# Is there a modern accurate formula for the length of the sidereal year?

Newcomb (1898) gives the following formula for calculating the tropical year:

$$365.24219879 - 0.00000614T$$

And the following for calculating the sidereal year:

$$365.25636042 + 0.00000011T$$

Where T is the number of Julian centuries (36525 ephemeris days) since J1900.0.

Since then, more accurate orbital elements for the planets have been calculated, like Simon et al. (1994). Several better formulae for the tropical year have been published since, such as in Borkowski (1991), which gives:

$$365.242189669781 - 0.000006161870T - 0.000000000644T^2$$

With T as Julian centuries since J2000.0. An even more accurate formula can be found in McCarthy & Seidelmann (2018, p. 267), which claims to have in turn calculated it from Laskar (1986):

$$365.2421896698 - 0.00000615359T - 0.000000000729T^2 + 0.000000000264T^3$$

With T as Julian centuries since J2000.0. A slightly different version can be found in Meeus & Savoie (1992):

$$365.242189623 - 0.000061522T - 0.0000000609T^2 + 0.00000026525T^3$$

Where this time, T is Julian millennia (365250 ephemeris days) since J2000.0.

However: I cannot find a single similar modern formula for the sidereal year. It seems almost incomprehensible to me that nobody would have calculated one in 100+ years.

Does such a formula exist in literature? If not, how could it be calculated?

Citations:
Borkowski, K. M. (1991). The tropical year and solar calendar. Journal of the Royal Astronomical Society of Canada, 85, 121.
Laskar, J. (1986). Secular terms of classical planetary theories using the results of general theory. Astronomy and astrophysics, 157, 59-70.
McCarthy, D. D., & Seidelmann, P. K. (2018). Time: from Earth rotation to atomic physics. Cambridge University Press.
Meeus, J., & Savoie, D. (1992). The history of the tropical year. Journal of the British Astronomical Association, 102(1), 40-42.
Newcomb, S. (1898). Astronomical Papers prepared for the use of the American Ephemeris and Nautical Almanac (Vol. 6). Bureau of Navigation, Navy Department.
Simon, J. L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., & Laskar, J. (1994). Numerical expressions for precession formulae and mean elements for the Moon and the planets. Astronomy and Astrophysics, 282, 663-683.

The length of the sidereal year is a function of the tropical year, and does not exist independently. The ratio of the length of the sidereal year to the length of the tropical year is (Barbieri & Bertini 2021, p. 148):

$$1296000/(1296000 - P)$$

Where P is the yearly rate of the procession along the ecliptic in seconds longitude. A formula for the precession constant is given in Liu & Capitaine (2017) as:

$$5028.796891T − 1.1111298T^2 + 0.01695523T^3 − 0.000020031T^4 − 0.000000017T^5$$

Where T is the number of Julian centuries (36525 ephemeris days) since J2000.0. However, we need the rate of change of this formula, which means taking the first derivative. This is:

$$5028.796891 − 2.2222596T + 0.05086569T^2 − 0.000080124T^3 − 0.000000085T^4$$

Since this formula is per century and not per year, the other parts of our first equation must be multiplied by 100 to match. Thus the ratio of the lengths of the sidereal and tropical years is:

$$129600000/(129600000 - 5028.796891 + 2.2222596T - 0.05086569T^2 + 0.000080124T^3 + 0.000000085T^4)$$

Now, let's use a more recent and accurate formula for the length of the tropical year. The one in Meeus & Savoie (1992) is based off VSOP87. One based off VSOP2013 is given in Institut de Mécanique (2021, p. 869-873) as:

$$365.2421904482 – 0.00006116623T – 0.000000065922T^2 + 0.0000002667909T^3 + 0.000000000947396T^4 + 0.000000000106233T^5 + 0.000000000371993T^6$$

But this one has T as Julian millennia since J2000.0. We should convert it to centuries so that it matches our other formula. This is:

$$365.2421904482 – 0.000006116623T – 0.00000000065922T^2 + 0.0000000002667909T^3 + 0.0000000000000947396T^4 + 0.00000000000000106233T^5 + 0.000000000000000371993T^6$$

Thus our formula for the length of the sidereal year is:

$$129600000/(129600000 - 5028.796891 + 2.2222596T - 0.05086569T^2 + 0.000080124T^3 + 0.000000085T^4)(365.2421904482 – 0.000006116623T – 0.00000000065922T^2 + 0.0000000002667909T^3 + 0.0000000000000947396T^4 + 0.00000000000000106233T^5 + 0.000000000000000371993T^6)$$

Which simplifies to:

$$(47335387882.08672 - 792.7143408T - 0.085434912T^2 + 0.03457610064T^3 + 0.00001227825216T^4 + 0.000000137677968T^5 + 0.0000000482102928T^6)/(129600000 - 5028.796891 + 2.2222596T - 0.05086569T^2 + 0.000080124T^3 + 0.000000085T^4)$$

The Maclaurin series of this formula to the 6th degree is approximately:

$$365.2563632882 - 0.000012380178T + 0.00000014270314T^2 + 0.0000000000409688T^3 - 0.0000000000001447616T^4 + 0.00000000000000106231T^5 + 0.000000000000000372007T^6$$

