The Accuracy of the Goddess Calendar as a Lunar Calendar
by Peter Meyer

Many calculations of the accuracy of a proposed calendar assume that the average motion of the Moon and of the Earth remain constant over time, which is true to some degree but is not absolutely true. Due to gravitational and other effects, the dynamics of the Sun-Earth-Moon system is changing very slowly. To investigate the question of whether the Goddess Calendar in fact keeps in sync with the lunar cycle over a period of at least several millennia we first need to look at the variation of the synodic month (the average time between successive new moons, i.e. between successive conjunctions of the Sun and the Moon) over the last few thousand years and in the millennia to come.

There is an expression for the synodic month (based on the lunar theory of Chapront-Touzé according to L. E. Doggett, taking his information from the Explanatory Supplement to the Astronomical Almanac). (Simon Cassidy has shown that this article contains an error; see Error in Statement of Tropical Year.) This expression is as follows:

S = 29.5305888531 + 0.00000021621 T - 3.64E-10 T^2 (days)

where S is the synodic month and

T = (JD - 2,451,545) / 36,525

where JD is the Julian day number (JDN 2,451,545 = 2000-01-01 G).

It is important to note the precise sense of "days" in the formula for S. Doggett writes:

In the preceding formulas, T is measured in Julian centuries of Terrestrial Dynamical Time (TDT), which is independent of the variable rotation of the Earth. Thus, the lengths of the tropical year and synodic month are here defined in days of 86400 seconds of International Atomic Time (IAT).

There are two senses of the word "day" which are relevant here. Firstly there is the "fixed-length" day, which is defined as 86,400 seconds of International Atomic Time (the sense of "day" used in the formula above). Secondly there is the "real day", which is the time of one rotation of the Earth about its axis (which, of course, must be defined more exactly before it can be measured precisely). In terms of the fixed-length day (a certain number of atomically-defined seconds) the length of the real day is slowly changing.

The slowing of the Earth's rotation is such that if the Earth were a clock we could say that it has lost about twelve hours in the last 4,000 years, or on average about eleven (atomic) seconds per year. Were the Earth's rotation to continue to slow at this rate then (neglecting any effect of the change in the Moon's orbital velocity) we would find that calendars which ignore this factor would (neglecting other sources of inaccuracy) be off by about one day after 8,000 years.

A lunar month is the time taken for the Moon to orbit the Earth divided by the number of rotations of the Earth which occur in this time. The synodic month is the mean lunar month; thus there are two factors operating to produce the value of the synodic month. These factors are operating in opposite directions. The Moon's angular velocity (the speed of its motion around the Earth) is decreasing, implying that the length of the lunar month is increasing (as shown by the formula for S above, in terms of fixed-length days). Counter to this, however, is the fact that the Earth's rotational speed is decreasing (even more significantly), implying that the length of the mean solar day is increasing. If we measure the lunar month in mean solar days then, assuming the orbital velocity of the Moon remains unchanged, the increase in the length of the mean solar day will produce a decrease in the length of the mean lunar month. Whether the synodic month is in fact increasing or decreasing depends on whether the Moon is slowing down in its orbit significantly faster than the Earth is slowing down in its rotation. According to some scholars it is even possible that the Earth's rotational velocity will increase, in which case the synodic month would also increase.

The length of the real day is given by the formula:

D = ( K * T * 36,525 + s ) / s

where
s = the number of seconds in a fixed-length day, namely 86,400
T = (JD - 2,451,545.0) / 36,525 where JD is the Julian day number
K = the rate at which the Earth is slowing in its rotation, namely, 3.4224 x 10^-7 revolutions/second^2 (this is derived from the above-mentioned loss of one day in 8000 years) and
D = the real day expressed in fixed-length days (thus the real day and the fixed-length day are equal only at 2000-01-01, when each is 29.53058885 days long).

We may use this formula to calculate the length of the real day during the period -1000 through 5000 C.E. We may then define the real synodic month as the synodic month (as calculated in fixed-length days using the formula above) divided by the length of the real day. We can compare this to the average length of a month in the Goddess Calendar (namely, 29.53058880 days) and, by taking the reciprocal of the difference, obtain the number of calendar months required for the calendar to diverge from the average lunar cycle by one day.

Year	Synodic month  in fixed- length days	Real day  in fixed- length days	Real synodic  month	Difference from  GC month length	Solar years  required for  1-day divergence
-1000	29.53058269	0.999995660	29.53071087	0.00012207	662
-500	29.53058368	0.999996383	29.53069048	0.00010168	795
0	29.53058467	0.999997106	29.53067012	0.00008132	994
500	29.53058569	0.999997830	29.53064978	0.00006098	1325
1000	29.53058673	0.999998553	29.53062945	0.00004065	1988
1500	29.53058778	0.999999277	29.53060914	0.00002034	3974
2000	29.53058885	1.000000000	29.53058885	0.00000005	1522637
2500	29.53058994	1.000000723	29.53056858	-0.00002022	3998
3000	29.53059105	1.000001447	29.53054833	-0.00004047	1997
3500	29.53059218	1.000002170	29.53052809	-0.00006071	1331
4000	29.53059332	1.000002894	29.53050788	-0.00008092	999
4500	29.53059449	1.000003617	29.53048768	-0.00010112	799
5000	29.53059567	1.000004340	29.53046750	-0.00012130	666

(This data was obtained using the GLCDIV.C program.)

It is interesting to compare this to the average length of the month in the Goddess Calendar as calculated above, namely, 29.53058880 days, a difference of only 0.00000005 days, or 0.00000017%. If the synodic month were absolutely constant then we could expect the Goddess Calendar to require about 20,000,000 synodic months (1/0.00000005), that is, about one and a half million years, to drift out of sync with the lunar cycles (on average) by about a day. But when we take into account the decrease in the real synodic month as shown by the table above, we can see that the Goddess Calendar can be expected to remain accurate only for a few thousand years.

However, as Simon Cassidy (to whom I am indebted for providing some of the astronomical information presented in this article) has pointed out (in Comment on Future Accuracy), predictions of accuracy for calendars are probably not possible beyond a few millennia, due to our inability to predict with sufficient accuracy the dynamics of the Earth/Moon/Sun system. Thus no rule-based calendar, however accurate at present, can be expected to remain accurate for longer than a few thousand years.

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