Ian Wilson: Connecting the 208 Year de Vries Cycle with the Earth-Moon System

Posted: August 22, 2013 by tallbloke in Astronomy, Astrophysics, cosmic rays, Cycles, data, Natural Variation, Solar physics, solar system dynamics

sun-moon-earth

Reposted for discussion from Ian Wilson’s blog Astro-Climate Connection

UPDATED & CORRECTED 23/08/2013

Direct instrumental observations of the Sun since 1610 have shown that the level of sunspot activity on the Sun has a mean periodicity of 22.3 years, known as the Hale cycle. In addition, these observations of the Sun have shown that there are longer-term periodicities present in the level of solar activity.

One of the most prominent long-term cycles that have been identified is the ~210 year de Vries (Suess) cycle. However, because of the limited time over which instrumental observations have been available, the confirmation of the de Vries cycle [1] has required the use of proxies such as de-trended δC14 from tree rings [2,3], Be10 levels in the GRIP ice cores [4,5,6], and dust profiles in GISP2 ice cores [7].  These proxy observations have indicated that:

a) the de Vries cycle amplitude varies with a period of about 2200 years [6]. In other words, its appearance is intermittent in nature.

b)  the largest amplitude of the de Vries cycle are found near Hallstatt cycle minima centered at 8,200, 5,500, 2,500 and 800 B.P .[6]

c) grand solar minima occur preferentially at minima of the Hallstatt cycle that are characterized by large de Vries cycle amplitudes [6].

d) the cycle length is somewhere in the range 205 – 210 years, with the more precise estimates being in the range 207-208 years.

Abreu et al. (2012) [5] have identify a 208 year period in a 9400 year reconstruction of the solar modulation potential that is derived from C14 and Be10 observations taken from ice cores. The solar modulation potential is thought to be a good indicator the strength of the solar magnetic field that is responsible for the deflection of cosmic ray, and so a good proxy of the overall level of past solar magnetic activity. Abreu et al. (2012) [5] also show that there is a 208 year period in the planetary induced torques that could act upon any asymmetric structure in the boundary layer known as the solar tachocline. These authors propose that it these planetary induced torques that could be responsible for modulating the long-term solar magnetic activity on the Sun.

Abreu et al. (2012) [5] do not identify the specific physical mechanism that is responsible for producing the 208 year period in the planetary torques, although it is reasonable to assume that it must be linked in some way with the synodic interactions between orbital period of Jupiter [the main source of planetary torque] and one or more of the other planets.

However, it can be shown that there is a natural 208 year periodicity associated with the position of the Earth in its orbit when it is observed at intervals separated by half the precession cycle of the Lunar line-of-apse, in a reference frame that is fixed with respect to the stars.

The following diagram shows the angle that the Earth in its orbit about the Sun forms with a fixed direction in a sidereal reference frame, at time steps of half the precession period of the lunar line-of-apse (= 4.42558131 sidereal years for 2000.0). This angle is plotted as a function of time measured in sidereal years.

[Notes:
A: The lunar line-of-apse is a line passing through the centre of the Earth that connects the perigee and apogee of the lunar orbit.
B: The lunar line-of-apse precesses about the Earth once every 8.85116364 sidereal years, when measured with respect to the fixed stars.
C: The reference direction in the sidereal frame that was used (as T = 0 years) is that of the Earth on January 1st 2000.0 at 00:00 UT.
D: The following values for the Anomalistic month = 27.554549878-(0.00000001039*T) and the Sidereal month = 27.321661547+(0.000000001857*T) were used for all calculations, where T is the number of sidereal years since 2000.0.]

