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# Mean lunar and solar periods

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Below the mean values of the most important lunar and solar periods are provided. A web site that also elaborate on cycles is here.

to calculate the periods for Jan. 1st 12:00 UTC [Julian calendar] (fill in the astronomical year and between -4000 [4001 BCE] and 2500 [2500 CE])

Most periods presented, which are shorter or equal to a year's period, have to be averaged over a few periods/cycles (due to variations in orbital periods). Values which are much bigger (like the Precession of the equator) can of course not be seen in one human generation, so one has to do extrapolation in these cases (assuming that it is a circular path).

### Reference system

The Aristotle's view of the universe (earth in middle and moon, sun and firmament of fixed stars around it) is being used. This is of course not the present day astronomical view, but it describes the view perhaps quite well for neolithic man.
Of course the orbits of the earth(/sun) and moon are not circular, but for this discussion the approximation of a circle is accurate enough (see Conclusion). The stars are also not fixed (our solar system moves through space), but for this discussion they are assumed fixed.
I am using the following reference systems:
1. the earth-sun system
Astronomically determined by how the sun is observed from the earth (sidereal period and geocentric). Of course there are small changes due to other planets/moon (which are purely related to Newton/Kepler laws).
The reference point here are the fixed stars.
2. the earth-moon system
Astronomically determined by how the moon is observed from the earth (sidereal period and geocentric). Of course there are changes due to other planets/sun (which are purely related to Newton/Kepler laws).
The reference point here are the fixed stars.
3. the time system
All cycles are expressed in the stable time system of Terrestrial Time (TT: Years, Days and Hours; defined by standard organization). Some cycles though are also expressed in Solar days (UT, Universal Time), this makes comparing the values easier with some sources (like Thom) and perhaps it is more related to the perception of people in former times. For this web page the Terrestrial Time and Ephemeris Time (ET) are equivalent.
The relation between TT and UT is DeltaT: TT = UT + DeltaT.
One can easily change a reference system like taking the stars as the reference instead of the sun-earth system or using the Mean Solar day as the basis of time reckoning instead of a scientific reference (the atomic clock). This would also be a valid starting point (as long as one makes it known).

#### Calendar convention

For input date I use Julian calendar dates, because:
• The input date () is normally coming (in the archaeocosmology environment) from monument construction dates that are either carbon dates or guesses. I assume that these dates have an accuracy of around 50 Years.
• Under the assumption that carbon dating uses the Julian year (or Year) definition (365.25 Days), there is no error introduced by this conversion.
• The difference between Gregorian calendar date and Julian calendar date is at 2000 CE (over 500 years since start of Gregorian calendar) around 4 Days.
So even if one inputs a Gregorian calendar date (for dates from 1582 CE to now) into the Change date field, the error in the periods is still very small (not visible in the 6 decimal).

