« Return to Thread: Find Solar Calendar Seasons spreadsheet posted

Re: Find Solar Calendar Seasons spreadsheet posted

by Irv Bromberg :: Rate this Message:

On 2009.03.11, at 12:24 , Palmen, KEV (Karl) wrote:

From: Irv Bromberg Sent: 11 March 2009 15:01
Although the mean year of any particular leap cycle may not exactly match the mean year of an equinox or solstice, each reasonably accurate leap cycle has a mean year that is a stable match to one or two points in the annual solar cycle, which we can call "calendar season(s)" for that leap cycle. In the present era, such calendar seasons are stable for calendar mean years that are as short as about 365 days 5 hours 47 minutes 52 seconds or about 365+43/178 days to as long as about 365 days 5 hours 49 minutes 31 seconds or 365+25/103 days, a range of only about 1 minute 39 seconds, and that stability endures for about 10 millennia.

Karl says: I dispute the lower end of this range 365+43/178 days , which I don’t believe has a stable season today (mean year about 365.241573).

Irv replies: Try it in my spreadsheet with middle year 2000 and step size 100 years. The curves all converge at aphelion, which means that at that date in the calendar year the leap cycle nicely approximates the solar longitude of aphelion (with such a small step size the advance of aphelion is also small). However, I believe that the basis for Karl's dispute is that the lines don't reverse with respect to each other. After converging they diverge beyond aphelion, in the same order that they were arranged before they converged. My quoting of this mean year was intended to give an outer limit. No cycle with a shorter mean year can have a stable calendar seasons in the present era. I was also trying to find the cycle that has its calendar season at aphelion. Cycles that have a slightly longer mean year have improved stability of the calendar season and develop the crossover reversals that Karl is looking for, but the calendar season will be at a solar longitude prior to aphelion. Cycles with a slightly shorter mean year converge incompletely at ecliptic longitudes that are after aphelion, without touching the baseline middle year line.

A leap cycle will not have any calendar seasons in an era in which its mean year is a few seconds shorter than the mean year at the ecliptic longitude of aphelion.

Karl says: I don’t believe that any leap cycle will have any calendar seasons in any era in which its mean year is shorter than the mean year at the ecliptic of aphelion

Irv replies: Karl has not said anything different from what I said. Perhaps he thinks that the leap cycle will never have a stable calendar season? That is not so, as tidal slowing of Earth's rotation rate progresses, we count fewer mean solar days per year, and eventually, provided that the leap cycle mean year was only slightly too short, a stable calendar season will emerge at aphelion and over the years split and migrate as described.

Karl continued: ...but may have a season of drift slow enough to appear like a calendar season if the SD method is used to find it.

Irv replies: The SD method identifies the solar longitudes of curve convergences, whereas line crossover analysis will identify more stable calendar seasons, especially if curve sequence reversal (rank) is a required criterion. Curve rank analysis is on the "To Do" list.

Irv wrote: The region near perihelion having the longest mean year spans about 45° of ecliptic longitude, ranging from about 15° before to about 30° after the ecliptic longitude of perihelion, so both calendar seasons disappear before ever reaching perihelion.

Karl continued: Actually I think both calendar seasons speed up much as they approach perihelion, so become unstable enough not be detected by the SD method even if the step period (exposure time) is reduced.

Irv replies: Nice analysis. In such an era they are not at all detectable by the line crossover method because the lines converge near perihelion without crossing each other, without crossing the baseline year, and without reversing their sequence. The SD method can at least detect the solar longitude of their closest convergence, but that is not a true calendar season as we define it. Nevertheless, even in such years on that day relative to the New Year Moment the leap cycle's approximation of the solar longitude is within a fraction of a degree of the corresponding point in the middle baseline year, so it is possible to continue using the cycle for calendrical purposes even for many centuries afterward. How much drift is acceptable for calendrical purposes? Less than 24 hours? The chart shows ±48 hours on the primary y-axis. With a small step size it is worth reducing the y-axis range accordingly. For example, with the 178-year cycle, 100-year step size and middle year 2000, try changing the primary y-axis to ±16 hours, with 2 as the major unit. The curves for years prior to 2000 are then plotted with thin lines, indicating that they don't cross the baseline year, to be expected because prior to the present era there is no point in the solar cycle that has such a short mean year. The curves for years after 2000 are all medium-thick, indicating that they do cross the baseline year. Thus I classify this as a stable calendar season starting in the present era at aphelion.

