☼
Tropical Events:
the solstices and equinoxes
This presentation is a simple layman‟s study of some interesting aspects of Earth‟s orbital history, especially concerning
solstices and equinoxes, and the changing lengths of the days, seasons, and years.
The charts below are from an Excel
spreadsheet, created with an open source Excel Add-In, both of which I am making available at the end of this webpage: as I
would have loved to have started out with something similar 10 years ago, when I first became interested in “the motions of the
heavens” and the workings of the solar system.
My first inquiry concerned the “mean tropical year”: why does it change?
chart 21
mty.png
days per mean tropical year
in dynamical days of 86400 SI seconds
days per mty
365.242750
365.242625
365.242500
365.242375
365.242125
days
365.242250
365.242000
365.241875
365.241750
365.241625
100,000
80,000
60,000
40,000
20,000
0
-20,000
-40,000
-60,000
-80,000
-100,000
365.241500
calendar year
It is mostly due to the changing speed of the precession!
The rate of precession is shown below in arcseconds per Julian year (365.25 days), a standard time increment used by
astronomers.
chart 2
rate of precession
precess.png
arcseconds per Julian year
quasi-periodic approximation, Laskar et al 1993
degree 7 polynomials from data La93(0,1)
Laskar 1986 (±10T)
53.00
52.75
52.50
52.25
52.00
51.75
51.50
51.25
50.75
50.50
50.25
50.00
arcseconds
51.00
49.75
49.50
49.25
49.00
48.75
48.50
48.25
100,000
80,000
60,000
40,000
20,000
0
-20,000
-40,000
-60,000
-80,000
-100,000
48.00
calendar year
Comparison with the next chart shows us the relationship of the precession rate to the obliquity of the ecliptic (the tilt of
our spin axis in relation to our orbital plane):
chart 4
obliq.png
quasi-periodic approximation, Laskar et al 1993
obliquity
degree 7 polynomials from data La93(0,1)
24.7500
24.5000
24.2500
24.0000
23.7500
23.2500
23.0000
22.7500
22.5000
22.2500
calendar year
-2-
100,000
80,000
60,000
40,000
20,000
0
-20,000
-40,000
-60,000
-80,000
-100,000
22.0000
degrees
23.5000
The precession and the obliquity are from Jacques Laskar (et al 1993), an awesome astronomer at the Bureau des
Longitudes in France who applies the concepts of Newton‟s equations of motion to the entire solar system for millions and
millions of years! He then approximates his results with one single equation (called a quasi-periodic approximation) that is very
close to the “true solution”, at least for a few millions of years. This one for the precession and obliquity gives good results for 18
million years in the past, according to his solution usually designated La93(0,1), and it also gives good approximate results for 2
million years into the future. The Add-In can calculate with either the quasi-periodic approximation formulae, or from 7th-order
polynomial expressions that interpolate the data from his La93(0,1) solution. The spreadsheet used to create these charts also uses
the formulae from Laskar 1986 (valid for ±10,000 years from J2000), providing a comparison during the time period “closer to
home”.
My next question concerned the lengths of the seasons: “why are they changing?”
-3-
chart 1
ecc1.png
Earth eccentricity
eccentricity
quasi-periodic approximation (Laskar et al, 2004)
degree 7 polynomials from data La93(0,1)
Laskar 1986 (±10 T)
0.0210
0.0190
0.0170
0.0150
0.0130
0.0110
0.0090
0.0070
0.0050
0.0030
30,000
25,000
20,000
15,000
10,000
5,000
0
-5,000
-10,000
-15,000
-20,000
-25,000
-30,000
0.0010
calendar year
chart 19
seasons1.png
Lengths of the Seasons (TD)
(northern hemisphere) in days of 86400 SI seconds
Winter
Spring
Summer
Fall
95
94
93
91
90
89
88
calendar year
It is mostly due to the changing eccentricity and motion of the perihelion!
-4-
30,000
25,000
20,000
15,000
10,000
5,000
0
-5,000
-10,000
-15,000
-20,000
-25,000
-30,000
87
days per season
92
As the eccentricity approaches zero (circular orbit), the seasons become nearly equal.You can detect the motion of the
perihelion from the previous chart, as it moves from season to season, causing that season to be “shortest”. The next chart shows
the speed of the perihelion‟s motion as the length of the anomalistic year:
chart 6
anom_yr.png
anomalistic year (TD)
days per anomalistic year
quasi-periodic approximation (Laskar et al, 2004)
degree 7 polynomials from data La93(0,1)
Laskar 1986 (±10 T)
365.277636
365.275636
365.273636
365.271636
365.269636
365.265636
days
365.267636
365.263636
365.261636
365.259636
365.257636
30,000
25,000
20,000
15,000
10,000
5,000
0
-5,000
-10,000
-15,000
-20,000
-25,000
-30,000
365.255636
calendar year
Laskar (et al 2004) has developed another approximation formula for both the eccentricity and perihelion, valid from
−15M (million years ago) to +5M (million years in the future). Again the Add-In has retained good accuracy by interpolating
Laskar‟s 1993 data with polynomial expressions. The method used is on p.32 in the book “Astronomical Algorithms” by Jean
Meeus (1998), referred to as “Lagrange‟s interpolation formula”.
-5-
The relationship between the eccentricity and the lengths of the seasons is very interesting to study:
chart 1
ecc2.png
Earth eccentricity
eccentricity
quasi-periodic approximation (Laskar et al, 2004)
degree 7 polynomials from data La93(0,1)
Laskar 1986 (±10 T)
0.0500
0.0450
0.0400
0.0350
0.0300
0.0250
0.0200
0.0150
0.0100
0.0050
200,000
175,000
150,000
125,000
100,000
75,000
50,000
25,000
0
-25,000
-50,000
-75,000
-100,000
-125,000
-150,000
-175,000
-200,000
0.0000
calendar year
chart 19
seasons2.png
Lengths of the Seasons (TD)
(northern hemisphere) in days of 86400 SI seconds
Winter
Spring
Summer
Fall
102
100
98
96
92
90
88
86
84
200,000
175,000
150,000
125,000
100,000
75,000
50,000
25,000
0
-25,000
-50,000
-75,000
-100,000
-125,000
-150,000
-175,000
-200,000
82
calendar year
The method and formulae used for charting these lengths of the seasons is documented on page 6 of TEequation.pdf
-6-
days per season
94
According to Laskar (2004), the eccentricity can reach as high as 0.063. This shows the variations for ±1 million years:
chart 1
ecc3.png
Earth eccentricity
eccentricity
quasi-periodic approximation (Laskar et al, 2004)
degree 7 polynomials from data La93(0,1)
Laskar 1986 (±10 T)
0.0600
0.0550
0.0500
0.0450
0.0400
0.0350
0.0300
0.0250
0.0200
0.0150
0.0100
0.0050
1,000,000
875,000
750,000
625,000
500,000
375,000
250,000
125,000
0
-125,000
-250,000
-375,000
-500,000
-625,000
-750,000
-875,000
-1,000,000
0.0000
calendar year
One of the most interesting things that I‟ve learned is that not only is there the “mean tropical year”, there are 4 separate
“true” tropical years, measured from vernal equinox to vernal equinox (VE to VE on the chart below), summer solstice to summer
solstice (SS to SS), autumnal equinox to autumnal equinox (AE to AE), and winter solstice to winter solstice (WS to WS). The
“names” of the seasons (and the tropical events) on these charts are those of the northern hemisphere (southern hemisphere
readers should of course swap the names “winter” and “summer”, and also the names representing “spring” and “fall”).
-7-
chart 21
days_yr1.png
days per tropical year
in dynamical days of 86400 SI seconds
(tropical events of northern hemisphere)
days per aty
VE to VE
SS to SS
AE to AE
WS to WS
days per mty
365.243250
365.243000
365.242750
365.242500
365.242000
days
365.242250
365.241750
365.241500
365.241250
50,000
40,000
30,000
20,000
10,000
0
-10,000
-20,000
-30,000
-40,000
-50,000
365.241000
calendar year
You can see from this chart how the “true” tropical years vary about the mean tropical year (mty), and how the mean
tropical year varies about the “average” tropical year (aty). The average tropical year is determined from the “average” length of
the sidereal year (~365.256362) and the “average” rate of precession (50.4712 arcseconds per Julian year, by use of the solution
La93(0,1)), so that, for this study, the average length of the tropical year is ~365.242138.
