Ian Wilson: Are the Strongest Lunar Perigean Spring Tides Commensurate with the Transit Cycle of Venus?

Posted: December 6, 2014 by tallbloke in Analysis, Astronomy, Astrophysics, Celestial Mechanics, Cycles, solar system dynamics
Tags: , ,

venus-transit-2012Congratulations to Astrophysicist Ian Wilson who has had a new paper published at Pattern Recognition in Physics:
Discussion of this paper is going to be in the form of a workshop with specific objectives, and comments will be strictly moderated for relevance. The objectives will be announced by the main participants, Ian Wilson and Paul Vaughan, in their opening comments. Basically, unless you have something to contribute to the mathematical exposition, please sit this one out and watch.

This new peer-reviewed paper is available for (free) download at: http://www.pattern-recognition-in-physics.com/pub/prp-2-75-2014.pdf . This post reproduces the one at Ian’s blog.

Are the Strongest Lunar Perigean Spring Tides Commensurate with the Transit Cycle of Venus?
Ian Wilson PhD

Abstract

This study identifies the strongest Perigean spring tides that reoccur at roughly the same time in the seasonal calendar and shows how their repetition pattern, with respect to the tropical year, is closely synchronized with the 243 year transit cycle of Venus. It finds that whenever the pentagonal pattern for the inferior conjunctions of Venus and the Earth drifts through one of the nodes of Venus’ orbit, the 31/62 year Perigean spring tidal cycle simultaneously drifts through almost exactly the same days of the Gregorian year, over a period from 1 to 3000 A.D. Indeed, the drift of the 31/62 year tidal cycle with respect to the Gregorian calendar almost perfectly matches the expected long-term drift between the Gregorian calendar and the tropical year. If the mean drift of the 31/62 Perigean spring tidal cycle is corrected for the expected long-term drift between the Gregorian calendar and the tropical year, then the long-term residual drift between: a) the 243 year drift-cycle of the pentagonal pattern for the inferior conjunctions of Venus and the Earth with respect to the nodes of Venus’s orbit and b) the 243 year drift-cycle of the strongest seasonal peak tides on the Earth (i.e. the 31/62 Perigean spring tidal cycle) with respect to the tropical year is approximately equal to -7 ± 11 hours, over the 3000 year period. The large relative error of the final value for the residual drift means that this study cannot rule out the possibility that there is no long-term residual drift between the two cycles i.e. the two cycles are in perfect synchronization over the 3000 year period. However, the most likely result is a long-term residual drift of -7 hours, over the time frame considered.

Fig. 13a

Figure 13a

Figure 13b

Figure 13b

Figure 13. The red curve in [a] shows the difference (in hours) between the tropical year and the Gregorian calendar year (measured from J2000), as calculated from equation (2) versus the year. This difference is subtracted from the measured mean drift displayed [a] to determine the long-term residual drift (in hours) versus the year, which is re-plotted in [b]. The ± 95 % confidence intervals for the measured mean drift [a] and the long-term residual drift [b] are displayed – see text for details.

Conclusion

This study identifies the strongest Perigean spring tides that reoccur at roughly the same time in the seasonal calendar and shows how their repetition pattern, with respect to the tropical year, is in near-resonance with the 243 year transit cycle of Venus.

A single representative time is determined for each of the transits (or transit pairs) of Venus, over the period from 1 to 3000 A.D., in order to delineate the 243 year transit cycle. The representative time chosen for the transit cycle is the precise time of passage of the drifting pattern for the inferior conjunctions of Venus and the Earth (i.e. the pentagram pattern seen in figure 1), through a given node of Venus’ orbit.

Two methods are used to determine the dates of these particular events, over the 3000 year period of the study:

1. The first involves finding the date on which the percentage fraction of the circular disk of Venus that is illuminated by the Sun (as seen by a geocentric observer) is a minimum.

2. The second method involves using the transits (or near transits) on either side of a given node of Venus’ orbit to determine the temporal drift rate (in solar latitude) for the pattern of inferior conjunctions of Venus and the Earth. This is then used to calculate the date on which the pattern crosses the solar equator.

A selection process is set up to identify all new/full moons that occur within ± 20 hours of perigee, between the (Gregorian) calendar dates of the 14th of December and the 11th of January, spanning the years from 1 A.D. to 3000 A.D. This process successfully identifies all of the spring tidal events with equilibrium ocean tidal heights greater than approximately 62.0 cm, over the time interval chosen. These events are designated as the sample tidal events or the sample tides. Four distinct peak tidal cycles with periodicities less than 100 years are identified amongst the sample tides.

Investigations of these peak tidal cycles reveal that the 31/62 year tidal cycle is best synchronized to the seasonal calendar, over centennial time scales. Sequential events in this tidal cycle move forward through the seasonal calendar by only 2 – 3 days every 31 years, and the number of hours between new/full moon and perigee (a measure of their peak tidal strength) only changes by ~ 0.6 hours every 31 years.

An analysis of the 31/62 lunar peak tidal cycle shows that the sample tidal events reoccur on almost the same day of Gregorian (seasonal) calendar after 106 years, and then they reoccur on almost the same day after another 137 years. This produces a two-stage long-term repetition cycle with a total length of (106 + 137 years =) 243 years.

Remarkably, this means that, whenever the pentagonal pattern for the inferior conjunctions of Venus and the Earth drifts through one of the nodes of Venus’ orbit, the 31/62 year Perigean spring tidal cycle simultaneously drifts through almost exactly the same days of the Gregorian year, over a period of almost three thousand years. Indeed, the drift of the 31/62 year tidal cycle with respect to the Gregorian calendar almost perfectly matches the expected long-term drift between the Gregorian calendar and the tropical year. If the mean drift of the 31/62 Perigean spring tidal cycle is corrected for the expected long-term drift between the Gregorian calendar and the tropical year, then the long-term residual drift between:

1. the 243 year drift-cycle of the pentagonal pattern for the inferior conjunctions of Venus
and the Earth with respect to the nodes of Venus’s orbit

and

2. the 243 year drift-cycle of the strongest seasonal peak tides on the Earth (i.e. the 31/62 Perigean spring tidal cycle) with respect to the tropical year

is approximately equal to -7 ± 11 hours, over a 3000 years period. The large relative error of the final value for the residual drift means that this study cannot rule out the possibility that there is no long-term residual drift between the two cycles i.e. the two cycles are in perfect synchronization over the 3000year period from 1 to 3000 A.D. However, the most likely result is a long-term residual drift of -7 hours, over the time frame considered. Finally, there is one speculative extrapolation that could encourage others to further investigate this close synchronization on much longer time scales. If these future investigations show that the long-term residual drift rate of -7 hours over 3000 years is valid over much longer time scales then this close synchronization may highlight a mechanism that might be responsible for the Earth’s 100,000 year Ice-Age cycle. This comes from the fact that the strongest Perigean spring tides would be in close synchronization with (i.e. ± half a day either side of) the date of the Earth’s Solstice (on or about December 21st) for a period (24/7) × 3000 years =10,300 years. In addition, this close synchronization would be re-established itself after the 31/62 peak tidal pattern drifted backward through the Tropical calendar by ~ 9.7 days (i.e. the average vertical spacing between sequences in figs 12a & b) such that after ((9.7 × 24) / 7) × 3000 years = 99,800 years.

Hence, the close synchronization discovered in this study lasts for ~10,000 years, with each period of close synchronization being separated from its predecessor by ~100,000 years. This is very reminiscent of the inter-glacial/glacial period that is characteristic of the Earth’s recent Ice-Age cycles.

Comments
  1. Ian Wilson says:

    Rog,

    Thank you again for giving me the privilege and opportunity to present my work here at Tallbloke’s Talkshop. The main conclusion of my paper are presented above and I will try to field questions that people might have about this work. I would respectfully ask that the questions be directly related to the contents of the paper. I will not be responding to broad general questions about factors affecting climate.

    What happens’ next is up to Roger but I am lead to believe that once we get the initial questions raised by this paper out of the way, a second Work Shop blog post will be presented by Paul Vaughan where he will discuss some of the interesting things he has found that are related to this general topic.

  2. tallbloke says:

    Ian, since your result is speaking strongly for itself with such a close commensurability, I’m hoping this will in fact be the ‘workshop thread’ If Paul is willing to go along with that idea. I’d like Paul to firstly address your result, and then lay out his work so the two of you (and anyone with a directly relevant input) can then develop the theme together.

  3. Ian Wilson says:

    Roger, no problem, I will go with the flow if Paul is also happy with the idea. I hope that my comments have not deterred anyone from asking questions and making comments. The Tallbloke Talkshop is widely know around the climate skeptic community as place you can go and safely discuss new and potentially innovative ideas.

    Just a heads up. I will have limited contact with the web for three days starting next Monday. This will mean that any discussion that we do have in the workshop will necessarily be spread over a period of at least a week or so at minimum.

  4. Ian Wilson says:

    Here is my take on the guidelines for the workshop [These guidelines were given earlier and I am repeating them here for emphasis].

    In this upcoming workshop, we not trying comprehensively solve all of the problems of the climate universe. Indeed if contributors demand this then it will just introduce a fog which will hinder (and possibly kill) the general discussion. All we get is a he says-she says “debate” where each contributor will defend to their last drop of blood, their particular view of the climate universe.

    The work shop purpose is to discuss a particular revelation that Paul has about the results of my most recent paper. Paul will present a number of issues that he wants us to digest and discuss. Obviously, there will be times that we have to clarify some points about his presentation, and there will be times where we want to contribute our own ideas. However, it is important to emphasis the following:

    a) The workshop will not progress far if Paul is continually held up trying to explain (or even debate) simple definitions and concepts – in fact it would be good if others could step in and give a brief explanation to a quizzical contributor so that the discussion can continue to flow.

    b) The workshop will fail if someone spends all of their time just trying to push their own broad philosophical climate agenda. If there is someone doing this, I believe that they should be called out and asked to start their own blog post or thread elsewhere.

    c) All contributors must realize that we live in a real (off-line) world and that it not possible for everyone to be permanently chained to their computer 24/7 – ready to give an immediate response to a any contributor.

    Above all, the workshop is primarily (but not solely) based around an exchange between Paul and myself about a specific set of topics (i.e. revelations/comments by Paul to issues that arise out of the general results in my most recent paper).

    It is my genuine hope that contributors will respect these guidelines – if they do, I believe that they may be rewarded for their efforts.

  5. Ian Wilson says:

    I will start off my contribution with this comment on the paper:

    “It is very likely that the lunar orbit partakes in the ~ 26,000 year precession of the Earth’s rotation axis with respect to the stars. When this ~ 26,000 year precession is couple with the ~ 71,000 year wobble of the Earth’s orbit (with respect to the plane of the ecliptic), it produces the well known ~ 41,000 year nodding of the Earth’s obliquity (i.e. tilt of the rotation axis with respect to the plane of the ecliptic). Hence, it is very possible that what I have really found in my paper is that the ~ 26,000 year precession of the Earth’s axis of rotation with respect to the stars is synchronized with the slow drift of the 13:8 year Venus/Earth commensurability pattern with respect to the nodes of Venus’ orbit. Support for this proposal is provided by the fact that it takes ~ 41,000 years for the calender dates for the transit of Venus to cycle once through the seasonal (tropical) year.

    Of course, even if the above is true, it does not distract from the point that the peak tides experienced here on the Earth appear to be linked to commensurable ratios of the orbital periods of the Earth and Venus.”

  6. J Martin says:

    So reduced tidal ranges every 10,300 years is the background driver to glaciations ? That would make plans to put turbines on the sea bed a rather bad idea. In the 41k world obliquity was the driver, now Venus but no one knows what caused the change.

  7. Ian Wilson says:

    J Martin,

    Thank you for your comment and welcome to the conversation.

    At no point in the paper did I say that a reduced tidal range every 10,300 years was responsible for glaciation.

    In the above post, I state the reason for ~ 41,000 year variation in the Earth’s obliquity. It is just produced by a combination of ~ 26,000 year precession of the Earth’s axis of rotation with respect to the stars and the ~ 71,000 year wobble of the Earth’s orbit with respect to a fixed Earth-Sun plane.
    No mystery here.

    What I am commenting about [post publication of the paper] is that the lunar orbit (which governs the long term pattern of peak lunar tides) almost certainly precesses in the same manner as the Earth’s rotation axis.

    Hence, the link that I have found in the drift of the peak lunar tides with respect to the tropical year, may be just a link between slow nodding of the Earth’s rotation axis with respect to the tropical year which takes ~ 41,000 years AND the slow drift of the dates of the transit cycle of Venus with respect to the tropical year which also takes ~ 41,000 years.

  8. Paul Vaughan says:

    I’ve rapidly developed tons of material to share (attractor frequency algebra, derivations, illustrations, etc.), but I have inadequate free time to communicate it in a clear, timely manner.

    I suspect discussion will move at a snail’s pace. A month might do. I have no sizable blocks of free time on the horizon, so my contributions will be guided by the Pareto Principle on the fly. Some topics will be too deep to address at all in the time available, so they’ll have to be skipped.

    Crucially Important:
    I need to know that participants have independently isolated the annual lunisolar aliasing envelopes mentioned in the previous thread.

    I gave exhaustive details on how to get the output from NASA JPL Horizons. I would advise TB that if workshop participants can’t independently find the envelopes in Horizons output using the cookbook recipe I shared, there’s little reason to be optimistic about their ability to even know what’s up for discussion.

    Ian has made an important contribution and I was very disappointed with the last thread, to the point where I almost walked away from those disrupting the thread with their ignorance & politics.