Which, for convenience, we can convert back to a per millennia value rather than per century:

$$365.2563632882 - 0.00012380178T + 0.000014270314T^2 + 0.0000000409688T^3 - 0.000000001447616T^4 + 0.000000000106231T^5 + 0.000000000372007T^6$$

This polynomial approximation is only appropriate for short term calculations, over the next century or so. Here are some years and the amount of time that the approximation is off by from the original formula:

2020 - 0.00000000000000000003 seconds
2050 - 0.000000052 seconds
2100 - 0.000053 seconds
2200 - 0.0133 seconds
2300 - 0.196 seconds
2500 - 14.41 seconds
3000 - 8.04 hours

Adding more terms to the Maclaurin series, while technically making the result more accurate, would significantly increase the amount of false precision.

However, this is all a secondary concern, since both formulas' divergence is much faster from whatever the true value is than from one another. For 2500, they give a length of over 371 days, and for 3000 they give over 738 days, which are obviously wrong results. These formulas for the sidereal year should be good for the short term, but should really only be used until more accurate versions of the procession and tropical year formulas are published in the coming decades.

It feels incredibly good to finally find an answer over a year and a half after I originally asked the question.

Citations:
Barbieri, C., & Bertini, I. (2021). Fundamentals of Astronomy. 2nd ed. CRC Press.
Institut de Mécanique Céleste et de Calcul des Éphémérides. (2021). Introduction aux éphémérides et phénomènes astronomiques.
Liu, J. C., & Capitaine, N. (2017). Evaluation of a possible upgrade of the IAU 2006 precession. Astronomy & Astrophysics, 597, A83.

• Might I suggest scientific notation? Commented Sep 15, 2022 at 19:44
• Question : is it possible to use this formula for past time, e.g. the last 2000 years ? Commented Jan 29, 2023 at 15:13

A synodic or a day is sunrise to next sunrise and assigned or divided in $$24$$ hours as an average. While a sidereal day is one complete rotation of earth and equal to $$1$$ degree on ecliptic.

Now in a day, there is difference of $$1$$ degree of earth's rotation between sun and $$1$$ rotation of earth. As duration of a day is $$1$$ unit then duration of a sidereal day in terms of the day is,

$$\text{P}_{sid} = 1 - \frac{1}{360} = \frac{359}{360}$$ days per degree

is amount of a sidereal day in terms of a day, that is $$24$$ hours. So convert it into minutes gives,

$$\text{P}_{sid} = \frac{359}{360}\times24\times60 = 1,436$$ minutes per sidereal day.

This is about $$4$$ minutes less than a day that is $$1,440$$ minutes.

For calculation of a sidereal year, multiply $$\text{P}_{sid}$$ by $$360$$. A sidereal day is $$1$$ degree on ecliptic, so a sidereal year consists of $$360$$ sidereal day,

Number of days in a sidereal year = $$\frac{359}{360}\times360 =$$359 days

As there is no fraction of days in a sidereal year and a complete circle align with both its position to stars as a sidereal year and to sun as a tropical year. So there is no difference in tropical and sidereal year, thus there is no backward motion of stars. This eliminates any calculation for precision of equinox.

As calculated above, there is recession of a sidereal day by $$4$$ minutes to a day which accumulates as a whole day in a year. Now this recession in term of angular measurement is given by difference of angular movement of a day and a sidereal day that is,

$$\frac{360}{359} - 1 = \frac{1}{359}$$ degree per day $$= \frac{3,600}{359}$$ arcseconds per day $$= 10.0278551532$$ arcseconds per day.

**

Time Equivalence of the Tropical Year and the Sidereal Year

**

"We scientists would claim that in the absence of precession, the tropical year and the sidereal year would be equal." Prof. Douglas P. Hube, Dept. of Physics University of Alberta1

ABSTRACT

The sidereal year is said to be the truest measure for a complete period of revolution of the Earth around the sun. By 1952 physicists had measured the precise time interval of the sidereal year in order to establish a definition for the unit 'second'. However, astronomers argued that the length of the sidereal year depends upon the adopted value of the precession. According to the theory of "the precession of Earth", Earth's axis of rotation gradually changes its orientation in space over a period of about 25800 years. This phenomenon causes a continuous displacement of the equinoctial points with respect to inertial space and with respect to the position of the sun. As a result, the sidereal year is supposed to be about 1223 seconds longer than the tropical year. Such a yearly time difference must be scientifically substantiated. This paper will prove that the time intervals of a tropical year and a sidereal year are, in fact, equal. Hence, the theory of "the precession of Earth" will be refuted. KEYWORDS: sidereal year, tropical year, rotation of Earth.

DEFINITIONS

Sidereal year: the period during which Earth makes a complete 360° revolution in its orbit around the sun, as measured with respect to the position of the fixed stars or inertial space.

Tropical year: the period during which Earth makes a complete 360° revolution in its orbit around the sun with respect to the position of the vernal equinox. The defined time interval of the tropical year for 1900.0 is 31,556,925.97474 seconds.