These previous charts are all in dynamical time, based on the well defined scientific second determined by atomic clocks,
and counted in days of dynamical time equal to 86400 seconds per day. Let‟s look at the lengths of the years during a longer time
span and compare dynamical time to Universal time, which measures mean solar days of Earth rotations, known to be lengthening
as Earth‟s spin rate slows down, mostly due to tidal friction. To avoid confusion between the dynamical time scale of days and the
Universal time scale of days, these charts have used days (with a lower case d) to designate dynamical time (TD), and Days (with
a capital D) are used to designate mean solar days of Universal time (UT).
-8-
chart 21
days_yr2.png
days per tropical year
in dynamical days of 86400 SI seconds
(tropical events of northern hemisphere)
days per aty
VE to VE
SS to SS
AE to AE
WS to WS
days per mty
365.25250
365.25125
365.25000
365.24875
365.24750
365.24625
365.24500
365.24250
365.24125
365.24000
365.23875
365.23750
365.23625
365.23500
365.23375
calendar year
-9-
100,000
80,000
60,000
40,000
20,000
0
-20,000
-40,000
-60,000
-80,000
-100,000
365.23250
days
365.24375
chart 34
Days_yr3.png
Days per tropical year
(tropical events of northern hemisphere)
Days per aty
VE to VE
SS to SS
AE to AE
WS to WS
Days per mty
365.25250
365.25125
365.25000
365.24875
365.24750
365.24625
365.24500
365.24250
365.24125
365.24000
365.23875
365.23750
365.23625
365.23500
365.23375
calendar year
- 10 -
100,000
80,000
60,000
40,000
20,000
0
-20,000
-40,000
-60,000
-80,000
-100,000
365.23250
Days
365.24375
What a difference! In this chart the dynamical time is converted to Universal time by showing the length of day (LOD)
to be currently increasing by 1.7 ms/cy (milliseconds per century), a value determined by F. R. Stephenson (1997) from studies of
eclipse records during the past 2700 years. This particular “scenario” in the spreadsheet shows the rate of change of LOD to be
slowly increasing to a more scientifically expected rate of 2.3 ms/cy at ±100,000 yrs.
I also like to study the calendar, and the dates and times of solstices and equinoxes. These next charts show the dates
and times of the vernal equinox every 10 years from −500 (501 BC) to +2000 (2000 AD) in both the Julian calendar and the
Gregorian calendar, in the time zone of Rome (+1 hrs from Greenwich).
- 11 -
chart 26
trueVE_Julian.png
trueTE
Julian calendar
true Solstice or Equinox (-4000 to +8000, Bretagnon and Simon, 1986)
"true" Vernal Equinox
year increment → 10
month of March
time zone: +1 hrs from Greenwich
31
30
29
28
27
26
25
24
23
20
19
18
17
16
15
14
Day of month
22
21
13
12
11
10
9
8
7
6
5
4
3
2
2000
1750
1500
1250
1000
750
500
250
0
-250
-500
1
Julian calendar year
chart 26
trueVE_Gregorian.png
trueTE
Gregorian calendar
true Solstice or Equinox (-4000 to +8000, Bretagnon and Simon, 1986)
"true" Vernal Equinox
year increment → 10
month of March
time zone: +1 hrs from Greenwich
31
30
29
28
27
26
25
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
Gregorian calendar year
- 12 -
2000
1750
1500
1250
1000
750
500
250
0
-250
-500
2
1
Day of month
24
23
It is easy to see why we changed from the Julian to the Gregorian calendar! The dynamical times have been calculated
by the method in the wonderfully compact book “Planetary Programs and Tables” by Pierre Bretagnon and Jean-Louis Simon
(1986). They are supposed to be accurate to 0.0009 degree (within about 1⅓ minutes of time) from −4000 to +8000 AD. These
times include the nutation in longitude (caused mainly by the Moon), and the aberration of light, so that they are the “observed”
times of the tropical events. The calendar dates are calculated by algorithm from Meeus (1998). The value of ΔT (TD − UT) can
be either from Morrison and Stephenson (2004), Espenak and Meeus (from NASA), a sum of sines, or any custom combination.
The spreadsheet is currently set to use the Espenak and Meeus polynomials from −404 to +2003.5. C₂ + r t² + ∑⁴ Sin(a t + c)/a + d
is used from −6106 to −404 and from 2050 to 2668. To transition to this sum of 4 sines, from 2003.5 to 2050 a polynomial is
used that also follows the data from the International Earth Rotation and Reference Systems Service (IERS) from 2000 to 2010:
63.9+0.164954*t-0.00281933*t^2+0.000879724*t^3-0.0000104809*t^4 where t = (JD -2451544.5)/365.2425
C₁ + r t² + ∑² Sin(a t + c)/a + d is used from −21,410 to −6106 and from 2668 to 18,749. Morrison and Stephenson (2004) is used
from −4.5M to +4.5M, and the integral of an equation given by Deubner (1990) is used from −4.5G until +5G.
The sum of 14 sines is still under development, but a preliminary result is charted below, and the formula is documented
on page 8 of TEequation.pdf
chart 23b
deltaT.png
360
∆T (spreadsheet settings)
340
Morrison & Stephenson (2004)
∆T = −20 + 32 ((y − 1820)/100)²
320
300
McCarthy & Babcock (1986)
280
table from R. H. van Gent
260
Espenak and Meeus polynomials
240
C₃ + r t² + ∑¹⁴ b Sin(a t + c)/a + d
220
C₂ + r t² + ∑⁴ b Sin(a t + c)/a + d
180
160
seconds
200
C₁ + r t² + ∑² b Sin(a t + c)/a + d
140
120
100
80
60
40
20
0
-20
2160
2140
2120
2100
2080
2060
2040
2020
2000
1980
1960
1940
1920
1900
1880
1860
1840
1820
1800
1780
1760
1740
1720
1700
1680
1660
1640
1620
1600
1580
1560
1540
1520
1500
1480
-40
calendar year
I also enjoy studying this rate of slowing for the entire history of the Earth-Moon system, from −4.5G (billions of years
ago) until +5G (billion years in the future), when the Sun is expected to become a Red Giant, boiling the oceans and scorching the
Earth! Intricately tied to the slowing of Earth‟s spin rate is the Moon, due to the tidal friction that it causes. From our laws of
physics (conservation of angular momentum), as the Earth‟s spin angular momentum decreases, the Moon‟s orbital angular
momentum must increase, so that the total angular momentum is conserved. In order for the Moon‟s orbital angular momentum
to increase, the Moon must increase its distance from Earth, and that is why we measure the Moon to be moving away from the
Earth at a rate of 3.82 ±0.07 centimeters per year (Dickey et al, 1994).
- 13 -
chart 8
E-M_km.png
average Earth-Moon distance
for geologic time scale use only
r = (1+Y/C)^(n/m), scenario 9a, 2 past era's
500,000
450,000
400,000
350,000
250,000
200,000
kilometers
300,000
150,000
100,000
50,000
5,000,000,000
4,000,000,000
3,000,000,000
2,000,000,000
1,000,000,000
0
-1,000,000,000
-2,000,000,000
-3,000,000,000
-4,000,000,000
-5,000,000,000
0
calendar year
chart 9
Lunar_retreat.png
Lunar retreat rate
centimeters per Julian year, scenario 9a, 2 past era's
80.000
70.000
60.000
50.000
30.000
20.000
calendar year
- 14 -
5,000,000,000
4,000,000,000
3,000,000,000
2,000,000,000
1,000,000,000
0
-1,000,000,000
-2,000,000,000
-3,000,000,000
-4,000,000,000
-5,000,000,000
10.000
0.000
centimeters
40.000
chart 37
LOD1.png
Length of solar day
LOD (solar)
60
56
52
48
44
40
36
28
24
20
hours (TD)
32
16
12
8
0
5,000,000,000
4,000,000,000
3,000,000,000
2,000,000,000
1,000,000,000
0
-1,000,000,000
-2,000,000,000
-3,000,000,000
-4,000,000,000
-5,000,000,000
4
calendar year
chart 23
deltaT2.png
1,400,000,000,000
∆T in days at aTE (1820 system)
1,200,000,000,000
ms/Julian cy (long term) @J2000: 2.569
ms/cy (short term) @ Y = 0: 1.720
1,000,000,000,000
days
800,000,000,000
600,000,000,000
400,000,000,000
200,000,000,000
- 15 -
5,000,000,000
4,000,000,000
3,000,000,000
2,000,000,000
calendar year
1,000,000,000
0
-1,000,000,000
-2,000,000,000
-3,000,000,000
-4,000,000,000
-5,000,000,000
0
chart 34
Days_yr4.png
Days per tropical year
(tropical events of northern hemisphere)
Days per aty
VE to VE
SS to SS
AE to AE
WS to WS
Days per mty
1850
1800
1750
1700
1650
1600
1550
1500
1450
1400
1350
1300
1250
1200
1150
1100
1050
950
900
850
800
750
700
650
600
550
500
450
400
350
300
250
200
150
calendar year
- 16 -
5,000,000,000
4,000,000,000
3,000,000,000
2,000,000,000
1,000,000,000
0
-1,000,000,000
-2,000,000,000
-3,000,000,000
-4,000,000,000
-5,000,000,000
100
Days
1000
These are of course speculative, as the true history of Earth-Moon distance and length of day (LOD) remains unknown.