    Other obligations are calling me away right now, but meanwhile here’s some JEV background material to consider:

    1. “coupling in the eccentricities of Venus and Earth”

    A survey of near-mean-motion resonances between Venus and Earth
    Bazsó, Á.; Dvorak, R.; Pilat-Lohinger, E.; Eybl, V.; Lhotka, Ch. (2010)
    Celestial Mechanics and Dynamical Astronomy 107(1-2), 63-76
    http://www.researchgate.net/publication/226874223_A_survey_of_near-mean-motion_resonances_between_Venus_and_Earth/file/d912f50a65e261bb8d.pdf
    (If hyperlink to researchgate.net pdf fails, copy/paste into new browser tab — that seems to reliably work when researchgate.net hyperlinks fail)

    earlier version (2009):
    An Overview of the 13:8 Mean Motion Resonance between Venus and Earth
    http://arxiv.org/pdf/0911.2357.pdf

    2. excellent Video

    – –

    Workshop Participants:
    Let me know when you’ve isolated the envelopes.
    Once everyone has done that independently, I have material to start sharing on a slow-release schedule…

    Best Regards

  9. tallbloke says:

    I think this is the location of the “Horizons recipe” Paul mentions:
    https://tallbloke.wordpress.com/2014/11/15/evidence-that-strong-el-nino-events-are-triggered-by-the-moon/comment-page-1/#comment-92954
    Now we have to work out what Paul wants us to do. First we need to answer the question of what an “annual lunisolar aliasing envelope” is.
    Perhaps Ian can help.

  10. Ulric Lyons says:

    Note the slip in the 243yr Transit series:
    http://eclipse.gsfc.nasa.gov/transit/catalog/VenusCatalog.html

  11. Paul Vaughan says:

    tallbloke (December 6, 2014 at 8:19 pm) suggested:
    “First we need to answer the question of what an “annual lunisolar aliasing envelope” is.”

    Actually no, the definition is baked into the recipe. Just follow the recipe. Simply graph the 9 variables and observe the envelopes (for which I’ve already outlined frequency algebra).

    It’s literally that simple.

  12. tallbloke says:

    Thanks Paul,
    OK, participants need to go to http://ssd.jpl.nasa.gov/horizons.cgi and set the parameters as Paul has them here:
    https://tallbloke.wordpress.com/2014/11/15/evidence-that-strong-el-nino-events-are-triggered-by-the-moon/comment-page-1/#comment-92954
    Generate the ephemeris
    Slap the output into notepad and save as .csv rather than .txt
    Open with excel or other spreadsheet and graph the 9 variables against time.

  13. tallbloke says:

    Here’s the sort of output I’m getting

  14. tchannon says:

    “Step-size : 1 calendar years”
    What checks have been done to show that aliasing is not an issue?

  15. Ian Wilson says:

    Tim,
    We are looking how lunar tidal strengths, at the same point in the tropical year, change over time. The investigation is based upon the premise that peaks in the tidal strengths that are aligned with the seasons (i.e. that are applied at the same point in the seasonal (tropical) calendar) have more affect on the climate than peaks in tidal strengths that systematically drift through the seasons.

    So yes, we are acutely aware of the difficulties of aliasing.

  16. Ian Wilson says:

    Paul,
    Looking at the envelop of the annually aliased lunar perigee distance between 1580 and 2120 A.D., I see a series of two or three very distinct peaks in closest perigee that are separated from each other by almost precisely 31 years. These peaks are modulated (i.e. fade in and out) over a much more slowly varying ~ 121 year cycle.

    (Note: each of the closest perigees is separated from the previous one by 4 years).
    (Note: The envelop that matters is the annually aliased lunar perigee distance since this corresponds to the strongest lunar tides. This is the lower envelop of annually aliased lunar distances.)

  17. Ian Wilson says:

    Paul,

    I can confirm a roughly 27.0 year envelop for the annually aliased lunar eccentricity between 1580 and 2120 A.D.

    You are asking us to find a:

    a) 26.72660385 year eccentricity envelope
    b) 30.02778018 year range envelope

    i assume that this is because the synodic (beat) product of these two periods is ~ 243.107 years.

    I agree that a) is ~ 27.0 years but I am not fully convinced that b) is ~ 30.0 years.

    To me, b) appears more like a 31.0 year cycle modulated by a much longer ~ 121 year cycle.

    Note: I used auto-correlation to confirm the 27.0 year period in the annually aliased ellipticity signal but this method does not work for the annually aliased lunar distance signal. Any suggestions?

  18. Paul Vaughan says:

    tchannon (December 7, 2014 at 12:27 am) read “Step-size : 1 calendar years” and asked:
    “What checks have been done to show that aliasing is not an issue?”

    We’re exploring physical aliasing. That’s what this is all about. The same day of year next year doesn’t get a random sample. It gets a systematic sample. It’s incorrect to falsely assume days of the year are random.

    Absolutely no offense intended, but I want everyone to know that I interpret such questions as deliberate distortion artistry, no matter who they’re from. I don’t have time for this sort of thing …from anyone, including people I respect highly for their talents aside from the finer subtleties of aggregation criteria.

    Regards

  19. tallbloke says:

    Hmm, 121 years eh?
    2J*JS/(2J-JS)=~122yrs

  20. Paul Vaughan says:

    Ian, I assure you the frequency algebra I’ve given is on lock-down.
    Later on I’m going to spell it all out with derivations and illustrations.

    You’ll discover that some of the first-glance notes you’ve just posted are incorrect upon more tedious inspection. Take note of the addendum about 34 later in the last thread. It’s not just about 27 & 30. Also, be careful inspecting 243 patterns that upon first-glance look like 121.5 patterns. There are reversals of the 34 wave with each 121.5 wave.

    There’s no 31.

    When I start seeing graphs of (a) 26 alone and (b) the 243 / 34 combo with calendar years clearly labeled, I’ll know it’s time to start sharing more, including illustrations that summarize dozens upon dozens of alignments — along with their strengths — on single images. This reduces perceptual complexity by orders of magnitude. I confess: I was misled into thinking lunar cycles are complicated. They are not. They’re dead simple.

    Regards

  21. Paul Vaughan says:

    Everyone remember:

    Events don’t get a pass from central limit theorem (CLT).
    CLT applies to all parent distributions (except Cauchy, in which case the median is the robust estimator, a technicality which is totally irrelevant in the context at hand).

  22. Paul Vaughan says:

    tallbloke (December 6, 2014 at 9:08 pm) wrote:
    “Slap the output into notepad and save as .csv rather than .txt
    Open with excel or other spreadsheet and graph the 9 variables against time.”

    Maybe easier:
    Just copy/paste-special-as-values unicode-text to Excel then go Data Text-to-Columns comma-delimited.

    Remember that Jan. 1 was just for an example.
    With good reason I’m recommending comparative exploration of:
    Mar 21, Jun 21, Sep 21, Dec 21

    Recommended variable priorities:
    [6.4] Z z-component of position vector (AU)
    [243 / 34] RG Range; distance from coordinate center (AU)

    [27] EC Eccentricity, e
    [27] QR Periapsis distance, q (AU)
    [27] AD Apoapsis distance (AU)

    Alert: Explorers will have to make some minor tweaks to Horizons settings to get EC, QR, & AD.
    I’ll post more recipes for those who can’t handle the task independently — maybe 1 every day.

    Start-to-finish it should take no more than 2 or 3 minutes to have all the graphs and see all the envelopes.

  23. Paul Vaughan says:

    Recipe #2:

    =
    Ephemeris Type : ELEMENTS
    Target Body : Moon [Luna] [301]
    Center : Geocentric [500]
    Time Span : Start=1580-01-01, Stop=2120-01-01, Step=1 Y
    Table Settings : reference plane=FRAME; CSV format=YES
    Display/Output : plain text
    *******************************************************************************
    Revised: Mar 11, 1998 Moon / (Earth) 301

    PHYSICAL PROPERTIES:
    Radius, km = 1737.53+-0.03 Mass, 10^20 kg = 734.9
    Density, gm cm^-3 = 3.3437 Geometric albedo = 0.12
    V(1,0) = +0.21 GM, km^3/s^2 = 4902.798+-.005
    Earth/Moon mass ratio = 81.300587 Surface gravity = 1.62 m s^-2
    Nearside crust. thick.= 58+-8 km Farside crust. thick. = ~80 – 90 km
    Heat flow, Apollo 15 = 3.1+-.6 mW/m^2 Heat flow, Apollo 17 = 2.2+-.5 mW/m^2
    Mean crustal density = 2.97+-.07g/cm^3 k2 = 0.0302+-.0012
    Induced magnetic mom. = 4.23×10^22Gcm^3 Magnetometer moment = 435+-15

    DYNAMICAL CHARACTERISTICS:
    Mean angular diameter = 31’05.2″ Orbit period = 27.321582 d
    Obliquity to orbit = 6.67 deg Eccentricity = 0.05490
    Semi-major axis, a = 384400 km Inclination = 5.145 deg
    Mean motion, rad/s = 2.6616995×10^-6 Nodal period = 6798.38 d
    Apsidal period = 3231.50 d Mom. of inertia C/MR^2= 0.3935+-.0011
    beta (C-A/B), x10^-4 = 6.31(72+-15) gamma (B-A/C), x10^-4 = 2.278(8+-2)
    *******************************************************************************

    *******************************************************************************
    Ephemeris / WWW_USER Sat Oct 11 12:46:36 2014 Pasadena, USA / Horizons
    *******************************************************************************
    Target body name: Moon (301) {source: DE-0431LE-0431}
    Center body name: Earth (399) {source: DE-0431LE-0431}
    Center-site name: BODY CENTER
    *******************************************************************************
    Start time : A.D. 1580-Jan-01 00:00:00.0000 CT
    Stop time : A.D. 2120-Jan-01 00:00:00.0000 CT
    Step-size : 1 calendar years
    *******************************************************************************
    Center geodetic : 0.00000000,0.00000000,0.0000000 {E-lon(deg),Lat(deg),Alt(km)}
    Center cylindric: 0.00000000,0.00000000,0.0000000 {E-lon(deg),Dxy(km),Dz(km)}
    Center radii : 6378.1 x 6378.1 x 6356.8 km {Equator, meridian, pole}
    System GM : 8.9970113901998714E-10 au^3/d^2
    Output units : AU-D, deg, Julian day number (Tp)
    Output format : 10
    Reference frame : ICRF/J2000.0
    Output type : GEOMETRIC osculating elements
    Coordinate systm: Earth Mean Equator and Equinox of Reference Epoch
    *******************************************************************************
    JDCT , , EC, QR, IN, OM, W, Tp, N, MA, TA, A, AD, PR
    *******************************************************************************
    $$SOE
    $$EOE
    *******************************************************************************
    Coordinate system description:

    Earth Mean Equator and Equinox of Reference Epoch

    Reference epoch: J2000.0
    xy-plane: plane of the Earth’s mean equator at the reference epoch
    x-axis : out along ascending node of instantaneous plane of the Earth’s
    orbit and the Earth’s mean equator at the reference epoch
    z-axis : along the Earth mean north pole at the reference epoch

    Symbol meaning [1 au=149597870.700 km, 1 day=86400.0 s]:

    JDCT Epoch Julian Date, Coordinate Time
    EC Eccentricity, e
    QR Periapsis distance, q (AU)
    IN Inclination w.r.t xy-plane, i (degrees)
    OM Longitude of Ascending Node, OMEGA, (degrees)
    W Argument of Perifocus, w (degrees)
    Tp Time of periapsis (Julian day number)
    N Mean motion, n (degrees/day)
    MA Mean anomaly, M (degrees)
    TA True anomaly, nu (degrees)
    A Semi-major axis, a (AU)
    AD Apoapsis distance (AU)
    PR Sidereal orbit period (day)

    Geometric states/elements have no aberration corrections applied.

    Computations by …
    Solar System Dynamics Group, Horizons On-Line Ephemeris System
    4800 Oak Grove Drive, Jet Propulsion Laboratory
    Pasadena, CA 91109 USA
    Information: http://ssd.jpl.nasa.gov/
    Connect : telnet://ssd.jpl.nasa.gov:6775 (via browser)
    telnet ssd.jpl.nasa.gov 6775 (via command-line)
    Author : Jon.Giorgini@jpl.nasa.gov
    *******************************************************************************
    =

  24. Paul Vaughan says:

    Recipe #3 — like #2, but different reference plane (but same ~27 year envelope for EC, QR, & AD)

    =
    Ephemeris Type : ELEMENTS
    Target Body : Moon [Luna] [301]
    Center : Geocentric [500]
    Time Span : Start=1580-01-01, Stop=2120-01-01, Step=1 Y
    Table Settings : CSV format=YES
    Display/Output : plain text
    *******************************************************************************
    Revised: Mar 11, 1998 Moon / (Earth) 301

    PHYSICAL PROPERTIES:
    Radius, km = 1737.53+-0.03 Mass, 10^20 kg = 734.9
    Density, gm cm^-3 = 3.3437 Geometric albedo = 0.12
    V(1,0) = +0.21 GM, km^3/s^2 = 4902.798+-.005
    Earth/Moon mass ratio = 81.300587 Surface gravity = 1.62 m s^-2
    Nearside crust. thick.= 58+-8 km Farside crust. thick. = ~80 – 90 km
    Heat flow, Apollo 15 = 3.1+-.6 mW/m^2 Heat flow, Apollo 17 = 2.2+-.5 mW/m^2
    Mean crustal density = 2.97+-.07g/cm^3 k2 = 0.0302+-.0012
    Induced magnetic mom. = 4.23×10^22Gcm^3 Magnetometer moment = 435+-15

    DYNAMICAL CHARACTERISTICS:
    Mean angular diameter = 31’05.2″ Orbit period = 27.321582 d
    Obliquity to orbit = 6.67 deg Eccentricity = 0.05490
    Semi-major axis, a = 384400 km Inclination = 5.145 deg
    Mean motion, rad/s = 2.6616995×10^-6 Nodal period = 6798.38 d
    Apsidal period = 3231.50 d Mom. of inertia C/MR^2= 0.3935+-.0011
    beta (C-A/B), x10^-4 = 6.31(72+-15) gamma (B-A/C), x10^-4 = 2.278(8+-2)
    *******************************************************************************

    *******************************************************************************
    Ephemeris / WWW_USER Fri Oct 10 19:40:35 2014 Pasadena, USA / Horizons
    *******************************************************************************
    Target body name: Moon (301) {source: DE-0431LE-0431}
    Center body name: Earth (399) {source: DE-0431LE-0431}
    Center-site name: BODY CENTER
    *******************************************************************************
    Start time : A.D. 1580-Jan-01 00:00:00.0000 CT
    Stop time : A.D. 2120-Jan-01 00:00:00.0000 CT
    Step-size : 1 calendar years
    *******************************************************************************
    Center geodetic : 0.00000000,0.00000000,0.0000000 {E-lon(deg),Lat(deg),Alt(km)}
    Center cylindric: 0.00000000,0.00000000,0.0000000 {E-lon(deg),Dxy(km),Dz(km)}
    Center radii : 6378.1 x 6378.1 x 6356.8 km {Equator, meridian, pole}
    System GM : 8.9970113901998714E-10 au^3/d^2
    Output units : AU-D, deg, Julian day number (Tp)
    Output format : 10
    Reference frame : ICRF/J2000.0
    Output type : GEOMETRIC osculating elements
    Coordinate systm: Ecliptic and Mean Equinox of Reference Epoch
    *******************************************************************************
    JDCT , , EC, QR, IN, OM, W, Tp, N, MA, TA, A, AD, PR
    *******************************************************************************
    $$SOE
    $$EOE
    *******************************************************************************
    Coordinate system description:

    Ecliptic and Mean Equinox of Reference Epoch

    Reference epoch: J2000.0
    xy-plane: plane of the Earth’s orbit at the reference epoch
    x-axis : out along ascending node of instantaneous plane of the Earth’s
    orbit and the Earth’s mean equator at the reference epoch
    z-axis : perpendicular to the xy-plane in the directional (+ or -) sense
    of Earth’s north pole at the reference epoch.

