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Find Solar Calendar Seasons spreadsheet posted
Dear Calendar People:
I have today posted my Excel spreadsheet with Visual Basic macro for automatically finding solar calendar seasons.
It is publicly accessible, but not yet linked into any web page at my web site. The URL is:
File size = 660 KB.
Be sure to enable execution of macros when you launch it.
(Macros are not supported by Excel 2008 for Macintosh. If you want to see this run on a Mac, either run the Windows version of Excel if you have the facility to do that on your Mac, or use Excel 2004 for Macintosh, which can execute the macro, albeit rather slowly.)
The "Setup" page lets you enter the leap days per cycle, years per cycle, step size (in years between plotted curves), and the middle year number (the one that is subtracted from all the rest). The user can monkey with the Delta T multiplier if desired, or just leave that = 1. The bottom half of the "Setup" page offers a continuous fraction calculator to help find useful leap cycles. After entering the desired cycle info, click on the "Update Chart" button and in a few seconds the report will be ready on the "Chart" page.
The chart shows 10 calendar drift curves and a horizontal baseline for the middle year.
The secondary logarithmic y-axis on the right side shows the standard deviation of the plotted curves at each elapsed day count, and the two minima of that curve are marked as the solar calendar seasons, and labelled at the bottom left of the chart legend.
There are also color-coded vertical lines on the chart that indicate the positions of the equinoxes, solstices, perihelion, and aphelion in the middle year.
The user can click on the "Shift to Past" or "Shift to Future" buttons to cause a recalculation and update.
A really neat feature that I just implemented today is the "Shift to Selected Curve" button: just click on any one of the plotted drift curves to select it (little square handles will appear along the curve) and then click on this button to cause it to recalculate with that year as the middle year. This provides a very quick way of getting to any desired era.
The "Cycles" page shows a few cycles that I have been experimenting with, and the "Pattern" page shows a plot of their calendar season solar longitude (as ±180°) vs. calendar fractional mean year, with both linear and 4th-order polynomial regressions shown. Longer or shorter mean years don't yield stable calendar seasons for the present era, meaning that there is no point in the solar cycle that has a longer or shorter mean year.
The baseline middle year is thick black. Curves that cross it are medium thickness, in color. Curves that miss it (don't cross it) are plotted as thin lines. Karl would probably prefer that only lines that cross the middle year AND reverse their respective ranks should be medium thickness, but so far I haven't figured out how to do that, nor am I convinced it is necessary.
Karl is still objecting to my SD method for finding the calendar seasons so I have also implemented a "Cross" sheet which contains a list of every curve crossover point, whether it was a crossing of the middle year (Drift=0) or a crossing at some non-zero drift value. Crossovers are detected by the difference in drift estimates changing sign. This list is sorted by elapsed day counts. I used simple linear interpolation to find the crossover moments:
x-coordinate = elapsed day count y-coordinate = hours of drift
Let (Ax, Ay) be the point prior to the crossover of a lower year. Let (Bx, By) be the point on that curve after the crossover. Between them is line AB, whose slope and intercept are calculated.
Let (Cx, Cy) be the point prior to the crossover of a higher year. Let (Dx, Dy) be the point on that curve after the crossover. Between them is line CD, whose slope and intercept are calculated.
The intersection of AB with CD is calculated, its coordinates are the fractional elapsed day count and drift of the crossover point. The fractional elapsed day is also expressed as the month day time. The solar longitude of the lower year at that point is also calculated and expressed normally (0-<360°) and also as ±180°. This solar longitude should be the same in the higher year, that is why their curves crossed at that point. The crossovers list also shows the lower and higher year numbers whose curves crossed. The information in the crossovers list could be used to calculate the advance of the calendar season as years pass, but that is not yet done. The user can auto-filter the crossovers list as desired.
Another way to detect crossovers would be to rank the drift of each curve, and look for adjacent curves switching ranks, and also look for pattern reversals. The Excel RANK() worksheet function could be used, or a loop could inspect all drift values in each row and determine ranks by counting. This latter approach lends itself better to making a rank string, for example where "A" is the first rank, "B" is the second, etc. Crossovers could then easily be detected when the string changes, and reversals when the string contains the reverse of what it contained earlier -- presumably each reversal pair would need to be detected and reported.