Mean sidereal day: the period during which Earth makes a complete rotation on its axis (absolute rotation). The time interval of the mean sidereal day is 86164.0905382 seconds.

MATHEMATICAL PROOF

The orientation of Earth's axis of rotation in space has no influence on Earth's complete period of revolution or on its absolute rotation. In one complete orbit, Earth makes exactly one less complete rotation of 1296000 arc-seconds on its axis with respect to the position of the sun than with respect to an outer fixed frame of reference. Mean solar time is based on the motion of a hypothetical sun traveling at an even rate throughout the year, and it is obtained in practice from observations of stars. The difference between the mean solar day of 86400 seconds (s) and the mean sidereal day is exactly 235.9094618 s. The rigorous mathematical relationship that exists between the mean solar day and the mean sidereal day is expressed by the equation:

365.24219878 × 86400 s = 366.24219878 × 86164.0905382 s = 31,556,925.9747 s

This equation describes Earth's complete 360° period of revolution of 31,556,925.9747 s relative to a fixed frame of reference, implying that the position of the vernal equinox remains fixed with respect to the orientation of Earth's axis in space. The total number of rotations of Earth in such a complete orbit is expressed by the equations: 1 ÷ (1- (86164.0905382 s ÷ 86400 s)) = 366.24219878 86400 s ÷ 235.9094618 s = 366.24219878

The time period of the Gregorian, or civil, calendar of 365.2425 mean solar days is about 26.03 seconds longer than the tropical year. In other words, the civil calendar deviates by only one day in almost 3320 tropical years. Since the reference frame of inertial space moves relative to the orientation of Earth's axis in space, Earth's period of rotation* with respect to a fixed direction in space is said to be about 9.12 milliseconds (0.1368") longer than the mean sidereal day (absolute rotation).

If the difference of 9.12 milliseconds (ms) per rotation were due to a precession of the axis, the same value must apply to the mean solar day. This is not the case, according to Newcomb's tables.3 For more than a century the mean solar day has essentially remained a constant, implying a non-precessing or fixed axis of rotation with respect to the position of the sun and the equinoctial points.

Based on the theory of "the precession of Earth", the equinoctial points retrograde around the sun by about 3.34 s per rotation of the Earth. In other words, due to precession the mean solar day would have to be 9.12 ms longer and also 3.34 s shorter. However, a difference of 3.34 s per rotation - between the moving origin and the non-moving frame of reference - is not being measured in practice.

Supposedly, the length of the sidereal year - i.e. the actual number of rotations of the Earth in one complete 360° orbit period - depends upon the adopted value of the precession. Hence, the following equation is said to represent the accumulating daily difference of 3.34 s over a period of one year:

365.256361 × 86400 s = 366.256361 × (86164.0905382 + 0.00912) s = 31,558,149.59 s

In reality this equation describes a slightly larger, but non-existing 360° orbit period for the Earth, implying in fact a non-precessing or fixed axis of rotation.

*** "No special name has been given to this kind of day, and although of theoretical interest, it is not used in practice."2**

DOCUMENTED PROOF

In its 1955 Transaction Report,4 the International Astronomical Union (IAU) passed the resolution to substitute the time interval of the tropical year for 1900.0 for the sidereal year for 1900.0 to define the unit 'second', as it had already been proposed in 1952:

"The General Assembly in Rome in 1952 adopted the recommendation that 'dans tous les cas où l' on juge que la variabilité de la seconde de temps solaire moyen s'oppose à son emploie comme unité de temps, l' année sidérale pour 1900.0 soit adoptée comme unité de temps'.

It was subsequently pointed out [...] that the tropical year is more fundamental than the sidereal year. The length of the tropical year is derived from Newcomb's tables of the Sun, whereas the length of the sidereal year depends upon the adopted value of the precession. The tropical year should therefore be substituted for the sidereal year in the resolution above."

Furthermore, the report states: "The second is the fraction 1:31556925.975 of the tropical year for 1900.0. [...] The proposed unit had, in effect, been agreed at the Rome meeting of the I.A.U. and all that was needed now was a minor correction by the substitution of 'tropical year' for 'sidereal year'."

CONCLUSIONS

A) The time interval of the sidereal year for 1900.0 is NOT about 1223 seconds longer than the time interval of the tropical year for 1900.0.

B) The civil calendar is NOT being corrected for an additional time discrepancy of about 1223 seconds per complete orbit period of Earth.

C) The complete orbit period of Earth is NOT about 1223 seconds longer than the fundamental time interval of the tropical year for 1900.

DISCUSSION

The scientific arguments presented here have conclusively proven that the time intervals of the tropical year and the sidereal year are indeed equivalent. It is has been mathematically verified that the mean time interval of 31,556,925.97474 s is Earth's true 360° orbit period, implying that the equinoctial points do NOT retrograde around the sun. Hence, we can no longer subscribe to the erroneous conclusion that the 'precession of Earth' is a scientific fact. However, the gradual displacement of the equinoctial points relative to inertial space does require a scientific explanation. [...]

Further Research and Observations are needed.