    Symbol meaning [1 au=149597870.700 km, 1 day=86400.0 s]:

    JDCT Epoch Julian Date, Coordinate Time
    EC Eccentricity, e
    QR Periapsis distance, q (AU)
    IN Inclination w.r.t xy-plane, i (degrees)
    OM Longitude of Ascending Node, OMEGA, (degrees)
    W Argument of Perifocus, w (degrees)
    Tp Time of periapsis (Julian day number)
    N Mean motion, n (degrees/day)
    MA Mean anomaly, M (degrees)
    TA True anomaly, nu (degrees)
    A Semi-major axis, a (AU)
    AD Apoapsis distance (AU)
    PR Sidereal orbit period (day)

    Geometric states/elements have no aberration corrections applied.

    Computations by …
    Solar System Dynamics Group, Horizons On-Line Ephemeris System
    4800 Oak Grove Drive, Jet Propulsion Laboratory
    Pasadena, CA 91109 USA
    Information: http://ssd.jpl.nasa.gov/
    Connect : telnet://ssd.jpl.nasa.gov:6775 (via browser)
    telnet ssd.jpl.nasa.gov 6775 (via command-line)
    Author : Jon.Giorgini@jpl.nasa.gov
    *******************************************************************************
    =

  25. Paul Vaughan says:

    It’s literally a game of “Spot the Differences”.

    Recipe #4 — like #1, but with different reference plane, which preserves RG 243/34, but (IMPORTANT – TAKE NOTE) swaps Z 6.4 (envelope summary graph for Jan. 1) for 18.6. Please get a solid handle on this folks. Otherwise sensible discussion is totally impossible.

    =
    Ephemeris Type : VECTORS
    Target Body : Moon [Luna] [301]
    Coordinate Origin : Geocentric [500]
    Time Span : Start=1580-01-01, Stop=2120-01-01, Step=1 Y
    Table Settings : reference plane=FRAME; labels=YES; CSV format=YES
    Display/Output : plain text
    *******************************************************************************
    Revised: Mar 11, 1998 Moon / (Earth) 301

    PHYSICAL PROPERTIES:
    Radius, km = 1737.53+-0.03 Mass, 10^20 kg = 734.9
    Density, gm cm^-3 = 3.3437 Geometric albedo = 0.12
    V(1,0) = +0.21 GM, km^3/s^2 = 4902.798+-.005
    Earth/Moon mass ratio = 81.300587 Surface gravity = 1.62 m s^-2
    Nearside crust. thick.= 58+-8 km Farside crust. thick. = ~80 – 90 km
    Heat flow, Apollo 15 = 3.1+-.6 mW/m^2 Heat flow, Apollo 17 = 2.2+-.5 mW/m^2
    Mean crustal density = 2.97+-.07g/cm^3 k2 = 0.0302+-.0012
    Induced magnetic mom. = 4.23×10^22Gcm^3 Magnetometer moment = 435+-15

    DYNAMICAL CHARACTERISTICS:
    Mean angular diameter = 31’05.2″ Orbit period = 27.321582 d
    Obliquity to orbit = 6.67 deg Eccentricity = 0.05490
    Semi-major axis, a = 384400 km Inclination = 5.145 deg
    Mean motion, rad/s = 2.6616995×10^-6 Nodal period = 6798.38 d
    Apsidal period = 3231.50 d Mom. of inertia C/MR^2= 0.3935+-.0011
    beta (C-A/B), x10^-4 = 6.31(72+-15) gamma (B-A/C), x10^-4 = 2.278(8+-2)
    *******************************************************************************

    *******************************************************************************
    Ephemeris / WWW_USER Sat Oct 11 12:39:55 2014 Pasadena, USA / Horizons
    *******************************************************************************
    Target body name: Moon (301) {source: DE-0431LE-0431}
    Center body name: Earth (399) {source: DE-0431LE-0431}
    Center-site name: BODY CENTER
    *******************************************************************************
    Start time : A.D. 1580-Jan-01 00:00:00.0000 CT
    Stop time : A.D. 2120-Jan-01 00:00:00.0000 CT
    Step-size : 1 calendar years
    *******************************************************************************
    Center geodetic : 0.00000000,0.00000000,0.0000000 {E-lon(deg),Lat(deg),Alt(km)}
    Center cylindric: 0.00000000,0.00000000,0.0000000 {E-lon(deg),Dxy(km),Dz(km)}
    Center radii : 6378.1 x 6378.1 x 6356.8 km {Equator, meridian, pole}
    Output units : AU-D
    Output format : 03
    Reference frame : ICRF/J2000.0
    Output type : GEOMETRIC cartesian states
    Coordinate systm: Earth Mean Equator and Equinox of Reference Epoch
    *******************************************************************************
    JDCT , , X, Y, Z, VX, VY, VZ, LT, RG, RR,
    *******************************************************************************
    $$SOE
    $$EOE
    *******************************************************************************
    Coordinate system description:

    Earth Mean Equator and Equinox of Reference Epoch

    Reference epoch: J2000.0
    xy-plane: plane of the Earth’s mean equator at the reference epoch
    x-axis : out along ascending node of instantaneous plane of the Earth’s
    orbit and the Earth’s mean equator at the reference epoch
    z-axis : along the Earth mean north pole at the reference epoch

    Symbol meaning [1 au=149597870.700 km, 1 day=86400.0 s]:

    JDCT Epoch Julian Date, Coordinate Time
    X x-component of position vector (AU)
    Y y-component of position vector (AU)
    Z z-component of position vector (AU)
    VX x-component of velocity vector (AU/day)
    VY y-component of velocity vector (AU/day)
    VZ z-component of velocity vector (AU/day)
    LT One-way down-leg Newtonian light-time (day)
    RG Range; distance from coordinate center (AU)
    RR Range-rate; radial velocity wrt coord. center (AU/day)

    Geometric states/elements have no aberration corrections applied.

    Computations by …
    Solar System Dynamics Group, Horizons On-Line Ephemeris System
    4800 Oak Grove Drive, Jet Propulsion Laboratory
    Pasadena, CA 91109 USA
    Information: http://ssd.jpl.nasa.gov/
    Connect : telnet://ssd.jpl.nasa.gov:6775 (via browser)
    telnet ssd.jpl.nasa.gov 6775 (via command-line)
    Author : Jon.Giorgini@jpl.nasa.gov
    *******************************************************************************
    =

  26. Paul Vaughan says:

    Recipe #5 — 243 in a number of variables here, including DEC_(a-app)

    =
    Ephemeris Type : OBSERVER
    Target Body : Moon [Luna] [301]
    Observer Location : Geocentric [500]
    Time Span : Start=1580-01-01, Stop=2120-01-01, Step=1 Y
    Table Settings : QUANTITIES=1-43; angle format=DEG; CSV format=YES
    Display/Output : plain text
    *******************************************************************************
    Revised: Mar 11, 1998 Moon / (Earth) 301

    PHYSICAL PROPERTIES:
    Radius, km = 1737.53+-0.03 Mass, 10^20 kg = 734.9
    Density, gm cm^-3 = 3.3437 Geometric albedo = 0.12
    V(1,0) = +0.21 GM, km^3/s^2 = 4902.798+-.005
    Earth/Moon mass ratio = 81.300587 Surface gravity = 1.62 m s^-2
    Nearside crust. thick.= 58+-8 km Farside crust. thick. = ~80 – 90 km
    Heat flow, Apollo 15 = 3.1+-.6 mW/m^2 Heat flow, Apollo 17 = 2.2+-.5 mW/m^2
    Mean crustal density = 2.97+-.07g/cm^3 k2 = 0.0302+-.0012
    Induced magnetic mom. = 4.23×10^22Gcm^3 Magnetometer moment = 435+-15

    DYNAMICAL CHARACTERISTICS:
    Mean angular diameter = 31’05.2″ Orbit period = 27.321582 d
    Obliquity to orbit = 6.67 deg Eccentricity = 0.05490
    Semi-major axis, a = 384400 km Inclination = 5.145 deg
    Mean motion, rad/s = 2.6616995×10^-6 Nodal period = 6798.38 d
    Apsidal period = 3231.50 d Mom. of inertia C/MR^2= 0.3935+-.0011
    beta (C-A/B), x10^-4 = 6.31(72+-15) gamma (B-A/C), x10^-4 = 2.278(8+-2)
    *******************************************************************************

    *******************************************************************************
    Ephemeris / WWW_USER Sat Oct 11 12:59:16 2014 Pasadena, USA / Horizons
    *******************************************************************************
    Target body name: Moon (301) {source: DE-0431LE-0431}
    Center body name: Earth (399) {source: DE-0431LE-0431}
    Center-site name: GEOCENTRIC
    *******************************************************************************
    Start time : A.D. 1580-Jan-01 00:00:00.0000 UT
    Stop time : A.D. 2120-Jan-01 00:00:00.0000 UT
    Step-size : 1 calendar year
    *******************************************************************************
    Target pole/equ : IAU_MOON {East-longitude +}
    Target radii : 1737.4 x 1737.4 x 1737.4 km {Equator, meridian, pole}
    Center geodetic : 0.00000000,0.00000000,0.0000000 {E-lon(deg),Lat(deg),Alt(km)}
    Center cylindric: 0.00000000,0.00000000,0.0000000 {E-lon(deg),Dxy(km),Dz(km)}
    Center pole/equ : High-precision EOP model {East-longitude +}
    Center radii : 6378.1 x 6378.1 x 6356.8 km {Equator, meridian, pole}
    Target primary : Earth
    Vis. interferer : MOON (R_eq= 1737.400) km {source: DE-0431LE-0431}
    Rel. light bend : Sun, EARTH {source: DE-0431LE-0431}
    Rel. lght bnd GM: 1.3271E+11, 3.9860E+05 km^3/s^2
    Atmos refraction: NO (AIRLESS)
    RA format : DEG
    Time format : CAL
    EOP file : eop.141010.p150101
    EOP coverage : DATA-BASED 1962-JAN-20 TO 2014-OCT-10. PREDICTS-> 2014-DEC-31
    Units conversion: 1 au= 149597870.700 km, c= 299792.458 km/s, 1 day= 86400.0 s
    Table cut-offs 1: Elevation (-90.0deg=NO ),Airmass (>38.000=NO), Daylight (NO )
    Table cut-offs 2: Solar Elongation ( 0.0,180.0=NO ),Local Hour Angle( 0.0=NO )
    Table format : Comma Separated Values (spreadsheet)
    *******************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************
    Date__(UT)__HR:MN, , ,R.A._(ICRF/J2000.0), DEC_(ICRF/J2000.0), R.A._(a-app), DEC_(a-app), dRA*cosD,d(DEC)/dt, Azi_(a-app), Elev_(a-app), dAZ*cosE,d(ELV)/dt, X_(sat-prim), Y_(sat-prim), SatPANG, L_Ap_Sid_Time, a-mass, APmag, S-brt, Illu%, Def_illu, ang-sep, v, Ang-diam, Obsrv-lon,Obsrv-lat, Solar-lon,Solar-lat, SN.ang, SN.dist, NP.ang, NP.dist, hEcl-Lon,hEcl-Lat, r, rdot, delta, deldot, 1-way_LT, VmagSn, VmagOb, S-O-T,/r, S-T-O, T-O-M,MN_Illu%, O-P-T, PsAng, PsAMV, PlAng, Cnst, CT-UT, ObsEcLon, ObsEcLat, N.Pole-RA, N.Pole-DC, GlxLon, GlxLat, L_Ap_SOL_Time, 399_ins_LT, RA_3sigma,DEC_3sigma, SMAA_3sig,SMIA_3sig, Theta,Area_3sig, POS_3sigma, RNG_3sigma,RNGRT_3sig, DOP_S_3sig, DOP_X_3sig, RT_delay_3sig, Tru_Anom, L_Ap_Hour_Ang, phi, PAB-LON, PAB-LAT,
    *******************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************
    $$SOE
    $$EOE
    *******************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************
    Column meaning:

    TIME

    Prior to 1962, times are UT1. Dates thereafter are UTC. Any ‘b’ symbol in
    the 1st-column denotes a B.C. date. First-column blank (” “) denotes an A.D.
    date. Calendar dates prior to 1582-Oct-15 are in the Julian calendar system.
    Later calendar dates are in the Gregorian system.

    Time tags refer to the same instant throughout the universe, regardless of
    where the observer is located.

    The dynamical Coordinate Time scale is used internally. It is equivalent to
    the current IAU definition of “TDB”. Conversion between CT and the selected
    non-uniform UT output scale has not been determined for UTC times after the
    next July or January 1st. The last known leap-second is used over any future
    interval.

    NOTE: “n.a.” in output means quantity “not available” at the print-time.

