Accuracy of representation of astronomical periods or ratios by gear trains comprises several components. These include:

1. Accuracy versus precision in measurements (see Accuracy versus precision)
2. Accuracy of the mathematical expressions used to represent the phenomena as given in the reference document (see Accuracy of mathematical expressions)
3. Time intervals of validity of these mathematical expressions (see Validity intervals of mathematical expressions)
4. Accuracy of the calculation of the reductions provided by selected gear trains, and the related accuracy of the calculated error between the calculated reduction ratio and the target (desired) ratio which the calculated ratio approximates (see Accuracy of calculations)
5. References

As given in Seidelmann [1], the current recognized authority and reference for descriptions of motions and aspects of astronomical objects in the solar system, these expressions are usually truncated series expansions in powers of time elapsed after a reference epoch, a Taylor series truncated to a few terms. The time argument, T, is usually given in Julian centuries (see Synodic month for an explanation of this unit of time). The Taylor series were derived by fitting coefficients of their terms to a series of (weighted) observations taken over some period of time. The Taylor series expansions are then used to predict the value of the period or ratio of interest into the future.

Astronomical phenomena of interest in horology include

1. The period of phases of the Moon, the synodic month,
2. The length of the year of seasons to which the calendar attempts to conform, the tropical year, and
3. The ratio of the length of the mean solar day to that of the mean sidereal day.

Synodic Month

The expression for the length of the synodic month (in days) is:

See Synodic month for the definition of T and further discussion of this expression and its source, Seidelmann [1].

Tropical Year

The expression for the length of the tropical year (in days) is:

The tropical year is the time between two successive passages of the Sun through the Vernal Equinox. The Vernal Equinox is a (nearly) fixed point in the sky corresponding to that intersection between the celestial equator (the apparent great circle of intersection between the plane of the Earth’s equator and the apparent celestial sphere) and the ecliptic (the apparent great circle of the Sun‘s annual path through the sky) at which the Sun passes from the southern side of the celestial equator to the northern; the Sun does this each year at the beginning of Spring, on or about 21 March. Again, see Synodic month for the definition of T and further discussion of this expression‘s source, Seidelmann [1].

Ratio of lengths of Mean Solar to Mean Sidereal Day

The mean solar day can be thought of as the average over the year of the time interval between two successive passages of the Sun across the visible meridian of a fixed observer on the Earth. The visible portion of an observer‘s meridian is an arc which extends from the north point on his horizon through the point directly overhead (the zenith) to the south point on his horizon.

The mean sidereal day is the average time interval of one full rotation of the Earth with respect to the Vernal Equinox. Due to the Earth‘s motion in its orbit about the Sun, the Sun appears to move eastward across the sky with respect to the (nearly) fixed background stars, making one complete turn in a tropical year from the Vernal Equinox back to the Vernal Equinox. The Earth rotates eastward on its axis, so when it has made a full turn with respect to the Vernal Equinox, a mean sidereal day, it will have to rotate just a bit farther to catch the (mean) Sun, a mean solar day. Thus, the mean solar day is slightly longer than the mean sidereal day, about 3m 56s longer. The ratio of lengths of the mean solar to the mean sidereal day (or minutes or seconds) is:

Again, see Synodic month for the definition of T and further discussion of this expression‘s source, Seidelmann [1].

at the standard epoch of J1900 (1900 January 0.5 = JD 241 5020.0 [JD is the abbreviation for Julian Date. See Synodic month for a discussion of this time scale.]). T₁₉ is the number of Julian centuries after this epoch, which is calculated as:

T₁₉ is analogous to T as discussed in Synodic month, but applies to times with respect to JD 2415020.0, the standard epoch near the beginning of the 20^th century, rather than to JD 2451545.0 (2000 January 1.5, J2000), the current standard epoch (see Synodic month for further discussion of this topic).

It is interesting to propagate the expressions in Reference [1] to J1900 to compare current values of these quantities with those given in the Reference [2].