This is the reason for the different “scenarios” in the spreadsheet. They are easily changeable, so that you might find one to better
suit your own ideas (as I am constantly changing my own!). Many of the parameters are also changeable, so that you can create
your own custom scenario to depict a realistic Earth-Moon history. Some of the “targets” I like to consider are:
Days per year
source
−4500M
Earth-Moon distance
in kilometers
19,000 to 38,000
1800 to 1600
Canup (2004)
−2450M
330,000
512
Walker and Zahnle (1986)
−2450M
337,000 to 360,000
465 ±15
G.E. Williams (1989)
−900M
344,000 to 346,000
481 ±4
Sonett et al (1996)
−900M
364,700
419
G.E. Williams (2000)
−620M
369,100 to 372,900
400 ±7
G.E. Williams (2000)
−250M
372,600
397
Laskar et al (2004)
+250M
392,400
339
Laskar et al (2004)
+5000M
453,600?
153.4 ±1.8
Néron de Surgy and Laskar (1997)
year
Most of the scenarios use an equation for Earth-Moon distance given by Walker and Zahnle (1986), in which you can
change the parameters. Then the Earth-Moon distance is applied to an equation given by Deubner (1990) to obtain Days per year.
It is supposed to obey the conservation of angular momentum and include the effects of the solar tides. Some parameters of this
equation are also changeable. The integral of Deubner‟s equation is then used to count the accumulated number of solar days
during any time span. Then the difference (dynamical days since 1820) − (mean solar days since 1820) = ΔT in days. Multiply
by 86400 to obtain seconds, and subtract 20 seconds to correspond to the dynamical time scale of the Julian day numbers, which
is set so that dynamical time = Universal time at ~1900 AD.
To continue with our study of solstices and equinoxes beyond ±6000 yrs from J2000 (the range of validity for the times
calculated by the method of Bretagnon and Simon, 1986), the spreadsheet uses the “mean orbital elements” from Laskar (1986,
1993). The accuracy then reduces to about ±20 minutes of dynamical time. That is, as long as the orbital elements are “correct”.
The reason for the ±20 minute “error”, even when using correct mean orbital elements, is that this assumes the Earth to be in a
purely elliptical unperturbed Keplerian orbit at the Earth-Moon barycenter. The nutation (caused mainly by the Moon) accounts
for ±10 minutes, and the other ±10 minutes is due to the perturbations of the planets (especially Jupiter!). The “constant” of
aberration (20.5 arcseconds, amounting to ~8.31 minutes of time) has been included, so that the variations throughout the year
(±8.5 seconds) are negligible (at the accuracy of 20 minutes!).
By use of the quasi-periodic approximation formulae (instead of the data) the error can of course be much greater. For
±300,000 yrs the difference in precession can amount to a ±20 hour error, while use of the approximation formula for eccentricity
and perihelion can accrue an additional error of ±5 hours. For 18 million years in the past, the quoted range of validity for the
quasi-periodic approximation of the precession, and until 2 million years in the future the error can amount to ±30 hours. Using it
for the entire range of Laskar‟s data (−20M to +10M) shows a maximum error of ±60 hours. By use of the approximation
formula for eccentricity and perihelion from −15M to +5M (its quoted range of validity) there can be an additional error of ±60
hours, while using it from −20M to +10M shows a maximum additional error of ±100 hours, or about 4 days. These stated
“errors” are given in dynamical time (TD), and are actually quite insignificant when compared to the unknown quantity ΔT (TD −
UT), which at −20M quite possibly reaches more than 20,000,000 days!
- 17 -
The following charts plot the dates and times of the 4 “tropical events” in the Gregorian calendar for ±30,000 yrs. They
are calculated with “correct” mean orbital elements (using Laskar‟s data beyond ±10,000 yrs). I have labeled these “eccentric”
tropical events (eTE) to reflect that they are calculated from pure ellipse. Whereas I use “true” tropical event for a solstice or
equinox calculated by the method of Bretagnon and Simon (1986), as these include the effects of our true orbit (perturbed by the
Moon and planets).
Summer Solstice
eVE_Gregorian1.png
31
30
29
29
28
27
28
27
26
26
25
24
25
24
23
23
22
22
21
20
21
20
19
19
17
16
18
17
16
15
14
13
14
13
12
12
11
11
10
9
10
9
8
8
7
6
7
5
5
4
4
3
3
2
2
1
1
Gregorian calendar year
eAE_Gregorian1.png
30000
25000
20000
15000
31
30
29
29
28
27
28
27
26
26
25
24
25
24
23
23
22
22
21
20
21
20
19
19
18
17
16
15
14
13
14
13
12
12
11
11
10
9
10
9
8
8
7
6
7
5
5
4
4
3
2
3
2
1
1
30000
25000
20000
15000
10000
5000
0
-5000
-10000
6
-30000
30000
25000
20000
15000
10000
5000
0
-5000
-10000
time zone: 0 hrs from Greenwich
30
15
-15000
month # of the eWS , 1st data entry = 11 (November)
31
16
-20000
year increment: 240.00
-15000
time zone: 0 hrs from Greenwich
eWS_Gregorian1.png
Gregorian calendar
date and time of the "eccentric" Winter Solstice
-20000
month # of the eAE , 1st data entry = 8 (August)
eTE
-25000
year increment: 240.00
Day of month
Gregorian calendar
date and time of the "eccentric" Autumnal Equinox
18
17
-25000
10000
Winter Solstice
chart 28
Gregorian calendar year
Gregorian calendar year
We see that the solstices and equinoxes start occurring in the “wrong” months after about 20,000 years (the red lines
indicate the month#). This is mostly due to the slowing of Earth‟s rotation rate, but why are the shapes of each curve different?