    R.A._(J2000.0)_DEC. =
    J2000.0 astrometric right ascension and declination of target center.
    Adjusted for light-time. Units: DEGREES

    R.A._(a-appar)_DEC. =
    Airless apparent right ascension and declination of the target center with
    respect to the Earth’s true-equator and the meridian containing the Earth’s
    true equinox-of-date. Adjusted for light-time, the gravitational deflection of
    light, stellar aberration, precession and nutation. Units: DEGREES

    dRA*cosD d(DEC)/dt =
    The rate of change of target center apparent RA and DEC (airless).
    d(RA)/dt is multiplied by the cosine of the declination.
    Units: ARCSECONDS PER HOUR

    Azi_(a-appr)_Elev =
    Airless apparent azimuth and elevation of target center. Adjusted for
    light-time, the gravitational deflection of light, stellar aberration,
    precession and nutation. Azimuth measured North(0) -> East(90) -> South(180) ->
    West(270) -> North (360). Elevation is with respect to plane perpendicular
    to local zenith direction. TOPOCENTRIC ONLY. Units: DEGREES

    dAZ*cosE d(ELV)/dt =
    The rate of change of target center apparent azimuth and elevation
    (airless). d(AZ)/dt is multiplied by the cosine of the elevation angle.
    TOPOCENTRIC ONLY. Units: ARCSECOND/MINUTE

    X & Y satellite coordinates & position angle =
    Satellite differential coordinates WRT the primary body along with the
    satellite position angle. Differential coordinates are defined in RA as
    X=[(RA_sat – RA_primary)*COS(DEC_primary)], in DEC as Y=(DEC_sat-DEC_primary).
    Non-Lunar satellites only. “SatPANG” is the angle from the North Celestial
    Pole measured counter-clockwise (CCW, or east) to a line from primary/planet
    center to satellite center.
    Units: ARCSECONDS (X & Y) and DEGREES (position angle)

    L_Ap_Sid_Time =
    Local Apparent Sidereal Time. The angle measured westward in the body
    true-equator of-date plane from the meridian containing the body-fixed
    observer to the meridian containing the true Earth equinox (defined by
    intersection of the true Earth equator of date with the ecliptic of date).
    TOPOCENTRIC ONLY. Units: HH.fffffffffff (decimal hours)

    a-mass =
    RELATIVE optical airmass; a measure of extinction. The ratio between
    the absolute optical airmass at the target’s refracted CENTER-POINT to the
    absolute optical airmass at zenith. Based on work of Kasten and Young
    (Applied Optics, vol. 28 no. 22, 15-Nov-1989). AVAILABLE ONLY FOR
    TOPOCENTRIC SITES WITH THE TARGET ABOVE THE HORIZON. Unitless.

    APmag S-brt =
    Moon’s approximate apparent visual magnitude & surface brightness. When
    phase angle Target->Observer angle; the interior vertex angle at
    target center formed by a vector to the apparent center of the Sun at
    reflection time on the target and the apparent vector to the observer at
    print-time. Slightly different from true PHASE ANGLE (requestable separately)
    at the few arcsecond level in that it includes stellar aberration on the
    down-leg from target to observer. Units: DEGREES

    T-O-M/Illu% =
    Target-Observer-Moon/Illuminated percentage. The apparent lunar elongation
    angle between target body CENTER and the Moon’s CENTER, seen from the observing
    site, along with fraction of the lunar disk illuminated by the Sun. A negative
    lunar elongation angle indicates the target center is behind the Moon.
    Units: DEGREES & PERCENT.

    O-P-T =
    Observer-Primary-Target angle; apparent angle between a target satellite,
    its primary’s center and an observer, at observing location, at print time.
    Units: DEGREES

    PsAng PsAMV =
    The position angles of the extended Sun->target radius vector (“PsAng”)
    and the negative of the target’s heliocentric velocity vector (“PsAMV”),
    as seen in the observer’s plane-of-sky, measured CCW (east) from reference
    frame North Celestial Pole. Computed for small-bodies only (and primarily
    intended for ACTIVE COMETS), “PsAng” is an indicator of the comet’s gas-tail
    orientation in the sky (being in the anti-sunward direction) while “PsAMV”
    is an indicator of dust-tail orientation.
    Units: DEGREES.

    PlAng =
    Angle between observer and target orbital plane, measured from center
    of target at the moment light seen at observation time leaves the target.
    Positive values indicate observer is above the object’s orbital plane, in
    the direction of reference frame +z axis. Small-bodies only. Units: DEGREES.

    Cnst =
    Constellation ID; the 3-letter abbreviation for the name of the
    constellation containing the target center’s astrometric position,
    as defined by IAU (1930) boundary delineation. See documentation
    for list of abbreviations.

    CT-UT =
    Difference between uniform Coordinate Time scale and Earth-rotation
    dependent Universal Time. Prior to 1962, the difference is with respect
    to UT1 (CT-UT1). For 1962 and later, the delta is with respect to UTC
    (CT-UTC). Values beyond the next July or January 1st may change if a
    leap-second is introduced. Units: SECONDS

    ObsEcLon ObsEcLat =
    Observer-centered Earth ecliptic-of-date longitude and latitude of the
    target center’s apparent position, adjusted for light-time, the gravitational
    deflection of light and stellar aberration. Although centered on the observer,
    the values are expressed relative to coordinate basis directions defined by
    the Earth’s true equator-plane, equinox direction, and mean ecliptic plane at
    print time. Units: DEGREES

    N.Pole-RA N.Pole-DC
    ICRF/J2000.0 Right Ascension and Declination (IAU2009 rotation model)
    of target body’s North Pole direction at the time light left the body to
    be observed at print time. Units: DEGREES

    GlxLon GlxLat =
    Observer-centered Galactic System II (post WW II) longitude and latitude
    of the target center’s apparent position. Adjusted for light-time,
    gravitational deflection of light, and stellar aberration. Units: DEG DEG

    L_Ap_SOL_Time =
    Local Apparent SOLAR Time at observing site. This is the time indicated by
    a sundial. TOPOCENTRIC ONLY. Units: HH.fffffffffff (decimal angular hours)

    399_ins_LT =
    Instantaneous light-time of the station with respect to Earth center at
    print-time. The geometric (or “true”) separation of site and Earth center,
    divided by the speed of light. Units: MINUTES

    RA_3sigma DEC_3sigma =
    Uncertainty in Right-Ascension and Declination. Output values are the formal
    +/- 3 standard-deviations (sigmas) around nominal position. Units: ARCSECONDS

    SMAA_3sig SMIA_3sig Theta Area_3sig =
    Plane-of-sky (POS) error ellipse data. These quantities summarize the
    target’s 3-dimensional 3-standard-deviation formal uncertainty volume projected
    into a reference plane perpendicular to the observer’s line-of-sight.

    SMAA_3sig = Angular width of the 3-sigma error ellipse semi-major
    axis in POS. Units: ARCSECONDS.

    SMIA_3sig = Angular width of the 3-sigma error ellipse semi-minor
    axis in POS. Units: ARCSECONDS.

    Theta = Orientation angle of the error ellipse in POS; the
    clockwise angle from the direction of increasing RA to
    the semi-major axis of the error ellipse, in the
    direction of increasing DEC. Units: DEGREES.

    Area_3sig = Area of sky enclosed by the 3-sigma error ellipse.
    Units: ARCSECONDS ^ 2.

    POS_3sigma =
    The Root-Sum-of-Squares (RSS) of the 3-standard deviation plane-of-sky error
    ellipse major and minor axes. This single pointing uncertainty number gives an
    angular distance (a circular radius) from the target’s nominal position in the
    sky that encompasses the error-ellipse. Units: ARCSECONDS.

    RNG_3sigma RNGRT_3sig =
    Range and range rate (radial velocity) formal 3-standard-deviation
    uncertainties. Units: KM, KM/S

    DOP_S-sig DOP_X-sig RT_delay-sig =
    Doppler radar uncertainties at S-band (2380 MHz) and X-band (8560 MHz)
    frequencies, along with the round-trip (total) delay to first-order.
    Units: HERTZ and SECONDS

    Tru_Anom =
    Apparent true anomaly angle of the target’s heliocentric orbit position;
    the angle in the target’s instantaneous orbit plane from the orbital periapse
    direction to the target, measured positively in the direction of motion.
    The position of the target is taken to be at the moment light seen by the
    observer at print-time would have left the center of the object. That is,
    the heliocentric position of the target used to compute the true anomaly is
    one down-leg light-time prior to the print-time. Units: DEGREES

    L_ap_Hour_Ang =
    Local apparent HOUR ANGLE of target at observing site. The angle between the
    observer’s meridian plane, containing Earth’s axis of-date and local zenith
    direction, and a great circle passing through Earth’s axis-of-date and the
    target’s direction, measured westward from the zenith meridian to target
    meridian along the equator. Negative values are angular times UNTIL transit.
    Positive values are angular times SINCE transit. Exactly 24_hrs/360_degrees.
    EARTH TOPOCENTRIC ONLY. Units: sHH.fffffffff (decimal angular hours)

    phi PAB-LON PAB-LAT =
    “phi” is the true PHASE ANGLE at the observer’s location at print time.
    PAB-LON and “PAB-LAT” are the J2000 ecliptic longitude and latitude of the
    phase angle bisector direction; the outward directed angle bisecting the arc
    created by the apparent vector from Sun to target center and the astrometric
    vector from observer to target center. For an otherwise uniform ellipsoid, the
    time when its long-axis is perpendicular to the PAB direction approximately
    corresponds to lightcurve maximum (or maximum brightness) of the body. PAB is
    discussed in Harris et al., Icarus 57, 251-258 (1984).

    Units: DEGREES, DEGREES, DEGREES, DEGREES

    Computations by …
    Solar System Dynamics Group, Horizons On-Line Ephemeris System
    4800 Oak Grove Drive, Jet Propulsion Laboratory
    Pasadena, CA 91109 USA
    Information: http://ssd.jpl.nasa.gov/
    Connect : telnet://ssd.jpl.nasa.gov:6775 (via browser)
    telnet ssd.jpl.nasa.gov 6775 (via command-line)
    Author : Jon.Giorgini@jpl.nasa.gov

    *******************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************
    =

  27. Paul Vaughan says:

    Typo Alert:
    Paul Vaughan (December 7, 2014 at 2:57 am) wrote:
    “There are reversals of the 34 wave with each 121.5 wave.”

    Should read:
    “There are reversals of the 30 wave with each 121.5 wave.” (which is why it’s 243/34, not 121.5/30)

    None of this is going to mean much to anyone who’s not bothering to look independently at graphs of all of the envelopes for all of the variables for all of the reference plane / ephemeris type combos outlined above. I trust that at least Ian will be doing so and Ian’s participation alone is enough to make the exercise worthwhile (so long as distortion artistry isn’t allowed to creep into the discussion).

    Important:
    Ian, please acknowledge:
    a) ~27 EC, QR, & AD
    b) 6.4 vs. 18.6 Z = function of reference plane
    c) RG 243/~34 (which some may initially perceive as 30-envelopes reversing envelope-phase every 121.5)

    I’ll proceed with algebra derivations & complexity-reducing illustrations after acknowledgement.

    Regards

  28. Ian Wilson says:

    Here is the meaning of the Central Limit Theorem:

    Taking a sample that contains a large number of observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic average of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the computed values of the average will be distributed according to the normal distribution (commonly known as a “bell curve”).

  29. Ian Wilson says:

    Paul,

    I can confirm the following:

    A. 27.0 year envelop for EC

    B. 27.0 year envelop for apoapsis (AD) although the mean (as opposed to the envelop) has a ~ 34.0 year period which realigns with the 27.0 envelop roughly every 3.5 x 34.0 year cycles (since 3.5 x 34.0 =119 years is almost the same as 4.5 x 27.0 years = 121.5 years).

    C. 27.0 year envelop for periapsis (QR) – although there appear to slippages of 1/2 x 27 years every now and then.

    D. Change in the envelop period from 6.4 years to 18.6 years if you change from Ecliptic and mean equinox reference frame to Earth mean equator and equinox reference frame [=FRAME]

    E. I can clearly see the 243 year envelop in the lunar distance RG but I DO NOT see any 34 year repeat cycle – If there is one it is either 30 or 31 years.

    I will have to believe you on recipe number 5 as I have spend a couple of hours on this so far.

  30. Paul Vaughan says:

    It’s not 27.0 & 34.0.
    It’s only 27 & 34 when rounded off.
    I gave the more precise (not rounded off) values in the previous thread.

    As for (e):
    With 30 there are lobe phase-reversals every 121.5.
    34 is the average spacing of the large lower-lobes.
    Be sure to check solstice/equinox physical sampling effect on lobe phase.
    34 & 30 are rounded-off values.
    I gave the more precise (not rounded off) values in the previous thread.

    We’re making sufficient progress that I’ll start organizing illustrations and derivations.
    Initially I won’t try to illustrate everything. I’ll start with a few examples.

    Much later on a complete taxonomy could be tabulated in exhaustive comparative detail with algebra, derivations, & illustrations, but I’m not sure I’ll be the one volunteering for the tedium of organizing the presentation, as I already have the awareness and there are other demands on my time. What I know though is that if such a tabulation existed it would ease by orders of magnitude an introduction to lunar cycles. Oldschool methods of visualizing event series are dreadfully cumbersome. Superior visualizations are needed to expedite introduction. Oldschool lunar cycle summary methods misled me to believe that lunar cycles are orders of magnitude more complex than they actually are.

    Eventually we’ll get to Venus-Earth frequency algebra for comparison. A familiar issue (one OB is familiar with) arises.

  31. Ian Wilson says:

    Paul,

    Sorry Paul, I overstated my precession when I posted 27.0 and 24.0, I should have written ~ 27 years and ~ 34 years.

  32. tallbloke says:

    An aside, but I’ll put it here so I don’t lose it.

    Metonic cycle x Venus transit cycle = ~ ‘the grand synod’ of 4628 years (=27U-N =233J-S)
    i.e. 19 x 243 = 4617 years. One solar cycle short…

  33. Paul Vaughan says:

    Caution: There’s a slip cycle on longer timescales:
    1/2530.131668 years = 13A/2+5Y-148T

    (Keep in mind that DE430 covers only years 1550 to 2650.)

  34. Paul Vaughan says:

    I have illustrations prepared, but not time to post them….
    will do so later….

