Summer Insolation Forcing

For more information on Milankovitch Cycles and the role that changes in the Earth's orbital elements play in global warming see Huybers' 2006 paper on summer insolation forcing.

credits: Peter Huybers, NOAA National Climate Data Center

Approximate Formula For #Days From V Equinox to A Equinox, Take 2

I was able to compute an expression for the remaining constant in B(e) by expanding the integral for the difference in time as a linear function of the eccentricity near e = 0.

The constant is the derivative of the integral w.r.t. e at e = 0.


The formula gives a good approximation for the time it takes to go from the Vernal Equinox to the Autumnal Equinox if the orbital period, T₀ (tropical year), the eccentricity, e, and the angle, θ, are known. There are higher order terms but the error is about 30 minutes or less.

Variations in the Period of Annual Heating

Below is a plot of the curves for the formula for the number of days between the equinoxes in the previous blog with the eccentricity of the Earth's orbit, e, changing by steps of 0.01. The large red dot marks where the Earth is currently.


Note that the number of days between the equinoxes can vary by as much as 25 days. The image below is from the Wikipedia article on Milankovitch cycles.


The first curve (blue) shows the changes in the Inclination of the Ecliptic, ε. The second curve (green) indicates the much slower changes in the Earth's eccentricity, e. And, the third curve (purple) gives the changes in the angle of perihelion.

Local heating depends on the angular distance of the Sun from the local zenith and so is a function of the Sun's apparent inclination and the number of hours of daylight. At any given time the Earth may not be exactly in equilibrium with the current heating conditions. Looking at the "number of days of heating" and the changes that are taking place might give a better picture of what is happening than one might get from the integrals in the Wikipedia article on insolation.

Supplemental (Aug 10): The calculation didn't take into consideration changes in the semi-major axis, a, which would affect the mean motion and the length of the year. In perturbation theory the semi-major axis is considered an adiabatic invariant. This would justify the first coefficient in the formula being a constant rather than a function of the eccentricity, e.

Empirical Formula for #Days Between V. Equinox & A. Equinox

I used different values for the eccentricity of the Earth's orbit, e, to get a set of values for the coefficients of the formula found last time. A turned out to be a constant equal to half the number of days in the year and B was a simple function of e. θ is the angle of Earth's perihelion from the direction of the Winter Solstice in the Earth's orbital plane or 90° behind the Vernal Equinox.


I labeled the function for the number of days from the Vernal Equinox to the Autumnal Equinox VEq2AEq. The empirical error bound is a simple cubic function within the range of e for the formula, 0 < e < 0.07.

Supplemental (Aug 10): The polynomial fit for B(e) had some near zero values so I tried doing a least squares fit for just odd powers of e and got a better fit.
The rms error was 2·10^-10. The function appears to be,
   B(e) = 232.520·(e - e³/3! - 3e⁵/5! + ···).

First Order Approximation For #Days of Summer

To first order the number of days in Summer function for apsidal precession is a constant plus a cosine function. ΔΔt, the error for the approximation, has a magnitude less than 0.0003 days.


As before θ is the angle between the Earth's perihelion and the Winter Solstice. Apsidal precession has to be considered as a contribution to global warming. If there is net annual heating due to apsidal precession then this contribution needs to be deducted before the effect of greenhouse gases can be determined. Only the change in the angle of angle of perihelion was taken into consideration and the other orbital elements were assumed to remain constant.

Conclusion: 30° Months Probably Not Practical

Defining a month as the time that it takes for the Earth to move 30° in its orbit is likely not practical since it would require frequent changes to keep the months accurate. The dates of apsides, equinoxes and solstices are something better to relocate to an almanac. The same for the number of days of net heating which I have referred to as "summer". It's probably best not to overcomplicate the calendar. One can just as easily draw attention to the perihelion by making it an "Earth day" holiday.

So, what to do about the months? The 7/12ths trick might be useful. Simplifying the day-of-year function would make it easier to compute the number of days between two dates such as the time between one equinox and the next.

When making changes that affect a lot of people it's probably best to adopt a minimalist approach.

The Equinoxes divide the year into two parts. When the Sun is above the Equator there is relatively more warming during this "half" of the year than when it is below it. There doesn't seem to be any collective terms for Spring/Summer and Fall/Winter. But the Earth's heat balance is positive for the former since the temperature rises and negative for the later since it decreases as a study of the Earth's insolation and climate models show. What is often overlooked or downplayed is that there are indeed long term changes over time.

Adjusting the Months to Fit the Earth's Orbidal Motion

If we wanted the Tropical Calendar to be a "Global Warming Calendar" we might adjust the number of days in each month to better fit the orbital motion of the Earth. We could make the length of each month correspond to a change of 30° relative to the Vernal Equinox. We would want the year to start in mid winter so the year would have to begin at the Winter Solstice or perhaps the next day.


Each of the months above fit this pattern with a Solstice or Equinox separating a block of three months. We could add the leap day at the end of the year where it was at one time or perhaps add it just before the Vernal Equinox to mark its significance as the date which we use to synchronize the calendar with the tropical year.

As the date of perihelion moves about in the year during the apsidal precession cycle periodic "calendar reforms" would be necessary to correct the lengths of the months to maintain the pattern of 30° monthly steps. It might be practical to do this every 1/12th of the 20,934 tropical year apsidal cycle which is about 1744 tropical years*. Leaving the calendar design somewhat open ended would allow changes to be made as our understanding of these changes increases and keep the calendar from becoming something rigidly preordained.

Supplemental: Basing the lengths of the months on the Earth's orbital motion was intended to focus attention of the Earth's heat balance and changes over time. There is a seasonal lag so temperatures do not exactly match the "seasons" but the highs and lows are delayed somewhat.

*Supplemental: Perihelion moves about 8.8° in 512 tropical years and 2.2° in 128 years. 512 years might be a little long to keep the system accurate and there is the problem of keeping people's interest in updating the calendar over that long a span of time. 128 years is more convenient because it could be done when the leap year is skipped but rearranging the calendar that often is a problem for historians.

Apsical Precession & the Length of Summer

Apsidal precession causes small changes in the length of Summer as the images below show. The plot gives the number of days of from the Vernal Equinox to the Autumnal Equinox as a function of the angle of perhelion from the Winter Solstice. The result is a peak-to-peak variation of about 8 days.


This is only a 2% change but could have a greater affect at higher latitudes where thermal cycling is the greatest.


If the perihelion is at midsummer one might expect a slightly greater intensity of solar radiation since the Earth would be closer to the Sun at this time of year. Winters in the Northern Hemisphere may on average be slightly milder at this time than at other times in the 21,000 year apsidal precession cycle. The apsidal precession is barely noticeable since it is masked by the shifts caused by a leap day every four years. It takes on average about 57.5 years for the date of perihelion to move one day backwards in the year so it may be just noticeable within the span of a lifetime.

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.

(click to enlarge)
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.

Click to enlarge
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.)