## Orbital periods

Mean orbital periods of the earth and moon are (be aware of possible variations when looking at actual values):
• Earth (reference fixed stars are reference point)
• Eccentricity cycle
~ - ~ and ~ Years (Berger, [1991], page 307)
Eccentricity: ~5
Determined by
: Changing eccentricity (e) of the earth's path around the sun.
Seen in the climate
: Duration can more or less be determined by time periods between the glacial maxima during the Quaternary (the '100 kyear' cycle of the Milankovitch cycles)
• Earth's perihelion cycle
~, ~ and ~ Years (Loutre, M.F., pers. comm. [2003])
Determined by: Changing Earth-Sun perihelion angle with regard to the stars.
• Obliquity cycle
~ Years
Obliquity: ~ Degrees6
Determined by: The angle between the earth-sun plane and equator of the earth.
Seen in the sky: Duration can be measured by determining when the sun gets back to the same winter solstice position on the horizon.
Seen in the climate: Determines the 41 kyear cycle in the Milankovitch cycles
At least recognized in:
480 BCE by Oenopides (Greek), obliquity has been recognized, its cycle not.
• Precession of the equator (luni-solar precession or precession of equinoxes or equinoctial precession or platonic year)
~ Years7
Determined by: The wobble of the earth's axis measured with regard to stars.
Seen in the sky:
Duration can be measured by determining when the turning point ('pole' star) of the sky is again at the same star position.
At least recognized in: 150 BCE by Hipparchus (Greek) [Wilson, 1997, page 13] or earlier.
• Sunspot cycle
• ~ Years (is quite variable: between 9 and 13 Tropical years)
• Sidereal year
~ Days1 ~ Solar days
Seen in the sky: Duration can be measured by determining when the sun-earth conjunction arrives at the same star (on the ecliptic).
At least recognized in: 1700 BCE by Babylonians [Wilson, 1997]
• Year (julian year)
Days
Seen in: Astronomy (John Herschel, 1849 CE)
At least recognized in: 1583 CE by Julian Scaliger (France)
• Day
Hours ~ Solar days
• Earth's rotationnote
~ Hours
Seen in the sky: The length of time for the stars to return to the meridian
Seen in the sky: Duration can be measured by determining when the sun is again at the same point against the star.
• Hour
Sec
Determined by: Sec (SI Sec) is measured by atomic time at sea level and standardized in Terrestrial Time.
• Moon (reference fixed stars are reference point)
• ICRS Lunar nodal cycle
• ~ Years3
Seen in the sky: Duration can be measured by determining when the ascending lunar node is against at the ICRS equinox of J2000.0.
Remark: Sometimes this cycle is referenced from the equinox of date (on this website called: Lunar nodal cycle so without the 'ICRS'), which gives a slight different!
• ICRS Lunar apse cycle
~ Years8
• Determined by: The rotation of the long axis of the Moon's elliptical orbit reference from the ICRS equinox of J2000.
Remark: Sometimes this cycle is referenced from the equinox of date (on this website called Lunar apse cycle, so without the 'ICRS'), which gives a slight difference!
At least recognized in: 1618 CE by Kepler (German) [Wilson, 1997]
• Sidereal month
~ Days4 ~ Solar day
Seen in the sky: Duration can be measured by determining when the moon returns to the same star back ground position from ICRS equinox of J2000.
• Lunar orbit's inclination
~ Degrees
Determined by: This is the inclination between lunar orbit's plane and the ecliptic plane. This seemed not to have changed over time.

## Other 'simple' periods: Harmonic Sum

The other lunar and solar mean periods can be calculated from the above in a simple way. The basic method of calculating them is in literature sometimes called: synodic period of two elements. A reference is Copernicus (De revolutionibus orbium coelestium, [1543]) who devised a mathematical formula to calculate a planet's sidereal period from its synodic period. This formula can also be found in Text-Book on Spherical Astronomy, W. M. Smart, 1965, 5th edition, equation 158, on page 131.
But I propose a more generic term Harmonic Sum: $HS ( A , B ) = 1 1 A + 1 B = A ⋅ B A + B$, as put forward by Dr. Math on my query (which is building upon the concept of the Harmonic mean).

A derivation of this formula is as follows (Draconic month is taken as an example):

The two circular movements (of the Lunar nodal cycle (A) and the Tropical month (B)) make that the Draconic month is shorter than the Tropical month. Lets assume that the Draconic month is x days.
The moon has moved $360 ⋅ x T r o p i c a l m o n t h$ degrees and the Lunar nodal cycle has moved: $360 ⋅ x L u n a r n o d a l c y c l e$ degrees.  Both are after x days at the same position, so:

$360 ⋅ ( 1 - x T r o p i c a l m o n t h ) = 360 ⋅ x L u n a r n o d a l c y c l e$
$1 - x T r o p i c a l m o n t h = x L u n a r n o d a l c y c l e$
$1 T r o p i c a l m o n t h + 1 L u n a r n o d a l c y c l e - 1 x = 0$
$1 T r o p i c a l m o n t h + 1 L u n a r n o d a l c y c l e - 1 D r a c o n i c m o n t h = 0$
In general:
$1A + 1 B - 1 C = 0$
$H S ( A , B ) = C = A ⋅ B A + B$

