.

For the sake of simplicity, we will discuss the solar cycle for the Julian calendar, which has exactly 365 days in most years and a 366-day leap year every fourth year. The solar cycle for the Julian calendar is 28 years long.  If March 3 was a Friday on one year, it will be a Friday 28 years later, and every other day of the year will also fall on the same day of the week that it did 28 years ago.

Where does the 28 come from? Begin by ignoring leap years for the moment. There are 7 days in a week.  If 365 were evenly divisible by seven, every day would always fall on the same day of the week every year.  But 365 ÷ 7 = 52 (weeks) with 1 day left over.  That leftover day will push 1 January of the next year forward by one day of the week each year.  If New Year's is a Monday one year, next year it will be on a Tuesday, the year after that on a Wednesday, and so on.  In seven years, New Year's will be back at Monday, where it started. Every other date would also be on the same day of the week as it was seven years ago, and the solar cycle would be 7 years long.

However, every 4th year is a leap year of 366 days, and an extra day is inserted at the end of February.  This pushes the subsequent 1 January forward two weekdays instead of one, but before that it also pushes all the days from 1 March onwards forward by one weekday.  As a result, when New Year's gets back to Monday after 5 or 6 years, the days from March 1 onwards will not be back where they were.

Three regular years followed by one leap year is a cycle, like the cycle of the 7 weekdays. For all the days, before and after 1 March, to get back to the weekday they originally were, we must wait for both cycles to complete in the same year: 4, 8, 12, 16, 20, 24, 28, and 7, 14, 21, 28. The solar cycle for the Julian calendar is the product of these two cycles, 4 times 7 is 28. The solar cycle for other calendars can be found similiarly, by computing the product of the lengths of their cycles, removing common factors.