     The above diagram clearly shows that there is a natural 208 year periodicity in the alignment of the Earth with respect to the fixed with the stars when it is observed every half precession cycle of the Lunar line-of-apse. It also shows that the 208 year periodicity in alignment slowly drifts out synchronization over a period ~ 650 years. Hence, this Earth/Lunar alignment pattern exhibits two characteristics that mimic those of the de Vries cycle, namely a periodicity of 208 years and slow loss of synchronicity on millennial time scales.
     An alternative way to display the 208 year Earth/Lunar cycle that is based upon ephemeris data rather than extrapolated mean lunar orbital data is shown in the following diagram [8].
     This diagram shows the number of hours perigee is away from New or Full Moon, when the Moon is at perigee between the 27th of December and the 10th of January, plotted against time in years.
  The limits that have been placed upon the dates of perigee are designed to restrict the observational window to +/- seven days either side of a nominal fixed date in the seasonal calender (in this case the 3rd of January which roughly corresponds to modern day Perihelion). In essence, they are restricting the observational window to +/- a quarter of a lunar orbit either side of a point in the Earth’s orbit that is “fixed” with respect to the stars.
[Skip down to ******* bar if you want to avoid the following details]
     In this diagram, you see long diagonal bands from the upper left to the lower right of the diagram. The dots of the same colour in any given long diagonal band lie inside the +/- seven day calender window. Each dot is separated from its immediate neighbor (of the same colour) by almost exactly 31 years. If you move down and to the right along the long diagonal bands, from a dot of one colour, to a dot of the next colour that has the same date in the seasonal calender, you jump 137 years.
     There are three separate colour groupings in this diagram. The first grouping, vertically from the top to the bottom of the diagram, is green, red and black. The second is yellow, blue, and brown and the third is orange and purple. As you move vertically down from one colour to the next in a given colour grouping, you advance by almost exactly four years e.g. if you pick a set of green dots near the top of the diagram, the set of red dots immediately below it are shifted forward in time by four years, and the set black dots immediately below the red dots are shifted forward by a further four years.
****************
     The important point to note is that the symbols in this diagram form long diagonal lines or waves that are separated by almost exactly 208 years [see the red spacing bars in the above diagram that link points that are separated by 208 years].
      Detailed investigations show that paired points in the above diagram that are separated by 208 years, occur almost exactly 2 1/3 days apart in the seasonal calendar. This accounts for the change between the observed time of perigee and the time of New/Full Moon that occur between the paired points separated by 208 years (i.e. it accounts for the slope of the red line and, hence, the tilt of the diagonal waves).
     It turns out that the 2 1/3 day slippage backward in the seasonal dates of the alignments between New/Full Moon and the lunar perigee every 208 years, corresponds to a westward slippage of ~ 40 arc seconds per year. This close to the westward drift of the equinoxes by 50.3 arc seconds per year that is caused by the precession of the equinox.
     So, what it is telling us is that if we correct the above diagram for the effects of the precession of equinoxes (i.e. correct for the drift between our co-ordinate frame and the fixed stars) we get a Earth/Lunar repetition cycle for the position of the Earth in its orbit (with respect to the stars) of 208 years.    
.
IMPORTANT IMPLICATIONS FOR THE INTEGRATED EARTH/MOON/VENUS SYSTEM  
 

Interestingly, the Earth/Venus pentagram alignment pattern resets itself with respect to the Sun and the fixed stars once every 149.5 VE alignments (of 1.59866 years) = 238.9996251 years ( with an error of 0.134964 degrees).

[see the updated and corrected blog post at:
http://astroclimateconnection.blogspot.com.au/2013/08/the-vej-tidal-torquing-model-can.html  and extensive discussion at the talkshop here: https://tallbloke.wordpress.com/2013/08/12/ian-wilson-the-vej-tidal-torquing-model-can-explain-many-of-the-long-term-changes-in-the-level-of-solar-activity-part-2]

Similarly, the relative position of the Moon in its orbit about the Earth compared to the Lunar line-of-apse reset themselves with respect to the Sun and the fixed stars almost exactly once every 31.0 sidereal years (actually closer to 31.0 sidereal years + 2 days). This comes about because:

31.0 sidereal years _________= 11322.94725312 days
383.5 synodic lunar months ___= 11324.980825 days
411.0 anomalistic lunar months _= 11324.92000 days
27.5 Full Moon Cycles _______= 11324.071833 days

This means that that if you have a New Moon at closest perigee, 31.00 sidereal years (+ 2 days) later, you will have a Full Moon at closest perigee, on almost the same day of the calender year.

Now, it seems quite remarkable that:

a) The position of the Earth in its orbit, as seen once every half precession cycle of the Lunar line-of-apse (= 4.42558131 sidereal years for 2000.0), resets itself with respect to the stars once every 208.0 sidereal years.

b) The relative position of the Moon in its orbit about the Earth compared to the Lunar line-of-apse reset themselves with respect to the Sun and the fixed stars almost exactly once every 31.0 sidereal years.

c) 208.0 sidereal years + 31.0 sidereal years = 239.0 sidereal years 

and that

d) the Earth/Venus pentagram alignment pattern resets itself with respect to the Sun and the fixed stars once every 149.5 VE alignments (of 1.59866 years) = 238.9996251 years.