Let‟s look at a “fictitious” Gregorian calendar, one that assumes the day is not lengthening, for a longer time span and see if we
can uncover the answer. The tropical events are shown below from −35,000 (35,001 BC) to 65,000 AD. The lines appear so
smooth because one is plotted every 400 years, corresponding to our 400 year “leap year schedule”. The red line again indicates
the month#. For example, the Vernal Equinox is shown below to be occurring in April (month# 4) at −30,000, and in February
(month# 2) at +60,000:
- 18 -
Day of month
Autumnal Equinox
-30000
5000
Gregorian calendar year
chart 28
eTE
0
-5000
-30000
6
30000
25000
20000
15000
10000
5000
0
-5000
-10000
-15000
-20000
15
-25000
time zone: 0 hrs from Greenwich
30
18
-30000
month # of the eSS , 1st data entry = 5 (May)
31
-10000
time zone: 0 hrs from Greenwich
eSS_Gregorian1.png
Gregorian calendar
year increment: 240.00
-15000
month # of the eVE , 1st data entry = 2 (February)
eTE
date and time of the "eccentric" Summer Solstice
-20000
year increment: 240.00
-25000
Gregorian calendar
date and time of the "eccentric" Vernal Equinox
Day of month
eTE
chart 28
Day of month
Vernal Equinox
chart 28
Summer Solstice
chart 28
fictVE.png
31
30
29
29
28
27
28
27
26
26
25
25
24
24
23
23
22
22
21
20
21
20
19
19
16
18
17
16
15
14
13
14
13
12
12
11
11
10
9
10
9
8
8
7
6
7
5
5
4
4
3
2
3
2
1
1
Autumnal Equinox
65000
55000
d'ays
65,000
28
27
26
25
24
25
23
23
22
22
21
20
21
20
19
19
17
16
15
14
13
12
11
11
10
9
10
9
8
8
7
6
7
5
5
4
4
3
2
3
1
1
6
65000
55000
45000
35000
25000
15000
5000
-5000
2
Gregorian calendar year
calendar year
calendar year
- 19 -
65,000
55,000
45,000
35,000
25,000
15,000
5,000
-5,000
-15,000
-25,000
-35,000
d'ays
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
-13
-14
-15
d'ays
delta_eWSplus.png
∆eWS + ∆mTE + ∆bTE (TD)
65,000
55,000
45,000
35,000
25,000
chart 31
Day of month
18
65000
55000
45000
35000
25000
24
12
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
-13
-14
-15
15,000
55,000
29
26
delta_eAEplus.png
∆eAE + ∆mTE + ∆bTE (TD)
5,000
45000
30
Gregorian calendar year
chart 31
-5,000
45,000
31
28
27
14
13
15000
time zone: 0 hrs from Greenwich
29
15
5000
month # of the eWS , 1st data entry = 1 (January)
30
16
-5000
year increment → 400.00
31
17
-15000
fictWS.png
"fictitious" Gregorian calendar
date and time of the "eccentric" Winter Solstice
-15000
time zone: 0 hrs from Greenwich
-25000
month # of the eAE , 1st data entry = 10 (October)
-35000
year increment → 400.00
eTE
Day of month
date and time of the "eccentric" Autumnal Equinox
-15,000
35,000
Winter Solstice
chart 28
fictAE.png
"fictitious" Gregorian calendar
18
-25000
25,000
calendar year
chart 28
-25,000
15,000
5,000
-5,000
-15,000
-25,000
-35,000
d'ays
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
-13
-14
-15
calendar year
-35000
delta_eSSplus.png
∆eSS + ∆mTE + ∆bTE (TD)
65,000
55,000
45,000
35,000
25,000
15,000
5,000
-5,000
-15,000
-25,000
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
-13
-14
-15
-35,000
chart 31
delta_eVEplus.png
∆eVE + ∆mTE + ∆bTE (TD)
-35,000
35000
Gregorian calendar year
Gregorian calendar year
chart 31
eTE
25000
-35000
6
65000
55000
45000
35000
25000
15000
5000
-5000
-15000
15
-25000
time zone: 0 hrs from Greenwich
30
18
17
-35000
month # of the eSS , 1st data entry = 7 (July)
31
15000
time zone: 0 hrs from Greenwich
year increment → 400.00
5000
month # of the eVE , 1st data entry = 4 (April)
fictSS.png
"fictitious" Gregorian calendar
date and time of the "eccentric" Summer Solstice
-5000
year increment → 400.00
-15000
date and time of the "eccentric" Vernal Equinox
eTE
-25000
"fictitious" Gregorian calendar
Day of month
eTE
Day of month
Vernal Equinox
chart 28
We can see from this short study that the solstices and equinoxes would remain in their “proper” months for about
50,000 years in the future, if Earth‟s rotation rate did not slow down. The reason they still fall out of their respective months,
even with this “fictitious” assumption that ΔT = 0, is of course that our calendar depicts the tropical year to be 365
97
/400
(365.2425) days in length: slightly longer than an average tropical year of 365.242138.
The general shapes of the curves are seen to correspond to the sum of 3 values. For the vernal equinox I‟ve labeled them
ΔeVE + ΔmTE + ΔbTE. (TD) indicates that these values are measured in days of dynamical time.
ΔeVE measures the effects of the eccentricity and position of the perihelion upon the vernal equinox. It is the difference,
in time, between a vernal equinox calculated in elliptical orbit, and one calculated in circular orbit. For an unperturbed elliptical
-
orbit it is the value
, where C is the “equation of center” at a vernal equinox, and nt is the “tropical” mean motion, measured
as 360° per mean tropical year.
ΔeSS is
-
ΔeAE is
-
ΔeWS is
-
where C is the equation of center at a summer solstice
where C is the equation of center at an autumnal equinox
where C is the equation of center at a winter solstice
ΔmTE measures the effects of the changing speed of the precession upon a solstice or equinox (tropical event, TE). It is
the difference, in time, between a solstice or equinox calculated with the “true” precession in longitude, and one calculated with a
long-term “average” precession rate. By the use of the solution La93(0,1) this long-term average rate is also constant, 50.4712
arcseconds per Julian year. This allows our simple study to remain quite simple (other of his solutions show this long-term
average rate to be slowly decreasing).
ΔbTE measures the effects of the changing length of the sidereal year upon a solstice or equinox. It is the difference in
time between a solstice or equinox calculated with a changing sidereal year, and one calculated with a constant long-term
“average” sidereal year. Its effects are supposed to be very minor.
The values charted separately from −35,000 to +65,000 look like this:
- 20 -
chart 24
∆eTE
delta_eTE1.png
change in the time of a Tropical Event due to the changing eccentricity and perihelion
in dynamical days of 86400 SI seconds
∆eVE
∆eSS
∆eAE
∆eWS
15
12
9
6
0
days
3
-3
-6
-9
-12
65,000
55,000
45,000
35,000
25,000
15,000
5,000
-5,000
-15,000
-25,000
-35,000
-15
calendar year
chart 22
delta_mTE1.png
∆mTE
change in the time of a Tropical Event due to the changing speed of the precession
∆mTE in days of 86400 SI seconds (TD)
15
12
9
6
0
-3
-6
-9
-12
calendar year
- 21 -
65,000
55,000
45,000
35,000
25,000
15,000
5,000
-5,000
-15,000
-25,000
-35,000
-15
days
3
chart 20
delta_bTE1.png
∆bTE (TD)
change in the time of a Tropical Event due to the changing sidereal mean motion
∆bTE in days of 86400 SI seconds (TD)
15
12
9
6
3
days
0
-3
-6
-9
-12
65,000
55,000
45,000
35,000
25,000
15,000
5,000
-5,000
-15,000
-25,000
-35,000
-15
calendar year
These values charted together look like this:
chart 25
delta_eTEplus.png
∆eTE + ∆mTE + ∆bTE in dynamical time (TD)
∆eVE + ∆mTE + ∆bTE
∆eSS + ∆mTE + ∆bTE
∆eAE + ∆mTE + ∆bTE
∆eWS + ∆mTE + ∆bTE
12
11
10
9
8
6
5
4
3
2
1
calendar year
- 22 -
65,000
55,000
45,000
35,000
25,000
15,000
5,000
-5,000
-15,000
-25,000
-35,000
0
days
7
The reason ΔmTE is not currently centered around zero is that the precession in longitude shows general trends of
“slightly faster” than average, and “slightly slower” than average, in regular cycles of ~4 million years. The next chart shows
ΔmTE from −20M to +10M, using 1500 data points (all of the previous charts have used 250). Once again, it is the difference
between the “true” precession in longitude, and an “average” precession in longitude described by 360 ° per 25678 Julian years
(50".4712 per yr), converted to time and expressed as „days‟ of dynamical time.
chart 22
delta_mTE2.png
∆mTE
change in the time of a Tropical Event due to the changing speed of the precession
∆mTE in days of 86400 SI seconds (TD)
15
12
9
6
0
days
3
-3
-6
-9
-12
10,000,000
8,000,000
6,000,000
4,000,000
2,000,000
0
-2,000,000
-4,000,000
-6,000,000
-8,000,000
-10,000,000
-12,000,000
-14,000,000
-16,000,000
-18,000,000
-20,000,000
-15
calendar year
What this chart is telling us is that about 3 million years ago the variations to the precession in longitude caused solstices
and equinoxes to occur about 12 days sooner than average, and 1 million years in the future about 12 days later than average. The
precession is currently causing them to occur about 8 days later than the long-term average.
To continue our study of solstices and equinoxes beyond 20,000 years, the spreadsheet creates a calendar, including the
formulae you use to determine ΔT, so that it includes the effects of the slowing of Earth‟s rotation rate. It‟s basic “rules” are quite
simple:
Place the “average” vernal equinox “mid-March”.
Place the “average” summer solstice “mid-June”.
Place the “average” autumnal equinox “mid-September”.
Place the “average” winter solstice “mid-December”.
By another “rule” it defines “mid-month” to currently be the 15th.
The next chart shows the values of ΔeTE + ΔmTE + ΔbTE converted to Universal time (or our estimation of UT) for
±100,000 years. It gives us a general idea of the changes in the day of the month that each solstice and equinox will occur in this
calendar.
