<blockquote>
<p>Calculating the exact time of the vernal equinox is essential for many astronomical calculations.</p>
</blockquote>
<p>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.</p>
<blockquote>
<p>My question: <em>Is the above quoted algorithm the most accurate way of calculating the vernal equinox for times far in the future or past?</em></p>
</blockquote>
<p>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.</p>
<p>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.</p>

CC BY-SA 4.0
2021-04-06T11:13:41.657Z
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.
James K
CC BY-SA 4.0
2021-04-06T16:13:48.400Z
@JamesK Just to clarify: Yes, it was Easter recently, and I did read about [Gauss Easter formula](https://en.m.wikipedia.org/wiki/Gauss%27s_Easter_algorithm), 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
B--rian
CC BY-SA 4.0
2021-04-06T10:43:09.007Z
2021-04-07T12:38:15.540Z
2021-04-07T12:38:15.540Z
David Hammen
<p>The <a href="https://data.giss.nasa.gov/modelE/ar5plots/solar.html" rel="nofollow noreferrer">climate modelers</a>' <code>VERNAL</code> is equivalent to this formula,
where Y is an integer, ΔY = Y - 2000,
and JD<sub>0</sub> is epoch J2000 = JD 2451545.0 = 2000-01-01 12:00 TT:</p>
<p><span class="math-container">$$ \text{JD}_\unicode{x2648}(Y) = \text{JD}_0
    + 78.813 + 365.24250~{\Delta Y} $$</span></p>
<p>We can tune the coefficients to fit JPL DE431 for years -5000 to 9000:</p>
<p><span class="math-container">$$ \text{JD}_\unicode{x2648}(Y) = \text{JD}_0
    + 79.414 + 365.24228~{\Delta Y} $$</span></p>
<p>but the estimated dates still have 11.6 hours RMS difference from dates computed using DE431.</p>
<p>A higher order polynomial can fit more closely:</p>
<p><span class="math-container">$$\begin{align}
\text{JD}_\unicode{x2648}(Y) = \text{JD}_0
    &amp;+ 78.814 + 365.24236~{\Delta Y}
    + 5.004\times 10^{-8}~{\Delta Y}^2 \\
    &amp;- 2.87 \times 10^{-12}~{\Delta Y}^3
    - 4.5 \times 10^{-16}~{\Delta Y}^4
\end{align}$$</span></p>
<p>Estimated this way, March equinox dates for years -5000 to 9000 have only 21.5 minutes RMS difference from DE431.</p>
<p><a href="https://shopatsky.com/products/astronomical-algorithms-2nd-edition" rel="nofollow noreferrer">Meeus <em>Astronomical Algorithms</em></a> 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.</p>

CC BY-SA 4.0
2021-04-09T23:51:55.747Z
2023-12-14T13:50:33.787Z
2023-12-14T13:50:33.787Z
Mike G
<p>Calculating the exact time of the <a href="https://en.wikipedia.org/wiki/March_equinox" rel="nofollow noreferrer">vernal equinox</a> is essential for many <a href="https://en.wikipedia.org/wiki/Ephemeris" rel="nofollow noreferrer">ephemeris calculations</a>. <a href="https://data.giss.nasa.gov/modelE/ar5plots/srvernal.html" rel="nofollow noreferrer">NASA Goddard Institute for Space Studies</a> uses (at least for some purposes) the following, rather short source code:</p>
<pre><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
</code></pre>
<p>What that means becomes clear when reading the comment inside the code:</p>
<blockquote>
<p>For a given year, VERNAL calculates an approximate time of vernal equinox in days measured from 2000 January 1, hour 0.</p>
<p>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].  <strong>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.</strong></p>
<p>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.</p>
</blockquote>
<p>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?</p>
<p>My question: <strong>Is the above quoted algorithm the most accurate way of calculating the vernal equinox for times far in the future or past?</strong> How would you do it accurately for time scales of 1000s of years, say with e.g. for an application for archeology?</p>
<p>PS: <a href="https://www.wolframalpha.com/input/?i=vernal+equinox+in+5000+years" rel="nofollow noreferrer">WolframAlpha calculates vernal equinoxes in the distant future</a> <em>with giving hours and minutes</em> and without any error bars which seems too good to be true.</p>