  35. oldbrew says:

    As there are several mentions of number 34 in this thread I’ll venture some lunar data:

    11186 synodic months (34 x 329) =
    12139 draconic months (34 x 357, -1) =
    953 draconic years (34 x 28, +1)
    [12139 – 11186 = 953]

    Time period is 330329 days = 55 x 21 x 13 x 22, -1

    Lunar data from here: http://eclipse.gsfc.nasa.gov/SEsaros/SEsaros.html

  36. Ian Wilson says:

    Paul,

    Offline for a couple of days as well. I am looking forward to seeing the next stage.

  37. oldbrew says:

    ‘This study identifies the strongest Perigean spring tides that reoccur at roughly the same time in the seasonal calendar and shows how their repetition pattern, with respect to the tropical year, is in near-resonance with the 243 year transit cycle of Venus.’

    243 years x 33 = 8450 draconic years
    8450 = 13² x 5² x 2

  38. Paul Vaughan says:

    I’m changing course as I’m noticing new things.

    Reminder:
    I cautioned everyone that calendar month sampling of JEV would alias. At the time I wasn’t clear on what it was aliasing, but I determined for sure that it was aliasing by reproducing my earlier (late 2009) JEV investigation with daily Horizons output. In the context of the current discussion, consider this with 20/20 hindsight:

    (Tip: Chandler wobble phase reversals)

  39. Paul Vaughan says:

    In case anyone doesn’t get what happened there: The wavelet detected the Venus transit cycle in the JEV aliasing pattern. No one recognized the pattern back whenever I originally shared the graph (or if they did (I can’t remember) none of us clued in then). The envelope goes undetected with dense sampling. That’s one of the really useful features of multi-grain multi-extent wavelets: Due to their 2-dimensional tuning capacity they can work magic with envelopes even if data are severely noisy. New status of old mysterious result: resolved, no longer even slightly mysterious.

    There’s a clear reversal in the annual lunisolar aliasing at the same timescale — and it’s perfectly synchronized with the Chandler wobble phase reversal near ~1930. Illustrations when time permits.

    I’ve redoubled the checks on 34. It’s definitely 34.

  40. Paul Vaughan says:

    Community Alert: Don’t trust NASA JPL Horizons lunisolar polynomial integration before 1550.

  41. tallbloke says:

    Hi Paul,
    as well as the 243yr transit cycle period there is the 239.8yr 1/5 V-E synodic cycle precession.
    Is it possible to definitively deduce which is represented in the envelope you’ve produced?

  42. oldbrew says:

    Re the J-E-V setup and 34:

    170 J-V (34 x 5) = 101 J-E (34 x 3, -1) = 69 V-E (34 x 2, +1) in 110.304~ years.

  43. Ulric Lyons says:

    This is the series for Perigee full Moon at close to the same calendar date, in years:

    62
    106
    168
    212
    239
    274
    345
    372
    407
    451
    478

    Where Ian’s 137 is 274/2, i.e at new Moon.

    478 = 243 + 235.

  44. Ulric Lyons says:

    Interestingly at 372 years there are 392 eclipse years, while at a further 19 years at 391 years, there are 372 Lunar years (of 13 Lunations), and a closer fit to the eclipse years.

  45. TLMango says:

    Ian,
    Excellent paper!!!
    You are gradually solidifying the lunar/ocean climate connection.

    Here are some observations regarding the 31/62 year cycles:

    J = 11.862242 :::::::: S=29.457784
    J(5) S(2) / (J(5) – S(2)) = 1472.018317(6) ‘ Bond cycle
    J(14) S(5) / (J(14) – S(5)) = 18.6043369(70) ‘ Lunar cycle
    S(25) J(25) / (S(25) – J(25)) = 62.06035075(8) ‘ Lunar cycle

    62.06035075(8) 18.6043369(25) / (62.06035075(8) – 18.6043369(25) = 1472.018317(5)

    Keep up the good fight.

  46. Ulric Lyons says:

    It looks like perigee spring tides, and eclipse years, synchronise with the tropical year well at 239 and 242 years respectively. 242*239 / 12.5 = 4627.04.

  47. Paul Vaughan says:

  48. Paul Vaughan says:

  49. Paul Vaughan says:

    Zooming in on the blue curve from the last figure:

  50. Paul Vaughan says:

    When the 243.107457826 year envelope allows lobe elongation, spacing stabilizes at 31 years, but the longer-term central limit is 34.25937839287 years. More details another day.

  51. Paul Vaughan says:

    I made a misleading comment above (about JEV aliasing). I’ll need to re-address JEV ~= J+N & JEV-J ~= N when time permits. The J+N & N frequency algebra has more than 1 physical interpretation. It’s about inclination & eccentricity (nodes & perihelion/aphelion). The risk of misinterpretation is high. (No time to handle this topic properly today…)

  52. Ulric Lyons says:

    Paul, on first chart above, surely 478 should be below 482?

  53. oldbrew says:

    Linking tropical years (TY) to full moon cycles (FMC) to lunar apo/perigee (~8.85y):

    1973 TY = 1750 FMC = 223 perigee (1973 – 1750 = 223)
    1973 / 223 = 8.847533632 TY

    The number of FMC in 8.847533632 TY is one less i.e.:
    1750 / 223 = 7.847533632
    ————
    There could be a link to Saros cycles here because 7 x 1973 TY is almost 766 Saros.
    Note 223 is also a Saros number i.e. 223 synodic months = 1 Saros.
    Confirmed: 7 Perigee = 766 synodic months.
    Therefore 766 x 223 SM = 766 Saros = 7 x 1973 TY (13811) = 7 x 1750 FMC (12250)

  54. Paul Vaughan says:

    Ulric Lyons (December 9, 2014 at 6:56 pm) asked:
    “Paul, on first chart above, surely 478 should be below 482?”

    According to some other aggregation criteria perhaps.
    I’ll be spelling out the aggregation criteria used when time permits (probably tomorrow).
    Meanwhile if you have some other aggregation criteria in mind and you spell them out in technical detail, I can easily explore them. I’m exploring a wide variety of aggregation criteria. I’ve been hoping to eventually find time to share some examples for comparison & contrast.

  55. oldbrew says:

    Re earlier comment: December 9, 2014 at 7:02 pm

    That period is also 2079 (21 x 99) Draconic years:
    1973 Tropical Years = 1750 Full Moon Cycles = 223 perigee = 2079 Draconic years

    2079 DY – 1973 TY = 106
    1973 / 106 = 18.613207 tropical years = lunar nodal precession
    (2079 / 106 = 19.613207 draconic years = TY + 1)

  56. Ulric Lyons says:

    Paul says “According to some other aggregation criteria perhaps.”

    The fit is better with 478 so it should be lower, as 239 is lower than 243.

  57. oldbrew says:

    Another way of looking at it:
    31 Venus transit cycles = 31 x 243y = 93y x 81 = 98 Draconic years x 81

    A lunar standstill occurs every 18.6 years so five every 93 years.
    So 31 VTC = 405 lunar standstills

    http://en.wikipedia.org/wiki/Lunar_standstill

  58. Paul Vaughan says:

    Ulric Lyons (December 10, 2014 at 12:57 pm) wrote:
    “The fit is better with 478 so it should be lower, as 239 is lower than 243.”

    For what aggregation criteria?

    We are discussing dozens of variables output from NASA Horizons.
    They don’t all get their differing patterns from the same aggregation criteria.

  59. Ulric Lyons says:

    Perigee-Syzygy Solar Year alignment of course Paul.

  60. Paul Vaughan says:

    (29.530589) / 2
    = 14.7652945 solar days

    (27.55455) / 2
    = 13.777275 solar days

    (27.55455)*(14.7652945) / (27.55455 – 14.7652945)
    = 31.81194133 solar days

    (31.81194133)*(13.777275) / (31.81194133 – 13.777275)
    = 24.30218869 solar days

    nearest tropical year harmonic:
    (365.242189) / 15
    = 24.34947927 solar days

    (24.34947927)*(24.30218869) / (24.34947927 – 24.30218869)
    = 12512.97036 solar days

    (12512.97036) / 365.242189
    = 34.25937839 tropical years

  61. Paul Vaughan says:

    3A-2Y = 1 / 24.30218869 solar days
    3A-2Y-15T = 1 / 34.25937839 tropical years

  62. Paul Vaughan says:

    (27.55455)*(14.7652945) / (27.55455 + 14.7652945)
    = 9.613717876 solar days

    nearest tropical year harmonic:
    (365.242189) / 38
    = 9.611636553 solar days

    (9.613717876)*(9.611636553) / (9.613717876 – 9.611636553)
    = 44396.55003 solar days

    (44396.55003) / 365.242189
    = 121.5537289 tropical years

    2Y+A = 1 / 9.613717876 solar days
    38T-(2Y+A) = 38T-2Y-A = 1 / 121.5537289 tropical years

    – – – –

    harmonic mean:
    (27.55455)*(14.7652945) / ( (27.55455 + 14.7652945) / 2 )
    = 19.22743575 solar days

    nearest tropical year harmonic:
    (365.242189) / 19
    = 19.22327311 solar days

    (19.22743575)*(19.22327311) / (19.22743575 – 19.22327311)
    = 88793.10006 solar days

    (88793.10006) / 365.242189
    = 243.1074578 tropical years

    Y+A/2 = 1 / 19.22743575 solar days
    19T-(Y+A/2) = 19T-Y-A/2 = 1 / 243.1074578 tropical years

  63. oldbrew says:

    PV says: (12512.97036) / 365.242189 = 34.25937839 tropical years

    361 Draconic years = 125129.84 days

    PV: (88793.10006) / 365.242189 = 243.1074578 tropical years

    6 x 88793.1 = 532758.6
    1537 DY = 532755.06 days

  64. Ian Wilson says:

    Paul,

    Community Alert: Don’t trust NASA JPL Horizons lunisolar polynomial integration before 1550.

    That’s why I verified my calculations in my paper with those of Ray and Cartwright. They used DE406 Ephemeris. The times and dates for the peak tidal events agreed to within less than 10 minutes over the period from 1 to 3000 A.D.

  65. Paul Vaughan says:

    Folkner, W.M.; Williams, J.G.; Boggs, D.H.; Park, R.S.; & Kuchynka, P. (2014). The Planetary and Lunar Ephemerides DE430 and DE431. JPL Interplanetary Network Progress Report 42-196.
    http://ipnpr.jpl.nasa.gov/progress_report/42-196/196C.pdf

  66. Paul Vaughan says:

    Take care to differentiate conceptually.

    Envelope stats are aggregate measures.
    They summarize the central limits of event clusters.

  67. Paul Vaughan says:

    Everyone should realize that regularly slipping 31 year events & 34.25937839287 year event envelopes are not contradictory. Events and event clusters differ conceptually, as do their stats. There have been a lot of unnecessary misunderstandings / misinterpretations / misrepresentations stemming from conceptual ignorance of comparative definitions. The community needs to do better.

  68. Paul Vaughan says:

    Apogee, perigee, & eccentricity annual aliasing envelopes are shaped by:

    1.
    apogee versus perigee extremes every
    (27.55455) / 2
    = 13.777275 solar days

    2.
    new or full moon
    versus
    neither new nor full moon
    hitting extremes every
    (29.530589) / 4
    = 7.38264725 solar days

    beat of 1 & 2:
    (13.777275)*(7.38264725) / (13.777275 – 7.38264725)
    = 15.90597066 solar days

    nearest tropical year harmonic determines annual aliasing & integration:
    (365.242189) / 23
    = 15.88009517 solar days

    (15.90597066)*(15.88009517) / (15.90597066 – 15.88009517)
    = 9761.683295 solar days

    (9761.683295) / 365.242189
    = 26.72660385 tropical years

    4Y-2A = 1 / 15.90597066 solar days
    23T-(4Y-2A) = 23T-4Y+2A = 1 / 26.72660385 tropical years

  69. Paul Vaughan says:

    beat:
    (34.25937839)*(26.72660385) / (34.25937839 – 26.72660385)
    = 121.5537289 tropical years

    harmonic mean:
    (34.25937839)*(26.72660385) / ( (34.25937839 + 26.72660385) / 2 )
    = 30.02778018 tropical years

    resultant lobe alternation cycle:

    (30.02778018)*(26.72660385) / (30.02778018 – 26.72660385)
    = 243.1074578 tropical years

    (34.25937839)*(30.02778018) / (34.25937839 – 30.02778018)
    = 243.1074578 tropical years

  70. Ian Wilson says:

    DE406 is a long ephemeris that approximately covering the years from -3000 to +3000 and it is not available on JPL’s website.

    The DE406 ephemeris was released with DE405 in 1998. As a long ephemeris, it was the condensed version of DE405, covering 3000 BC to AD 3000 with the same limitations as DE404.

    In other words, DE406 is older version of DE431 and not DE430, and so it designed to cover the period from 3000 BC to 3000 AD.

    Some comments on ephemeris accuracy for those who might be interested.

    Every few years, JPL has the habit of releasing “accurate” ephemeris covering the period period from about 1580-90 A.D. through to maybe 2500 A.D. This immediately followed by a longer version that uses a less accurate ephemeris to cover a more extended period e.g. from -3000 to 3000 CE. (Note:
    “less accuracy” may mean that lunar libration and nutation may be left out reduce the computation load, with little or no effect upon the precise location of the Moon.)

    Since 1995 ( i.e. DE403) , the JPL ephemeris have used higher-accuracy radar-ranging of the planets, radio-ranging of spacecraft, and very-long-baseline-interferometric (VLBI) observations of spacecraft, especially for the four inner planets. Telescopic observations remained important for the outer planets because of their distance, hence the inability to bounce radar off of them, and the difficulty of parking a spacecraft near them. Other refinements that have been included since DE403 have included, better values of the planets’ masses and improved lunar laser ranging accuracy, giving better positions of the Moon.

    If you think about it for a second you realize that, since 1995, there have not been too many more closer/improved observations of factors that could genuinely affect the Moon’s predicted position (e.g. Jupiter and Saturn’s orbital positions and masses) other than those from the tracking of the Galileo and Cassini probes and maybe some small technological improvements through the establishment of the GPS system post-1995.

    Indeed, despite all the bells and whistles that have been added in subsequent ephemeris since 1995, the biggest source of uncertainty in predicting the orbital position of the Moon (backwards or forwards in time) is caused by the adopted models for the the tidal interactions between the Earth and the Moon.