Another example, now using an old fashioned watch, so with a minute and hour hand;-)
The minute hand takes 1 hour (C) per revolution, while the hour hand takes 12 hours (B) to make a revolution.
The question is now how much time does it take when minute and hour hand are at the same point (A):

$1 A + 1 B - 1 C = 0$
$H D ( B , C ) = A = B ⋅ C B - C = 12 ⋅ 1 12 - 1 = 1 h 5.45 min$

The above formula works analogous for all of the below mean periods (be aware of possible variations when looking at actual values):

• Earth
• Climatic precession (precession of perihelion, Berger, [1991], page 307)
Calculated by Earth's perihelion cycle (A) and Precession of the equator (B)
Important cycles: ~ and ~ Years (=A*B/(A+B)), composite ~ Years9
Time of perihelion: ~ Days
Seen in the sky: Duration when the perihelion and equinox coincidence
Seen in the climate: This is the '21 kyear' cycle in the Milankovitch cycles
At least recognized in: 1618 CE by Kepler (German) [Wilson, 1997]
• Tropical year (or solar year)
Calculated by Precession of the equator (A) and Sidereal year (B)
~
Days ( =A*B/(A+B)) ~ Solar days
Determined by: The tropical year is the time needed for the Sun's mean longitude to increase by 360° (Danjon, A. [1959])
Seen on the horizon: Duration can be measured by determining when the sun is every second time at its equinox position (not fully correct, see here).
At least recognized in: 2800 BCE by Egyptians [Wilson, 1997, page 16]
• Anomalistic year
Calculated by solar Climatic precession (A) and Tropical year (B)
~ Days ( =A*B/(A-B)) ~ Solar days
Seen in the sky: Duration can be measured by determining when the solar disc size returns to the same size.
Seen at sea: Duration can be measured by determining when an extra higher/lower tide than normal is happening.
• Ecliptic year (or eclipse year)
Calculated by Lunar nodal cycle (A) and Tropical year (B)
~ Days ( =A*B/(A+B)) ~ Solar days
• Solar day (LOD: length of day)
Can be calculated by Tropical year (A) and Sidereal day (B): (=A*B/(A-B))
~ Hours2
Seen in the sky: The length of time for the Sun to return to the meridian
Seen in planetary programs: The calendar dates one inputs is strongly related to Solar day (the computer program calculates the real Days by incorporating DeltaT)
At least recognized in: 1798 CE by Laplace (Beutler 2004)
• Sidereal daynote
Calculated by Precession of the equator (A) and Earth's rotation (B)
~ Hours(=A*B/(A+B))
Determined by:
The length of time for the vernal equinox to return to the meridian.
Note: I would call this Tropical day, but it is called Sidereal day in Explanatory Supplement to the Astronomical Almanac ([1992], page 48) and through feedback of people on the HASTRO-L (you need to use your HASTRO-L membership e-mail address and password). And in the same spirit; the Explanatory Supplement calls Earth's rotation what I would call Sidereal day.
• Moon
• Lunar nodal cycle
Calculated by Precession of equator (A) and ICRS Lunar nodal cycle (B)
~ Years (=A*B/(A-B))
Seen in the sky: Duration can be measured by determining when the ascending lunar node is against at the mean equinox of date.
• Lunar apse cycle (or apsis or line of apsides cycle)
Calculated by Precession of equator (A) and ICRS Lunar apse cycle (B)
~ Years (=A*B/(A+B))
Seen in the sky: The rotation of the long axis of the Moon's elliptical orbit reference from the mean equinox of date.
• Lunation length period
Calculated by Lunar apse cycle (A) and Tropical year (B)
~ Days ( =A*B/(A-B))
Seen in the sky: The Babylonians saw this by measuring the difference between rise and set times of Sun and Moon around Full Moons (the Lunar Four: ŠÚ, na, ME and GE6)
Defined as: Time between two consecutive alignments of the major axis in the direction of the Sun..
Seen in the sky: Duration can be measured by determining when the lunation length has maximum or minimum length.
At least recognized in: 600 BCE by Babylonians [Brack-Bernsen, 1999]
• Synodic month (or lunar month)
Calculated by Sidereal year (A) and Sidereal month (B) (or both A and B are Tropical)
~ Days ( =A*B/(A-B)) ~ Solar day
Defined as: Period between the repetition of the same position between earth, sun and object.
Seen in the sky: Duration can be measured by determining when the moon is again in the same lunar phase.
At least recognized in: 1700 BCE by Babylonians [Wilson, 1997]
• Day on the moon
Calculated by Sidereal year (A) and Sidereal month (B) (or both A and B are Tropical)
~ Days ( =A*B/(A-B))
• Anomalistic month
Calculated by Lunar apse cycle (A) and Tropical month (B)
~ Days ( =A*B/(A-B)) ~ Solar day
Seen in the sky: Duration can be measured by determining when the lunar disc size returns to the same size.
Seen at sea: Duration can be measured by determining when a higher/lower tide than normal is happening.
• Moon's rotation
Calculated by Sidereal  month
~ Days ~ Solar day
Seen in the sky: The moon keeps facing to the earth with more or less the same surface.
• Tropical month
Calculated by Precession of equator (A) and Sidereal month (B)
~ Days ( =A*B/(A+B)) ~ Solar days
Determined by: A full revolution of the moon around the earth reference from equinox of date (equinoxes are the points of the object's path that cross the earth's equator plane)
Seen on the horizon: Duration can be measured by determining when the moon is again at a maximum (or minimum) declination position.
• Draconic month (or nodical month)
Calculated by Lunar nodal cycle (A) and Tropical month (B)
~ Days ( =A*B/(A+B)) ~ Solar day
Seen in the sky: Duration can me measured by determining when the moon returns every second time to its node on the ecliptic (the nodes are the points of the object's path that cross the ecliptic plane).
• Lunar day (or Diurnal tide) (which is different then Day on the moon!)
Calculated by Tropical month (A) and Sidereal day (B)
~ Hours ( =A*B/(A-B))
Seen at sea: Duration can be measured by determining when every second time high/low tide is happening.

