<p>Maybe this is only a partial answer, but the post was getting too long as it anyway.</p>
<p>I’ve since realised that there are major pitfalls to my method of trying to interpret peaking frequencies in the discrete Fourier transform. The closest analogy I can think of is <a href="https://en.wikipedia.org/wiki/Aliasing#Folding" rel="nofollow noreferrer">aliasing/folding</a> due to the discrete nature of the Fourier transform.</p>
<p>In short, if I see a peak at frequency <span class="math-container">$0.37 y^{-1}$</span> (period of 2.7 years) that could have been from an underlying frequency of <span class="math-container">$(n+0.37) y^{-1}$</span> or <span class="math-container">$(n+0.63) y^{-1}$</span> for any integer <span class="math-container">$n$</span>. As if on cue, the synodic and sidereal lunar month periods correspond to the frequencies <span class="math-container">$12.37 y^{-1}$</span> and <span class="math-container">$13.37 y^{-1}$</span> so it’s probably true that “a lot of the variation is just because the Earth &amp; Moon orbit their barycentre” as pointed by PM2Ring.</p>
<p>I still don’t know why I had convinced myself that high frequencies when sampled at a low rate would be pseudo-random. Of course they would not be, it would just appear to be a lower, different frequency. This whole doubt could have been avoided if I did not have this erroneous mental model. I just wasted my own and other people’s time, sorry about that</p>
<p>As for the other peaks, even though I don’t know exactly what they are from, they’re less interesting in light of this. The peak at <span class="math-container">$0.25 y^{-1}$</span> could be due to a 4 year process or 4/3 years or 4/5 or 4/7 etc.</p>

CC BY-SA 4.0
2024-04-16T09:36:37.953Z
The cause of the peak at $0.25\,\text{y}^{-1}$ is obvious: Every fourth year between 1901 and 2099 is a leap year.
David Hammen
CC BY-SA 4.0
2024-04-16T10:06:28.037Z
@DavidHammen The leap year cycle certainly affects the calendar date of the equinox. But why should it affect the tropical year length (which anonymous is reading from a table on TimeAndDate)?
PM 2Ring
CC BY-SA 4.0
2024-04-16T05:11:11.537Z
2024-04-16T05:11:11.537Z
anonymous
<p>By a &quot;march-equinox year&quot; I mean the time between successive march/vernal equinoxes. I understand that this is different from the mean tropical year, and I also understand why this would be different from (say) a &quot;june-solstice year&quot;.</p>
<p><a href="https://astronomy.stackexchange.com/a/36256/56000">This</a> discusses long term variations in the march equinox year but I am looking at (much to my own plight) the short-term variations (year by year)</p>
<p>It seems that the mean tropical year is, over the short term of say 200 years, <a href="https://en.wikipedia.org/wiki/Tropical_year#Mean_tropical_year_current_value" rel="nofollow noreferrer">roughly constant</a> (variation of the order of 1 second).</p>
<p>However, the march equinox year can vary sporadically by upto 15 minutes from the previous year as evident from <a href="https://www.timeanddate.com/astronomy/tropicalyearlength.html" rel="nofollow noreferrer">this time-and-date article</a> (note that even though they title the table as 'tropical' year, each row is unambiguously the march-equinox year and not the mean tropical year). I tried to plot these values:</p>
<p><a href="https://i.sstatic.net/MinSURpB.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/MinSURpB.png" alt="enter image description here" /></a></p>
<p>First of all, why does it look a little different between ~1955 and ~1985? Does it have something to do with difference between observed values and predicted values or is it something simple and unrelated like the higher frequency terms form some kind of <a href="https://en.wikipedia.org/wiki/Beat_(acoustics)" rel="nofollow noreferrer">'beat pattern'</a> that vanish near ~1970.</p>
<p>Second of all, what causes this variation? For instance I understand that the earth's nutation (which has a period of 18.6 years) has a component in the same, erm, direction (?) as axial precession so that would be one part of this variation but that part has a period too large to explain a year by year variation of 10-15 minutes</p>
<p>I guessed that <em>somehow</em> it must be the combined effect of all the other planets' gravity on earth but if I try to get a sense of the dominant frequencis of this variation via a discrete fourier transform I see the following:</p>
<p><a href="https://i.sstatic.net/mLcJezbD.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/mLcJezbD.png" alt="enter image description here" /></a></p>
<p>Where the leftmost peak does indeed happen at (or, well, near) the familiar nutation period of 18.6 years. But from the 2 large peaks on the right, it seems the year by year variation is largely caused by 'something' of period ~4 years and ~2.7 years. None of the planets have an orbital period close to that so I don't have an intuition for this.</p>
<p>I am not particularly curious about <em>every single</em> peak in the above graph but for the sake of completeness the other three major peaks in between correspond to time periods of ~11.76 years (Jupiter!), ~8 years and ~5.9 years.</p>
<p>Can anyone help?</p>
<hr />
<p>Edit: To summarise some relevant discussions in the comments, @PM2Ring pointed out that since the earth orbits the EMB in a very dynamic orbit it has atleast some effect on, for example, when <em>exactly</em> equinox is achieved. This would be similar to how the &quot;perihelion year&quot; (time between successive perihelions) can vary by upto a day from the previous year.</p>
<p>But I had always assumed that since the orbit of the earth around EMB has periods much lesser than 1 year, its effects observed on a yearly 'sampling rate' (i.e. its influence on time of perihelion) should be somewhat pseudo-random. And by extension I assumed that the observed variation of the order of 1 day in a perihelion-year would be pseudo-random.</p>
<p>But it's not! (I think, not completely atleast. Maybe I've been abusing terminology by calling it pseudo-random to begin with. Anyway:)</p>
<p>From <a href="http://www.astropixels.com/ephemeris/perap/perap1901.html" rel="nofollow noreferrer">here</a> and <a href="http://www.astropixels.com/ephemeris/perap2001.html" rel="nofollow noreferrer">here</a> I gather the perihelion-year durations from 1900-2100 and plot them as:</p>
<p><a href="https://i.sstatic.net/kkwns8b8.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/kkwns8b8.png" alt="enter image description here" /></a></p>
<p>Whose discrete fourier transform is:</p>
<p><a href="https://i.sstatic.net/bbQj8IUr.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/bbQj8IUr.png" alt="enter image description here" /></a></p>
<p>Where the rightmost peak corresponds to a period of ~2.7 years same as before! From @PM2Ring's suggestion it could have something to do with other small lunar cycles (which is something I'll try and look into when I next get free time), but now I'm not even completely convinced that it's not somehow an artefact of performing DFTs that I might be overlooking.</p>
<p>Either way it's not yet clear why there is a peak for 2.7 years in both cases.</p>