    The method of projection relies upon the specific tidal interaction model that is adopted to generate a set of functions that are used to determine the Moon’s position for those epochs were observations are not available. Of course, the inherent error in the ability of these models to accurately predict the Moon’s position, slowly accumulates over time, limiting their relative accuracy to a specific time period centered on the present.

  71. Ian Wilson says:

    Ulric Lyons said:
    December 7, 2014 at 9:19 pm

    Thanks Ulric. It looks very interesting – I will save it for a future date.

  72. Ian Wilson says:

    oldbrew said:
    December 10, 2014 at 1:19 pm

    Thank you oldbrew – that is a little gem.

  73. Ulric Lyons says:

    I posted this earlier but it has been overlooked. Venus transit intervals are occasionally 235 years. Over this range that happens twice:
    eclipse.gsfc.nasa.gov/transit/catalog/VenusCatalog.html
    So multiples of 19 or 31 transit cycles do not occur.

  74. Ian Wilson says:

    Paul Vaughan said:
    December 11, 2014 at 8:39 am

    Your 34 year period comes about because the beat period between the 1.5 x synodic month and 1 x anomalistic month is almost exactly 1/5th of a tropical year.

    Hence, if you beat the beat period between the 1.5 x synodic month and 1 x anomalistic month with the nearest sub-multiple (1/5th) of a tropical year you get 102.77813518 years = 3 x 34.25937839 tropical years.

    In order for this to make physical sense, you have to explain why you need to beat 1.5 synodic months with 1 anomalistic month? What physical meaning does it have?

  75. Paul Vaughan says:

    Unfortunately something’s going more than a little wrong here.

    Ian Wilson (December 12, 2014 at 2:38 am) wrote:
    “Your 34 year period […]”
    “In order for this to make physical sense […]”
    “What physical meaning does it have?”

    It’s not “mine”.
    We’re discussing
    “RG Range; distance from coordinate center (AU)”
    from NASA Horizons.

    Ian Wilson (December 12, 2014 at 2:38 am) wrote:
    “[…] comes about because the beat period between the 1.5 x synodic month and 1 x anomalistic month is almost exactly 1/5th of a tropical year. Hence, if you beat the beat period between the 1.5 x synodic month and 1 x anomalistic month with the nearest sub-multiple (1/5th) of a tropical year you get 102.77813518 years = 3 x 34.25937839 tropical years. […] , you have to explain why you need to beat 1.5 synodic months with 1 anomalistic month?”

    We’re discussing the intersection of simple straight lines, the realm of mathematical proof.

    Please review the simple geometric proof I gave above.

    Ian: In the last thread you admitted the 31 year pattern slips, but there was something you still found “counterintuitive”. It clarified exactly where your thinking about the 34 year envelope is going off-track.

    Sincerely

  76. Paul Vaughan says:

    Paul Vaughan (December 8, 2014 at 4:11 pm) wrote:
    “Community Alert: Don’t trust NASA JPL Horizons lunisolar polynomial integration before 1550.”

    Ian: Tight synchronization of 2 models before 1550 doesn’t change this.

  77. Chaeremon says:

    @oldbrew (December 12, 2014 at 10:13 am):

    1973 TY is an eclipse cycle (80 Inex -19 Saros; ~3 members according to R.H. van Gent) with, in the middle opposite syzygy eclipse (and many have opposite apse in the middle); checked 12086 pairs (7ka time-frame) and found that 7542 match😎 But it’s clear that draconic runs out of steam😦

    Now, when comparing all syzygies (not just eclipses) it’s clear that 1973 TY endpoints and middle syzygy can have virtually same apsidal distance (near ~average) and therefore not perigee😦

  78. oldbrew says:

    1973 TY = 720622.84 days
    2079 DY = 720623.14 days
    1750 full moon cycles = 720622.57 days

    0.3 of a day difference (DY-TY) in nearly 2000 years is a problem?

    720623 / 6798.331* = 106 = 2079 – 1973
    (* = the period of a complete revolution of the Moon’s ascending node around the ecliptic: 18.612815932 Julian years (6798.331019 days; at the epoch J2000.0):

    http://en.wikipedia.org/wiki/Year#Astronomical_years

    1973 TY x 7 = 766 Saros
    By definition that’s 766 x 223 synodic months: number of days matches 7 x 1973 TY.
    1973 – 1750 = 223 perigee (1973 / 223 = 8.8475336 TY)

    Speculation: 1973 TY / 329 = 5.99696 TY
    ‘One last major variation is the rotation of the long axis of the ellipse within the wobbling orbital plane. This takes 8.85 years to travel around the earth, or 6 years relative to the wobbling plane.’

    http://www.cyclesresearchinstitute.org/cycles-astronomy/lunar.shtml

    2079 – 1750 = 329 (= 106 + 223)

    Also, re ‘1973 TY x 7 = 766 Saros’: 7 perigee = 766 synodic months

  79. Chaeremon says:

    @oldbrew: Saros series is [by definition] from 69 to 87 eclipses, not 766 eclipses, because draconic runs out of steam, simple that😉

    B.t.w. I’m working [OT] on connecting pairs of Saros series, which all have duo eclipses (same month) at both ends of the series, so that extended series are practically unlimited. But this implies that, between pairs of series there is one extra draconic month.

    IMHO after one Saros series you need to resync draconic … I think that mathematically 0.3 of a day difference does not help much (but please convince me with something observable).

  80. oldbrew says:

    @ Chaeremon : I’m just saying what the numbers add up to. No theory.

  81. Chaeremon says:

    @oldbrew: yes I know and appreciate, but let’s try to not push the observable limit😉

    Alas, in the above 1973 TY (odd number) equals 24403 synodic (odd number) but also equals 1750 FMC (even number). As a consequence of evenness, FMC cannot have opposite apside in the middle (but see what I reported from running numerical integration): contradiction that this interval is commensurate with FMC.

  82. oldbrew says:

    Earlier I said re 766 Saros:
    ‘By definition that’s 766 x 223 synodic months: number of days matches 7 x 1973 TY.’

    766 x 223 (170818) SM = 183068 anomalistic months
    183068 – 170818 = 12250

    That’s the number of full moon cycles in 7 x 1973 tropical years i.e. 7 x 1750 = 12250.

    Also if 766 Saros = 360 degrees re the full Moon then 1 Saros = 0.4699738 degrees:
    ‘after one Saros eclipse cycle of 18.031 years, the node moves about one half degree east in relation to the full Moon

    http://www.idialstars.com/osec.htm

  83. Ulric Lyons says:

    Ulric Lyons says:
    December 12, 2014 at 2:12 am

    typo, that should have been “235 years”. [amended – mod]

  84. Paul Vaughan says:

    Progress Review:
    So far we’ve covered less than 15% of what I hoped we’d cover in this workshop.

  85. Ian Wilson says:

    Paul said:

    Paul Vaughan (December 8, 2014 at 4:11 pm) wrote:
    “Community Alert: Don’t trust NASA JPL Horizons lunisolar polynomial integration before 1550.”

    Ian: Tight synchronization of 2 models before 1550 doesn’t change this.

    Paul: No, but there are no other ways to project the lunar orbit to earlier dates, so we are left with no other option. The projections are made with the best tidal models available. Every one knows that there are limitation to the models and estimations can be made of the build up of cumulative errors over time. It is just silly (and unscientific) to say that nothing can be trusted prior to 1550.

  86. Ian Wilson says:

    Oldbrew, Chaeremon and Ulrich,

    We need to give Paul a bit of space here. Your posts while fascinating are not helping Paul to outline his main results. Can I suggest that if you want to persist in putting up these posts that you ask Roger start a parallel thread. This is the fog that I warned everyone about.

    I am trying to follow what Paul is trying to say and it really not helping of all of you guys are heading off on a related but tangential topic.

  87. Ian Wilson says:

    Paul said:

    “We’re discussing the intersection of simple straight lines, the realm of mathematical proof.”

    OK, I can live with that. However, at some point [may be a bit further down the track] it has to be linked back to physical reality. It is quiet possible that we may not be able to link the 34 year pattern to a underlying physical principle at this stage, so I will hold off asking for this link.

  88. Ian Wilson says:

    Paul Vaughan said:
    December 11, 2014 at 5:32 pm

    This post above makes a lot of sense. Thanks

  89. Ian Wilson says:

    Paul,

    If a 243.1074578 year envelop amplitude modulates a 30.02778018 year pattern
    it will produce side lobes at 34.25937839 and 26.72660385 years.

    (243.1074578)*(30.02778018) / ( 243.1074578 – 30.02778018) = 34.25937839

    and

    (243.1074578)*(30.02778018) / ( 243.1074578 + 30.02778018) = 26.72660385

    Is this another way of stating what you mean?

  90. oldbrew says:

    IW: ‘It is quite possible that we may not be able to link the 34 year pattern to a underlying physical principle at this stage, so I will hold off asking for this link.’

    I did suggest one earlier: December 12, 2014 at 10:13 am

  91. Paul Vaughan says:

    Paul Vaughan (December 8, 2014 at 4:11 pm) wrote:
    “Community Alert: Don’t trust NASA JPL Horizons lunisolar polynomial integration before 1550.”

    Paul Vaughan (December 12, 2014 at 6:28 am ) wrote:
    “Tight synchronization of 2 models before 1550 doesn’t change this.”

    Ian Wilson (December 12, 2014 at 5:20 pm) wrote:
    “Paul: No, but there are no other ways to project the lunar orbit to earlier dates, so we are left with no other option. The projections are made with the best tidal models available. Every one knows that there are limitation to the models and estimations can be made of the build up of cumulative errors over time. It is just silly (and unscientific) to say that nothing can be trusted prior to 1550.”

    Ian, you’re consistently underestimating the extent of my awareness.
    Encumbering the discussion like this isn’t helpful.

    =
    Nov 07, 2009:
    — Version 3.35a
    Fixed a bug that caused no output when using calendrical output
    stepping to step by month into a non-existent range of date labels
    (i.e., the Gregorian calendar switch-over point in October 1582).
    =

    http://ssd.jpl.nasa.gov/?horizons_news

    Guess who alerted them of the problem? (Hint: Paul Vaughan)

    =
    GREGORIAN AND JULIAN CALENDAR DATES:

    Input calendar dates 1582-Oct-15 and after are taken to be expressed in the extended Gregorian calendar system. Prior dates are assumed to be in the Julian proleptic calendar.

    Historically, not all regions switched calendars at the same time (or even in the same century). Thus, the user must be aware of which calendar was in effect for a particular historical record. It should NOT be assumed this system’s calendar automatically correlates with a date from an arbitrary historical document.

    Here is the progression near the calendar switch point:

    Calendar Type Calendar Date Julian Day Number
    ————- ————- —————–
    Julian 1582-Oct-03 2299158.5
    Julian 1582-Oct-04 2299159.5 —>
    (skipped) “1582-Oct-05” 2299160.5 |
    (skipped) “1582-Oct-06” 2299151.5 |
    (skipped) “1582-Oct-07” 2299152.5 |
    (skipped) “1582-Oct-08” 2299153.5 |
    (skipped) “1582-Oct-09” 2299154.5 |
    (skipped) “1582-Oct-10” 2299155.5 |
    (skipped) “1582-Oct-11” 2299156.5 |
    (skipped) “1582-Oct-12” 2299157.5 |
    (skipped) “1582-Oct-13” 2299158.5 |
    (skipped) “1582-Oct-14” 2299159.5 |
    Gregorian 1582-Oct-15 2299160.5 <—
    Gregorian 1582-Oct-16 2299161.5
    Gregorian 1582-Oct-17 2299162.5

    Note that Julian (calendar) dates are different than (and unrelated to) Julian day numbers.

    Examination of this table shows that the date labels from Oct 5, 1582 through Oct 14, 1582 don't exist. Of course, the days themselves do, as is shown in the continuous Julian day number column; it's just a matter of what one calls them. If you specify a non-existent calendar date label that was "skipped", this program will automatically use a day number, as shown above, that maps into the previous Julian calendar system. For example, requesting a date of 1582-Oct-14 (skipped) is the same as requesting the Julian calendar date 1582-Oct-04.
    =

    http://ssd.jpl.nasa.gov/?horizons_doc

    So we canNOT sensibly explore approximations of tropical year annual lunisolar aliasing across the entire Horizons record using a sampling step size of 1 calendar year.

    Pre-1582 problems will be crystal clear to sensible, sober, competent data explorers doing careful diagnostics.

    – –

    Progress Update:
    No progress since last progress update since precious time & energy were hijacked, diverted, and drained by unnecessary peripheral distractions. Not impressed.

    We need to expeditiously raise the level of discussion above myriad turbulently meandering peripheral eddies that threaten to keep our explorations permanently encumbered.

  92. Paul Vaughan says:

    Ian Wilson (December 12, 2014 at 5:33 pm) wrote:
    “[…] at some point [may be a bit further down the track] it has to be linked back to physical reality. It is quiet possible that we may not be able to link the 34 year pattern to a underlying physical principle at this stage, so I will hold off asking for this link.”

    We’re discussing
    “RG Range; distance from coordinate center (AU)”
    from NASA Horizons.

  93. Paul Vaughan says:

    Ian Wilson (December 12, 2014 at 5:33 pm) wrote:
    “It is quiet possible that we may not be able to link the 34 year pattern to a underlying physical principle”

    Some kind of extreme misunderstanding / misinterpretation / misrepresentation at play here.

    https://tallbloke.wordpress.com/2014/11/29/paul-pukite-sloshing-model-for-enso/comment-page-1/#comment-94218

    We’re discussing
    “RG Range; distance from coordinate center (AU)”
    from NASA Horizons.

    This is getting a little creepy.

  94. Paul Vaughan says:

    Slip & alias are systematic.
    We’re in the realm of mathematical proof (mere intersection of simple straight lines).

    Unfortunately the discussion has been reduced to easing correction of counter intuitive misconceptions about 24.30218869 solar days:

    X_____________	Y_______	+beat____	-beat____	harmean
    27.55455	14.7652945	31.81194133	9.613717876	19.22743575
    27.55455	31.81194133	205.8922145	14.7652945	29.530589
    27.55455	205.8922145	31.81194133	24.30218869	48.60437737
    31.81194133	24.30218869	102.9461072	13.777275	27.55455
    31.81194133	13.777275	24.30218869	9.613717876	19.22743575
    

    Simple facts:
    a) 205.8922145 is NOT a subharmonic of 27.55455.
    b) 205.8922145 is NOT a subharmonic of 14.7652945.