- 23 -
chart 35
date of aTE + ∆eTE + ∆mTE + ∆bTE
March equinox
June solstice
( ≈ calendar date of the Tropical Event )
September equinox
calendar_dates.png
December solstice
31
30
29
28
27
26
25
24
23
22
21
20
19
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
calendar year
- 24 -
100,000
80,000
60,000
40,000
20,000
0
-20,000
-40,000
-60,000
-80,000
-100,000
1
Day of month
18
The next charts separately plot the date and time of each solstice and equinox in this calendar. I enjoy labeling this a
“Copernican” calendar, as it assumes we are in “circular” orbit, making the “seasons” (the calendar quarters) equal in length. It
also assumes that the length of the sidereal year is constant (unchanging), and that the rate of precession is constant, so that the
length of the (average) tropical year is a constant length of dynamical time. I also rather like this label because he is often
described as the progenitor of modern scientific advancement, as in the “Copernican Revolution”. One of the most profound
modern discoveries (as far as a calendar is concerned) is that the Earth‟s spin rate is slowing down, mostly due to tidal friction.
To include this phenomenon in a calendar, and attribute it‟s discovery to the Copernican Revolution, provides further enjoyment
in labeling this a “Copernican calendar”. You, of course, can name it anything you like, change some of the “rules”, and create a
calendar of your own. The “average” tropical events are also plotted, to make sure that they are currently occurring on the 15th of
the month (in Greenwich).
- 25 -
This is the vernal equinox in the month of March:
chart 29
eVE_Copernican1.png
Copernican calendar
date and time of the "eccentric" Vernal Equinox
Days per Mar, Jun, Sep, & Dec, 1st data entry = 31
year increment → 400.00
date and time of the "average" Vernal Equinox
month # of the eVE , 1st data entry = 3 (March)
time zone: 0 hrs from Greenwich
31
30
29
28
27
26
25
24
23
22
21
19
18
17
16
15
Day of month '
20
14
13
12
11
10
9
8
7
6
5
4
3
2
50,000
40,000
30,000
20,000
10,000
0
-10,000
-20,000
-30,000
-40,000
-50,000
1
calendar year
This is the summer solstice in the month of June:
chart 29
eSS_Copernican1.png
Copernican calendar
date and time of the "eccentric" Summer Solstice
Days per Mar, Jun, Sep, & Dec, 1st data entry = 31
year increment → 400.00
date and time of the "average" Summer Solstice
month # of the eSS , 1st data entry = 6 (June)
time zone: 0 hrs from Greenwich
31
30
29
28
27
26
25
24
23
22
21
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
calendar year
- 26 -
50,000
40,000
30,000
20,000
10,000
0
-10,000
-20,000
-30,000
-40,000
-50,000
1
Day of month '
20
This is the autumnal equinox in the month of September:
chart 29
eAE_Copernican1.png
Copernican calendar
date and time of the "eccentric" Autumnal Equinox
Days per Mar, Jun, Sep, & Dec, 1st data entry = 31
year increment → 400.00
date and time of the "average" Autumnal Equinox
month # of the eAE , 1st data entry = 9 (September)
time zone: 0 hrs from Greenwich
31
30
29
28
27
26
25
24
23
22
21
19
18
17
16
15
Day of month '
20
14
13
12
11
10
9
8
7
6
5
4
3
2
50,000
40,000
30,000
20,000
10,000
0
-10,000
-20,000
-30,000
-40,000
-50,000
1
calendar year
And this is the winter solstice in the month of December:
chart 29
eWS_Copernican1.png
Copernican calendar
date and time of the "eccentric" Winter Solstice
Days per Mar, Jun, Sep, & Dec, 1st data entry = 31
year increment → 400.00
date and time of the "average" Winter Solstice
month # of the eWS , 1st data entry = 12 (December)
time zone: 0 hrs from Greenwich
31
30
29
28
27
26
25
24
23
22
21
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
calendar year
- 27 -
50,000
40,000
30,000
20,000
10,000
0
-10,000
-20,000
-30,000
-40,000
-50,000
1
Day of month '
20
It is interesting to inspect this calendar far into the past and future. The next chart shows the winter solstice 3 million
years ago, when it can dip into the end of November by a few days:
chart 29
Copernican calendar
eWS_Copernican2.png
date and time of the "eccentric" Winter Solstice
Days per Mar, Jun, Sep, & Dec, 1st data entry = 31
year increment → 400.00
date and time of the "average" Winter Solstice
month # of the eWS , 1st data entry = 12 (December)
time zone: 0 hrs from Greenwich
31
30
29
28
27
26
25
24
23
22
21
19
18
17
16
15
14
Day of month '
20
13
12
11
10
9
8
7
6
5
4
3
2
-2,950,000
-2,960,000
-2,970,000
-2,980,000
-2,990,000
-3,000,000
-3,010,000
-3,020,000
-3,030,000
-3,040,000
-3,050,000
1
calendar year
And this charts the summer solstice 1 million years in the future, where it can occasionally occur during the beginning of
July:
chart 29
eSS_Copernican2.png
Copernican calendar
date and time of the "eccentric" Summer Solstice
Days per Mar, Jun, Sep, & Dec, 1st data entry = 31
year increment → 400.00
date and time of the "average" Summer Solstice
month # of the eSS , 1st data entry = 6 (June)
time zone: 0 hrs from Greenwich
31
30
29
28
27
26
25
24
23
22
21
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
calendar year
- 28 -
1,050,000
1,040,000
1,030,000
1,020,000
1,010,000
1,000,000
990,000
980,000
970,000
960,000
950,000
1
Day of month '
20
This shows the long term “ups and downs” of the vernal equinox for the entire time span (−20M to +10M) of the orbital
elements from Laskar‟s 1993 data:
chart 29
Copernican calendar
eVE_Copernican2.png
date and time of the "eccentric" Vernal Equinox
Days per Mar, Jun, Sep, & Dec, 1st data entry = 31
year increment → 20,000
date and time of the "average" Vernal Equinox
month # of the eVE , 1st data entry = 3 (March)
time zone: 0 hrs from Greenwich
31
30
29
28
27
26
20
19
18
17
16
15
14
13
Day of month '
25
24
23
22
21
12
11
10
9
8
7
6
5
4
10,000,000
7,500,000
5,000,000
2,500,000
0
-2,500,000
-5,000,000
-7,500,000
-10,000,000
-12,500,000
-15,000,000
-17,500,000
-20,000,000
3
2
1
calendar year
Beyond −20M to +10M the solstices and equinoxes that the spreadsheet calculates are of course no longer valid (even
though it continues to calculate them from the quasi-periodic approximation formulae), but it continues to count the days per year
and divide the 12 months according to the “rules”. This particular scenario shows 482 days per year 686,000,000 years ago, and
divides the 12 months into 38 days each, with mid-month defined as the 19th. 5 billion years in the future it shows 152 days per
year, 13 days per month, with mid-month defined as the 6th.