I invite everybody to try using this spreadsheet and hope you find it useful and enjoyable. Suggestions and comments will be most appreciated! -- Irv Bromberg, Toronto, Canada
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Re: Find Solar Calendar Seasons spreadsheet posted
On 2009.03.08, at 23:10 , Irv Bromberg wrote: I have today posted my Excel spreadsheet with Visual Basic macro for automatically finding solar calendar seasons.
It is publicly accessible, but not yet linked into any web page at my web site. The URL is:
File size = 660 KB.
Be sure to enable execution of macros when you launch it.
Irv adds, red-faced: Ooops, inadvertently I had left some of the user-entry fields on the "Setup" page in a "protected" state, which the user can't change.
Either use the "Unprotect Sheet" command to let you make changes (there is no password), or download the replacement posted today (down to 588 KB for some reason), which has those fields unprotected. -- Irv Bromberg, Toronto, Canada
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Re: Find Solar Calendar Seasons spreadsheet posted
Dear Calendar People:
Includes discussion and simple arithmetic, links to spreadsheet and example PDF.
Added more cycles to the spreadsheet, better filling in the pattern chart (stable solar longitude vs. mean year fraction).
-- Irv Bromberg, Toronto, Canada
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Re: Find Solar Calendar Seasons spreadsheet posted
On 2009.03.10, at 01:47 , Irv Bromberg wrote:
Dear Calendar People:
I posted a new version last night, now improving the behavior at the remote past and future limits, extended both limits by a millennium. Then I was able to better characterize the evolution of calendar seasons: In any given era the longest cycle mean year having a stable
calendar season will match the mean year (in terms of mean solar days) at the
ecliptic longitude of the Earth orbital aphelion, and the shortest cycle mean
year having a stable calendar season will match the mean year at the ecliptic
longitude of the Earth orbital perihelion. The advance of perihelion (and
aphelion) together with the tidal slowing of the Earth rotation rate cause calendar
seasons to evolve and migrate as the millennia pass. 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. With tidal slowing of the Earth rotation rate, however,
eventually the mean year at aphelion will equal the cycle mean year, so a
calendar season will appear at the ecliptic longitude of aphelion. With
progressive tidal rotation slowing that calendar season will split into a more
stable season that will migrate ahead of aphelion (to earlier solar longitudes)
as well as a less stable season that will migrate after aphelion (to later
solar longitudes). As tidal rotation slowing continues, eventually the average
length of the solar cycle will approximately equal the cycle mean year, and
then both calendar seasons will be optimally stable and perihelion and aphelion
will be situated approximately midway between them, with perihelion having the
less stable calendar season behind it (prior solar longitude) and the more
stable calendar season ahead of it (later solar longitude). Further tidal
rotation slowing will cause the calendar seasons to converge towards
perihelion, disappearing in later years when the mean year at the ecliptic
longitude of perihelion becomes shorter than the cycle mean year.
-- Irv Bromberg, Toronto, Canada
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Re: Find Solar Calendar Seasons spreadsheet posted
On 2009.03.11, at 09:45 , Irv Bromberg wrote: I posted a new version last night, now improving the behavior at the remote past and future limits, extended both limits by a millennium. Then I was able to better characterize the evolution of calendar seasons:
<snip>
Oops, I really mucked up that previous message, disregard it please, here is the corrected version:
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.
In any given era the longest cycle mean year having a stable calendar season will match the mean year (in terms of mean solar days) at the ecliptic longitude of the Earth orbital perihelion, and the shortest cycle mean year having a stable calendar season will match the mean year at the ecliptic longitude of the Earth orbital aphelion. The advance of perihelion (and aphelion, always 180° away) together with the tidal slowing of the Earth rotation rate cause calendar seasons to evolve and migrate as the millennia pass.