CC BY-SA 4.0
2024-04-15T09:54:49.510Z
Here's a plot I did recently using Horizons data. I used TT time, and the geocenter. https://i.sstatic.net/4CHtg.png
PM 2Ring
CC BY-SA 4.0
2024-04-15T09:58:01.063Z
My guess is that the 8 years is from the 8:13 "near-resonance" of Venus. https://astronomy.stackexchange.com/a/36489/16685
PM 2Ring
CC BY-SA 4.0
2024-04-15T10:16:28.853Z
I expect that a lot of the variation is just because the Earth & Moon orbit their barycentre. And that orbit is fairly dynamic because of the lunar precessions, even discounting the perturbations due to Jupiter & Venus, etc. https://astronomy.stackexchange.com/a/55112/16685
PM 2Ring
CC BY-SA 4.0
2024-04-15T11:28:41.540Z
The mean tropical year (taking the mean of the equinox & solstice years) has a bit less variation. https://i.sstatic.net/65YQb3CB.png The mean year mentioned on Wikipedia is averaged over a span of many years.
PM 2Ring
CC BY-SA 4.0
2024-04-15T12:15:00.660Z
@PM2Ring Thanks for replying! [reply to 4] Interesting, the nutation period is much more pronounced and easily visible. [reply to 3] I actually had discarded that line of reasoning when I first thought about it because I figured that such an effect from the moon would cause much larger variations than 15 minutes as it does in the case of time between successive perihelions (it can be off by more than a day from the previous year). But just now I made a similar DFT for time between successive perihelions and there is an even bigger peak for ~2.7 yrs! Am I reading into it or could it be related?
anonymous
CC BY-SA 4.0
2024-04-15T12:27:54.913Z
FWIW, I have some info (with graphs) on the variation of perihelion due to Earth orbiting the EMB: https://astronomy.stackexchange.com/a/49605/16685 I expect the effect on equinox & solstice times is similar, except that at perihelion the ecliptic longitude speed is maximum, so it magnifies the effect.
PM 2Ring
CC BY-SA 4.0
2024-04-15T12:42:31.353Z
All that makes sense except it's difficult to interpret the DFTs in this context. I can get behind the fact that the earth orbiting EMB causes variations but this variation could not be what the ~2.7 year period peak we observe in the DFT (in both cases). Infact our 'sampling rate' is 1 year because of which the EMB effect (whose period of variation is much less than 2 times 1 year) should effectively be pseudo-random, right? maybe I should add onto the post if I'm being unclear
anonymous
CC BY-SA 4.0
2024-04-15T12:53:12.700Z
Yeah, I have no idea where that 2.7 y peak is coming from. But there are lots of small-ish lunar cycles, as mentioned in  https://en.wikipedia.org/wiki/Eclipse_cycle Maybe try some of the lunar months (or small multiples) as your sampling rate, to see what patterns they bring up. Especially the synodic, anomalistic, and draconic months. https://en.wikipedia.org/wiki/Lunar_month
PM 2Ring
CC BY-SA 4.0
2024-04-15T13:29:55.113Z
Time And Date is a great site. But I can't find what exact definition they're using for equinox / solstice. Are they using the geocenter, or a surface site (eg, Greenwich)? Are they using UTC / GMT in their year length calculations, or an ephemeris time (preferably TT)? BTW, there were subtle changes to timekeeping from the mid-1950s to 1972, with the intoduction of atomic time, and UTC with leap seconds (and a few even more subtle tweaks to atomic time after that), but that doesn't explain the "different" section in your graph over that period.
PM 2Ring
<solar-system><celestial-mechanics><equinox>
What causes the short term variation in the length of a march-equinox year
anonymous
# What causes the short term variation in the length of a march-equinox year