Karl continued: Also I expect the initial spilt to be faster than the final merger and so a stable season would start (become stable) further away from aphelion than it ends (ceases being stable) from the perihelion.

Irv replies: That is not what my spreadsheet shows. When the calendar season emerges at aphelion, it rapidly becomes stable before it has migrated much from aphelion. I haven't carried out migration rate analysis over the long term. It seems to me that to do so one would have to employ the same middle year, for example 2000, and calculate many more curves to the past and future, perhaps spaced at each century, finding all of the baseline year crossovers and from that the migration rates over the ages. The spreadsheet could be configured to calculate those extra columns of data without plotting them (at the expense of extra computing time of course), but to include them in the crossover analysis. Alternatively, the spreadsheet could always calculate each century from its minimum to maximum year range, but only plot the range that the user asked for. This would execute appreciably more slowly, but ought to be valuable for migration rate analyses.

Karl continued: This is because I’d expect a plot of the tropical year length beginning at a given time of year to resemble a sine wave, but have a sharper peak to the perihelion and a flatter dip at the aphelion.

Irv replies: The term "tropical year length" is ambiguous in this context. You can see examples of the mean equinoctial and solstitial year lengths on my "Lengths of the Seasons" web page at <http://www.sym454.org/seasons/>, based on SOLEX numerical integration.

Without tidal slowing of the Earth rotation rate (without subtracting Delta T), working in terms of atomic days instead of mean solar days, the pattern is quasi-sinusoidal, for example see <http://individual.utoronto.ca/kalendis/solar/Solar_Year_Lengths_30K.pdf>. The amplitude of the wave decreases with decreasing Earth orbital eccentricity.

With correction for tidal slowing of the Earth rotation rate (subtracting Delta T), working in terms of mean solar days, the pattern looks like a round-edged descending staircase, for example see <http://individual.utoronto.ca/kalendis/solar/Mean_Solar_Years_15K_L.pdf> or the longer-term <http://individual.utoronto.ca/kalendis/solar/Mean_Solar_Years_50K.pdf>. In this latter plot especially observe the correlation of the wave amplitude with the Earth orbital eccentricity, shown in lavender as labelled on the secondary y-axis at the right.

The positions of perihelion and aphelion with respect to the equinoxes and solstices are also shown on these cited charts.

A similar pattern could be obtained for any other desired point in the solar cycle, but it is much more convenient to carry out such analysis for the equinoxes and solstices, because SOLEX can automatically find them. It could be done by having SOLEX log the daily solar longitude, producing a very large file if integrating over tens of thousands of years, and then use non-linear interpolation to calculate the moment of the desired solar longitude in each year.

Anyhow, I'm not too sure about what Karl is expecting. If you look at the "Patterns" chart in my spreadsheet, it is essentially a straight line from the shortest to the longest mean year, whereas if Karl's expectation were correct then there should be deviations from linearity at the extremes. On the other hand, I generated the data for the "Patterns" chart in a way that Karl disputes as valid, so once we can all agree on what is the valid procedure to follow, I hope to redo it and see if the linear relationship still holds up. My plot includes middle years that in some cases are far away from the present era. It would be better to strictly limit the plot to calendar seasons that are stable in the present era.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/leap/>

<http://www.sym454.org/seasons/>

« Return to Thread: Find Solar Calendar Seasons spreadsheet posted