Synodic month

The value given in expression (4) is the result of direct calculation from the theory of the motion of the Moon current in the mid 1950s. Reference [2] notes that the change in the mean length of the synodic month "does not exceed a few hundredths of a second per century, and depend(s) partly upon the variations in the rate of rotation of the Earth". Probably due to the lack of high stability atomic clocks at that time and the difficulty of obtaining accurate (and precise) measurements of the position of the Moon because it is an extended body (not a point), no time-varying terms are provided in Reference [2] for this value.

The appropriate value of T in expressions (1), (2), and (3) for propagating from J2000 to J1900 is T = -1.0. Substituting this value in expression (1) yields the following value:

This propagated value agrees with that given in expression (4) to the number of significant figures given in the latter expression. It is interesting to note how many more significant figures the length of the synodic month can now be determined compared with a half-century ago.

Tropical year

The theory of the apparent motion of the Sun is far less complex than that for the Moon, and in 1960, at the time of the publication of Reference [2], this theory was substantially unchanged since about 1900. The length of the tropical year at J1900 will be the constant term in expression (5), 365.24219879 days. The value for that epoch calculated from expression (2), again using T = -1.0, is:

This value differs by about 300 in the eighth (last) decimal place in expression (5), which is a surprise. As noted under Accuracy versus precision, it is usual to publish measured values of quantities with the number of significant figures such that the uncertainty in the value of the least significant figure published, here the eighth decimal place, is of the order of ±10. This difference is thirty times that. It is unlikely that improved observational and timing techniques over the past 50 years would have brought such a change. A possible explanation would be that there is a misprint in one or the other of References [1] or [2], but a not-exhaustive investigation failed to reveal any misprint. A mystery.

Ratio of the lengths of the mean solar to the mean sidereal day

This ratio is also determined from the theory of the apparent motion of the Sun. The value of this ratio at J1900 as given in Reference [2] will be the constant term in expression (6), 1.002737909265. The value for that epoch calculated from expression (3), again using T = -1.0, is:

The constant term in expression (6) has 13 significant figures, while that in expression (3) has 16. Since this propagation was made using a pocket calculator which displays only 12 significant figures, the last digit in expression (10) is uncertain by ±0.5 or so. The agreement in the 12^th significant figure is within ±3, which is reasonable.

So, this analysis suggests that expressions (1) through (3) are likely to be valid for at least ±100 years from J2000, their common epoch.

Another way of estimating the interval of validity of a given series would be to assume that it is unlikely to extend substantially longer into the future (or the past) than the time at which the value of a term of a given power of T has the same magnitude as the corresponding value of the next term in the series, of some higher power of T.

On this basis, for expression (1), the magnitude of the terms in T and T² will be equal in a bit less than 25 Julian centuries, or about 2,500 years. Because the change in the length of the synodic month is substantially linear, it might be safe to assume that expression (1) would be a reasonable representation of the true length of the synodic month for at least 10% of this period, or about 250 years.

For expression (2), the magnitude of the terms in T² and T³ will be equal for T = ±2.76 Julian centuries, or ±276 years from J2000. Because the coefficients of these terms are so nearly equal in magnitude, they are likely not strongly determined from the available data. However, the term in T dominates the higher order terms by a factor of about at least 8,400 in this expression and differs relatively little from that in expression (5) from reference [2]. Thus, the terms in T and T² will be equal in about 8,400 Julian centuries, or about 840,000 years. All this suggests that expression (2) would likely be useful for a few centuries or so into the future or past.

For expression (3), the magnitude of the terms in T and T² will be equal in almost exactly 10,000 Julian centuries, or about 1 million years. From the discussion above, this would suggest that this ratio is well-determined, and would also likely be useful for a few centuries or so into the future or past.

In any case, it should be recognized that expressions (1) through (3) referred to above should be used with caution in trying to determine the value of the period or ratio they represent for times substantially before or after their reference epoch. Also, in any calculation using these expressions to determine the relevant length or ratio, the resulting value should be rounded to the number of significant figures given in the constant term of each expression.

[1] Seidelmann, P. Kenneth "Explanatory Supplement to the Astronomical Almanac", The Nautical Almanac Office [now: Astronomical Applications Department ], U.S. Naval Observatory ; University Science Books , Sausalito, CA 1992

[2] ____, "Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac", Her Majesty’s Stationery Office, London, 1961