<p>Using PyEphem, I calculated the time difference between consecutive summer solstices in the south hemisphere for the last 4000 years, and plotted them in a graph. It seems like random noise around the mean, but the amplitude seems to oscilate with a period of around ~161 years. What explains this behavior?</p>
<p><a href="https://i.sstatic.net/PNlZz.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/PNlZz.png" alt="enter image description here" /></a></p>
<p>Code used:</p>
<pre><code>import ephem

date = ephem.Date(&quot;2023/01/01&quot;)
solstice = ephem.previous_winter_solstice(date)
for i in range(4000):
    previous_solstice = ephem.previous_winter_solstice(solstice)
    print(solstice - previous_solstice)
    solstice = previous_solstice
</code></pre>

CC BY-SA 4.0
2023-06-04T15:44:20.890Z
I *suspect* that it's due to some cycle involving the Moon, similar to what happens with perihelion & aphelion, as I explained [here](https://astronomy.stackexchange.com/a/49605/16685).
PM 2Ring
CC BY-SA 4.0
2023-06-04T15:45:36.597Z
OTOH, just playing with various mean month & year lengths I can't make a simple 161 year cycle. The best I can do is 239 years, just under 1½ of your cycles. Roughly, 2956 synodic months = 3168 anomalistic months = 239 tropical years = 87293 days. https://sagecell.sagemath.org/?z=eJxVjs0OgyAQhO8kvMMetSGmgovQpM9ijNqUWH4CXPTpS20Pdg9z2G8yM9a7_BzS5vxsJrgD1w2KKyqlZEuJPejovB1fH9g3iB12ulBKtmWMQ44-mOmgQmLDO94q3VMSlmj8nMrbnhsYnCMZ_GVQkoou6ZiBkoFopWLAhabk4SOsDDIYB7sJ1dfJ4NdT3yiBciEal6sVLpDrN873Qm8=&lang=python
PM 2Ring
CC BY-SA 4.0
2023-06-04T16:15:11.863Z
The paper [Fedorov, V.M. "Interannual variations in the duration of the tropical year."](https://go.gale.com/ps/i.do?id=GALE%7CA344393169&sid=googleScholar&v=2.1&it=r&linkaccess=abs&issn=1028334X&p=AONE&sw=w&userGroupName=anon%7E3956bc72&aty=open+web+entry) Seems to be about this. The abstract says that it is due to "perturbations" but the article is behind a paywall.
James K
CC BY-SA 4.0
2023-06-04T16:17:50.653Z
If I include the synodic period of Jupiter (398.88404 days), then I get 6011 synodic, 6442 anomalistic, 445 Jupiter, 486 Tropical ~= 177508 days, which is intriguing. ;)
PM 2Ring
CC BY-SA 4.0
2023-06-04T16:37:20.253Z
FWIW, I used the LLL algorithm from the answer to your old question: https://math.stackexchange.com/q/4412720/207316
PM 2Ring
CC BY-SA 4.0
2023-06-04T16:37:43.653Z
You should probably add a little more detail to this question. What time scale are you using (UTC, Terrestrial Time, JPL's TDB, etc)? What exact definition for solstice are you using? Maybe post your Python code, or at least a minimal version that people can use to reproduce your results.
PM 2Ring
CC BY-SA 4.0
2023-06-04T19:18:49.373Z
@PM2Ring I edited to include the code I used.

With regard to finding a cycle that is close to 161 years, the only one I was able to find is the orbital period of Neptune, but it most certainly doesn't have any influence on the tropical year.
ordptt
CC BY-SA 4.0
2023-06-04T22:14:07.903Z
@JamesK: I have read the article, but it mentions no 161-year period. I suspect, as PM 2Ring does, that it has to do with Moon cycles. Multiplying the apsidal precession period (8.61 years) by the nodal precession period (18.6 years) gives 164.61 years, which is close to 161 years…
Pierre Paquette
CC BY-SA 4.0
2023-06-05T03:32:27.197Z
Check the algorithm used to compute it.  Generally the solstice accounts for precession, but not nutation.  So you're likely looking at combined effects of precession and Earth's orbit.  So, to find the source, you'd want to remove one of those variables.  I doubt the result would be anything other than "the math just happens to produce that pattern".
Greg Miller
CC BY-SA 4.0
2023-06-05T14:13:02.773Z
@GregMiller Comparing the solstice times of the algorithm with other sources, they agree within the minute, the difference usually being of a couple of seconds. The variations in the graph are of several minutes, so I tend to think it is not related to an imprecision of the algorithm.
ordptt
CC BY-SA 4.0
2023-06-13T20:56:20.667Z
The 161 year cycle is consistent with Fedorov's paper which notes a decreased variation in the years 1960-2000, but since their Figure 2 only looks at 150 years they don't see the full cycle.  The cycle is visible in the 400 years accessible from the US Naval Observatory at https://aa.usno.navy.mil/data/Earth_Seasons.  The cycle occurs for all solstices and equinoxes, but shifted by 40 years for each step as you go from summer to fall to winter to spring. e.g. When the variations are largest in the summer-solstice "year", they are smallest in the winter-solstice "year".
David Bailey
<orbit><solar-system><time><orbital-elements>
Why does the tropical year have a ~161 years cycle?
ordptt
# Why does the tropical year have a ~161 years cycle?