(Asked by: username_0 on 2024-04-15)

By a "march-equinox year" I mean the time between successive march/vernal equinoxes. I understand that this is different from the mean tropical year, and I also understand why this would be different from (say) a "june-solstice year".

[This](https://astronomy.stackexchange.com/a/36256/56000) discusses long term variations in the march equinox year but I am looking at (much to my own plight) the short-term variations (year by year)

It seems that the mean tropical year is, over the short term of say 200 years, [roughly constant](https://en.wikipedia.org/wiki/Tropical_year#Mean_tropical_year_current_value) (variation of the order of 1 second).

However, the march equinox year can vary sporadically by upto 15 minutes from the previous year as evident from [this time-and-date article](https://www.timeanddate.com/astronomy/tropicalyearlength.html) (note that even though they title the table as 'tropical' year, each row is unambiguously the march-equinox year and not the mean tropical year). I tried to plot these values:

[![enter image description here](https://i.sstatic.net/MinSURpB.png)](https://i.sstatic.net/MinSURpB.png)

First of all, why does it look a little different between ~1955 and ~1985? Does it have something to do with difference between observed values and predicted values or is it something simple and unrelated like the higher frequency terms form some kind of ['beat pattern'](https://en.wikipedia.org/wiki/Beat_(acoustics)) that vanish near ~1970.

Second of all, what causes this variation? For instance I understand that the earth's nutation (which has a period of 18.6 years) has a component in the same, erm, direction (?) as axial precession so that would be one part of this variation but that part has a period too large to explain a year by year variation of 10-15 minutes

I guessed that *somehow* it must be the combined effect of all the other planets' gravity on earth but if I try to get a sense of the dominant frequencis of this variation via a discrete fourier transform I see the following:

[![enter image description here](https://i.sstatic.net/mLcJezbD.png)](https://i.sstatic.net/mLcJezbD.png)

Where the leftmost peak does indeed happen at (or, well, near) the familiar nutation period of 18.6 years. But from the 2 large peaks on the right, it seems the year by year variation is largely caused by 'something' of period ~4 years and ~2.7 years. None of the planets have an orbital period close to that so I don't have an intuition for this.

I am not particularly curious about *every single* peak in the above graph but for the sake of completeness the other three major peaks in between correspond to time periods of ~11.76 years (Jupiter!), ~8 years and ~5.9 years.

Can anyone help?

---

Edit: To summarise some relevant discussions in the comments, @PM2Ring pointed out that since the earth orbits the EMB in a very dynamic orbit it has atleast some effect on, for example, when *exactly* equinox is achieved. This would be similar to how the "perihelion year" (time between successive perihelions) can vary by upto a day from the previous year.

But I had always assumed that since the orbit of the earth around EMB has periods much lesser than 1 year, its effects observed on a yearly 'sampling rate' (i.e. its influence on time of perihelion) should be somewhat pseudo-random. And by extension I assumed that the observed variation of the order of 1 day in a perihelion-year would be pseudo-random.

But it's not! (I think, not completely atleast. Maybe I've been abusing terminology by calling it pseudo-random to begin with. Anyway:)

From [here](http://www.astropixels.com/ephemeris/perap/perap1901.html) and [here](http://www.astropixels.com/ephemeris/perap2001.html) I gather the perihelion-year durations from 1900-2100 and plot them as:

[![enter image description here](https://i.sstatic.net/kkwns8b8.png)](https://i.sstatic.net/kkwns8b8.png)

Whose discrete fourier transform is:

[![enter image description here](https://i.sstatic.net/bbQj8IUr.png)](https://i.sstatic.net/bbQj8IUr.png)

Where the rightmost peak corresponds to a period of ~2.7 years same as before! From @PM2Ring's suggestion it could have something to do with other small lunar cycles (which is something I'll try and look into when I next get free time), but now I'm not even completely convinced that it's not somehow an artefact of performing DFTs that I might be overlooking.

