A Puzzle Regarding the Time Between Consecutive March Equinoxes

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Buzz Bloom
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Summary:

I have recently become curious about the calculation of the GMT time of the March equinox (and also for September, but to a lesser extent since it is now March). I found the following source.
http://www.astropixels.com/ephemeris/soleq2001.html
Out of curiosity I calculated (from the cited source using a spreadsheet) the time in minutes between consecutive March equinoxes, and I noticed a pattern of irregularities. (See graph in the post body.)

Main Question or Discussion Point

Unfortunately, the only source I could find on the internet is the one I cited in the summary. This source has the date and GMT time for the two annual equinoxes and the two solstices for a hundred years: 2001-2100.

The 99 time difference values, Ai, shown in the graph below were calculated as described later.

EquinoxChart.png


As I considered the possible causes of this irregularity, I came up with the following candidates.
1. The Earth-Moon barycenter.
2. The circular rotation of the Earth's axis.
3. The general relativity precession of Earth's elliptical orbit.
4. The gravitational influence of other planets. In particular the most likely are Venus and Jupiter.

Of these four, only the first seemed reasonably plausible to me. However, the radius of the barycenter is 4670 km. The Earth's orbital velocity around the sun is approximately 29.79 km/s. This implies that the barycenter effect is limited to a variation in the time between consecutive March equinoxes within an approximate range of +/- 3 minutes. The chart shows a variation range of approximately +/- 13 minutes.

The following is a summary of my calculations.
t0 = midnight (00:00) March 19 for a non-leap-year, and
t0 = midnight (00:00) March 20 for a leap year.
ty = the equinox day in March for year y together with the equinox GMT time of day
My = ty - t0 (in minuites)
i = y - 2001
If y is not a leap year then
Di = ith DELTA time (in minutes) = My+1 - My
If y is a leap year then
Di = ith DELTA time (in minutes) = 24 x 60 + My+1 - My
A = Average[i=1..99](Di)
Ai = Di - A
The chart of Ai values shows the variability of the Di times relative to the average Di time.

If anyone knows of a reliable reference that discusses the causes if the variability of time between consecutive equinoxes, I would very much appreciate it. I searched the internet but failed to find any. If anyone finds an error in my calculations, i would also appreciate that information, as well as any other comments.

I have not included a picture of my spreadsheet, but I will post it if anyone requests it.
 
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Answers and Replies

  • #2
256bits
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a reliable reference
Well, I tried wiki and got,
https://en.wikipedia.org/wiki/Equinox
where they state the march equinox us derived from the
Since the Moon (and to a lesser extent the planets) causes Earth's orbit to slightly vary from a perfect ellipse, the equinox is officially defined by the Sun's more regular ecliptic longitude rather than by its declination. The instants of the equinoxes are currently defined to be when the apparent geocentric longitude of the Sun is 0° and 180°.
and a discussion at,
https://en.wikipedia.org/wiki/Ecliptic_coordinate_system
which if you can easily make sense of that, ...
 
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  • #3
Buzz Bloom
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Well, I tried wiki and got,
https://en.wikipedia.org/wiki/Equinox
where they state the march equinox us derived from the
. . .
and a discussion at,
https://en.wikipedia.org/wiki/Ecliptic_coordinate_system
which if you can easily make sense of that, ...
Hi 256:

Thank you much for your response. I started my search at the Wikipedia article you cite, and went to its reference #2 which is reference I gave.

As I understand the Wikipedia article about the Ecliptic coordinate system, it describes a variation due to the Earth's axis nutation with a period of about 25,700 years. That would produce a much much smaller variation between annual equinoxes than the effect due to the Earth-Moon barycenter, which I mentioned as candidate #2.

Regards,
Buzz
 
  • #4
anorlunda
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Did you read the discussion as @256bits said? Go to the Wikipedia article and click on the "Talk" tab.
 