(Asked by: username_0 on 2023-06-04)

Using PyEphem, I calculated the time difference between consecutive summer solstices in the south hemisphere for the last 4000 years, and plotted them in a graph. It seems like random noise around the mean, but the amplitude seems to oscilate with a period of around ~161 years. What explains this behavior?

Code used:
import ephem

date = ephem.Date("2023/01/01")
solstice = ephem.previous_winter_solstice(date)
for i in range(4000):
    previous_solstice = ephem.previous_winter_solstice(solstice)
    print(solstice - previous_solstice)
    solstice = previous_solstice

**Tags:** orbit, solar-system, time, orbital-elements
> **Comments:**
> **username_1**: I *suspect* that it's due to some cycle involving the Moon, similar to what happens with perihelion & aphelion, as I explained [here](https://astronomy.stackexchange.com/a/49605/16685).  
> **username_1**: OTOH, just playing with various mean month & year lengths I can't make a simple 161 year cycle. The best I can do is 239 years, just under 1½ of your cycles. Roughly, 2956 synodic months = 3168 anomalistic months = 239 tropical years = 87293 days. https://sagecell.sagemath.org/?z=eJxVjs0OgyAQhO8kvMMetSGmgovQpM9ijNqUWH4CXPTpS20Pdg9z2G8yM9a7_BzS5vxsJrgD1w2KKyqlZEuJPejovB1fH9g3iB12ulBKtmWMQ44-mOmgQmLDO94q3VMSlmj8nMrbnhsYnCMZ_GVQkoou6ZiBkoFopWLAhabk4SOsDDIYB7sJ1dfJ4NdT3yiBciEal6sVLpDrN873Qm8=&lang=python  
> **username_2**: The paper [Fedorov, V.M. "Interannual variations in the duration of the tropical year."](https://go.gale.com/ps/i.do?id=GALE%7CA344393169&sid=googleScholar&v=2.1&it=r&linkaccess=abs&issn=1028334X&p=AONE&sw=w&userGroupName=anon%7E3956bc72&aty=open+web+entry) Seems to be about this. The abstract says that it is due to "perturbations" but the article is behind a paywall.  
> **username_1**: If I include the synodic period of Jupiter (398.88404 days), then I get 6011 synodic, 6442 anomalistic, 445 Jupiter, 486 Tropical ~= 177508 days, which is intriguing. ;)  
> **username_1**: FWIW, I used the LLL algorithm from the answer to your old question: https://math.stackexchange.com/q/4412720/207316  
> **username_1**: You should probably add a little more detail to this question. What time scale are you using (UTC, Terrestrial Time, JPL's TDB, etc)? What exact definition for solstice are you using? Maybe post your Python code, or at least a minimal version that people can use to reproduce your results.  
> **username_0**: @PM2Ring I edited to include the code I used.  With regard to finding a cycle that is close to 161 years, the only one I was able to find is the orbital period of Neptune, but it most certainly doesn't have any influence on the tropical year.  
> **username_3**: @JamesK: I have read the article, but it mentions no 161-year period. I suspect, as PM 2Ring does, that it has to do with Moon cycles. Multiplying the apsidal precession period (8.61 years) by the nodal precession period (18.6 years) gives 164.61 years, which is close to 161 years…  
> **username_4**: Check the algorithm used to compute it.  Generally the solstice accounts for precession, but not nutation.  So you're likely looking at combined effects of precession and Earth's orbit.  So, to find the source, you'd want to remove one of those variables.  I doubt the result would be anything other than "the math just happens to produce that pattern".  
> **username_0**: @GregMiller Comparing the solstice times of the algorithm with other sources, they agree within the minute, the difference usually being of a couple of seconds. The variations in the graph are of several minutes, so I tend to think it is not related to an imprecision of the algorithm.  
> **username_5**: The 161 year cycle is consistent with Fedorov's paper which notes a decreased variation in the years 1960-2000, but since their Figure 2 only looks at 150 years they don't see the full cycle.  The cycle is visible in the 400 years accessible from the US Naval Observatory at https://aa.usno.navy.mil/data/Earth_Seasons.  The cycle occurs for all solstices and equinoxes, but shifted by 40 years for each step as you go from summer to fall to winter to spring. e.g. When the variations are largest in the summer-solstice "year", they are smallest in the winter-solstice "year".  

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## 0 answers