Either way it's not yet clear why there is a peak for 2.7 years in both cases.

**Tags:** solar-system, celestial-mechanics, equinox
> **Comments:**
> **username_1**: Here's a plot I did recently using Horizons data. I used TT time, and the geocenter. https://i.sstatic.net/4CHtg.png  
> **username_1**: My guess is that the 8 years is from the 8:13 "near-resonance" of Venus. https://astronomy.stackexchange.com/a/36489/16685  
> **username_1**: I expect that a lot of the variation is just because the Earth & Moon orbit their barycentre. And that orbit is fairly dynamic because of the lunar precessions, even discounting the perturbations due to Jupiter & Venus, etc. https://astronomy.stackexchange.com/a/55112/16685  
> **username_1**: The mean tropical year (taking the mean of the equinox & solstice years) has a bit less variation. https://i.sstatic.net/65YQb3CB.png The mean year mentioned on Wikipedia is averaged over a span of many years.  
> **username_0**: @PM2Ring Thanks for replying! [reply to 4] Interesting, the nutation period is much more pronounced and easily visible. [reply to 3] I actually had discarded that line of reasoning when I first thought about it because I figured that such an effect from the moon would cause much larger variations than 15 minutes as it does in the case of time between successive perihelions (it can be off by more than a day from the previous year). But just now I made a similar DFT for time between successive perihelions and there is an even bigger peak for ~2.7 yrs! Am I reading into it or could it be related?  
> **username_1**: FWIW, I have some info (with graphs) on the variation of perihelion due to Earth orbiting the EMB: https://astronomy.stackexchange.com/a/49605/16685 I expect the effect on equinox & solstice times is similar, except that at perihelion the ecliptic longitude speed is maximum, so it magnifies the effect.  
> **username_0**: All that makes sense except it's difficult to interpret the DFTs in this context. I can get behind the fact that the earth orbiting EMB causes variations but this variation could not be what the ~2.7 year period peak we observe in the DFT (in both cases). Infact our 'sampling rate' is 1 year because of which the EMB effect (whose period of variation is much less than 2 times 1 year) should effectively be pseudo-random, right? maybe I should add onto the post if I'm being unclear  
> **username_1**: Yeah, I have no idea where that 2.7 y peak is coming from. But there are lots of small-ish lunar cycles, as mentioned in  https://en.wikipedia.org/wiki/Eclipse_cycle Maybe try some of the lunar months (or small multiples) as your sampling rate, to see what patterns they bring up. Especially the synodic, anomalistic, and draconic months. https://en.wikipedia.org/wiki/Lunar_month  
> **username_1**: Time And Date is a great site. But I can't find what exact definition they're using for equinox / solstice. Are they using the geocenter, or a surface site (eg, Greenwich)? Are they using UTC / GMT in their year length calculations, or an ephemeris time (preferably TT)? BTW, there were subtle changes to timekeeping from the mid-1950s to 1972, with the intoduction of atomic time, and UTC with leap seconds (and a few even more subtle tweaks to atomic time after that), but that doesn't explain the "different" section in your graph over that period.  

---

## 1 answers

### Answer by username_0 on 2024-04-16. Score: 0

Maybe this is only a partial answer, but the post was getting too long as it anyway.

I’ve since realised that there are major pitfalls to my method of trying to interpret peaking frequencies in the discrete Fourier transform. The closest analogy I can think of is [aliasing/folding](https://en.wikipedia.org/wiki/Aliasing#Folding) due to the discrete nature of the Fourier transform.

In short, if I see a peak at frequency $0.37 y^{-1}$ (period of 2.7 years) that could have been from an underlying frequency of $(n+0.37) y^{-1}$ or $(n+0.63) y^{-1}$ for any integer $n$. As if on cue, the synodic and sidereal lunar month periods correspond to the frequencies $12.37 y^{-1}$ and $13.37 y^{-1}$ so it’s probably true that “a lot of the variation is just because the Earth & Moon orbit their barycentre” as pointed by PM2Ring.

I still don’t know why I had convinced myself that high frequencies when sampled at a low rate would be pseudo-random. Of course they would not be, it would just appear to be a lower, different frequency. This whole doubt could have been avoided if I did not have this erroneous mental model. I just wasted my own and other people’s time, sorry about that

As for the other peaks, even though I don’t know exactly what they are from, they’re less interesting in light of this. The peak at $0.25 y^{-1}$ could be due to a 4 year process or 4/3 years or 4/5 or 4/7 etc.

> **Comments:**
> **username_2**: The cause of the peak at $0.25\,\text{y}^{-1}$ is obvious: Every fourth year between 1901 and 2099 is a leap year.  
> **username_1**: @DavidHammen The leap year cycle certainly affects the calendar date of the equinox. But why should it affect the tropical year length (which anonymous is reading from a table on TimeAndDate)?  

---