  • #5
Buzz Bloom
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Well, I tried wiki and got,
https://en.wikipedia.org/wiki/Equinox
where they state the march equinox us derived from the
. . .
and a discussion at,
https://en.wikipedia.org/wiki/Ecliptic_coordinate_system
which if you can easily make sense of that, ...
Hi 256:

Thank you much for your response. I started my search at the Wikipedia article you cite, and went to its reference #2 which is reference I gave.

As I understand the Wikipedia article about the Ecliptic coordinate system, it describes a variation due to the Earth's axis nutation with a period of about 25,700 years. I mentioned that as candidate #2, but that would produce a much much smaller variation between annual equinoxes than the effect due to the Earth-Moon barycenter.

Regards,
Buzz
 
  • #6
Buzz Bloom
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Did you read the discussion as @256bits said? Go to the Wikipedia article and click on the "Talk" tab.
Hi anorlunda:

Thank you for your post.

I am not able to figure out what you are suggesting that I look at in one or the other of two Wikipedia articles. Please suggest which article and which sections you are recommending. I did a quick scan and did nor find anything relevant to the issue I was asking about.

https://en.wikipedia.org/wiki/Talk:Ecliptic_coordinate_system
1 Celestial latitude and longitude
2 conversion algorithm
3 Rectangular coordinate system (ecliptic)
4 Diagram is needed
5 Spherical Trigonometry
6 Units of measurement
7 This was "dumped" in the article. I relocated it here.
7.1 Ecliptic latitude and longitude
8 Removed incorrect addition

https://en.wikipedia.org/wiki/Equinox
Season-specific equinox redirects
RfC on season-specific redirects
Contour plot mis-labels antarctic and arctic circles

Regards,
Buzz
 
  • #7
256bits
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Hi 256:

Thank you much for your response. I started my search at the Wikipedia article you cite, and went to its reference #2 which is reference I gave.

As I understand the Wikipedia article about the Ecliptic coordinate system, it describes a variation due to the Earth's axis nutation with a period of about 25,700 years. I mentioned that as candidate #2, but that would produce a much much smaller variation between annual equinoxes than the effect due to the Earth-Moon barycenter.

Regards,
Buzz
Well am learning too.

Try,
Tropical Year
https://en.wikipedia.org/wiki/Tropical_year

And,
axial precession.
https://en.wikipedia.org/wiki/Axial_precession

In the first wiki it is interesting reading that Hipparchus in the 2nd century BC measured the yearly equinox interval, and also noticed the precession to be about 1 degree per century. How he did by just visually star gazing that is amazing.

The second gives a formula(s) farther down.
Pretty complicated stuff.
It seems you are quite correct in your original assumptions I would think.
 
  • #8
Buzz Bloom
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In the first wiki it is interesting reading that Hipparchus in the 2nd century BC measured the yearly equinox interval, and also noticed the precession to be about 1 degree per century. How he did by just visually star gazing that is amazing.

The second gives a formula(s) farther down.
Pretty complicated stuff.
It seems you are quite correct in your original assumptions I would think.
Hi 256:

Thanks for the additional links. I will have to study the articles in some detail. I will make an effort to determine what specifically the effect on the variability of Ai is due to the nutation of Earth's axis. I think a mystery will still remain, but I will try to keep an open mind.

ADDED

The earth's axis nutation takes approximately N = 15,700 years to complete 360 degrees. The time to complete a year's orbit is approximately Y = 365.2475 days = approximately = 526,669 minutes.
Y/N = approximately 33 minutes. This means that each year between the March equinoxes varies about plus or minus 33 minutes. I am not sure which. I need to study the articles in more detail. The chart shows an approximately periodic cycle of variation with 37 cycles in a hundred years. This is nothing like a 33 minute regular difference per year. So the source of these cycles must be due to another cause.