    Nothing “counter intuitive” about it.

  95. tallbloke says:

    All,
    To reiterate what Ian requested above. Please sit back and let Ian and Paul work on the technical issues specific to Paul’s exposition. I have as Paul suggested started a parallel thread for tangential observations and ideas that Ian and Paul can visit as they find time.
    https://tallbloke.wordpress.com/2014/12/12/eclipses-moon-cycles-and-inner-solar-system-observations-open-thread/

    Thanks for your co-operation. I’ll restrict this thread to Paul and Ian’s interaction from here. Comments from others will be unapproved so their owners can copy and paste them over onto the open thread.

  96. Ian Wilson says:

    Paul said,

    “Ian, you’re consistently underestimating the extent of my awareness.
    Encumbering the discussion like this isn’t helpful.”

    “So we canNOT sensibly explore approximations of tropical year annual lunisolar aliasing across the entire Horizons record using a sampling step size of 1 calendar year.

    Pre-1582 problems will be crystal clear to sensible, sober, competent data explorers doing careful diagnostics.”

    I will put these childish insults aside for the good of the debate.

  97. Ian Wilson says:

    Paul said,

    Slip & alias are systematic.
    We’re in the realm of mathematical proof (mere intersection of simple straight lines).

    Reply: OK got it. Let’s move on.

  98. Ian Wilson says:

    Paul,

    Here is my part summary of what I think you are trying to say – please correct me if I am wrong:

    1. ~ 27 year apogee, perigee, & eccentricity annual aliasing envelopes:

    = 26.72660385 tropical years
    4Y-2A = 1 / 15.90597066 solar days
    23T-(4Y-2A) = 23T-4Y+2A = 1 / 26.72660385 tropical years

    2. ~ 34 year RG Range (distance from coordinate center (AU)) annual aliasing envelop

    = 34.25937839 tropical years
    3A-2Y = 1 / 24.30218869 solar days
    3A-2Y-15T = 1 / 34.25937839 tropical years

    AND

    ~ 121.5 and 243.0 years RG Range (distance from coordinate center (AU)) annual aliasing envelop

    = 121.5537289 tropical years
    2Y+A = 1 / 9.613717876 solar days
    38T-(2Y+A) = 38T-2Y-A = 1 / 121.5537289 tropical years

    = 243.1074578 tropical years
    Y+A/2 = 1 / 19.22743575 solar days
    19T-(Y+A/2) = 19T-Y-A/2 = 1 / 243.1074578 tropical years

    Hence:

    beat:
    (34.25937839)*(26.72660385) / (34.25937839 – 26.72660385)
    = 121.5537289 tropical years

    harmonic mean:
    (34.25937839)*(26.72660385) / ( (34.25937839 + 26.72660385) / 2 )
    = 30.02778018 tropical years

    resultant lobe alternation cycle:

    (30.02778018)*(26.72660385) / (30.02778018 – 26.72660385)
    = 243.1074578 tropical years

    (34.25937839)*(30.02778018) / (34.25937839 – 30.02778018)
    = 243.1074578 tropical years

    And now I understand what you mean by your emphasis of the following:

    “Simple facts:
    a) 205.8922145 is NOT a subharmonic of 27.55455.
    b) 205.8922145 is NOT a subharmonic of 14.7652945.”

    Crystal clear.

  99. Paul Vaughan says:

    Paul Vaughan (December 12, 2014 at 9:06 pm ) advised:
    So we canNOT sensibly explore approximations of tropical year annual lunisolar aliasing across the entire Horizons record using a sampling step size of 1 calendar year.”

    This is not a “childish insult”.
    It’s a crucial technical point.

    The pre-1582 Horizons lunisolar records failed every diagnostic I threw at them. This suggests to me that scholars are struggling to:

    a) fit historical observational records to models due to calendaring issues, such as:

    “Historically, not all regions switched calendars at the same time (or even in the same century). Thus, the user must be aware of which calendar was in effect for a particular historical record. It should NOT be assumed this system’s calendar automatically correlates with a date from an arbitrary historical document.”http://ssd.jpl.nasa.gov/?horizons_doc

    b) stitch pre-1582 models to post-1582 models.

    It may be the best thing on offer, but it’s not good enough. I’m not open to “debating” this. I can’t use for serious research models that fail every diagnostic I throw at them. I can however acknowledge that patient scholars have on their hands an epically tedious task. I respect any serious, sufficiently patient, successful efforts they can make moving forward to better use historical observations to improve these currently defective models. I don’t expect the work will be finished to my satisfaction during my lifetime.

  100. Paul Vaughan says:

    I knew and advised early on that the risk of misunderstanding / misinterpretation was very high. At several points during the discussion I restrained myself to escalating the grievance incrementally. I wasn’t sure we would reach this point, but I figured the table might get the job done:

    Ian Wilson (December 13, 2014 at 3:04 am) wrote:
    “Crystal clear.”

    This restores trust that temporarily broke. This clearly distinguishes Ian Wilson from the following people, who NEVER did admit their FAILURE TO ACKNOWLEDGE GEOMETRIC PROOFS:

    [list omitted in case someone might be inclined to sue me for calling them out so harshly]

    As time permits, I’m now willing to share more of the remaining 85% of the material I have in mind (and not necessarily on file…)

    There is one brand new finding (from today) that I think TB in particular will appreciate (hint: JSEV Mn), but I will definitely be running serious diagnostics on that one before sharing it.

  101. Paul Vaughan says:

    Will Janoschka (December 13, 2014 at 3:34 am) requested:
    “Can you guys not slow down a wee bit so “some others”, can distinguish between “interesting” and “bull shit”?

    Will, you have correctly identified the challenge. Another constraint on the problem is that if we go that slow, we may only make it 10% of the way to sufficient understanding because we opted to instead spend 90% of our time (remember that we all eventually die — and maybe become fatally encumbered with other responsibilities sooner) tied up explaining the first 10% of the path. It won’t be my choice, but maybe you can find other competent scouts to translate (without misunderstanding, misinterpreting, misrepresenting, and harassing) while I blaze exploration trails voluntarily. Of course an alternative is to become a serious employer so you can tailor demands on employees to (expensively) meet (hopefully but not assured) your wishes.

    What you can reliably expect from me: Operations devoutly mindful of the Pareto Principle.

    Best Regards

  102. Paul Vaughan says: December 13, 2014 at 4:27 am

    (Will Janoschka (December 13, 2014 at 3:34 am) requested:
    “Can you guys )not slow down a wee bit so “some others”, can distinguish between “interesting” and “bull shit”?)

    “What you can reliably expect from me: Operations devoutly mindful of the Pareto Principle.
    Best Regards”

    Thank you Paul. I can not expect, but I must wish and wonder! -will-

  103. Paul Vaughan says:

    This can be summarized in a neat table later but for now here it is unrefined to avoid delays:

    tropical year harmonic nearest 27.55455
    (365.242189) / 13 = 28.095553

    (28.095553)*(27.55455) / (28.095553 – 27.55455) = 1430.972323
    (1430.972323) / 365.242189 = 3.917872487

    tropical year harmonic nearest 14.7652945
    (365.242189) / 25 = 14.60968756

    (14.7652945)*(14.60968756) / (14.7652945 – 14.60968756) = 1386.289965
    (1386.289965) / 365.242189 = 3.79553624

    (27.55455)*(14.7652945) / ( (27.55455 + 14.7652945) / 2 ) = 19.22743575
    (19.22743575)*(14.7652945) / (19.22743575 – 14.7652945) = 63.62388265
    (27.55455)*(19.22743575) / (27.55455 – 19.22743575) = 63.62388265

    tropical year harmonic nearest 63.62388265
    (365.242189) / 6 = 60.87369817

    (63.62388265)*(60.87369817) / (63.62388265 – 60.87369817) = 1408.276808
    (1408.276808) / 365.242189 = 3.855734225

    (27.55455)*(14.7652945) / (27.55455 – 14.7652945) = 31.81194133

    tropical year harmonic nearest 31.81194133
    (365.242189) / 11 = 33.20383536

    (33.20383536)*(31.81194133) / (33.20383536 – 31.81194133) = 758.8785019
    (758.8785019) / 365.242189 = 2.077740537

    harmonic of 3.917872487 nearest 2.077740537
    (3.917872487) / 2 = 1.958936244

    (2.077740537)*(1.958936244) / (2.077740537 – 1.958936244) = 34.25937839

    (3.917872487)*(3.79553624) / (3.917872487 – 3.79553624) = 121.5537289
    (3.917872487)*(3.79553624) / (3.917872487 + 3.79553624) = 1.927867112
    (3.917872487)*(3.79553624) / ( (3.917872487 + 3.79553624) / 2 ) = 3.855734225

    harmonic of 3.855734225 nearest 1
    (3.855734225) / 4 = 0.963933556

    (1)*(0.963933556) / (1 – 0.963933556) = 26.72660385

    harmonic of 3.79553624 nearest 2.077740537
    (3.79553624) / 2 = 1.89776812

    (2.077740537)*(1.89776812) / (2.077740537 – 1.89776812) = 21.90930044

    (34.25937839)*(21.90930044) / ( (34.25937839 + 21.90930044) / 2 ) = 26.72660385
    (34.25937839)*(21.90930044) / (34.25937839 – 21.90930044) = 60.77686446
    2*(60.77686446) = 121.5537289

    (3.917872487)*(3.855734225) / (3.917872487 – 3.855734225) = 243.1074578
    (3.855734225)*(3.79553624) / (3.855734225 – 3.79553624) = 243.1074578

    apogee versus perigee extremes every
    (27.55455) / 2 = 13.777275 solar days
    new or full moon versus neither new nor full moon hits extremes every
    (29.530589) / 4 = 7.38264725 solar days

    (13.777275)*(7.38264725) / (13.777275 – 7.38264725) = 15.90597066
    (19.22743575)*(15.90597066) / (19.22743575 – 15.90597066) = 92.07714693

    tropical year harmonic nearest 92.07714693
    (365.242189) / 4 = 91.31054725

    (92.07714693)*(91.31054725) / (92.07714693 – 91.31054725) = 10967.41216
    (10967.41216) / 365.242189 = 30.02778018

    (3.917872487)*(2.077740537) / (3.917872487 – 2.077740537) = 4.42377107
    (3.79553624)*(2.077740537) / (3.79553624 – 2.077740537) = 4.590848313

    (4.590848313)*(4.42377107) / (4.590848313 – 4.42377107) = 121.5537289
    (4.590848313)*(4.42377107) / ( (4.590848313 + 4.42377107) / 2 ) = 4.505761384

    (2.077740537)*(1.927867112) / (2.077740537 – 1.927867112) = 26.72660385
    (2.077740537)*(1.927867112) / (2.077740537 + 1.927867112) = 1
    (2.077740537)*(1.927867112) / ( (2.077740537 + 1.927867112) / 2 ) = 2

  104. Paul Vaughan says:

    8V-13E
    V = Venus phase
    E = Earth phase

    http://ssd.jpl.nasa.gov/horizons.cgi
    = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
    = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

    Ephemeris Type [change] : VECTORS
    Target Body : Venus [299]
    Coordinate Origin : Sun (body center) [500@10]
    Time Span : Start=0001AD-06-21, Stop=3000-06-21, Step=1 Y
    Table Settings : labels=YES; CSV format=YES
    Display/Output : plain text

    *******************************************************************************
    Revised: Jul 31, 2013 Venus 299 / 2

    GEOPHYSICAL DATA (updated 2014-Mar-13):
    Mean radius (km) = 6051.8(4+-1) Density (g cm^-3) = 5.204
    Mass (10^23 kg ) = 48.685 Flattening, f =
    Volume (x10^10 km^3) = 92.843 Semi-major axis =
    Sidereal rot. period = -243.0185 d Rot. Rate (x10^5 s) = -0.029924
    Mean solar day = 116.7490 d Polar gravity ms^-2 =
    Mom. of Inertia = 0.33 Equ. gravity ms^-2 = 8.870
    Core radius (km) = ~3200 Potential Love # k2 = ~0.25

    Grav spectral fact u = 1.5 Topo. spectral fact t = 23
    Fig. offset (Rcf-Rcm) = 0.19+-01 Offset (lat./long.) = 11/102 dg/dg
    GM (km^3 s^-2) = 324858.63 Equatorial Radius, Re = 6051.893 km
    GM 1-sigma (km^3 s^-2)= +-0.04 Mass ratio (Sun/Venus)= 408523.72

    Atmos. pressure (bar) = 90 Max. angular diam. = 60.2″
    Mean Temperature (K) = 735 Visual mag. V(1,0) = -4.40
    Geometric albedo = 0.65 Obliquity to orbit = 177.3 deg
    Sidereal orb. per. = 0.61519726 y Orbit vel. km/s = 35.0214
    Sidereal orb. per. = 224.70079922 d Escape vel. km/s = 10.361
    Hill’s sphere rad. Rp = 167.1 Planetary Solar Const = 2613.9 (Wm^2)
    *******************************************************************************

    *******************************************************************************
    Ephemeris / WWW_USER Sun Dec 7 04:56:42 2014 Pasadena, USA / Horizons
    *******************************************************************************
    Target body name: Venus (299) {source: DE-0431LE-0431}
    Center body name: Sun (10) {source: DE-0431LE-0431}
    Center-site name: BODY CENTER
    *******************************************************************************
    Start time : A.D. 0001-Jun-21 00:00:00.0000 CT
    Stop time : A.D. 3000-Jun-21 00:00:00.0000 CT
    Step-size : 1 calendar years
    *******************************************************************************
    Center geodetic : 0.00000000,0.00000000,0.0000000 {E-lon(deg),Lat(deg),Alt(km)}
    Center cylindric: 0.00000000,0.00000000,0.0000000 {E-lon(deg),Dxy(km),Dz(km)}
    Center radii : 696000.0 x 696000.0 x 696000.0 k{Equator, meridian, pole}
    Output units : AU-D
    Output format : 03
    Reference frame : ICRF/J2000.0
    Output type : GEOMETRIC cartesian states
    Coordinate systm: Ecliptic and Mean Equinox of Reference Epoch
    *******************************************************************************
    JDCT , , X, Y, Z, VX, VY, VZ, LT, RG, RR,
    *******************************************************************************
    $$SOE
    $$EOE
    *******************************************************************************
    Coordinate system description:

    Ecliptic and Mean Equinox of Reference Epoch

    Reference epoch: J2000.0
    xy-plane: plane of the Earth’s orbit at the reference epoch
    x-axis : out along ascending node of instantaneous plane of the Earth’s
    orbit and the Earth’s mean equator at the reference epoch
    z-axis : perpendicular to the xy-plane in the directional (+ or -) sense
    of Earth’s north pole at the reference epoch.