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. With tidal slowing of the Earth rotation rate, however, eventually the mean year at aphelion will equal the cycle mean year, so a calendar season will appear at the ecliptic longitude of aphelion. With progressive tidal rotation slowing that calendar season will split into a more stable season that will migrate ahead of aphelion (to earlier solar longitudes) as well as a less stable season that will migrate after aphelion (to later solar longitudes). As tidal rotation slowing continues, eventually the average length of the solar cycle will approximately equal the cycle mean year, and then both calendar seasons will be optimally stable and perihelion and aphelion will be situated approximately midway between them, with perihelion having the less stable calendar season behind it (prior solar longitude) and the more stable calendar season ahead of it (later solar longitude). Further tidal rotation slowing will cause the calendar seasons to converge towards perihelion, disappearing in later years when the mean year near the ecliptic longitude region of perihelion becomes longer than the cycle mean year. 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.
With that description, the reader might think that aphelion should eventually "catch up" to perihelion, but that can never happen because they both advance in unison around Sun, always 180° apart. As aphelion passes through each point of the solar cycle, however, that solar longitude has the shortest mean year, and the opposite solar longitude 180° away at perihelion has the longest mean year.
(I also have just posted another new version of the spreadsheet, extending the continued fraction convergent list from 10-->12, making it easier to get exact fractions. For experimenting with characterizing features such as those outlined above, it is helpful to use a small step size such as 100 years.)
-- Irv Bromberg, Toronto, Canada
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Re: Find Solar Calendar Seasons spreadsheet posted
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Dear Irv and Calendar People
From: East Carolina University Calendar
discussion List [mailto:CALNDR-L@...] On Behalf Of Irv
Bromberg
Sent: 11 March 2009 15:01
To: CALNDR-L@...
Subject: Re: Find Solar Calendar Seasons spreadsheet posted
On 2009.03.11, at 09:45 , Irv Bromberg wrote:
I posted a new version last night, now improving the
behavior at the remote past and future limits, extended both limits by a
millennium. Then I was able to better characterize the evolution of
calendar seasons:
Oops, I really mucked up that previous message, disregard it
please, here is the corrected version:
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.
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).
In any given era the longest cycle mean year having a stable
calendar season will match the mean year (in terms of mean solar days) at the
ecliptic longitude of the Earth orbital perihelion, and the shortest cycle mean
year having a stable calendar season will match the mean year at the ecliptic
longitude of the Earth orbital aphelion. The advance of perihelion (and
aphelion, always 180° away) together with the tidal slowing of the Earth rotation
rate cause calendar seasons to evolve and migrate as the millennia pass.
They’ll migrate from aphelion to perihelion in opposite
directions relative to them. The migration is fastest at start and end and
slowest in the middle of the lifetimes of the calendar season. One season is
stable because its migration towards the perihelion is in the opposite
direction to the precession of the perihelion. I see that Irv explains some of
this in more detail later on.
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.
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, but may have a season of drift slow
enough to appear like a calendar season if the SD method is used to find it.
There is no reversal of drift within the year nor does the graph of the present
year have any crossings with any nearby years.
With tidal slowing of the Earth rotation rate, however,
eventually the mean year at aphelion will equal the cycle mean year, so a
calendar season will appear at the ecliptic longitude of aphelion. With
progressive tidal rotation slowing that calendar season will split into a more
stable season that will migrate ahead of aphelion (to earlier solar longitudes)
as well as a less stable season that will migrate after aphelion (to later
solar longitudes). As tidal rotation slowing continues, eventually the average
length of the solar cycle will approximately equal the cycle mean year, and
then both calendar seasons will be optimally stable
I’d say are migrating at slowest rate towards perihelion (We
have seen that the stable season is not most stable at this time).
and perihelion and aphelion will be situated approximately
midway between them, with perihelion having the less stable calendar season
behind it (prior solar longitude) and the more stable calendar season ahead of
it (later solar longitude). Further tidal rotation slowing will cause the
calendar seasons to converge towards perihelion, disappearing in later years
when the mean year near the ecliptic longitude region of perihelion becomes
longer than the cycle mean year. 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.
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.
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. 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.
With that description, the reader might think that aphelion
should eventually "catch up" to perihelion, but that can never happen
because they both advance in unison around Sun, always 180° apart. As aphelion
passes through each point of the solar cycle, however, that solar longitude has
the shortest mean year, and the opposite solar longitude 180° away at
perihelion has the longest mean year.
