The Tropical Year and Solar Calendar The Journal of the Royal Astronomical Society of Canada
vol. 85, No. 3 (whole No. 630), June 1991, p. 121–130.

THE TROPICAL YEAR AND SOLAR CALENDAR

BY KAZIMIERZ M. BORKOWSKI
Toruñ Radio Astronomy Observatory,
Nicolaus Copernicus University, Toruñ, Poland

(Received October 5, 1990, revised March 20, 1991)


ABSTRACT

The expression for the length of tropical year, based on a modern theory of the motion of the Earth, is derived. The formula valid over about 8 000 years centered at the present reads in days:
t = 365.242189669781 - 6.161870·10-6·T - 6.44·10-10·T2,
where T is the time reckoned from J2000 and measured in Julian centuries of 36525 ephemeris days. A comparison of the Gregorian calendar with a perfect solar calendar suggests that the former will be adequate at least during the nearest one to two thousand years. Due to high uncertainty in the Earth rotation it is premature at present to suggest any reform that would reach further than a few thousand years into the future. An approach to calendrical analysis relying on the summation of the length of tropical years is shown to be methodologically incorrect.


RÉSUMÉ

Une formule la théorie moderne du mouvement de la Terre pour la durée de l'année tropicale est présentée. Cette formule, qui est valide pour une période de 8000 ans centrée sur le présent, se lit comme suit:
t = 365.242189669781 - 6.161870·10-6·T - 6.44·10-10·T2,
ou T est le temps écoulé depuis J2000.0 exprimé en siìcles juliens de 36525 jours du temps des éphémérides. Une comparison entre le calendrier grégorien et un calendrier solaire parfait suggìre que le calendrier grégorien sera adéquat pour les prochains un ou deux millénaires. Il est donc trop tôt pour suggérer une révision pouvant s'appliquer á une période plus étendue du futur, car notre connaissance de la rotation terrestre contient encore un haut niveau d'incertitude. Enfin, il est démonstré qu'une tentative de réviser l'analyse des calendriers reposant sur l'addition de la durée des années tropicales est méthodiquement érronée.

1  Introduction

Among those who studied calendar structures the opinion seems to prevail that our Gregorian calendar, introduced in 1582, although not perfect, does not require any revision as to its year length or rules of intercalation of leap days, still for thousands of years to come. Quite a usual approach to reach such a conclusion is a comparison of the mean length of the tropical year (taken either as a constant unit or one linearly diminishing) with the mean length of the calendar year, combined with certain summation of both. In the present study I shall demonstrate that the approach is methodologically incorrect, in spite of about correct results it normally yields.

In his recent article Peck (1990) has detaily analysed a possible reform of solar calendar in view of the changing length of the tropical year. Unfortunately, his ideas are based on wrong grounds and not only because of the mentioned incorrect approach. This author has also overlooked the limited time span the Newcomb formula is valid over and, equally importantly, has ignored the variable Earth rotation. My analysis shows that our present knowledge allows to plan solar calendars roughly for 2-3 thousands of years into the future, and going considerably beyond this range remains a pure speculation carrying little, if any, practical significance. Specifically, the Gregorian calendar rules, used now for over 400 years, serve their purpose very satisfactorily and will do so still for at least a thousand years or so.

2  Longitude of the Sun versus solar calendar

The natural basis for computing passing tropical years is the mean longitude of the Sun reckoned from the precessionally moving equinox (the dynamical equinox or equinox of date). Whenever the longitude reaches a multiple of 360° the mean Sun crosses the vernal equinox and a new tropical year begins. In a modern theory of the motion of Earth around the Sun, VSOP82* (Bretagnon 1982), the mean longitude of the Sun referred to the dynamical equinox is given by (see also e.g. Connaissance des Temps 1990)
L = 280°27¢59.2146¢¢ + 129602771.36329¢¢T + 1.093241¢¢T2 + 0.0000762¢¢T3,
(1)
where T is the uniform time measured in Julian centuries and is reckoned from the fundamental epoch J2000 (January 1.5, 2000), i.e.
T =  JD - 2451545

36525
,
(2)
with JD designating the Julian Day Number, a continuous count of days since 4713 BC, January 0.5 (of the proleptic Julian calendar). Here we shall specify the days to be of equal length of 86 400 SI seconds (this specification is usually associated with renaming of JD to JED, for Julian Ephemeris Day).

