The Earth's Apsides & Long Term Climate Change

The proposed calendar change would help fix the dates of the equinoxes and solstices better but the perihelion and aphelion of the Earth's orbit will still move about within the year. The process involved is known as apsidal precession. I've computed the rate of motion of the equinoxes within the Julian calendar and the rate of apsidal precession for the Julian, Gregorian and "Tropical" calendars. The "P"s are the periods or "days per year" and the "n"s are the rates of the mean motions.* One can use the relative difference in rates to find the period of the cycle since dividing by a rate is equivalent to multiplying by the length of its year.

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This prcession will affect the motion of the Sun within the year and consequently the length of the seasons since the Earth moves faster in its orbit near perihelion and slower near aphelion. The lengths of summer and winter are an important factor in long term climate change.

*Edit: The subscript "a" refers to the anomalistic year which is the time between one perihelion and the next. It is slightly different than the time between successive Vernal Equinoxes.

Supplemental: With apsidal precession one could "fix" the date of the Vernal Equinox but the Autumnal Equinox and the solstices would vary slightly relative to it over time due to changes in the length of the seasons. All the dates would still vary over a four year period because of the leap day.

Historical Values for the Tropical Year Converted to Alternating Series

If one converts the historical values for the tropical year into alternating unit fractions in which the numbers are multiples of each other it appears that Tycho Brahe was the first to get 365 + 1/4 - 1/128 days.

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Brahe's value for the tropical year, 365 days, 5 hours, 48 minutes, 45 seconds, can be found in Astronomiae Instauratae Progymnasmatum, Pt I (1602).

A More Practical Decision Tree

Division of the year by 128 is more difficult than division by 100 or 400 but in practice one could keep track of the year that is the next multiple of 128. Initially the next multiple is 2176.


As time passes and the year reaches the multiple the leap year is skipped and a new next multiple is determined. This simplifies the decision process by avoiding division by 128.

Simplied Decision Tree For Switch Between 2048 and 2100

I waw looking for a way to simplify the flowchart for the Alternative Calendar and noticed that there was a window of opportunity to adopt a simple decision tree between 2048 and 2100 since 2048 is exactly divisible by 128. The decision tree for the Gregorian Calendar is,


The simplified decision tree for the Alternative Calendar is,


The 52 year window allows us to use the year instead of a difference in years in the decision tree. Both decision trees give the same results within this window.

Comparision of the Flowcharts for Determining Leap Years

I used MS Word 2003 to do create the flowcharts needed to determine when a year is a leap year for the Gregorian Calendar and the proposed Alternative Calendar. There is one step less for deciding in the Alternative Calendar which shows that this calendar is simpler. At the decision points in the flowcharts the "mod" used is the modulo operator.


The Gregorian Calendar was introduced in late 1582. It was noticed that the Vernal Equinox was shifting relative to the Julian Calendar so a new calendar was devised. This was just prior to the year 1600 which was the last year of the 399th quadrennium, the four year leap year cycle. It was a convenient time to initiate the centennial leap year cycle.


There were other factors that were taken into consideration is developing the Gregorian Calendar. For example the number of days in the 400 year cycle of the calendar is exactly divisible by 7 and the cycle of the weeks will then repeat itself. The Alternative Calendar ignores lunar cycles but it may be possible do something similar for the Lunar Calendar and develop a simple set of rules for it too.

Edit: Corrected the errors involving the year in the Gregorian Calendar Flowchart

Supplemental: The procedure for the Alternative Calendar can be simplified by working in the new system and noting that 2048 is exactly divided by 128. See next blog.

A Simple Set of Rules for Leap Years

The Wikipedia article on the tropical year gives a more accurate value for the mean number of days in a year than was previously used. It would also be more convenient to make the period of a correction a multiple of the previous period. This was a fluke for the alternating series composed of unit fractions.


As one can see this works quite well for the mean length of the year. A limit on the usefulness of any calendar and its set of rules results from the changing length of the day which decreases by about 1.5 msec per century. Using the first two fractions would probably be the best correction to make at this time. In making changes we also need to consider the convenience of decision points and the ease of transition from one set of rules to another. We have a window of opportunity to make such a change before the year 2100. The rules for the Gregorian calendar are good for approximately 3200 years and the change isn't actually necessary. It would just be a matter of convenience. Should we give the World an opportunity to decide or just let it ride?

Efficient Rules & A Proposed Calendar Reform

Too much dissent is likely to be harmful to a nation and can disrupt the government's need to make decisions. But we also need to consider the quality of the decisions made. Let's assume that a given rule has a given lifetime. Change within that lifetime is unnecessary and arbitrary change may be considered a nuisance. So we might approach the problem from the perspective of least action.

Let us look at calandar reform as an example. The Julian Calendar adopted 365 + 1/4 days for the year. It was found that this led to a deviation from the natural year or the mean time between equinoxes. The Gregorian Calendar corrected this and adopted a mean year of 365 + 1/4 - 1/100 + 1/400 days. But it turns out that this system does not result in the most efficient set of rules.

The value that The Explanatory Supplement to the Astronomical Almanac gives for the mean number of days per year is 365.2421897. There are a number of ways of representing a fraction one of which is continued fractions. The number of terms in the continued fraction can be terminated after a given number of terms and can be converted to a mixed fraction. This simplifies the design of gear trains which can reproduce a given ratio. A simplier system is that of a series of alternating unit fractions. Each unit fraction tells us how frequently we have to break the set of rules defined by the subset of unit fractions. The sequence of alternating unit fractions that corresponds to the number of days in a year is 365 + 1/4 - 1/128 + 1/454545...


Evaluating the expressions for the various number of days in a year shows that even two terms of the alternating unit fraction provides a better approximation for the actual number of days in a year than that does the Gregorian Calendar. So we may ask if all the hassle of Y2K was really necessary. One might conclude that the Catholic Church, the scientific community and government are all "fallible" to some extent.

It might be wise to consider alternatives to this set of rules but if the world wanted to this calendar reform could be adopted by the United Nations and approved by individual nations. A nominal starting year would be the year 2001* since the new proposal and the Gregorian Calendar agree on leap years until the year 2100.

*Edit: You would need to count the number of years from the year 2000 and base the decision for the leap day on whether the number of years is zero modulo 4, 128 (, etc.)