HomeUp Search Mail
NEW

Mean lunar and solar periods

Browser should support JavaScript (see here for description of these functions)

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:

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 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):

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:
Relations between different lunar and solar cycles
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.
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:

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

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)
    27.32166155356 + T 1000 ( 0.000000216673 + T 1000 ( - 0.00000000031243 + T 1000 1.9989 E - 12 ) )  [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]

Disclaimer and Copyright
Home Up SearchMail

Last major content related changes: Feb. 23, 2001