Karl
10(06(15
Scanned by iCritical.
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Re: Find Solar Calendar Seasons spreadsheet posted
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.
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
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Re: Find Solar Calendar Seasons spreadsheet posted
On 2009.03.11, at 14:03 , Irv Bromberg wrote: 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.
Irv adds: OK, I've added an "Automatically Scale" checkbox option to the "Setup" page. It uses the list of axis scale settings that appear on the "Scale" worksheet. If the checkbox is unchecked then the macro leaves the primary y-axis scale alone.
This revealed an anomaly that I hadn't noticed before -- there are marked lunar oscillations when the step size is small and the y-axis scale is automatically reduced accordingly! These oscillations cause the calendar season crossovers to appear rather erratic.
I tried eliminating the lunar oscillations by removing nutation from the solar longitude calculation -- no effect. One would need to remove all lunar terms, but it would be better to have an algorithm specifically designed for the Earth-Moon barycenter. SOLEX can do that in the "planets only" mode, but that doesn't help me much, except possibly to develop such an algorithm. The mean solar longitude is needed instead of the actual solar longitude.
It turns out that there is a satisfactory workaround -- use a step size that is a reasonable multiple of the mean synodic month! Even a number like 19 works well! So do multiples of 19 and so on -- I have provided a list on the "Setup" page. -- Irv Bromberg, Toronto, Canada
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Re: Find Solar Calendar Seasons spreadsheet posted
On 2009.03.12, at 08:23 , Irv Bromberg wrote: Irv adds: OK, I've added an "Automatically Scale" checkbox option to the "Setup" page. It uses the list of axis scale settings that appear on the "Scale" worksheet.
Irv adds further: This way of handling the y-axis scale is crude, but effective, yet certainly far from general purpose. Can anybody point me in the direction of an algorithm or set of rules for general purpose selection of "nice" axis scales, where the axis span (or 1/2 span in this case) is divisible by the major interval, which in turn is divisible by a suitably small minor interval?
Anyhow, try this with my spreadsheet and automatic scaling switched on: For any leap cycle, choose a 20-year step size. You'll see wild monthly oscillations. (Some curves may be nearly smooth if they happen to be close to a multiple of the lunar cycle away from the middle year.) Change the step size to 19 years and update the chart, then the curves will all be quite smooth.
-- Irv Bromberg, Toronto, Canada
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Re: Find Solar Calendar Seasons spreadsheet posted
On 2009.03.12, at 09:32 , Irv Bromberg wrote: Anyhow, try this with my spreadsheet and automatic scaling switched on: For any leap cycle, choose a 20-year step size. You'll see wild monthly oscillations. (Some curves may be nearly smooth if they happen to be close to a multiple of the lunar cycle away from the middle year.) Change the step size to 19 years and update the chart, then the curves will all be quite smooth.
Irv adds further: I just think this is really cool. Try other multiples of 19 in comparison with nearby rounded numbers, for example 38 with 40, 57 with 60, 95 with 100. Those that are multiples of 19 yield smooth lines, the others yield wavy lines.
-- Irv Bromberg, Toronto, Canada
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Re: Find Solar Calendar Seasons spreadsheet posted
On 2009.03.12, at 14:33 , Irv Bromberg wrote: Irv adds further: I just think this is really cool. Try other multiples of 19 in comparison with nearby rounded numbers, for example 38 with 40, 57 with 60, 95 with 100. Those that are multiples of 19 yield smooth lines, the others yield wavy lines.
I posted a new version of the "Find Calendar Seasons" spreadsheet that now calculates the mean synodic month in the era of the specified middle year, then calculates the continued fraction of the ratio of that synodic month to the leap cycle mean year, highlighting the optimal choices (>=19 and <500) in bold red-face text. For the present era the choices are 19 and 296 but in the remote past or distant future there may be useful alternatives. I also added more auto-scale settings.
