# Accuracy of calculating the vernal equinox?

Calculating the exact time of the vernal equinox is essential for many ephemeris calculations. NASA Goddard Institute for Space Studies uses (at least for some purposes) the following, rather short source code:

Function VERNAL (IYEAR)
Implicit Real*8 (A-H,O-Z)
Parameter (EDAYzY=365.2425d0, VE2000=79.3125)
VERNAL = VE2000 + (IYEAR-2000)*EDAYzY
Return
End


What that means becomes clear when reading the comment inside the code:

For a given year, VERNAL calculates an approximate time of vernal equinox in days measured from 2000 January 1, hour 0.

VERNAL assumes that vernal equinoxes from one year to the next are separated by exactly 365.2425 days, a tropical year [Explanatory Supplement to The Astronomical Ephemeris]. If the tropical year is 365.2422 days, as indicated by other references, then the time of the vernal equinox will be off by 2.88 hours in 400 years.

Time of vernal equinox for year 2000 A.D. is March 20, 7:36 GMT [NASA Reference Publication 1349, Oct. 1994]. VERNAL assumes that vernal equinox for year 2000 will be on March 20, 7:30, or 79.3125 days from 2000 January 1, hour 0. Vernal equinoxes for other years returned by VERNAL are also measured in days from 2000 January 1, hour 0. 79.3125 = 31 + 29 + 19 + 7.5/24.

I highlighted the essential part of the quote. We know for sure that this algorithm will give the wrong date of the vernal equinox in the year 5333 when it will be off by more than 24 hours, but I suppose it can already be incorrect earlier if the time stamp for the vernal equinox is close enough to midnight. When would be the first time that this algorithm picks an incorrect day?

My question: Is the above quoted algorithm the most accurate way of calculating the vernal equinox for times far in the future or past? How would you do it accurately for time scales of 1000s of years, say with e.g. for an application for archeology?

PS: WolframAlpha calculates vernal equinoxes in the distant future with giving hours and minutes and without any error bars which seems too good to be true.

• Two comments: (1) Re Calculating the exact time of the vernal equinox is essential for many astronomical calculations: Citation needed. (2) Re WolframAlpha calculates vernal equinoxes in the distant future: Think of WolframAlpha as the Wikipedia of calculation. There are some places where Wikipedia is inaccurate, and a few where it is flat-out wrong. This also happens with WolframAlpha, but at a higher rate. Wikipedia has the benefit of lots of editors who freely volunteer their time to correct (but sometimes induce) errors. WolframAlpha does not have that benefit. Commented Apr 6, 2021 at 10:21
• Meeus (Astronomical Algorithms, Second Edition, 1998) devotes a full chapter (§27) to the calculation of the equinox and solstice dates. I will not quote it here, for copyright issues, but you can probably find it at a local library. Commented Apr 6, 2021 at 23:06
• @B--rian Re the claim "Calculating the exact time of the vernal equinox is essential for many ephemeris calculations." This is no longer true. Modern ephemerides are calculated using a frame based on quasars. The exact time of the vernal equinox does not come into play. Not-so-modern ephemerides were calculated using a frame based on the "fixed stars". The exact time of the vernal equinox once again did not come into play. Astronomers have moved away from using mean of date or true of date coordinates, in which the location of the Sun at the time of the vernal equinox did play a critical role. Commented Apr 7, 2021 at 12:46
• I would venture that nowadays, the only astronomical concern for which calculating the exact time of the vernal equinox is essential is calculating the exact time of the vernal equinox itself. Commented Apr 7, 2021 at 12:48
• @B--rian The equinox points, where the ecliptic crosses the celestial equator, are still significant. Ecliptic & equatorial coordinates use the First Point of Aries as a reference direction. But the exact time that the Sun is at those points isn't very important. See en.wikipedia.org/wiki/Celestial_coordinate_system & the articles linked there. Commented Apr 7, 2021 at 13:07

The climate modelers' VERNAL is equivalent to this formula, where Y is an integer, ΔY = Y - 2000, and JD0 is epoch J2000 = JD 2451545.0 = 2000-01-01 12:00 TT:

$$\text{JD}_\unicode{x2648}(Y) = \text{JD}_0 + 78.813 + 365.24250~{\Delta Y}$$

We can tune the coefficients to fit JPL DE431 for years -5000 to 9000:

$$\text{JD}_\unicode{x2648}(Y) = \text{JD}_0 + 79.414 + 365.24228~{\Delta Y}$$

but the estimated dates still have 11.6 hours RMS difference from dates computed using DE431.

A higher order polynomial can fit more closely:

\begin{align} \text{JD}_\unicode{x2648}(Y) = \text{JD}_0 &+ 78.814 + 365.24236~{\Delta Y} + 5.004\times 10^{-8}~{\Delta Y}^2 \\ &- 2.87 \times 10^{-12}~{\Delta Y}^3 - 4.5 \times 10^{-16}~{\Delta Y}^4 \end{align}

Estimated this way, March equinox dates for years -5000 to 9000 have only 21.5 minutes RMS difference from DE431.

Meeus Astronomical Algorithms table 27.B gives a similar polynomial optimized for years 1000 to 3000. The periodic terms in table 27.C, modeling perturbations by the Moon and other planets, add up to 29 minutes at most.

Calculating the exact time of the vernal equinox is essential for many astronomical calculations.

I dispute that claim. What is true is that calculating the exact time of the vernal equinox is essential for some astrological and religious calculations.

My question: Is the above quoted algorithm the most accurate way of calculating the vernal equinox for times far in the future or past?

Of course not. The value of 365.2425 days is an exact value; it is the average number of days per year per the Gregorian calendar. The Gregorian calendar repeats over a 400 span. In any 400 span, there will be 97 leap years (96 non-century leap years, plus one century leap year) with 366 days and 303 years with 365 days. That results in 146097/400 days in a year on average, or exactly 365.2425 days.

The Gregorian calendar was developed as a replacement for the Julian calendar because scholars attached to the Catholic Church were finding it increasingly more difficult to correctly calculate the date on which Easter should fall. The Julian calendar had a leap year every four years. That was a bit too much, and it resulted in an ever increasing number of days between the Jewish Passover and the Christian Easter. The switch to the Gregorian calendar, which is where that 365.2425 values comes from, was motivated by religion rather than astronomy.

• Its not that calculating the date of easter was becoming more difficult, it was that it was becomeing increasingly disconnected from the reality of the time of equinox and full moon. The church has always (since nicea) used a calculated not observed date of Easter, but the calculations were not matching the real objects that they were nominally meant to represent. Commented Apr 6, 2021 at 11:13
• @JamesK Just to clarify: Yes, it was Easter recently, and I did read about Gauss Easter formula, but I am indeed interested only in the astronomical part of it, namely the calculation of the vernal equinox including error bounds, from about 5000 B.C. onwards Commented Apr 6, 2021 at 16:13