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:
τ = 365.242189669781 –
6.161870·10^{–6}·T – 6.44·10^{–10}·T^{2}, 

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:
τ = 365.242189669781 –
6.161870·10^{–6}·T – 6.44·10^{–10}·T^{2}, 

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 23 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′′T^{2}
+ 0.0000762′′T^{3}, 
 (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.
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, p_{A}.
The longitude referred to the equinox of date, in which we are now interested,
is obtained as L = L′ + p_{A}.
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 timehonored 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.

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):
L_{τ} = 100.0021383976·T +
8.43550·10^{–7}·T^{2} +
5.88·10^{–11}·T^{3}.

 (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
L_{C} = 
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 L_{τ} – L_{C} by 365.2425, the number of days in the calendar year,
N′ = 0.03103369·T +
3.08100·10^{–4}·T^{2}
+ 2.147·10^{–8}·T^{3}.

 (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′ = 0.74^{d}. 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 0.5^{d} –
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 = 
T ∫
0

Ω dT =
36525·T – 
ΔT – ΔT_{o}
86400

, 

where Ω = Ω_{o} –
ω is the angular
speed of the Earth rotation, Ω_{o}
is equal to 1 rotation per 86 400 SI
seconds, exactly, ω = [(dΔT)/dT],
and ΔT and ΔT_{o}
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
ΔT_{o}
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
L_{C} = n/365.2425. This, of course, is equivalent to increasing
N′ by ΔT scaled to
days, so that a generalized version of equation (5) becomes:
N = [(2.147·10^{–8} T + 3.081·10^{–4}) T + 0.03103369] T + 
ΔT –
ΔT_{o}
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 ΔT
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/cy^{2}, in contrast to much greater differences between obtained
decelerations themselves.
One of the most recent compilations of determinations of
ΔT based on telescopic observations (McCarthy, Babcock 1986)
leads to the following formula (in seconds of time)
ΔT_{MB} = 48.75 + 48.1699·T + 13.3066·T^{2}, 
 (7) 
that predicts rather small values of ΔT.
The coefficient at T^{2}
has the formal statistical uncertainty of about 0.3 s/cy^{2}.
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:
ΔT_{SM} = 2177 + 408.6·T +
44.3·T^{2} 
 (8) 
with formal error of the order of 1·T^{2} 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 ΔT will lie generally between
ΔT_{MB} and ΔT_{SM}.
In that case in the year 4000 (T = 20, ΔT
between about 6000 and 28 000 seconds) we would have 0.8^{d}
< N < 1.1^{d} instead of
the earlier calculated N′ = 0.74^{d}.
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: 8^{d} < N < 12^{d}. 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.

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_{o} + aT
+ bT^{2} + cT^{3} + ... 
 (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:
τ = 
^{ }360·60·60′′

= 
1296000′′
a + 2bT + 3cT^{2}
+ ...

, 
 (10) 
where Å = 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, ...
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:
τ = 36525 
1296000
a

 ( 
1 – 2 
b
a

T – 3 
c
a

T^{2} – ... 
) 
, 
 (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)
τ_{N} = 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 uptodate expression for the length of the tropical year:
τ = 365.242189669781–
6.161870·10^{–6}·T –
6.44·10^{–10}·T^{2}. 
 (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 τ_{N}),
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 + ε(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
exactly. In terms of the tropical year count we would have
τ = 
1
Å 
=  1

≈
1 – 
dε(t)/dt 
 (14) 
(assuming dε(t)/dt « 1) and
N′′ = 
t ∫
0

τ dt –
t ≈  t ∫
0

[1 – dε(t)/dt]

dt – t
= –ε(t).

 (15) 
Thus, integrating the period, τ, 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 δ, shorter than
the calendar year then in q calendar years there would be approximately q + qδ
tropical years, and not q – qδ
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 time derivative of ε(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·L_{τ}(T),
ie. the difference between the total number of days and the count of
days contained in L_{τ} 365day 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_{o} =
–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:


n(T) – n(T_{o}) – 365·[L_{τ}(T) –
L_{τ}(T_{o})] 

(16)  

484.504 + 24.2195·T – 3.079·10^{–4}T^{2} – 2.15·10^{–8}T^{3}
– 
ΔT – ΔT′_{o}
86400

, 


where ΔT′_{o} stands for the value of ΔT
at the year numbered 0 (or at T_{o}).
In this equation we can substitute q·365.2422/36525 + T_{o} for
the argument T to approximately convert it to:
l = q[0.242313 – q(3.07·10^{–8} +
2.15·10^{–14}q)]
– 
ΔT – ΔT′_{o}
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_{o} 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):
l_{P}
= 0.24231545q – 3.07·10^{–8}q(q
+ 1). 
 (18) 
We see that, aside of the term ΔT, 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).
Because of large uncertainties in the length of day it is
a speculation to give new calendar rules for epochs removed more than some
23 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,
PL87100 Toruñ,
Poland
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
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