CC BY-SA 4.0
2021-04-06T10:21:04.497Z
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.
David Hammen
CC BY-SA 4.0
2021-04-06T10:35:42.287Z
@DavidHammen ad (1) No quote, just my experience, probably to unclear stated. I mean mainly *calculating ephemeris*. ad (2) I am well aware of the limitations of WolframAlpha, and I agree that is a much grayer box than Wikipedia. Nevertheless you can still contribute by asking the developers. I find it potentially very dangerous to claim exact dates in the future. The link to WA was meant as oracle, not as real reference.
B--rian
CC BY-SA 4.0
2021-04-06T10:39:19.303Z
@DavidHammen I edited my post slightly, if you know a better wording for my PS, please feel free to edit as well.
B--rian
CC BY-SA 4.0
2021-04-06T16:34:34.027Z
BTW, the vernal equinox year is slightly different to the mean tropical year. See https://calendars.wikia.org/wiki/Tropical_year
PM 2Ring
CC BY-SA 4.0
2021-04-06T23:06:10.983Z
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.
Pierre Paquette
CC BY-SA 4.0
2021-04-07T12:46:26.207Z
@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.
David Hammen
CC BY-SA 4.0
2021-04-07T12:48:55.940Z
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.
David Hammen
CC BY-SA 4.0
2021-04-07T12:53:00.660Z
@DavidHammen Thanks for the very insightful comment on ephemeris calculation and showing me that I am barking the wrong tree. I had some 1970s celestrial mechanics tutorial which I recently digged out again and which is mainly teaching what I claimed - that the vernal equinox is essential.
B--rian
CC BY-SA 4.0
2021-04-07T13:07:04.553Z
@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 https://en.wikipedia.org/wiki/Celestial_coordinate_system & the articles linked there.
PM 2Ring
CC BY-SA 4.0
2023-12-14T14:40:11.113Z
I should clarify that I'm *not* disagreeing with David Hammen. Modern ecliptic & equatorial coordinates refer to the quasar-based ICRF. That frame is very closely aligned with the equinox direction of J2000.0
PM 2Ring
CC BY-SA 4.0
2023-12-14T15:14:51.593Z
The term "Vernal Equinox" has different meanings.  Originally it was when the Sun passed over the equator from South to North.  But that would only be measurable at that specific instance, and there's no stars to compare it to.  So, even in ancient times, it was more of a computed value, rather than measured directly.  More modern usage defines it as ~"the intersection of the plane of Earth's equator with the plane of Earth's orbit".  And the instant of the equinox is when Earth's ecliptic longitude is 90deg.  IAU's SOFA libraries have algorithms to compute the instantaneous equinox.
Greg Miller
<mathematics><celestial-mechanics><ephemeris><equinox><archaeoastronomy>
Accuracy of calculating the vernal equinox?
B--rian
# Accuracy of calculating the vernal equinox?

(Asked by: username_0 on 2021-04-06)

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.

**Tags:** mathematics, celestial-mechanics, ephemeris, equinox, archaeoastronomy
> **Comments:**
> **username_1**: 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.  
> **username_0**: @DavidHammen ad (1) No quote, just my experience, probably to unclear stated. I mean mainly *calculating ephemeris*. ad (2) I am well aware of the limitations of WolframAlpha, and I agree that is a much grayer box than Wikipedia. Nevertheless you can still contribute by asking the developers. I find it potentially very dangerous to claim exact dates in the future. The link to WA was meant as oracle, not as real reference.  
> **username_0**: @DavidHammen I edited my post slightly, if you know a better wording for my PS, please feel free to edit as well.  
> **username_2**: BTW, the vernal equinox year is slightly different to the mean tropical year. See https://calendars.wikia.org/wiki/Tropical_year  
> **username_3**: 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.  
> **username_1**: @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.  
> **username_1**: 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.  
> **username_0**: @DavidHammen Thanks for the very insightful comment on ephemeris calculation and showing me that I am barking the wrong tree. I had some 1970s celestrial mechanics tutorial which I recently digged out again and which is mainly teaching what I claimed - that the vernal equinox is essential.  
> **username_2**: @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 https://en.wikipedia.org/wiki/Celestial_coordinate_system & the articles linked there.  
> **username_2**: I should clarify that I'm *not* disagreeing with David Hammen. Modern ecliptic & equatorial coordinates refer to the quasar-based ICRF. That frame is very closely aligned with the equinox direction of J2000.0  
> **username_4**: The term "Vernal Equinox" has different meanings.  Originally it was when the Sun passed over the equator from South to North.  But that would only be measurable at that specific instance, and there's no stars to compare it to.  So, even in ancient times, it was more of a computed value, rather than measured directly.  More modern usage defines it as ~"the intersection of the plane of Earth's equator with the plane of Earth's orbit".  And the instant of the equinox is when Earth's ecliptic longitude is 90deg.  IAU's SOFA libraries have algorithms to compute the instantaneous equinox.  

---

## 2 answers

### Answer by username_6 on 2021-04-09. Score: 6 | Accepted answer

The [climate modelers](https://data.giss.nasa.gov/modelE/ar5plots/solar.html)' `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*](https://shopatsky.com/products/astronomical-algorithms-2nd-edition) 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.


---

### Answer by username_1 on 2021-04-06. Score: 4

> 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.

> **Comments:**
> **username_5**: 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.  
> **username_0**: @JamesK Just to clarify: Yes, it was Easter recently, and I did read about [Gauss Easter formula](https://en.m.wikipedia.org/wiki/Gauss%27s_Easter_algorithm), 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  

---