Regards,
Buzz
 
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  • #9
256bits
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The earth's axis nutation takes approximately N = 15,700 years to complete 360 degrees. The time to complete a year's orbit is approximately Y = 365.2475 days = approximately = 526,669 minutes.
Y/N = approximately 33 minutes. This means that each year between the March equinoxes varies about plus or minus 33 minutes. I am not sure which. I need to study the articles in more detail. The chart shows an approximately periodic cycle of variation with 37 cycles in a hundred years. This is nothing like a 33 minute regular difference per year. So the source of these cycles must be due to another cause.
I am stuck on that part also. I would wager a guess that depending upon when the earth is in its complex cycles that span thousands of years, the length of the tropical year changes and the influence of any one cycle may predominate over another. Or at least combing to give differing lengths of the seasons measured from a solstice to an equinox, and the amount of solar radiation allotted to which hemisphere at particular times of the year.

There are at least three cycles to the wobble of the earth - one is the precession of the rotational axis every 26000 years, another is the change in tilt of the rotation axis from what we have now as 23.5 degrees between 22.1 and 24.5 egrees every approx 40000 years, as well as a nutation of the rotation axis every 18.6 ( coinciding with the tilt of the moon's orbit I beleve ).
As well, the earth's orbit is an ellipse with a perihelion shift as well as a change in eccentricity, both around 100,000 years or so.
For historical record computations all that must be a nightmare, taking into account all the different calendars used by civilizations.

Is the 33 minutes you calculated the difference in the lengths of the tropical year and the sidereal year?
( I do recall a N=15,700 from somewhere, but do not remember the exact details )

The N should then be be 26000 giving a difference of approx 20 minutes.
Actual values
sidereal year = 365.256 363 days.
tropical year = 365.242 190 402 days
where day is 86000 seconds,
giving a difference of 20. 408 minutes.

PS\Hopefully you find thus interesting as to some earth-sun-moon mechanics
http://farside.ph.utexas.edu/teaching/336k/Newton/node113.html
 
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  • #10
jim mcnamara
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@Buzz Bloom consider the library ( when things are safer), get:
Jean Meeus 'Astronomical Algorithms'

I think you can buy it, too. Maybe rent Kindle version (which runs in a free app on Windows)?

There is a lot going on you may not be aware of. I think 256 is hinting at this idea.
PS: both of you may be missing leap seconds, 86400 seconds is just a very close approximation. For example, POSIX standards for UNIX do not require leap seconds. Astronomers do.
 
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  • #11
Buzz Bloom
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@Buzz Bloom consider the library ( when things are safer), get:
Jean Meeus 'Astronomical Algorithms'
Hi Jim:

I found a source
https://bowlingacademyinc.com/astronomical-algorithms-by-jean-meeus-50/
of a free PDF of the Meeus book, but the interface is puzzling. It says I can download the PDF after making a free registration, but there does not seem to be a way to do this registration. I also do not use Kindle stuff.

Thanks for the recommendation. If our town library is ever functional again, I will try to get the book through it.

Regards,
Buzz
 
  • #12
davenn
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but the interface is puzzling. It says I can download the PDF after making a free registration, but there does not seem to be a way to do this registration.
click on the green download PDF button … takes you to another page, this one......

https://pdfdatabase.top/?id=astronomical+algorithms+by+jean+meeus

when you click on the download button there, a signup form is presented




Dave
 
  • #13
Buzz Bloom
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click on the green download PDF button … takes you to another page, this one......

https://pdfdatabase.top/?id=astronomical+algorithms+by+jean+meeus

when you click on the download button there, a signup form is presented

Dave
Hi Dave:

Thank you for your help. During the registration process I am asked to enter credit card information which I do not want to do. I will have to give up on this PDF download and wait for my town library to restart functioning.