## Conclusion

The Harmonic sum formula calculates the values for solar and lunar periods very good, because the calculated periods have the same value up to at least the 5th decimal digit as the linked literature values.
The following relations are calculated in the above sections:

Blue: related to earth
Green: related to moon
Yellow: related to stars
Underlined: my chosen reference system of orbits

Remember that a Harmonic Sum/Difference relation e.g. between Tropical-year/Ecliptic-year/Lunar-nodal-cycle, is defined between the three of them, so one can chose any two to calculate the third one. The + or - near a cycle name tells if that cycle is derived from the Harmonic Sum or Harmonic Difference of the two other linked cycles. As said earlier one can taken any other reference scheme (but the relation picture stays the same).

The above only explains what you 'observe in real live'. The fact stands that these observations can be made (how 'simple' they perhaps can be explained)!

Other epoch values can be less accurate, because of missing proper time series of periods/cycles/orbits. See the notes.

## Other elaborated relations

Beside the above method of the Harmonic Sums (HS), two other methods like Fitting Duration Method (FDM) and Period of Inequality (PoI) are used in many discussions on cycles.
These methods have different uses:
1. to determine the composite period, due to two interacting periods A and B (e.g. interaction Precession of the equator and Tropical year: Sidereal year).
This can be done with the above HS formula ( HS(A,B) = A*B/(A+B) ). This HS formula pops simply up, when two or more periodic signals interact. When using the time durations (n*A and/or m*B) as arguments for the HS, one must have sound scientific proof.
2. to check if multiple instances (n) of one period (A) fit multiple instance (m) of another period (B); like the Metonic cycle
3. An Excel spreadsheet (Fitting Duration Method: FDM) has been made, which calculates n*A and m*B time durations and determines when the difference ( FDM(n,A,m,B) = Residue = n*A-m*B ) between these two time durations becomes small(est).
A and B can of course be calculated based on the above HS.
This might have some relation to Kuttaka, an Indian algorithm from around 500 CE, which is also close to the Chinese remainder principle.
4. Period of Inequality (PoI).
This calculates the periodicy related to the FDM: PoI = A*B/FDM(n,A,m,B) = HS(A/m,-B/n)
The Great Inequality cycle of Jupiter and Saturn can be determined with this method. On this very web page the PoI is not really used.
So it is important to understand that 1), 2) and 3) are quite distinct methods.

The FDM is used below for for instance Saros and Metonic cycles.
• Metonic cycle
Determined by Sidereal months and Tropical years (using FDM).
G. S. Hawkins [1966, page 130] quotes Oldfather's translation (page 40) of Diodorus Siculus (Book II):
"... They also say that the moon, as viewed from this island, appears to be but a little distance from the earth and to have upon it prominences, like those of the earth, which are visible to the eye. The account is also given that the god visits the island every nineteen years, the period in which the return of the stars to the same place in the heavens is accomplished; and for this reason the nineteen-year period is called by the Greeks the year of Meton. ..."
In above definition one sees a link with Sidereal month (same star background), but one can also see a link between the Metonic cycle and the Synodic month (the same lunar phase), as described in Geminus's book Elementa Astronomiae (first centruy BCE, translated version by Carolus Manitius, 1898, Greek-> German, page 120-121).