     One is left with the feeling that this is more than just a coincidence.     

References

1. Rogers, M. L., Richards, M.T. and Richards, D. St. P. (2006), Long-term variability in the length of the solar cycle, preprint, arXiv: astro-ph/0606426v3
2. Peristykh, A.N. and Damon, P.E. (2003) Persistence of the gleissberg 88-year solar cycle over the last ~12,00 years: Evidence from cosmogenic isotopes. Journal of Geophysical Research 108, 1003.
3. Stuiver, M. and Braziunas, T.F. (1993) Sun, ocean, climate and atmospheric CO2: An evaluation of causal and spectral relationships. Holocene 3, 289-305
4. Wagner, G., Beer, J., Masarik, J., Muscheler, R., Kubik, P. W., Mende, W., Laj, C., Raisbeck, G.M. and Yiou, F., (2001), Presence of the Solar deVries cycle (≈205 years) during the last ice age, Geophysical Research Letters 28 (2), 303-306
5. Abreu J. A., Beer J., Ferriz-Mas A., McCracken K.G., and Steinhilber F., (2012), Is there a planetary influence on solar activity?, A&A 548, A88.
6. Steinhilber F., et al., (2012), 9,400 years of cosmic radiation and solar activity from ice cores and tree rings, PNAS,  vol. 109, no. 16, 5967–5971
7. Ram, M. and Stolz, M. R. (1999) Possible solar influences on the dust profile of the GISP2 ice core from Central Greenland, Geophysical Research Letters, 26 (8), 1043-1046

8. Lunar Perigee and Apogee Calculator: http://www.fourmilab.ch/earthview/pacalc.html
Comments
  1. MangoChutney says:

    could this be the reason the moon is moving away from the earth by 38mm per year?

  2. tallbloke says:

    ” It also shows that the 208 year periodicity in alignment slowly drifts out synchronization over a period ~ 860 years.”

    Close to the period of the translation of ‘star points’ on the Jupiter-Saturn tri-synodic cycle. Coupled with the 208 year variation in C14 and 10Be indicating a solar activity cycle of the same frequency, this is good indirect evidence of a J-S influence on solar activity variation.

  3. Ian Wilson says:

    Rog and Tim,

    Again, thank you for exposing my investigations to wider audience. You are so quick in re-blogging my posts that I don’t even have time to correct the typos [not that I am complaining].

    My current blog post has corrected most of the typos, and I now think that the ~ 850 year drift in synchronization in the first figure is more like ~ 630 years.

    If my claims stand-up to scrutiny, then the next question to ask is, does the Moon indirectly affect the accumulation rates Be10 and C14 in ice cores? Or is the Moon’s orbit just mimicking the 208 tidal torque that is evident in the planetary orbits?

  4. Ian Wilson says:

    Mungo Chutney,

    The movement of the Moon away from the Earth is cause by the tidal interactions which are transferring the Earth’s rotational angular momentum to the Moon’s orbital angular momentum.

  5. Ian Wilson says:

    Rog,

    The calculations of the Lunar orbit would have to be more detailed to precisely determine the re-synchronization period – assuming that is repeatable over longer time scales. All that I can say
    is that it ~ 600 – 800 years and that is probably reading too much into the quality of the data at this stage.

  6. tallbloke says:

    I think the general consensus on the reason the Moon is receding is due to a transfer of angular momentum from Earth to Moon caused by the tidal bulge it raises on Earth accelerating it via a quadrupole moment. That raises it to a higher orbit.
    http://www.bbc.co.uk/news/science-environment-12311119

  7. MangoChutney says:

    thanks guys = educated 🙂

  8. tallbloke says:

    Ian, OK, I’ll dial back on the J-S solar link being supported by Lunar timing for now. 🙂
    Interesting question about 10Be in ice cores and 14C in trees. What would the physical explanation relaate to in terms of the angles and timings of the appearance of the Moon from Earth? Do spallation rates have a diurnal component modulated by the Moon phase? A latitudinal component modulated by declination cycle? Is it measurable directly in the atmosphere with modern instruments?