Regards,
Buzz
 
  • #15
256bits
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There is a lot going on you may not be aware of. I think 256 is hinting at this idea.
PS: both of you may be missing leap seconds, 86400 seconds is just a very close approximation. For example, POSIX standards for UNIX do not require leap seconds. Astronomers do.
Thing is there us a whole bunch of time standards - GMT, UTC, solar time, sideral time, barycentric celestial time, geocentric cootdinate time, the superceded ephemeris time ( from 1972 ) or second to that of the SI second ( cesium clock )....
We should labour the point as to which measures the earth's year to return to its starting position with reference to celestial objects versus the return to the equinox.
Of course the SI second is used nowadays.
But as to the diagram in opening post 1, the variation between equinoxes ranges from approx -10 seconds to plus ten over the 99 year span.
And the question asks why does this happen.

The 20.408 minutes is the time difference if the tropical year from the sidereal year, the tropical year being that much shorter in duration.

I mentioned
as well as a nutation of the rotation axis every 18.6 ( coinciding with the tilt of the moon's orbit I beleve ).
Is that the pattern.
@Buzz Bloom
 
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  • #16
davenn
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I am asked to enter credit card information which I do not want to do.
I didn't go that far .... No, I wouldn't want to either 😮
 
  • #17
Baluncore
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  • #18
Buzz Bloom
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Hi @Baluncore:

Thank you for your post and also for your message. I want to get the Meeus book because I am hopeful it will contain explanations about the variability of the +/- approx 12 min time range difference between consecutive equinoxes that my chart demonstrates.

I have no clear idea about what help executable algorithms would be to me.

BTW: I have found an error in my previous calculation regarding the approx +/- 2.5 min range based on the time it takes the Earth to move the distance r in its orbit around he sun, where r is the radius of the Earth's orbit around the Earth-Moon paracenter. The assumption is wrong that the time for Earth to move twice this distance is relevant.

I recalculated this effect based on the following.

https://en.wikipedia.org/wiki/Year
sidereal year= YS 365.256363004 days (365 d 6 h 9 min 9.76 s)
Because of the Earth's axial precession, this year is about 20 minutes shorter than the sidereal year. The mean tropical year Y is approximately 365 days, 5 hours, 48 minutes, 45 seconds, using the modern definition.
(Georgean calendar year YG = 365.24219 days of 86400 SI seconds)
[My small correction YG = 365.2421875 days]
The difference D = YS and YG is
D = 365.256363004 days - 365.2421875 days = 0.014175504 days = 20.41272576 min
This is the contribution of Earth axis nutation.
The partial day to complete a siderial year following 2 lunar cycles is 0.2421875 days = 348.75 min.
(Note: the average calculation from the
http://www.astropixels.com/ephemeris/soleq2001.html
data is 348.82 min.)

https://en.wikipedia.org/wiki/Lunar_phase
The lunar phases gradually change over the period of a synodic month (about 29.53 days)...
(... 29.530588853 days the length of a synodic month).
12 lunar months = 354.367066236 days.
The incomplete lunar orbit time at the end of a year is
1 tropical year - 12 lunar months = 0.875121264 days.

https://en.wikipedia.org/wiki/Barycenter
Radius of Earth from barycenter with moon = 4670 km
Rotational velocity = 2 * 3.1415926536 *4670 km / 29.530588853 days
= 993.6298775 km/day
Assume that the 0.875121264 days preceding the the equinox in centered on a new moon, and also assume that the barycenter velocity of the Earth is approximately constant during this time.
The distance the Earth moves relative to the barycenter during this time is
D = 0.875121264 day * 993.6298775 km/day = 8869.5466 km

https://en.wikipedia.org/wiki/Earth's_orbit#Events_in_the_orbit
The earths orbital speed is
SE = Earth's orbital speed = 29.78 km/sec = 1786.8 km/min
The time it takes the Earth to move the distance D in its orbit is
T = D * SE = 4.9639 min.

So my new calculation of the range of differences shown in the chart is approx double that of my first try, but it is still less than half of the value needed.

Regards,
Buzz
 
  • #19
Buzz Bloom
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Hi @256bits, @anorlunda, @jim mcnamara. @davenn, @Baluncore:

Thank you all for participating in this thread.