"Sie hatten nämlich durch ihre Beobachtungen festgestellt, daß in 19 Jahren 6940 Tage oder 235 Monate [...] (Es hat also das Jahr nach ihrer Rechnung 365 5/19 Tage)."
This duality is due to the close relation between Synodic and Sidereal month. A study on the fitting of these periods in this cycle can be seen on this URL.
~254 Sidereal months ( Days)~ 19 Tropical years ( Days)~ 235 (254-19) Synodic months ( Days)
Seen in the sky
: Duration can me measured by determining when the moon is again on the same date, at appr. the same star background or appr. the same phase.
At least recognized in
: 1300 BCE by Chinese
• Nutation cycle
Determined by the Lunar nodal cycle.
~ Years
Seen in the sky: an extra deviation from the wobble of the earth axis around the pole star.
At least recognized in:
• Lunar major/minor standstill limit period
Determined by Lunar perturbation, Lunar parallax, Lunar nodal cycle and Tropical year
The period (when viewing it along the horizon) between major (or minor) standstill limit is ~18, ~18.5 or ~19 Years (it is not precisely the stated number of years, it can vary with a few days/weeks, sometimes though the 19 solar tropical years is a Metonic cycle and the 18 solar tropical years is sometimes a Saros cycle). On average it is around Years (Lunar nodal cycle)
Seen in the sky: Duration can be measured by determining when the moon is at its maximum azimuth.
• Saros cycle (Chaldean cycle)
Determined by Synodic months, Draconic months and Anomalistic months (using FDM)
A study on the fitting of these periods in this cycle can be seen on this URL.
~ 223 Synodic months ( Days)~ 242 Draconic months ( Days)~ 239 Anomalistic months ( Days) (~ Tropical years or ~ Ecliptic years)
Seen in the sky: Duration can me measured by determining intervals between eclipses.
At least recognized in: 600 BCE by Chaldean (Babylonian, using Lunar Six)
• Octaeteris cycle
Calculated by Synodic month and Tropical years (using FDM).
~ 99 synodic months ~ tropical years (~ 5 Venus synodic years = 7.99 tropical years)
At least recognized in: 500 BCE by Cleostratus of Tenedos (Greece)
• Earth rotations per sidereal year
Calculated by Sidereal year (A) and Earth's rotation (B)
~ ( =A/B)
• Julian calendar year
~ Days = Solar Days
Seen in: Julian calendar (Julius Caesar, 48 BCE)
At least recognized in: 48 BCE by Sosigenes  (Greece)
• Gregorian calendar year
~ Days  = Solar days
Seen in
: Gregorian calendar
At least recognized in: 1582 CE by Pope Gregory XIII (Vatican)
• Vernal equinox year
Calculated by Tropical year, Eccentricity cycle, Climatic precession and Anomalistic year
Determined by: A full revolution (360°) of the sun around the earth related to the mentioned reference point.
At least recognized in: 1582 CE by John Dee (England)
Calculation based on Length astronomical season:
Vernal equinox year: ~ Days ~ Solar days
Summer solstice year: ~ Days
Autumnal equinox year: ~ Days
Winter solstice year: ~ Days
A comparison between other values can be seen on the Length astronomical season page.
• Babylonian year (Babylon) and the Vague year (Egypt)
= (Solar days)
• Lunar perturbation
Determined by Ecliptic year, Synodic month and Draconic month.
~ 0.5 ecliptic year (~ Days) and with maxima at quarter lunar phases and minima at full/new lunar phases
At least recognized in: 920 CE by Aboul-Hassan-Aly-ben-Amajour (Arab) [Thom, 1973]
• Length astronomical season
Calculated by Tropical year, Eccentricity cycle, Climatic precession and Anomalistic year
~ avg. 0.25 Tropical year (~ Days)
• Spring length: Days
Summer length: Days
Autumn length: Days
Winter length: Days
• Lunar parallax
Determined by the Anomalistic month.
~ Days
At least recognized in: 300 BCE by Aristarchus (Greek) [Wilson, 1997]
• Many eclipse cycles
The above relations can be depicted with the following picture:

### Relations within Metonic cycle

Difference of number of Synodic and Sidereal months in the number of Tropical years. A numerical evaluation:
• The difference between Synodic and Sidereal months is (1+0.0808) (determined as above).
• 0.0808 is equal to Synodic month/Tropical year (1/12.37).
• n Tropical years equals to 12.37*n Synodic months and thus the number of Sidereal months equals to 12.37*n*(1+1/12.37) = 12.37*n + n = 13.37*n
• So the difference between Synodic (12.37*n) and Sidereal months (13.37*n) is equal to the number (n) of Tropical years.
A more formal way of the proof:
• A Synodic month is:
1/'Synodic month'= 1/'Tropical month' - 1/'Tropical year'
• Because Sidereal month is numerically almost the same as Tropical month, one can also write it as:
1/'Synodic month'= 1/'Sidereal month' - 1/'Tropical year'
• If we express all periods not in Days but in Tropical years we get:
1/'Synodic month'= 1/'Sidereal month' - 1
• If we now look over n number of Tropical years we get the formula:
n/'Synodic month'= n/'Sidereal month' - n
• So here you see again that the number of Synodic months in n Tropical years is equal to the number of Sidereal (or Tropical) months in n years minus the number of tropical years.
• If one use n=19, one gets the Metonic cycle.

### Short term variations

Remember that the actual duration can vary due to variations. The following short term variations give some examples in the different cycles lengths:
• Year
Solar variations in the Vernal equinox year from the mean can be in the order of plus or minus some 15 minutes (1 sigma ~ 5 min).

Data is coming from this site
• Month
The variations in e.g. the Synodic month are some 7 hours from the mean, (1 sigma ~ 140 min)
• Day
The variations in the Solar day are some 30 sec. from the mean (1 sigma ~ 10 sec) (Stephenson [1997, page 4]).