chart 29
Copernican calendar
eWS_Copernican3.png
date and time of the "eccentric" Winter Solstice
Days per Mar, Jun, Sep, & Dec, 1st data entry = 150
year increment → 38,000,431
date and time of the "average" Winter Solstice
month # of the eWS , 1st data entry = 12 (December)
time zone: 0 hrs from Greenwich
151
136
121
91
76
61
46
31
16
calendar year
- 29 -
5,000,000,000
4,000,000,000
3,000,000,000
2,000,000,000
1,000,000,000
0
-1,000,000,000
-2,000,000,000
-3,000,000,000
-4,000,000,000
-5,000,000,000
1
Day of month '
106
Currently the vernal equinox in the calendar looks like this (at the longitude of Greenwich):
chart 29
eVE_Copernican3.png
Copernican calendar
date and time of the "eccentric" Vernal Equinox
Days per Mar, Jun, Sep, & Dec, 1st data entry = 31
year increment → 1
date and time of the "average" Vernal Equinox
month # of the eVE , 1st data entry = 3 (March)
time zone: 0 hrs from Greenwich
31
30
29
28
27
26
25
24
23
22
21
Day of month '
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
2125
2100
2075
2050
2025
2000
1975
1950
1925
1900
1875
1
calendar year
The reason for the “regularity” is that this (currently) shows a “leap year” once every four years or once every five years,
whereas the Gregorian calendar shows once every four years or once every eight years:
chart 28
eTE
eVE_Gregorian3.png
Gregorian calendar
date and time of the "eccentric" Vernal Equinox
year increment → 1
month # of the eVE , 1st data entry = 3 (March)
time zone: 0 hrs from Greenwich
31
30
29
28
27
26
25
24
23
22
21
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
Gregorian calendar year
- 30 -
2125
2100
2075
2050
2025
2000
1975
1950
1925
1900
1875
1
Day of month
20
The different “scenarios” are best interpreted by the derivative of LOD, the rate that we are slowing, expressed as
milliseconds per century. In the spreadsheet I am currently working on scenario 14a, as I believe it is one of the most realistic:
chart 12c
ms_cy2.png
slowing of Earth's rotation
milliseconds per Julian century
Morrison & Stephenson (2004): constant 1.752 ms/cy
spreadsheet settings
2 r + ∑² -a b Sin(a t + c)
scenario 14a
3.200
3.000
2.800
2.600
2.400
2.000
1.800
milliseconds
2.200
1.600
1.400
1.200
1.000
25000
22500
20000
17500
15000
12500
10000
7500
5000
2500
0
-2500
-5000
-7500
-10000
-12500
-15000
-17500
-20000
-22500
-25000
0.800
calendar year
1.0 ms/cy is shown at −20,000 (20001 BC). It is supposed to represent the latter stage of the last ice-age. Our rate of
slowing is supposed to be much less during glaciations (ice-ages), as the sea levels are lower, and most of the tidal friction occurs
in the hydrosphere, which is mainly the oceans. From −20,000 to −3M the rate climbs slowly up to 1.4 ms/cy. This is intended to
show an average rate of 1.2 ms/cy during the last 3 million years. It is a value suggested by Lourens et al (2001), from their study
of core drillings in the Mediterranean Sea from 3 million years ago. This scenario then slowly increases to a more scientifically
expected rate of 2.4 ms/cy at −3.5M.
At −20,000 (the minimum of 1.0 ms/cy in the chart above) the Earth‟s shape is supposed to be in equilibrium. That is,
she no longer needs to shift any more solid earth-mass from the polar regions into the equatorial region to compensate for the
heavy ice loads at the poles and the lower sea level. The sharp increase to 3.0 ms/cy from −15,000 to −12,000 is supposed to
represent the melting of the glaciers and the rising of the sea level. Imagine Earth to be like a spinning ice-skater. When the huge
ice mass at the poles melts and sends the water mass towards the equator, this is like a spinning ice-skater slowly extending her
arms, decreasing her measured moment of inertia and causing her to spin more slowly. The peak at 3.0 is supposed to represent
the epoch in which the movement of water mass was at a maximum, and before “post-glacial rebound” was starting to have much
effect.
- 31 -
The period between −12,000 and −8,000 is supposed to represent a combination of continued sea level rising and postglacial rebound. From −8,000 to +22,000 (22,000 AD) the curve is supposed to represent our current era of post-glacial rebound
(in which the ice-skater is drawing her arms inward causing her to spin faster). The Earth‟s shape is shown to be back in
equilibrium at +22,000, no longer needing to shift any more mass from the equator to the poles. It then continues on into the
future at a scientifically expected rate of 2.4 ms/cy. The minimum of 1.3 is supposed to represent the epoch (about +5,000 in this
scenario) in which the rate of movement of solid earth-mass (from post-glacial rebound) is at a maximum.
These rates and the epochs in which they are shown to occur are of course speculative. Due to a study of ΔT given by
Morrison and Stephenson (2004), and the polynomials of Espenak and Meeus (from NASA) following that study, I am also
working on a “sum of sines”, charted above as “2 r + ∑² -a b Sin(a t + c)”. It comes from the derivative the first two terms of the
preliminary results of an LOD series I am working on, charted below as “2 r t + ∑¹⁴ b Cos(a t + c)”, a sum of cosines:
chart 37c
LOD2.png
Length of solar day
2 r t + ∑¹⁴ b Cos(a t + c)
2 r t + ∑⁴ b Cos(a t + c)
86400.03
2 r t + ∑² b Cos(a t + c)
86400.02
Espenak and Meeus polynomials
86400.01
Morrison & Stephenson (2004)
Long term derivative
86400.00
86399.99
86399.97
86399.96
86399.95
seconds (TD)
86399.98
86399.94
86399.93
86399.92
3000
2500
2000
1500
1000
500
0
-500
-1000
-1500
-2000
86399.91
calendar year
You can barely detect from this chart why I have chosen to use the Espenak and Meeus polynomials only until −404,
even though they are given to −500. The next chart exemplifies this more clearly, as Morrison and Stephenson have given data
(to be used with caution) until −2000:
- 32 -
chart 12b
ms_cy1.png
slowing of Earth's rotation
milliseconds per Julian century
Morrison & Stephenson (2004): constant 1.752 ms/cy
Espenak and Meeus polynomials
2 r + ∑¹⁴ -a b Sin(a t + c)
2 r + ∑⁴ -a b Sin(a t + c)
2 r + ∑² -a b Sin(a t + c)
9.000
8.000
7.000
6.000
4.000
3.000
2.000
milliseconds
5.000
1.000
0.000
-1.000
3000
2500
2000
1500
1000
500
0
-500
-1000
-1500
-2000
-2.000
calendar year
Scenario 9b charted below uses the accuracy of the “sum of sines” for about ±20,000 years, then shows a constant rate of
1.75 ms/cy during the past 4.5 million years. My reasoning is this: if the average rate during the past 3 million years (from
Lourens et al 2001) was 1.2 ms/cy and then the rate suddenly increased to as much as 2.6 ms/cy (from Laskar et al 2004), it
would probably take at least another 1½ million years for the scenario‟s rate of 1.75 to be showing the “correct” LOD at −4.5M.
At −4.5M it simply steps up to the more scientifically expected rate of 2.5 ms/cy. Not being able to predict that the future will be
much different than the past, it simply shows 1.75 ms/cy for 4.5 million years into the future, where it steps up to 2.5 ms/cy.
- 33 -
chart 12
ms_cy4.png
slowing of Earth's rotation
the change in LOD = the change in Days per Year per (Days per Year)^2
milliseconds per century (solar day)
scenario 9b, 2 past era's
9.00
8.00
7.00
6.00
4.00
3.00
2.00
milliseconds
5.00
1.00
0.00
-1.00
25,000
20,000
15,000
10,000
5,000
0
-5,000
-10,000
-15,000
-20,000
-25,000
-2.00
calendar year
chart 12
ms_cy3.png
slowing of Earth's rotation
the change in LOD = the change in Days per Year per (Days per Year)^2
milliseconds per century (solar day)
scenario 9b, 2 past era's
9.00
8.00
7.00
6.00
4.00
3.00
2.00
1.00
0.00
-1.00
calendar year
- 34 -
6,000,000
5,000,000
4,000,000
3,000,000
2,000,000
1,000,000
0
-1,000,000
-2,000,000
-3,000,000
-4,000,000
-5,000,000
-6,000,000
-2.00
milliseconds
5.00
Scenario 9c is the same as 9b, but uses a simpler fraction to represent the length of the average tropical year (aty). 9b
uses 365.2421378, while 9c uses 365 77/318 (365.24213836…). This is “allowed” for this spreadsheet, due to the fact that we
don‟t really know the true length of the long term average sidereal year (asy). This scenario uses 365.2563624… as the average
sidereal year so that the average precession rate is still 50".4712/Julian year. The spreadsheet requires you to enter the fractional
portions of asy and aty as proper fractions. 14a and 9b use 365 121069/500000 for the average tropical year so that the nicely rounded
decimal representation of 365.2421378 is achieved. Using proper fractions is just something I enjoy, so you‟ll have to put up
with this little “quirk” of mine. Rather than “rounding” to seven decimal places, I have fun “rounding” to the simplest fraction
that also rounds to the same seven decimals. A few scenarios use 365 1409/5819 for the average tropical year (= 365.24213782…).