I have looked through the Meeus Book Table of Contents and found that 9 of the 58 chapters (about 100 pages) appear to have some relevance to the equinox problem I am trying to understand.
Chapt. 10: Dynamical Time and Universal Time - page 77 (pdf page 85)​
Chapt. 12: Sidereal time at Greenwich - page 87 (pdf page 95)​
Chapt. 21: Precession - page 131 (pdf page 139)​
Chapt. 22: Nutation and Obliquity of the Ecliptic - page 143 (pdf page 151)​
Chapt. 24: Reduction of Ecliptical Elements from one Equinox to another one - page 159 (pdf page 167)​
Chapt. 27: Equinoxes and Solstices - page 177 (pdf page 185)​
Chapt. 33: Elliptic Motion - page 223 (pdf page 231)​
Chapt. 47: Position of the Moon - page 337 (pdf page 345)​
Chapt. 52: Maximum Declinations of the Moon - page 367 (pdf page 375)​
I also scanned through these 9 chapters and confirmed that there are many distinct influences, i.e., parameters, (at least 10) on the variability of the time between consecutive March equinoxes. I have not yet determined which of these influences are significant (a matter of minutes) or not (a matter of seconds), and I have concluded that it is very likely that I will not be able to complete this study.

Regards,
Buzz
 
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  • #20
Baluncore
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The equinox is an instant in time, independent of the observer's place on Earth.
It would be interesting to compare the direction defined (90° and 270°) astronomical equinoxes with an alternative situation, namely the two annual instants when the Earth's equatorial plane sweeps across the centre of the Sun.
That may help understand the variations.
 
  • #21
256bits
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Hi @256bits, @anorlunda, @jim mcnamara. @davenn, @Baluncore:

Thank you all for participating in this thread.

I have looked through the Meeus Book Table of Contents and found that 9 of the 58 chapters (about 100 pages) appear to have some relevance to the equinox problem I am trying to understand.
Chapt. 10: Dynamical Time and Universal Time - page 77 (pdf page 85)​
Chapt. 12: Sidereal time at Greenwich - page 87 (pdf page 95)​
Chapt. 21: Precession - page 131 (pdf page 139)​
Chapt. 22: Nutation and Obliquity of the Ecliptic - page 143 (pdf page 151)​
Chapt. 24: Reduction of Ecliptical Elements from one Equinox to another one - page 159 (pdf page 167)​
Chapt. 27: Equinoxes and Solstices - page 177 (pdf page 185)​
Chapt. 33: Elliptic Motion - page 223 (pdf page 231)​
Chapt. 47: Position of the Moon - page 337 (pdf page 345)​
Chapt. 52: Maximum Declinations of the Moon - page 367 (pdf page 375)​
I also scanned through these 9 chapters and confirmed that there are many distinct influences, i.e., parameters, (at least 10) on the variability of the time between consecutive March equinoxes. I have not yet determined which of these influences are significant (a matter of minutes) or not (a matter of seconds), and I have concluded that it is very likely that I will not be able to complete this study.

Regards,
Buzz
Would the position of Venus and Jupiter position ( since they would have the greatest influence ) wrt the earth's position have an affect on the particular sidereal year length ( tropical year would/should always be 20 minutes behind they say ). Venus should speed up the earth and Jupiter slow it down, although the influence is supposed to cancel out in the long run ( something to do with who is on the inner or outer orbit ).
You did mention that in the opening post.
 
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  • #22
Buzz Bloom
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It would be interesting to compare the direction defined (90° and 270°) astronomical equinoxes with an alternative situation, namely the two annual instants when the Earth's equatorial plane sweeps across the centre of the Sun.
Hi Baluncore:

I am unable to understand what this quote is describing. Earth's orbit is in the ecliptic plane which passes through the center of the sun.
What defines the 90o and 270o direction from the sun along this plane? Also, at what instant is the Earth's equatorial plane NOT across the center of the Sun.