  9. Chaeremon says:

    Ian,

    thanks for bringing the 208 years cycle and its lunar line-of-apse relation to attention 😎 I’ ve checked the 5kyrs eclipse canons, they show pairs of solarlunar eclipses at 1/3 of the cycle length (857.5 moons) for noticeably more than 50% of the cases and both combinations.

    Example (both eclipses observable from the pacific):
    2010 July 11 Total Solar Eclipse
    1941 March 13 Partial Lunar Eclipse

    So, is the draconic component involved as well?

  10. Ian Wilson says:

    Chaeremon,

    Impressive maps, however, I do not understand your question. What cycle length of 857.5 Moons are you referring to?

  11. tallbloke says:

    Chaeremon seems to be saying the 857.5 moons is 1/3 of the cycle he is referring to. If my calcs are right these periods would be 64.09yrs and 192.28yrs

  12. Chaeremon says:

    Ian, Rog,

    yes: 208 years make 2572.5 moons and 1/3 of that is 857.5 moons. At this 1/3 distance appear eclipses in solar/lunar pairs, therefore the draconic component seems involved in the whole 208 yrs cycle.

  13. tallbloke says:

    Chaeremon:

    Ah, ok, I’m calculating lunar orbital periods and you’re calculating lunations. I should have paid more attention to what you meant by ‘moons’, sorry. Ian is calculating relative to fixed stars so maybe we’d better stick to that frame of reference to avoid confusion. Or state which frame of reference you are using.

  14. Chaeremon says:

    Rog,

    please don’t get me wrong, I comment only for bettering communication (how dare I, you are mod 😉

    Due to the characteristics of the lunar orbit, x synodic lunations “moons” per year = x+1 sidereal lunations per year, and the mention of eclipses (syzygy) tells them apart. This “only” gets nasty if the precession of equinox (about one degree every 71.657 years) must be accounted for. But that is not really affecting a subinterval unless there are more than ~71.5 years in the equation, which is not the case for the above considered eclipse interval of 208/3 = 69+(1/3) years.

    With other words: Ian’s sidereal frame of reference in the whole cycle is not affected, he is always right 🙂

    I fully agree we always attempt to avoid confusion.

  15. tallbloke says:

    Hi Chaeremon: I see what you mean. So my original calculation of 192 years becomes 208 years when we add the 16 ‘extra’ lunations due to the sidereal/synodic multiplication factor. 208/192= around 1.08

  16. Ian Wilson says:

    Slight change of topic,

    Has anyone noticed that the transition from a La Nina (1788 – 1790) event to the great El Nino of 1791 – 1793 [1] took place 208 years before the transition from the great El Nino of 1997-98 to the strong La Nina events of 1998-2000.

    [1] http://climatehistory.com.au/wp-content/uploads/2009/12/Environmental-History-2010-Gergis-envhis_emq079.pdf

  17. tchannon says:

    About every 100 years, the infamous hide the spike just before 1880 and I am sure why GISS etc. start 1880. Each Hadcrut gets smaller. Fits with reports from various parts of the world as pretty severe.

  18. Ian Wilson says:

    Rog,

    A thousand apologies, but I have found that I have made some blunders in my blog post that had to be corrected. If you go to the corrected and updated blog post at:

    http://astroclimateconnection.blogspot.com.au/2013/08/connecting-208-year-de-vries-cycle-with.html

    you will find some very interesting claims about the integrated Earth/Moon/Venus system.

    [Reply] No problem. No-one expects real-time science to be without glitches. I’ll replace the headline post with your latest version.

  19. Paul Vaughan says:

    1st, minor nitpicking:
    Reference 1 “Rodgers” should read “Rogers”.

    next up:
    something more fun (give me a few minutes…)

  20. Ian Wilson says:

    Paul,

    Corrected – apparently their paper never did get into print – at least as far as my limited knowledge goes.

  21. Ian Wilson says:

    Paul,

    Off radar the next 24 hours or so..