Since the true average length of the sidereal year remains unknown, I allow myself values between 365.256361 and
365.256363. At the average precession rate of 50".4712/Jyr (Julian year) this results in an average tropical year between
365.242137 and 365.242139. I then like to make a list of some of the simplest fractions that fall within this range. This is easiest
to see from the “number line” that I use:
L4…
L3
L2
…2 …
7
(1:1)
L1
… 15
(4:1)
1
4
(3:1)
…3
8
… 512 …
… 7 12 …
(1:1)
(1:1)
2
5
3
7
…
…4
(2:1) (1:1) (1:2)
1
3
(2:1)
7
3
5
5
8
…
…5 …
7
(2:1) (1:1) (1:2)
1
2
(1:1)
(1:1)
2
3
(1:2)
3
4
(1:3)
4 …
5
(1:4)
L0
0
1
irrationals
I split the number line into specific separate number lines. L0 (level zero) is the integers. L1 (level one) contains the
simplest fractions. These are the same “set” of fractions comprising the “outside shell” of the “farey tree”. It helps considerably
to understand the construction of the farey tree by means of a farey sum in order to easily grasp this number line structure that I
use. It is in fact a “flattened” farey tree, pushed flat (and inverted). The values in parentheses below each fraction I refer to as the
ratio (x : y), where x designates the number of “left-hand parents” and y designates the number of “right-hand parents” used to
construct the fraction by means of a farey sum. For example, 4/7 is labeled (2:1). It belongs to the “set” of level 2 fractions
between 1/2 (the left-hand parent of this set) and 2/3 (the right-hand parent). It uses the left-hand parent twice and the right-hand
parent once (in a 2:1 ratio) to construct the fraction by means of a farey sum: ⁄
⁄
⁄
⁄ . All of the “sets” of fractions
on L2 (level two) are the “subshells” of the farey tree that are once removed from the outside shell. Each set of level 3 fractions is
a subshell twice removed, etc. Each “(1:1)” value is the mediant of each set, and is the apex of each “shell” of the farey tree. I
can “zoom in” to any portion of this structure to view different portions of this “number line”. This view zooms in to a portion
around our upper and lower limits for the average tropical year:
- 35 -
L6…
L5
L4
… 123
L3
23
54
318
(2:1)
223
(1:1)
85
116
147
351
479
607
(1:2)
(1:3)
(1:4)
…
31
128
95
(1:2)
L2
7
L1
77
100
413
508
(4:1) (3:1)
(1:3)
8
29
(1:6)
1
33
(1:7)
1
5
365.242137
(4:1)
365.242139
4
(3:1)
This view zooms in a little closer:
L7…
.
L6
.
1201
2341
4960
(1:1)
9668
(1:1)
L5
… 408
1685
(1:4)
L4
100
L3
L1
562
2321
(1:6)
639
2639
(1:7)
…
… 1209
4993
(15:1)
77
413
1132
4675
(14:1)
1055
4357
(13:1)
…
54
318
(2:1)
223
(1:1)
31
128
95
(1:2)
(1:3)
7
8
29
(1:6)
1
L0
2003
(1:5)
(3:1)
23
L2
485
33
(1:7)
1
5
4
(3:1)
(4:1)
365
366
365.242139
365.242137
- 36 -
This begins a list of some of the simplest fractions within our range. When one is familiar with the construction of the
farey tree it is quite simple to determine all of the fractions corresponding to the points not labeled above. The top portion of
sheet f! in the spreadsheet is a tool I use to navigate this number line, and is separate from the main portion of the spreadsheet. It
uses the following algorithm to find the placement of any rational number within this structure:
.
L = level, I = number rounded down to the nearest integer, R = remainder
Input: any rational number  N
D
1.
2.
3.
4.
5.
L = 0 , N D I R
, integer = I
D
if R = 0 , rational number = integer , end.
else num = R , den = D , a = 0 , b = 1 , c = 1 , d = 1
L  L  1 , x  b  den  d  num , y  c  num  a  den
if x  y , (GCF  x) , x  1 , y  1 , go to 5.
else if x  y , y  x  I  R
x
if R  0 , (GCF  x) , x  1 , y  I , go to 5.
else a  a  b  I , b  a  b , c  c  d  I , d  c  d , go to 3.
else x  y  I  R
y
if R  0 , (GCF  y) , x  I , y  1 , go to 5.
else b  a  I  b , a  a  b , d  c  I  d , c  c  d , go to 3.
integer  integer
numerator  a  x  b  y
denominator  c  x  d  y , end.
.
(from line 5.) is a fraction‟s “definition”, and will always be in it‟s proper “reduced” form, the same as all of the
fractions from the farey tree. (GCF) indicates the point in the algorithm where the greatest common factor can be determined,
even though it is not used to display the fraction in its proper reduced form.
is used. It is not necessary to understand this
in order to use the spreadsheet. The bottom portion of sheet f! does use this algorithm and is part of the main spreadsheet. I
could have more easily used a “continued fraction” algorithm for the bottom portion, but I just recently learned about continued
fractions, so for now I‟ve left it as it is. A continued fraction finds the closest fraction (to the number in question) on each
successive level of this structure, whereas this “construction algorithm” finds the two that are closest, a/c & b/d . It is of course
between the two, and in fact constructed (by means of a farey sum) of the two it is between, on each level. The Excel file
ConstructionAlgorithm(xn).xls is the same as the top portion of sheet f!, but with the precision of Xnumbers (a required Add-In
available from Foxes Team that can calculate to a precision of up to 250 significant digits), for anyone interested in delving into
deeper levels of this structure.
I am sure there are other “oddities” in this spreadsheet, as I am not an expert user of Excel (e.g. I just discovered the
“formula auditing toolbar”: how did I ever get along without it?!). I am also not by any means an expert in astronomy, so that
some things are undoubtedly “shortcuts”, while many more things are probably on the order of “taking the long way around”:
- 37 -
For example, when I first started this project I posed the question to myself: “if I were an ancient astronomer, how would
I build a calendar?” First I would ask the wisest and most knowledgeable high priests and astronomers all that they knew about
the length of the year and the lengths of the seasons. I would undoubtedly have found out that while the shortest season was
currently the fall season, our ancient ancestors might have measured the summer to be shortest (we had extensive records), and in
the future winter will probably be shortest (as it is today), and on and on. Desiring this calendar to work for many millennia, it
would become obvious that “equal quarters” would be the best solution, so that the solstices and equinoxes (the divisions of the
seasons) would differ from the calendar‟s equal “divisions of the quarters” by only a few days, sometimes sooner and sometimes
later, depending upon the changing lengths of the seasons. But how would we start this calendar “correctly”? Simply set the
vernal equinox on the same date as its representative “quarter division”? It seems that it should fall a day or so ahead or behind of
its calendar representation, just the same as the other tropical events. Today we can use the equation of center and convert it to
time, but the ancient astronomer would not have known Kepler‟s equation, and might not have deduced that we were in orbit, let
alone elliptical orbit! But simply from a “list” of the different lengths of the seasons he could easily have deduced the following
relationships:
W
V
S
F
=
=
=
=
winter
spring
summer
fall
eVE 
F  3W  3V  S
8
eSS 
W  3V  3S  F
8
eAE 
V  3S  3F  W
8
eWS 
S  3F  3W  V
8
And he would have been absolutely correct! You can check that these relationships are exact by making a “list” of the
lengths of the seasons, at any given instant (epoch), by use of Kepler‟s equation. If you use the “true” lengths of the seasons
(from our “perturbed” orbit), you will find that the ancient astronomer‟s method gives somewhat of an approximation to the
angular difference (true “perturbed” planet) − (mean planet), while the equation of center gives the angular difference (true
“unperturbed” planet) − (mean planet). His determination of the equation of center at a solstice or equinox is
ΔeTE
W+V+S+F
× 360°.
And of course I use this method in the spreadsheet, simply because it is fun! I also understand it better: calculating the lengths of
the seasons by use of Kepler‟s equation I simply look at as “magic”, as I have absolutely no idea how he came up with his famous
formula. I also look at Laskar‟s approximation formulae as “magic”. From my meager point of view (in regards to his
formulae‟s use of complex numbers):
“This high priest in France
makes all numbers dance…
to the square root of negative one!”