Regards,
Buzz
 
  • #23
Buzz Bloom
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Venus should speed up the earth and Jupiter slow it down, although the influence is supposed to cancel out in the long run
Hi 256:

I have concluded from my scanning of the Meeus book, and also from several Wikipedia articles, that there is an influence of the planets on the varying velocity of the Earth in its orbit about the Earth-Moon barycenter, and the velocity of the Earth-Moon barycenter about the barycenter of the sun together with its planets. However, it seems that these influences are at the seconds level rather than the minutes level. There are also some other second-type influences as well. Another example is planet influence on the rate at which the Earth axis nutates. That is, all these effects influence the variable time between consecutive March equinoxes, but these influences are too small to be significant to the problem I am exploring.

Regards,
Buzz
 
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  • #24
Baluncore
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What defines the 90° and 270° direction from the sun along this plane?
See; Meeus. Chapter 26. “The times of the equinoxes and solstices are the instants when the apparent geocentric longitude of the Sun (that is, calculated by including the effects of aberration and nutation) is a multiple of 90 degrees”.

The Earth's elliptical orbit in the ecliptic is not aligned with the Earth's axial tilt. So I would expect the solstice and equinox to be slightly different when specified by angular position in the orbit, as opposed to when the Sun is apparently directly above the equator.

Also, at what instant is the Earth's equatorial plane NOT across the center of the Sun.
I suspect you are confusing the Earth's equatorial plane with the Earth's ecliptic orbital plane.
 
  • #25
Buzz Bloom
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Hi @Baluncore:

I am experiencing some confusion about your post.

See; Meeus. Chapter 26. “The times of the equinoxes and solstices are the instants when the apparent geocentric longitude of the Sun (that is, calculated by including the effects of aberration and nutation) is a multiple of 90 degrees”.
I cannot find this quote in the Meeus book. The application that I am using cannot do searches. Please post the page number. Even of we are looking at different editions, the page number should be of some help.
The phrase "is a multiple of 90 degrees" seem wrong according to my understanding, which might be wrong.
The equinox occurs when the angle between (a) the axis of the earth and (b) the line connecting the center of the Earth with the center of the Sun is exactly 90o, not a multiple.
I also do not understand "the apparent geocentric longitude of the Sun". The most relevant that I could find is
https://en.wikipedia.org/wiki/Position_of_the_Sun
The position of the Sun in the sky is a function of both the time and the geographic location of observation on Earth's surface. As Earth orbits the Sun over the course of a year, the Sun appears to move with respect to the fixed stars on the celestial sphere, along a circular path called the ecliptic.​
When an equinox occurs, the longitude of the sun can be any value between 0 longitude and +/- 180 longitude.

The Earth's elliptical orbit in the ecliptic is not aligned with the Earth's axial tilt. So I would expect the solstice and equinox to be slightly different when specified by angular position in the orbit, as opposed to when the Sun is apparently directly above the equator.
https://en.wikipedia.org/wiki/Ecliptic
The ecliptic is the mean plane of the apparent path in the Earth's sky that the Sun follows over the course of one year; it is the basis of the ecliptic coordinate system.​
I agree with the first sentence: "The Earth's elliptical orbit in the ecliptic is not aligned with the Earth's axial tilt."
The second sentence confuses me. First, I need a definition for "angular position in the orbit". My guess is you are referring to the position of the sun in its apparent movement along the ecliptic. I agree that the position of the sun at an equinox is different that its position at a solstice, but the difference is far from "slightly different". The consecutive difference in the sequence from March equinox to June solstice to September equinox to December solstice are all approximately 90o with a range of variation.
I am also confused by what you mean by "... the solstice and equinox to be slightly different when specified by angular position in the orbit, as opposed to when the Sun is apparently directly above the equator."
I do not understand what you mean by "specified by angular position in the orbit".

I suspect you are confusing the Earth's equatorial plane with the Earth's ecliptic orbital plane.
I agree and apologize. I had carelessly mis-read your "Earth's equatorial plane sweeps across the centre of the Sun" as referring to the ecliptic.

Regards,
Buzz
 

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