-- Irv Bromberg, Toronto, Canada
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Re: Find Solar Calendar Seasons spreadsheet posted
On 2009.03.12, at 18:18 , Irv Bromberg wrote: I posted a new version of the "Find Calendar Seasons" spreadsheet that now calculates the mean synodic month in the era of the specified middle year, then calculates the continued fraction of the ratio of that synodic month to the leap cycle mean year, highlighting the optimal choices (>=19 and <500) in bold red-face text. For the present era the choices are 19 and 296 but in the remote past or distant future there may be useful alternatives.
I realize now that the 296 showed up instead of the expected 334 or 353 years because at the time that I wrote that I was experimenting with the 327-year cycle, which has one of the shortest useful mean years in the present era, with a calendar season near the north solstice. The continued fraction calculator uses the actual mean year of the specified leap cycle, so if its mean year is close to the northward equinoctial mean year then it will list 19 and then 353, or if its mean year is intermediate then it will list 19 and 334.
The longer step size is preferable. Although a 19-year step size yields smooth curves, they are not uniformly spaced as expected, because the 19-year cycle is enough of a mismatch to the mean synodic month that phase differences can sequentially add or sequentially subtract between a plotted curve and the baseline middle year, skewing that entire plotted curve to a higher or lower drift position.
If it is possible to find or develop a function that will calculate the solar longitude of the Earth-Moon barycenter over a satisfactory span of millennia then any arbitrary step size ought to work alright.
-- Irv Bromberg, Toronto, Canada
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Re: Find Solar Calendar Seasons spreadsheet posted
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Dear Irv and Calendar People
I’ll need to try out Irv’s latest spreadsheet before
commenting in detail.
Irv is correct in stating that the reversal of the lines is
essential for a calendar season as I define it . This indicates a reversal of
the drift direction across the season and hence a zero drift existing. I think
Irv has detected a very slow drift, which would lead to a low SD and the graphs
getting so close together that they may appear to cross.
What I think Irv has discovered here is that a calendar that is
close to having calendar seasons may be very accurate for a considerable range
of seasons. This seems to be the case for the 178-year cycle.
Irv states that the tropical year beginning at a given season is
ambiguous. If so, he should be able to give examples of two different things
that both may be called tropical year beginning at a given season. Actually, I
think you get a graph equivalent to the graph I’m talking about in this
context by choosing a step size that would give rise to a near constant drift
rate at each season. The graphs would then be all nearly identical except for
amplitude. The near constant drift rate of any season can then be subtracted
from the calendar mean year to get the tropical year length for a year beginning
at this season. I’ll try to get such graphs when I try out the
spreadsheet.
Karl
10(06(16 till noon
From: East Carolina University Calendar
discussion List [mailto:CALNDR-L@...] On Behalf Of Irv
Bromberg
Sent: 11 March 2009 18:03
To: CALNDR-L@...
Subject: Re: Find Solar Calendar Seasons spreadsheet posted
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.
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
Scanned by iCritical.
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Re: Find Solar Calendar Seasons spreadsheet posted
On 2009.03.13, at 05:35 , Palmen, KEV (Karl) wrote: Irv states that the tropical year beginning at a given season is ambiguous. If so, he should be able to give examples of two different things that both may be called tropical year beginning at a given season. Actually, I think you get a graph equivalent to the graph I’m talking about in this context by choosing a step size that would give rise to a near constant drift rate at each season. The graphs would then be all nearly identical except for amplitude. The near constant drift rate of any season can then be subtracted from the calendar mean year to get the tropical year length for a year beginning at this season. I’ll try to get such graphs when I try out the spreadsheet.
Irv asks: I'm confused.
I thought that the calendar season exists because the mean length of the solar year at that point in the solar cycle equals the leap cycle mean year at that point, but is Karl now saying that they are not equal at that point?
-- Irv Bromberg, Toronto, Canada
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Re: Find Solar Calendar Seasons spreadsheet posted
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Dear Irv and Calendar People
I’m writing about the graph for all seasons not just the
calendar seasons. For the calendar seasons, the constant drift rate would be
zero and so nothing would be subtracted from the calendar mean year to get the
tropical year beginning at that season. Both are then equal.
Karl
10(06(17
From: East Carolina University Calendar
discussion List [mailto:CALNDR-L@...] On Behalf Of Irv
Bromberg
Sent: 13 March 2009 12:27
To: CALNDR-L@...