As we shall soon see, the equation (1) is entirely sufficient to meet the demends of calendrical calculus. To assess its accuracy and range of validity one may refer to the Laskar (1986) paper where coefficients of polynomials of degree 10 in T are given for the longitude of the Sun referred to a fixed equinox, L¢, and for the precession in longitude, pA. The longitude referred to the equinox of date, in which we are now interested, is obtained as L = L¢ + pA. In the same paper we read that this more accurate expression is valid (within a few arcseconds) over 10 000 years. In Fig. 1 we have plotted the difference between the mean longitudes of the Sun calculated from the equation (1) and from the Laskar formulation. The reader may now discover how little the VSOP82 expression errs for remote epochs. While in the year 6000 (T = 40) equation (1) yields longitudes too small by only about 1¢ and in the year 12 000 (T = 100) the error is still on the order of 0.6°. However, further millenia bring about a rapid change. Forget not that so accumulated error of 1° corresponds to nearly 1 extra day. Since the time-honored formula of Newcomb differs but little from the modern equation (1), we clearly see how dangerous it is to extend its use beyond the range of 10 000 years. In fact, it is also risky to use Laskar's formulae over considerably more than 10 000 years, the range they were designed for. The following discussions will be limited to a few thousands of years in which equation (1) does not introduce any significant errors.

Trop1.gif
    Fig. 1. The difference between the mean longitude of the Sun defined in the French ephemeris (equation (1)) and in Laskar (1986). It shows that our third degree polynomial approximation adopted from the VSOP82 theory is very acceptable over a period of about 4000 years from the present. The longitude taken from Newcomb's theory behaves similarly to the one from the VSOP82, so much that at the scale of this figure they would not be distinguishable.

Through skipping of the constant term in equation (1) and dividing the numerical coefficients by 1 296 000¢¢, the number of seconds of arc in the complete revolution through 360°, we obtain a convenient expression for finding the number of rotations of the Sun around the ecliptic, i.e. the number of tropical years elapsed between J2000 and a given epoch (T):
Lt = 100.0021383976·T + 8.43550·10-7·T2 + 5.88·10-11·T3.
(3)
This number is to be compared with number of calendar years elapsed in the same period. In the Julian calendar there are simply 100·T calendar years over T centuries. For the Gregorian calendar, in which years are on average 365.2425 days long, we have
LC =  3652500

36524.25
T
(4)
calendar years in T Julian centuries.

Now, the difference between counts of tropical (equation (3)) and calendar years (equation (4)) can be expressed in days by multiplying Lt - LC by 365.2425, the number of days in the calendar year,
N¢ = 0.03103369·T + 3.08100·10-4·T2 + 2.147·10-8·T3.
(5)
The particular factor of 365.2425 seems entirely apropriate with the Gregorian calendar. However, it is easy to see that replacing it by any number between 365 and 366 would not make any significant difference, unless a calendar departs from the solar by more than, say, a third of a year. The above obtained formula tells us how far the Gregorian calendar advances ahead of the astronomically exact calendar, assuming the two were aligned at J2000. For example, with T = 20 (i.e. at the epoch J4000) N¢ = 0.74d. This can be interpreted to mean that after 2000 years from now the date of vernal equinox (say, 20 March) has shifted by approximately 1 day backwards in Gregorian calendar (say, to 19 March around AD 4000). Thus, around that epoch, or rather somewhat earlier - when N¢ has accumulated to 0.5d, one would correct the Gregorian calendar by redefining one of the leap years to be a common year (365 days long).