  22. Paul Vaughan says:

    Something I easily figured out during my first year of climate, solar, & solar system explorations:

    average period of absolute solar barycentric radial acceleration
    = Jupiter – Neptune = J-N = 12.78 years

    period of terrestrial polar motion group wave
    = 12.78 / 2 = 6.39 years

    Chandler wobble period = beat of 6.39 with terrestrial year
    = (6.39)*(1) / (6.39 – 1) = 1.1855

    __
    Not long thereafter I wondered about how that would beat with the nearest LNC harmonic…

    period of LNC (lunar nodal cycle) = 18.6 years

    harmonic of LNC nearest 6.39
    = 18.6 / 3 = 6.2

    beat of 6.2 year LNC harmonic with terrestrial year
    = (6.2)*(1) / (6.2 – 1) = 1.1923

    Beat with Chandler wobble =
    (1.1923)*(1.1855) / (1.1923 – 1.1855) = 208 years

    __
    The Chandler wobble phase reversal coincides with the middle of what Mursula proposes is the solar de Vries cycle:

    Mursula, K. (2007).
    Asymmetric sun viewed from the heliosphere.

    I’ve checked his work using a few approaches — e.g. _A_ & _B_.

    __
    The best exploratory insights are always the ones that raise more, interesting questions.

    Ian: Thanks for giving occasion to dig out some old notes.
    TB: Thanks for championing unobstructed solar/climate brainstorming.

    I have some other old astro-specs on 208 year de Vries. I’ll see if I can find time to dust them off during the next days…

    Cheers

  23. tallbloke says:

    Ian says: Rog, A thousand apologies, but I have found that I have made some blunders in my blog post that had to be corrected.

    No problem. No-one expects real-time science to be without glitches. I’ve replaced the headline post with your latest version

    Paul says: TB: Thanks for championing unobstructed solar/climate brainstorming.

    Thanks for making the time to share your insight. Unfortunately Ulric has been saying some very unpleasant things about the blog, other contributors and myself in email so we’ll have to proceed without his input.

  24. tallbloke says:

    Well, Ian’s update is a bit of a blockbuster. Stuart and I have also come across several of these ‘needs one extra’ type inter-relations in other parts of the solar system. It’s as if the system self adjusts to be as close as it can to its ‘preferred state’ (whatever underlies that; e.g. principle of least action, constructal law, entropy), in quantum steps. Food for thought.

    Apropos nothing, just leaving a note for later
    31^2/4=240.25 SQRT(240.25)=31/2

  25. crikey says:

    Ian said
    “Has anyone noticed that the transition from a La Nina (1788 – 1790) event to the great El Nino of 1791 – 1793 [1]”

    Here is some moon cycles from another source..
    No 208 cycle here? but a 177- 187 yr cycle?

    1974 – 1787 = 187yrs
    1610 to 1787 = 177yrs
    1974 to …? = ? yrs

    Source of info on Moon cycles..from . my copy from chiefos blog
    https://picasaweb.google.com/110600540172511797362/TIMESERIESAndTrends#5915308533328100802
    original
    http://chiefio.wordpress.com/2013/01/24/why-weather-has-a-60-year-lunar-beat/

    The max’ of the larger cycle corresponds to the the great climate shift of the mid 70’s( 1974 )
    and they have a lunar max cycle around 1787 close to your ENSO enquiry.
    A maximum in 1610?

    1787 – 1610 = 177yrs

    Why does the cycle change length?

    Why does the 208 cycle not appear here?

    are you able to specify exact dates for changes of phase or step shifts? in the 208yr cycle or in fact any cycle

    When does your 208 yr cycle reach the next maximum.. or is 208 yrs a mean value?
    —————————————————————————————————————–

    There is quite a bit on the web on this topic
    another at chiefo
    http://chiefio.wordpress.com/2013/01/29/arctic-standstill-tropical-saros/

    and tallbloke has some more in his archives
    https://tallbloke.wordpress.com/2012/10/22/de-vries-cycle-links-warming-rate-peaks-to-solar-system-frequencies/

    To be useful.
    The cycle timing needs to be mastered..
    When is the next De Vries minimum and maximum?

  26. oldbrew says:

    TB: Re. ‘needs one extra’ – a cycle that needs one extra (or ‘loses’ one) can be suspected of indicating a larger cycle lurking above/behind it, wherever such things lurk 😉

  27. suricat says:

    oldbrew says: August 23, 2013 at 3:11 pm

    Too true OB. From my own knowledge of analogue circuits, radio transmission and ‘Pilot Wave’ mass kinetic energy transmission, ‘wavelets’ get caught up in ‘waves’ and add to the wave’s energy.