## Acknowledgements

I would like to thank the following people for their help and constructive feedback: Simon Cassidy, A. Dutta, James Q. Jacobs, Vladimir Ladma, M.F. Loutre, Dr. Math, Jean Meeus, Tom Peters and all other unmentioned people. Any remaining errors in methodology or results are my responsibility of course!!! If you want to provide constructive feedback, let me know.

### Notes

For most cycles I would like to have a higher order time series (say around 5 to 7th order) than quoted below; if you have one, let me know.
In some cases the time series for the cycles/periods are derived from a known longitude/angle formula in the following way:
$L = p + q ⋅ t + r ⋅ t 2 + ... + s ⋅ t n$ [deg]

p, q, r, s: arguments of the time series
t: a time length; say of m Days (like in Julian ephemeris centuries, where m= 36525)

From this longitude one can determine the cycle length by differentiating the longitude and calculating the time it takes to do one cycle ( 360 degrees):
$360 q m + 2 ⋅ r m ⋅ t + ... + n ⋅ s m ⋅ t n - 1$
Furthermore:
T100 = time from J1900.5 [100 Year]
T1000 = time from J2000.0 [1000 Year]
T10000 = time from J1900.5 [10000 Year]
1. The time series (3rd order) for Sidereal year comes from (Chapront [2002], page 704)
$365.2563629530 + 0.0000001139 ⋅ T 1000 - 0.000000000076 ⋅ T 1000 2 - 0.00000000000169 ⋅ T 1000 3$ [Day]
2. The time series (1st order and sinus term) for Solar day is based on own formula derived mainly from data of Morrison&Stephenson [2004].
3. The time series (3rd order) for Lunar nodal cycle comes from (derived by T. Peter from Chapront [2002], page 704)
$6793.476501 + T 1000 ⋅ ( 0.0124002 + T 1000 ⋅ ( 0.000022325 - T 1000 ⋅ 0.00000013985 ) ) 365.25$ [Year]
4. The time series (3rd order) for Sidereal month comes from (derived by T. Peter from Chapront [2002], page 704)
[Day]
5. The time series (2nd order) for Eccentricity comes from (Nautical Almanac Office [1974], page 98).
$0.01675104 - 0.0000418 ⋅ T 100 - 0.000000126 ⋅ T 100 2$[-]
6. The time series (5th order) for Obliquity comes from (Hilton [2006], page 351):
$84381.406 + ( - 46.836769 + ( - 0.0001831 + ( 0.0020034 + ( - 0.000000576 + ( - 0.0000000434 ) ⋅ T 1000 ) ⋅ T 1000 ) ⋅ T 1000 ) ⋅ T 1000 ) ⋅ T 1000$ ["]
7. The time series (4th order) for Precession of equator comes from (Capitaine [2005], page 6).
$360 ⋅ 3600 ⋅ 100 ( 5028.796195 + T 1000 ⋅ ( ( 1.1054348 ⋅ 2 + T 1000 ⋅ ( 0.00007964 ⋅ 3 + T 1000 ⋅ ( - 0.000023857 ⋅ 4 - T 1000 ⋅ 0.0000000383 ⋅ 5 ) ) ) ) )$ [Year]
8. The time series (3rd order) for Lunar apse cycle comes from (Chapront [2002], page 704)
$3232.60542496 + T 1000 ⋅ ( 0.0168939 + T 1000 ⋅ ( 0.000029833 - T 1000 ⋅ 0.00000018809 ) ) 365.25$ [Year]
9. The time series (2nd order) for Climatic precession cycle comes from (Nautical Almanac Office [1974], page 98).
$360 365.25 ⋅ ( 0.0000470684 + 0.0000339 ⋅ 2 10000 ⋅ T 10000 + 0.00000007 ⋅ 3 10000 ⋅ T 10000 2 )$ [Year]