- 38 -
There is only one more (extreme) “oddity” that requires discussion, and this involves the unknown changes to our semimajor axis (and therefore the unknown changes to the length of the sidereal year). From (what portions I can understand of)
Laskar‟s publications, his model of the solar system is a “second order general theory”, and that it would require a third order
theory to accurately detect the changes to the semi-axes of the planets. And that it would be right next to impossible to even think
about the integration of the equations of motion to the third order with respect to the masses of the planets for millions and
millions of years! In order to derive his polynomial description for ±10,000 yrs (Laskar, 1986), (from what I can understand) the
third order from existing classical theory was considered. My first idea to extend the known changes beyond ±10,000 came from
the title of the article “fitting a line to a sine” by J Laskar and J-L Simon, 1988. My intention was to show (by a simple sine
wave) the semi-axis returning to an “average” at about ±35,000 and then simply continuing with the constant, unchanging average
beyond that range. This, however, produced an ugly “blip” in some of the charts: whereas the frequency of the integral is the
same, it is exactly out of phase. To continue with this same sine function for millions of years removed the “blip”, but was
endlessly boring, showing the same frequency and amplitude year after year after year. Simply to relieve my boredom, I decided
to use the frequencies from the inclination variables from the quasi-periodic approximation formula from Laskar et al (2004), as
they also show frequencies of about 70,000 yrs. It is important to note that this is not a scientific method, it is more on the order
of “pick a formula, any formula” and force it to fit. The amplitudes have been adjusted in an extremely unscientific manner,
simply to show some slight variation, and to “agree” with any reasonable choice for a “long term average sidereal year”. Once
again, the description of the sidereal year in this spreadsheet is to be considered fictitious beyond ±10,000 yrs. It is used merely
for interest, to depict a “possible” history of amplitudes, and to continue calculating in a “correct” manner, which would include
our changing semi-major axis. If you would rather not use this fictitious rendition, it is very easy to direct the spreadsheet to
calculate with the average sidereal year, and ∆bTE will simply always equal zero. The additional “error” in calculating a solstice
or equinox, during ±10,000 yrs from J2000, without using the “correct” length of the sidereal year (from Laskar 1986) is only
about 1½ hours of time.
chart 3
semi-axis.png
Earth semi-major axis
in astronomical units (AU)
semi-major axis
1.000001053700
1.000001048975
1.000001044250
1.000001039525
1.000001034800
1.000001030075
1.000001025350
1.000001020625
1.000001011175
1.000001006450
1.000001001725
1.000000997000
1.000000992275
1.000000987550
1.000000982825
calendar year
- 39 -
100,000
80,000
60,000
40,000
20,000
0
-20,000
-40,000
-60,000
-80,000
-100,000
1.000000978100
AU
1.000001015900
chart 5
sid_yr.png
sidereal year (TD)
sine wave sy
fictitious rendition
Laskar 1986 ( ±10 T )
365.256382
365.256377
365.256372
365.256367
days
365.256362
365.256357
365.256352
100,000
80,000
60,000
40,000
20,000
0
-20,000
-40,000
-60,000
-80,000
-100,000
365.256347
365.256342
calendar year
chart 20
delta_bTE2.png
∆bTE (TD)
change in the time of a Tropical Event due to the changing sidereal mean motion
∆bTE in days of 86400 SI seconds (TD)
0.25
0.20
0.15
0.10
0.05
days
0.00
-0.05
-0.10
-0.15
-0.20
calendar year
- 40 -
100,000
80,000
60,000
40,000
20,000
0
-20,000
-40,000
-60,000
-80,000
-100,000
-0.25
∆bTE involves the integral of the (changing) sidereal year. At ~1000 AD the sidereal year is shown to be at the average,
while ∆bTE is at a minimum, due to the previous 35,000 years, during which time the sidereal year was below average.
References
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Canup, R., 2004, Origin of the Terrestrial Planets and the Earth-Moon System, Physics Today 57, (April 2004) 56-61 (pdf)
Deubner, F.-L. : 1990, Discussion on Late Precambrian tidal rhythmites in South Australia and the history of the Earth‟s rotation,
J. Geol. Soc. London, v. 147, no. 6, 1083-1084
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Walker, J.C.G. & Zahnle, K.J., 1986, Lunar nodal tide and distance to the Moon during the Precambrian. Nature, 320, 600–602
Williams, G.E., 1989, Tidal rhythmites: geochronometers for the ancient Earth-Moon system. Episodes, v.12, no. 3, 162–171
Williams, G.E., 2000, Geological constraints on the Precambrian history of Earth‟s rotation and the Moon‟s orbit: Reviews of Geophysics, 38,
37–59 (pdf)
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Downloads and Links
Notice: due to a Windows security measure all of the zipped files that you download might be “blocked”. Before unzipping rightclick the .zip, choose “Properties”, then in the “General” tab click “Unblock”, “Apply”, and “OK”. Otherwise you might need to
perform these steps on each individual file (especially .chm helpfiles).
If you are interested in using the Excel Add-In for the interpolation of Laskar‟s La93(0,1) data from −20M to +10M, you
can read a description of the functions in OrbitalElements1.7.pdf, and download OE.xlam1.7 for Excel 2007/2010, or OE.xla1.7
for 97/2000/XP/2003. This newer version 1.7 has a bug fix for Excel 97 & 2000. If you would like to use the spreadsheet that
created the charts on this webpage, you can read the “short” instructions to start, and download ECT1.6.xlsm for Excel
2007/2010, or ECT1.6.xls for Excel 97/2000/XP/2003.
If you are interested in the method being used to calculate the
approximate times of solstices and equinoxes, download TEequation.pdf. This updated pdf also contains a new ΔT section with
readjusted “double precision” formulae for the entire time span −4.5G to +5G. There are two additional Add-Ins required to use
this workbook. The 1st is “Analysis Toolpak” (on your Office CD if it isn‟t already installed), in order to read the function
“SERIESSUM( )”. The 2nd is “Xnumbers v.6.0”, a remarkable math tool for Excel.
All of these Add-Ins are being distributed as open source, fully accessible to modify and redistribute as you desire, as
long as you give credit to the original authors, Leonardo Volpi from Foxes Team for the Xnumbers Add-Ins, and John Beyers for
the Orbital Elements Add-Ins.
For any other questions or comments, or (if you aren‟t too comfortable with Excel and/or all of these Add-Ins) to request
custom charts (in Word), and/or data to create them (in Excel) for specific time spans and parameters: email to
[email protected]. Please be very patient to receive a reply, as I am usually at home, a “little cabin in the woods”,
with my dog Tuxedo, far from the city lights (and an internet connection).
Thanks for your interest,
P.S. to help bring an “ancient astronomer” (or “old carpenter who uses fractions”) up to date in the modern world, don‟t
hesitate to donate to the solar system (electric) and satellite (internet connection) fund!
(back to top)
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Other interesting websites:
http://www.imcce.fr/Equipes/ASD/person/Laskar/Laskar.html
Laskar‟s website
ftp://ftp.imcce.fr/pub/ephem/sun/la93
has the ASCII files with the data (used in this spreadsheet) from La93(0,1), given every thousand Julian years,
−20M to +10M from J2000. They are in a compressed format, *.ASC.Z
http://adc.gsfc.nasa.gov/adc-cgi/cat.pl?/catalogs/6/6063/
has the same ASCII files in a different compressed format, *.ASC.dat.gz
ftp://cdsarc.u-strasbg.fr/pub/cats/VI/63
has the same, larger uncompressed ASCII files, *.ASC, easily imported to EXCEL
http://www.bowdoin.edu/~rdelevie/excellaneous/
Advanced Excel for scientists and engineers by Robert de Levie
http://www.astrosociety.org/education.html
Astronomical Society of the Pacific
http://www.astrosociety.org/education/publications/tnl/45/globe1.html
great educational site for astronomy
http://www.intute.ac.uk/sciences/astronomy/
another good educational site
http://www.astronomynotes.com/
and another
http://www.astrophysicsspectator.com/
The Astrophysics Spectator
Great learning site
http://www.amsat.org/amsat/keps/kepmodel.html
Keplerian Elements Tutorial
http://www.daviddarling.info/encyclopedia/C/celestial_mechanics_entries.html
awesome encyclopedia of celestial mechanics
http://individual.utoronto.ca/kalendis/seasons.htm
great charts of the seasons and years! Also a good variety of astronomical calendars
http://www.iol.ie/~geniet/eng/season.htm
lengths of astronomical seasons
http://webexhibits.org/calendars/
good info on astronomy and calendars
http://www.maa.mhn.de/Scholar/calendar.html
Astronomical Calendars: good basic info
http://www.ecben.net/calendar.shtml
huge list of different calendars
http://www.calendarhome.com/clink/general.html
good general calendar information
http://www.calendarhome.com/clink/reform.html
calendar reform
http://personal.ecu.edu/mccartyr/calendar-reform.html
calendar reform
http://www.astro.uiuc.edu/projects/data/
fun demos and animations
http://seds.lpl.arizona.edu/nineplanets/nineplanets/nineplanets.html
A Multimedia Tour of the Solar System
http://www.badastronomy.com/
interesting and widely referenced astronomy site
Send any correspondence to:
Steve Beyers
145 Electric Ave
Seal Beach, CA 90740
USA
last updated December 5, 2010
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