    It’s a ‘harmonics’ thingy. Half and quarter ‘wavelengths’ (double and quadruple ‘frequencies’) live well within the ‘main harmonic’ (dominant waveform), but ‘dis-harmonic’ ‘wavelengths’ (frequencies) that don’t fit the main harmonic have a tendency to ‘surrender their energy’ and ‘merge’ with the main harmonic.

    Hope that makes sense. 🙂

    Best regards, Ray.

  28. Ian Wilson says:

    oldbrew, Rog and suricat,

    Wheels within wheels……

    Using a “Golden” year that is located at the position of the golden ratio between the tropical and sidereal year i.e.

    365.24219 days + ((365.256363004 – 365.24219)/1.61803398875) = 365.25095 days

    If you want to advance by exactly 1/2 of a complete synodic period (i.e. adavance from New to Full Moon) once every 208 “Golden” years then this is equivalent to advancing by:

    (29.530588853/2) x 31/208 = 2.200596765 days once every 31 “Golden” years.

    Given that 31.0 “Golden” years = 11322.779431344 days
    and 383.5 synodic lunar months = 11324.980825 days.

    the addition of the 2.200596765 day advance to 31 “golden” years becomes:

    11322.779431344 days + 2.200596765 days = 11324.980028109 days

    which is extremely close to 383.5 synodic months.

    Hence, if an exact half-multiple (383.5) of the synodic month moves ahead of 31.0000 “Golden” years by 2.200596765 days (= 11324.980825 days – 11322.779431344 days), is just the right amount time-drift for that advance to accumulate to exactly half a synodic month after 208 “Golden” years.

  29. tallbloke says:

    Ian: Nice, the solar system’s hidden clockwork is being unveiled little by little.You should read Miles Mathis piece on Phi and the relative densities of Earth and Moon http://milesmathis.com/phi.html.

    By the way, Tim Channon has posted on his blog about his replication/development of RJ’s Sunspot series model.
    http://daedalearth.wordpress.com/2013/08/24/a-non-linear-solar-cycle-exploration/

  30. TLMango says:

    Ian Wilson says:
    “the DeVries cycle amplitude varies with a period of about 2200 years”

    Here is some evidence that there is a relationship between Abreu’s 104 & 208 year signals and the 2200 year Hallstatt cycle.

    J = 11.862242 : : S = 29.457784
    S/2 * J / (S/2 – J) = 60.94838271 ‘ Scafetta’s 60 year temperature cycle
    60.94838271 * (J * 5) / (60.94838271 – (J * 5)) = 2208.027476 ‘ 2200 year Hallstatt cycle

    (2208.027476 / 15) * 60.94838271 / ((2208.027476 / 15) – 60.94838271) = 104.0157081

  31. […] It are combined planetary cycles. 88.37 years is the triple Saturn cycle. It is also 8 solar cycles and 10 lunar precession cycles. The lunar precession cycle (8.85 years) is often overlooked, but actually quite important as it determines when we get spring tides and plays a role in weather and climate as well. The 208 year cycle is equal to 19 solar cycles and 11 Metonic cycles. It is also 11.5 Saros cycles. It is 10.5 times the Jupiter-Saturn synodic period (19.8589 years), and 15 times the Jupiter-Uranus synodic period (13.8119 years). Through a more complex interaction, a 208 year periodicity also emerges from the lunar cycle. See this article. […]

  32. oldbrew says:

    ‘the relative position of the Moon in its orbit about the Earth compared to the Lunar line-of-apse reset themselves with respect to the Sun and the fixed stars almost exactly once every 31.0 sidereal years (actually closer to 31.0 sidereal years + 2 days).’

    As far as I can see nobody mentioned that 31 years is 3:5 with the 18.6 year lunar node cycle i.e. 18.6 * 5 = 31 * 3.

    Note also that 125 LNC = 196 Jupiter orbits in a period of 2325 years which is within the range of the Hallstatt cycle/period, also 31 * 75 = 2325 (75:125 = 3:5).

  33. Chaeremon says:

    oldbrew cited: “the relative position of the Moon in its orbit about the Earth compared to the stars…”

    Have the UK shops run out of spectacles 😕 during ~31 yrs are ~415.5 sidereal revolutions of the Moon (against the fixed stars). This seems 180° off (it’s a bit short of 384.5 syzygy b.t.w.) Yet ~412 times line-of-apse looks commensurate.