The above considerations assumed a constant duration of a calendar day, equal to the ephemeris day of 86 400 atomic seconds. We know, however, that the Earth rotates nonuniformly on its axis and the duration of the calendar or cilvil day slowly but systematically lengthens. The true number of civil days elapsed between J2000 and an epoch given by T is the count of rotations of the Earth:
n = ó
õ
T

0 
W  dT = 36525·T -  DT - DT°

86400
,
where W = W° - w is the angular speed of the Earth rotation, W° is equal to 1 rotation per 86 400 SI seconds, exactly, w = [(dDT)/dT], and DT and DT° represent the difference between the Terrestrial Dynamical Time (or the Ephemeris Time) and the Universal Time accumulated up to the epoch T and J2000, respectively (the term DT° is included here for completeness even though we know it to be negligibly small, about 65 s). Since the difference is conventionally expressed in seconds of time we have explicitly divided it by 86 400 to convert it to days. The effect of variable diurnal rotation may be incorporated into our calendrical calculations by setting LC = n/365.2425. This, of course, is equivalent to increasing N¢ by DT scaled to days, so that a generalized version of equation (5) becomes:
N = ((2.147·10-8T + 3.081·10-4)T + 0.03103369)T +  DT - DT°

86400
.
(6)

The real problem is that presently we are unable to accurately predict the duration of the day even for a moderate future, not speaking about thousands or millions of years. Besides many regular components, the rotation of our planet exhibits irregular variations on different time scales. Many researchers attempted to fit a parabola to the measured DT values in order to determine the magnitude of deceleration of the Earth rotation. The results, when taken together, are rather discouraging. It is not unusual that formal errors of idividual determinations are on the order of 1 s/cy2, in contrast to much greater differences between obtained decelerations themselves. One of the most recent compilations of determinations of DT based on telescopic observations (McCarthy, Babcock 1986) leads to the following formula (in seconds of time)
DTMB = 48.75 + 48.1699·T + 13.3066·T2,
(7)
that predicts rather small values of DT. The coefficient at T2 has the formal statistical uncertainty of about 0.3 s/cy2. On the other hand, the study of historical observations recorded between 390 BC and AD 948, conducted also recently by Stephenson and Morrison (1984), gave entirely different picture of past behaviour of the Earth rotation:
DTSM = 2177 + 408.6·T + 44.3·T2
(8)
with formal error of the order of 1·T2 seconds.

Numerous other results generally fall between these two. Not knowing the true value of the deceleration, to proceed further in our analyses of the future calendars we may temporarily assume the Earth will rotate so that DT will lie generally between DTMB and DTSM. In that case in the year 4000 (T = 20,   DT between about 6000 and 28 000 seconds) we would have 0.8d < N < 1.1d instead of the earlier calculated N¢ = 0.74d. This result alone entitle us to state with confidence that our Gregorian calendar will not deviate from the exact solar calendar by significantly more than 1 day yet for some 2000 years to come.

An estimate made in a similar manner for 10 000 years from the present (T = 100; see Fig. 2) is much more disappointing: 8d < N < 12d. This uncertainty of 4 days, together with quite satisfactory behavior of the Gregorian calendar during one or two nearest millenia, renders approaches to a calendar reform both premature and unnecessary. Should our civilization survive probable social, scientific and technical revolutions of that far future it might for long had forgot about the desire to exactly synchronize its calendar with the Christian Church festivals (recall that the Gregorian reform aimed to fix the date of Easter in accord with the religious tradition). Predicting calendar rules further than several thousand years into the future is a task worth contrary speculation suggesting that future generations will control the Earth rotation to preserve proper agreement of a current civil calendar with astronomical phenomena.

Trop2.gif
    Fig. 2. The difference between counts of tropical years and Gregorian years calculated according to equation (6). The lower and upper curve corresponds to supposedly extreme scenarios of Earth's rotation expressed by equation (7) and (8), respectively.

3  The length of tropical year

Peck (1990) observes a somewhat curious lack of an explicit formula for the length of tropical year in the literature since the introduction of the new system of astronomical reference frames and constants. Using available astronomical literature I myself tried to locate an explanation on how to derive such an expression. Since none has been found in the following we shall derive one.