Subject: Re: Find Solar Calendar Seasons spreadsheet posted
On
2009.03.13, at 05:35 , Palmen, KEV (Karl) wrote:
Irv states that the tropical
year beginning at a given season is ambiguous. If so, he should be able to give
examples of two different things that both may be called tropical year
beginning at a given season. Actually, I think you get a graph equivalent to
the graph I’m talking about in this context by choosing a step size
that would give rise to a near constant drift rate at each season. The graphs
would then be all nearly identical except for amplitude. The near
constant drift rate of any season can then be subtracted from the calendar mean
year to get the tropical year length for a year beginning at this season.
I’ll try to get such graphs when I try out the spreadsheet.
I
thought that the calendar season exists because the mean length of the solar
year at that point in the solar cycle equals the leap cycle mean year at that
point, but is Karl now saying that they are not equal at that point?
-- Irv Bromberg, Toronto,
Canada
Scanned by iCritical.
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Re: Find Solar Calendar Seasons spreadsheet posted
Dear Calendar People:
Last night I posted an update of the Find Solar Calendar Seasons spreadsheet.
This version checks the mean year of the specified cycle and if it is beyond a plausible range it first checks if the user has intentionally or inadvertently entered a fraction corresponding to a leap week cycle, in which case it converts it to a leap day cycle and continues, otherwise it beeps and automatically sets the cycle to a maximum (365+1/4 days) or a minimum (365+4/17), neither of which will have converging lines. I also adjusted the maximum scale factors so that it won't crash when either of these extremes is used.
I've noticed that the secondary (less stable) calendar season seems to land a few weeks earlier than I expected:
For example, the 15/62 cycle secondary calendar season is about 3 weeks before the southward equinox, yet presently it is a useful leap cycle for that equinox, and although the secondary calendar season of the 71/400 cycle is 5-6 weeks before the south solstice, it is presently a useful cycle for that solstice as well as the Besselian New Year moment.
Because of the faster migration of the secondary calendar season, its most useful solar longitude range for a forward-looking calendar is indicated by its future curves. It is not possible to standardize on using a particular future curve, such as the last plotted curve, because it may be a wild curve, or the interval of years between curves may be insufficient or excessive.
Sometimes my minimum standard deviation (SD) method find the secondary calendar season a number of weeks away from where I would have put it, when there happen to be some wild curve paths in the region of the secondary calendar season. In such cases the problem can be avoided by choosing another era that avoids such wild curves. I've been thinking of ways to ignore such wild curves -- the spreadsheet automatically makes their plotted lines thinner, based on their failure to cross the middle year, but they are still included in the SD because it is calculated as a worksheet function, not within the macro. If I calculated the SD within the macro then I could ignore curves that don't cross the middle year, but that would have to be during a second or third pass through the drift values. This would be a lot of extra processing, but nevertheless probably would execute quickly enough that the user wouldn't notice a delay. I was deferring doing this in the hopes of developing a crossover analysis algorithm instead of the SD method.
-- Irv Bromberg, Toronto, Canada
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Re: Find Solar Calendar Seasons spreadsheet posted
Dear Calendarists:
Today I posted an update of the Find Solar Calendar Seasons spreadsheet.
This version displays the shortest, average, and longest solar year lengths (based on polynomials that you can see in the cell formulae), and the range of time between them, for the specified middle year number.
It also visually indicates if the specified leap cycle's mean year is OK (happy face), too short (down arrows), or too long (up arrows) relative to the available solar year range. This indication requires that your computer system has the Wingdings font installed (that is a standard font in Windows, usually present unless somebody removed it).
This version works slightly better with Excel 2007, but still works best with earlier versions of Excel. Font sizes and font attributes work significantly differently in Excel 2007, so the chart title comes out plain and smaller than intended, without the nice super/sub-scripting. To see how the chart is supposed to look, I have now posted links to several example PDFs just below the spreadsheet download link.
Since the last posting on this topic I've been making quite a few updates to the discussion of calendar seasons, see this topic on <http://www.sym454.org/leap/>.
-- Irv Bromberg, Toronto, Canada
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