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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. Good pages that also elaborate on the cycles is here.

to calculate the periods for Jan. 1st 12:00 UTC [Julian calendar] (positive for CE and negative for BCE and between 4000 BCE and 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 luni-solar precession) 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 four reference systems:
  1. the earth-sun system
    Astronomically determined by how the sun is observed from the earth (tropical period: geocentric). Of course there are small changes due to other planets/moon.
  2. the earth-moon system
    Astronomically determined by how the moon is observed from the earth (tropical period: geocentric). Of course there are changes due to other planets/sun. 
  3. the earth-star system
    Here the precession of the earth axis (which is due by the solar&lunar forces) becomes evident. This precession is specific to the earth and not for the whole solar system the earth belongs to.
    Using this with 2 and 3, one get the sidereal periods (which are purely related to Newton/Kepler laws).
  4. the time system
    All cycles are expressed in the stable time system of Terrestial 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.
    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).

Calandar convention

For input date I use Julian calendar dates, because:

Orbital periods

Mean orbital periods of the earth and moon are (be aware of possible variations when looking at actual values):

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 sumharmonic formula, 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 (lunar draconic month is taken as an example):

The two circular movements (of the 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/'tropical month' degrees and the nodal cycle has moved: 360*x/'nodal cycle' degrees.  Both are after a days at the same position, so:

360*(1 - x/'tropical month') = 360*x/'nodal cycle'
1 - x/'tropical month'= x/'nodal cycle'
  x= 1/(1/'tropical month' + 1/'nodal cycle')
'draconic month'='nodal cycle'*'tropical month'/('nodal cycle' + 'tropical month')
In general:
X=A*B/(A+B)
or
1/X = 1/B+1/A

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

X= A*B/(A-B)
X= 1*12/(12-1)=1h 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):

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:
Relatiosn between different lunar and solar cycles
Blue: related to earth
Green: related to moon
Yellow: related to stars
Much darker: the reference system's orbits/cycles


Remember that a 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.
I have chosen those orbits, which have the most relations with other orbits/cycles/periods, as my reference orbits/cycles/periods. 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 relations

The above relations can be depicted with the following picture:
Other relations in picture

Relations within Metonic cycle

Difference of number of synodic and sidereal months in the number of tropical years. A numerical evaluation: A more formal way of the proof:
This is like playing the number games in the school yard. Unbelievable in some way, but that is due to the strong relations between the variables.

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:
Variations in Vernal equinox year
Data is coming from this site

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 well known longitude/angle formula in the following way:
L = p + q*t + r*t2 + ... + s*tn [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):
cycle = 360/(q/m + 2*r/m*t + ... + n*s/m*tn-1)
  1. The time series (3rd order) for tropical year comes from this site.
  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 (2nd order) for lunar nodal cycle comes from (Nautical Almanac Office [1974], page 107)
  4. The time series (2nd order) for tropical month comes from (Nautical Almanac Office [1974], page 107)
  5. The time series (2nd order) for eccentricity comes from (Nautical Almanac Office [1974], page 98).
  6. The time series (7th order) for obliquity comes from (Bretagnon [1986], page 6).
  7. The time series (6th order) for luni-solar precession comes from (Bretagnon [1986], page 6).
  8. The time series (2nd order) for lunar apse cycle comes from (Nautical Almanac Office [1974], page 107).
  9. The time series (2nd order) for climatic precession cycle comes from (Nautical Almanac Office [1974], page 98).

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Last major content related changes: Feb. 23, 2001