The tropical year is an orbital period like any other of the many periods met in astronomy and associated with orbiting celestial bodies. It is essentially the reciprocal of the mean motion of the Sun. Thus, in general, if the mean longitude of the Sun can be expressed as
L = L° + aT + bT2 + cT3 + ¼
(9)
with T measured, as before, in Julian centuries, then the length of the tropical year (in units of Julian centuries) can be calculated from:
t =  360·60·60¢¢

×
L
=  1296000¢¢

a + 2bT + 3cT2 + ¼
,
(10)
where the dotted L or dL/dT is the rate of change (time derivative) of the longitude or centurial mean motion of the Sun, and the numerical factors were placed in the numerators assuming the coefficients a,  b,  c, ¼ are expressed in arcseconds. Since in practice the a term is much greater then the rest of the denominator, equation (10) can be rewritten in a more convenient form:
t = 36525  1296000

a
æ
è
1 - 2  b

a
T - 3  c

a
T2 - ¼ ö
ø
,
(11)
where the supplementary multiplier of 36525 converts the result from centuries to days.

Now, if we take the longitude of the Sun from the Newcomb theory (a = 129602768.13¢¢,  b = 1.089¢¢,  c = 0) we easily obtain his widely known formula (that formed the basis of Peck's paper)
tN = 365.24219879 - 6.14·10-6·T¢ = 365.24219265 - 6.14·10-6·T,
where T¢ = T + 1 (the original Newcomb theory is referred to the epoch J1900). At this point the reader may wish to consult the paper by Sôma and Aoki (1990) where this quantity is independently derived (the insignificant numerical differences arise from their modification of the Newcomb longitude).

Using coefficients of equation (1) in equation (11) we arrive at a more accurate and up-to-date expression for the length of the tropical year:
t = 365.242189669781- 6.161870·10-6·T - 6.44·10-10·T2.
(12)
This is the formula that replaces Newcomb's one and that Peck (1990) has unsuccessfully scanned the literature for. Note that here the days are each equal to 86 400 SI seconds, thus equation (12) is independent of the diurnal rotation of Earth. Observe also that if we adhere to sum the length of tropical years according to this formula (or the one of Newcomb, the expression for tN), the accumulated error will have the magnitude similar to that displayed in Fig. 1, in addition to errors introduced by the method itself.

Also, it is apparent now that the meaning of equation (12) is that the absolute length of the year is changing. The change is certainly not due to a frictional retardation of the Earth axial rotation, as Hope (1964) has convincingly suggested.

4  The summation of tropical year length is incorrect

In some calendrical discussions, to find the number of tropical years in a given period the summation of the lengths of tropical years is employed. Though the numerical results will be usually acceptable this is principally incorrect method and works only because the length changes very slowly indeed. To substantiate this statement suppose we rewrite our formula for longitude as L = t + e(t), where t is the time expressed in years of constant duration (e.g. 365.25 ephemeris days). Then the difference of time measured in tropical and constant years is
N¢ = L - t = e(t),
(13)
exactly. In terms of the tropical year count we would have
t =
 
 1

 .
L
=
 
1

1 +
 .
e
(t)
» 1 -
 
 .
e

 
(t)
 
(14)
 
(assuming
 .
e
<< 1) and
N¢¢ = ó
õ
t

0 
t   dt - t » ó
õ
t

0 
[1-
 .
e
 
(t)]dt - t = -e(t).
(15)
Thus, integrating the period, t, i.e. summing up the length of tropical years, yields almost the same result, save the reversal of the sign. This conclusion is also intuitively appealing, for if a tropical year were constant but slightly, say by d, shorter than the calendar year then in q calendar years there would be approximately q + qd tropical years, and not q - qd as the summation of lengths of tropical years alone would imply.

It should be born in mind, however, that the exactness of |N¢¢| depends on smallness of the dotted e(t), so that in general it is to be preferable to use directly the mean longitude of the Sun instead of the tropical year length.

5  Conclusion

The general formulae that Peck (1990) has derived and that he has found useful in practice, may be further generalized to conform to the methodologically correct approach and to the more accurate expression for the motion of the Sun. Clearly, the quantity n(T) - 365·Lt(T), ie. the difference between the total number of days and the count of days contained in Lt 365-day years, represents the desired number of leap years in a solar calendar between J2000 and the epoch T. Thus, using Peck's Year Zero, AD 0 (JD = 1721058, T° = -730487/36525), as the origin for counting years we easily obtain a formula that allows to find the number of leap years until epoch T, necessary in order that given calendar closely agrees with the Sun's motion:
l
=
n(T) - n(T°) - 365·[Lt(T) - Lt(T°)]
(16)
=
484.504 + 24.2195·T - 3.079·10-4T2 - 2.15·10-8T3 -  DT - DT¢°

86400
,
where DT¢° stands for the value of DT at the year numbered 0 (or at T°). In this equation we can substitute q·365.2422/36525 + T° for the argument T to approximately convert it to:
l = q(0.242313 - q(3.07·10-8 + 2.15·10-14q)) -  DT - DT¢°

86400
,
(17)
where now the argument is the year of the common era, q. Note that l can be expressed exactly as a function of q through the substitution T = (365q + l)/36525 + T° and then solving a qubic equation. However, such a solution is relatively complicated and the approximation we have made is very good indeed (over the range 0 – 12000 of q, the departure from the exact solution nowhere exceeds 0.002 of a day).

The equation (17) can be directly compared with the equation (5) in Peck (1990):
lP = 0.24231545q - 3.07·10-8q(q + 1).
(18)
We see that, aside of the term DT, there are differences essentially insignificant over the few thousands years over which one can rely on the mean longitude of the Sun. Subtracting the number of leap years actually introduced in a specified calendar from the equation (17) leads to another useful calendar formula analoguous to the equation (3) in Peck (1990).

To sum up, in this paper we have shown or observed that:

  • For comparison of a solar calendar with an exact count of tropical years the mean longitude of the Sun should be used instead of any summed lengths of tropical year.

  • Though there exist more accurate expressions for the mean longitude of the Sun, the simple formula used in the VSOP82 theory (Bretagnon 1982) is entirely adequate for calendrical calculus up to a few thousand years into the future (at the epoch J12000 it errs by somewhat more than half a degree).

  • Due to large uncertainties in the length of day it is a speculation to give new calendar rules for epochs removed more than some 2-3 thousand years from the present.

  • The mean length of tropical year can be calculated according to equation (12). This expression is a modern replacement of the similar but less accurate formula of Newcomb.

  • The number of tropical years passed between J2000 and any other epoch can be found using equation (3).

  • To obtain the number of days by which the Gregorian calendar advances the astronomical one, the use of equation (6) is suggested.

  • The calendar formula worked out by Peck (1990), supposed to be practically useful, is improved. The corrected version is given here in the form of equation (17). This equation tells us how many leap years should there be in a solar calendar up to the year q of the common era.

    K.M. Borkowski
    Toruñ Radio Astronomy Observatory,
    Nicolaus Copernicus University,
    ul. Chopina 12/18,
    PL-87-100 Toruñ,
    Poland

    REFERENCES
    Bretagnon, P., 1982, Astron. Astrophys., 114, 278.
    Connaissance des Temps, 1990, Éphémérides astronomiques 1991, Bureau des Longitudes, Paris.
    Hope, E.R., 1964, J. Roy. Astron. Soc. Can., 58, 3.
    Laskar, J., 1986, Astron. Astrophys., 157, 59.
    McCarthy, D.D., Babcock, A.K., 1986, Phys. Earth Planet. Interiors, 44, 281.
    Peck, P.A., 1990, J. Roy. Astron. Soc. Can., 84, 14.
    Sôma, M., Aoki, S., 1990, Astron. Astrophys., 240, 150.
    Stephenson, F.R., Morrison, L.V., 1984, Phil. Trans. Roy. Soc. London, A313, 47.


    Footnote:
    *I have often seen the letters "VSOP" on the side of a bottle of wine and I have wondered what the letters stand for. I understand that, in the context of this paper, VSOP82 refers to the 1982 version of Bretagnon's "Variations Séculaires des Orbites Planétaires". Whether this is what is meant on wine bottles remains a mystery to me. — Editor


    Tropic-author.gif


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