    Symbol meaning [1 au=149597870.700 km, 1 day=86400.0 s]:

    JDCT Epoch Julian Date, Coordinate Time
    X x-component of position vector (AU)
    Y y-component of position vector (AU)
    Z z-component of position vector (AU)
    VX x-component of velocity vector (AU/day)
    VY y-component of velocity vector (AU/day)
    VZ z-component of velocity vector (AU/day)
    LT One-way down-leg Newtonian light-time (day)
    RG Range; distance from coordinate center (AU)
    RR Range-rate; radial velocity wrt coord. center (AU/day)

    Geometric states/elements have no aberration corrections applied.

    Computations by …
    Solar System Dynamics Group, Horizons On-Line Ephemeris System
    4800 Oak Grove Drive, Jet Propulsion Laboratory
    Pasadena, CA 91109 USA
    Information: http://ssd.jpl.nasa.gov/
    Connect : telnet://ssd.jpl.nasa.gov:6775 (via browser)
    telnet ssd.jpl.nasa.gov 6775 (via command-line)
    Author : Jon.Giorgini@jpl.nasa.gov
    *******************************************************************************

    = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
    = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

    Ephemeris Type : VECTORS
    Target Body : Earth-Moon Barycenter [EMB] [3]
    Coordinate Origin : Sun (body center) [500@10]
    Time Span : Start=0001AD-06-21, Stop=3000-06-21, Step=1 Y
    Table Settings : labels=YES; CSV format=YES
    Display/Output : plain text

    *******************************************************************************
    Revised: Jul 31, 2013 Earth Barycenter 3

    Dynamical point:
    —————
    The common point about which the Earth and Moon revolve (center of mass). This
    is approximately 4671 km from the center of the Earth, or about 3/4 of the way
    to the surface. See 399 for Earth center, or 301 for Moon center ephemeris.
    *******************************************************************************

    *******************************************************************************
    Ephemeris / WWW_USER Sun Dec 7 05:03:33 2014 Pasadena, USA / Horizons
    *******************************************************************************
    Target body name: Earth-Moon Barycenter (3) {source: DE-0431LE-0431}
    Center body name: Sun (10) {source: DE-0431LE-0431}
    Center-site name: BODY CENTER
    *******************************************************************************
    Start time : A.D. 0001-Jun-21 00:00:00.0000 CT
    Stop time : A.D. 3000-Jun-21 00:00:00.0000 CT
    Step-size : 1 calendar years
    *******************************************************************************
    Center geodetic : 0.00000000,0.00000000,0.0000000 {E-lon(deg),Lat(deg),Alt(km)}
    Center cylindric: 0.00000000,0.00000000,0.0000000 {E-lon(deg),Dxy(km),Dz(km)}
    Center radii : 696000.0 x 696000.0 x 696000.0 k{Equator, meridian, pole}
    Output units : AU-D
    Output format : 03
    Reference frame : ICRF/J2000.0
    Output type : GEOMETRIC cartesian states
    Coordinate systm: Ecliptic and Mean Equinox of Reference Epoch
    *******************************************************************************
    JDCT , , X, Y, Z, VX, VY, VZ, LT, RG, RR,
    *******************************************************************************
    $$SOE
    $$EOE
    *******************************************************************************
    Coordinate system description:

    Ecliptic and Mean Equinox of Reference Epoch

    Reference epoch: J2000.0
    xy-plane: plane of the Earth’s orbit at the reference epoch
    x-axis : out along ascending node of instantaneous plane of the Earth’s
    orbit and the Earth’s mean equator at the reference epoch
    z-axis : perpendicular to the xy-plane in the directional (+ or -) sense
    of Earth’s north pole at the reference epoch.

    Symbol meaning [1 au=149597870.700 km, 1 day=86400.0 s]:

    JDCT Epoch Julian Date, Coordinate Time
    X x-component of position vector (AU)
    Y y-component of position vector (AU)
    Z z-component of position vector (AU)
    VX x-component of velocity vector (AU/day)
    VY y-component of velocity vector (AU/day)
    VZ z-component of velocity vector (AU/day)
    LT One-way down-leg Newtonian light-time (day)
    RG Range; distance from coordinate center (AU)
    RR Range-rate; radial velocity wrt coord. center (AU/day)

    Geometric states/elements have no aberration corrections applied.

    Computations by …
    Solar System Dynamics Group, Horizons On-Line Ephemeris System
    4800 Oak Grove Drive, Jet Propulsion Laboratory
    Pasadena, CA 91109 USA
    Information: http://ssd.jpl.nasa.gov/
    Connect : telnet://ssd.jpl.nasa.gov:6775 (via browser)
    telnet ssd.jpl.nasa.gov 6775 (via command-line)
    Author : Jon.Giorgini@jpl.nasa.gov
    *******************************************************************************

    = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
    = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
    http://ssd.jpl.nasa.gov/horizons.cgi

  105. Ian Wilson says:

    Paul,

    I know that the post directly above is a proof but is it possible to get a hint as to which aggregation it is referring to? Thanks

  106. Ian Wilson says:

    Sorry,

    I was referring to: Paul Vaughan says:
    December 13, 2014 at 10:35 am

  107. Ian Wilson says:

    JPL Horizon vectors done – plotted z and Rg versus time for both Earth-Moon Barycentre and Venus (with respect to the Sun) to investigate envelopes.

    ~ 243 year envelop seen in z Venus

  108. Paul Vaughan says:

    Ian, that’s referring to multiple Horizons variables and aggregation criteria. I trust that those exploring independently will figure it out.

    Later if/when time permits I’ll introduce a tabulation (or at least a partial &/or sample tabulation). Ultimately someone with more free time could organize an exhaustive taxonomy complete with illustrations added to the tabulation. This would reduce by an order of magnitude the complexity of an introduction to lunar cycle alignments with the tropical year.

    I suggest everyone give at least 3.855734225 & 3.917872487 tropical years some carefully tedious attention. They define the most dominant highest frequency annual aliasing spacing & alternation framework and this is at the root of a brief note I left above. I suspect this is at the root of Chandler wobble phase reversals. It would be helpful if someone could find some historical proxies to identify past Cw phase reversals as we’ll be waiting an impractical amount of time (relative to a human lifetime) to gather new samples. In the yellow & grey illustrations I posted above, I hope everyone noticed how locally-extremal spacing goes something like 31, 31, 31 before slipping (and reversing phase on a 3.855734225 tropical year framework). The central limit of the skewed (asymmetric) lobe density is ~34. Aggregate (statistical) properties of event clusters vary with aggregation criteria. There’s no contradiction — just different aggregation criteria that answer different questions. I want to give everyone more opportunity to explore independently rather than spell this out any more clearly, as clearer spelling diminishes the first hand learning opportunity afforded to curious, motivated readers.

  109. Paul Vaughan says:

  110. Paul Vaughan says:

    10 days is a big chunk of a lunar cycle.

  111. Paul Vaughan says:

    Julian year – Gregorian year = 365.25 – 365.2425 = 0.0075

    Tabulating accumulated cycle fractions:

    years	0.0075	27.55455	14.7652945	365.242189
    1	0.0075	0.0002722	0.0005079	0.0000205
    100	0.75	0.0002722	0.0005079	0.0000205
    500	3.75	0.0272187	0.0507948	0.0020534
    1000	7.5	0.1360937	0.2539739	0.0102672
    1500	11.25	0.2721874	0.5079479	0.0205343
    

    After 1500 years the tropical year is out by only 2%, but lunar anomalistic month aliasing is more than a quarter-cycle out of phase and lunar new or full moon phasing has reversed.

    I think the simplest solution would be that Horizons be upgraded to facilitate more sampling scheme flexibility. For example, for climate research it would be helpful to be able to sample once per tropical year, twice per tropical year, once per QBO, etc. — i.e. user-defined input box rather than drop-down menu selection.

    To clarify there are really 3 issues with pre-1582 vs. post.
    1. The Gregorian Reform changed the length of the year.
    2. The Gregorian Reform inserted a 10 day jump in 1582.
    3. Lunisolar polynomial integration (this is actually continuous rather than discrete so there’s nothing special about 1582, but NASA technical literature expresses concerns about extrapolation outside of 1550-2650).

    I think Horizons could easily enough be upgraded to remove 1 & 2 from the list. There are of course workarounds — e.g. using Julian dates for the reasons astronomers use them, but convenient flexible sampling access for novices makes teaching easier. I’m thinking mostly of the climate discussion where functional numeracy is mostly low to non-existent, but I’m sure there would be value in other contexts such as university course assignments.

    Some of my above comments may have misled readers not distinguishing between issues 1 & 2 versus 3. 1 & 2 are not ephemeris issues but rather software design limitations (although this statement does require some qualification as models are fit to historical observational records that are subject to regionally varying calendaring issues). I wanted to make the record more accurate by leaving this note. Time is pressed, but if/when possible I do try to come back to clarify.

    This is certain: We can’t use the easy recipes I gave above for pre-1582 without doing corrections for systematic aliasing slip.

  112. Paul Vaughan says:

    When time permits I’ll demonstrate an extension of the aggregation criteria to demand perigee, eclipses, and new or full moon all aligned with the terrestrial year. The simple tools I’ve developed can handle any clearly defined aggregation criteria. (An inconvenience in discussions here has been that some commenters only partially define the aggregation criteria they use. For example they don’t specify the periods they use (along with sources) all in a single convenient list and additionally they leave ambiguity such as “days” (solar or sidereal?), “years” (tropical or sidereal?), etc. If they would volunteer more convenient specificity, efficient exploration of their perspectives would become easily feasible.) Other things I aim to share when time permits: 2. JSEV Mn 3. EV frequency algebra derivation 4. tabulation of lunisolar frequency algebra.

  113. tallbloke says:

    To tidy columns in tables of numbers, nest the ‘pre’ tag inside the ‘code’ tag. I’ve tidied two tables in comments above.

  114. Paul Vaughan says:

    Ian Wilson (December 13, 2014 at 4:06 pm) wrote:

    “JPL Horizon vectors done – plotted z and Rg versus time for both Earth-Moon Barycentre and Venus (with respect to the Sun) to investigate envelopes.

    ~ 243 year envelop seen in z Venus”

    It’s unfortunate that Horizons doesn’t provide the convenience of sampling in Venus-orbit time-steps….

    ….but you can do ATAN2() (arctangent function) on xy to estimate phase and use frequency algebra.

  115. Paul Vaughan says:

    Helpful tip — thanks Rog.
    Just noticed there’s an error in that table — here’s the correction:

    years	0.0075	27.55455	14.7652945	365.242189
    1	0.0075	0.0002722	0.0005079	0.0000205
    100	0.75	0.0272187	0.0507948	0.0020534
    500	3.75	0.1360937	0.2539739	0.0102672
    1000	7.5	0.2721874	0.5079479	0.0205343
    1500	11.25	0.4082810	0.7619218	0.0308015
    

    The comments I made apply to 1000 (not 1500).

    For 1500:
    anomalistic: ~40% (not quarter-cycle)
    full/new moon: ~3/4 (not 1/2)
    tropical year: 3% (not 2%)

    Trying to go too fast to save time for hiking/kayaking…. (certainly worth the trouble of occasional hindsight corrections)

  116. tallbloke says:

    Paul: “It’s unfortunate that Horizons doesn’t provide the convenience of sampling in Venus-orbit time-steps”

    It does offer the option of step sizes in minutes, which can get you pretty close.

  117. Ian Wilson says:

    Paul,

    In my paper (which is at the core of this thread), I do two things to correct for the transition form the Julian to the Gregorian calendar:

    1. I only use Gregorian dates prior to October 15th 1582 i.e. is I convert all dates prior to 1582 from Julian calender dates and times to Gregorian calender dates and times.

    2. I confirmed my calculation by using Julian days for each lunar tidal event. In this analysis, all calculations (i.e. both differences and integrations were solely done in Julian days).

  118. Paul Vaughan says:

    Seidelmann (1992) http://ssd.jpl.nasa.gov/?planet_phys_par

    (1.0000174)*(0.61519726) / (1.0000174 – 0.61519726) = 1.598689623 sidereal years

    harmonic of 1.598689623 nearest 1.0000174
    (1.598689623) / 2 = 0.799344811 sidereal years

    (1.0000174)*(0.799344811) / (1.0000174 – 0.799344811) = 3.98339766 sidereal years

    harmonic of 3.98339766 nearest 1.0000174
    (3.98339766) / 4 = 0.995849415 sidereal years

    (1.0000174)*(0.995849415) / (1.0000174 – 0.995849415) = 238.9324164 sidereal years

    Frequency Algebra

    V-E

    harmonic of V-E nearest E
    2(V-E)

    2(V-E)-E = 2V-2E-E = 2V-3E

    harmonic of 2V-3E nearest E
    4(2V-3E)

    4(2V-3E)-E = 8V-12E-E = 8V-13E

    J2000 1800AD-2050AD http://ssd.jpl.nasa.gov/txt/p_elem_t1.txt
    241.1339144 sidereal years

    J2000 3000BC-3000AD http://ssd.jpl.nasa.gov/txt/p_elem_t2.txt
    222.4944428 sidereal years

    Comments Ian?
    Do these models all fit historical observations or not?

  119. Paul Vaughan says:

    Ian Wilson (December 14, 2014 at 2:44 am) wrote:
    “I only use Gregorian dates prior to October 15th 1582 i.e. is I convert all dates prior to 1582 from Julian calender dates and times to Gregorian calender dates and times.”

    I read about that today:
    http://en.wikipedia.org/wiki/Proleptic_Gregorian_calendar

  120. Paul Vaughan says:

    tallbloke (December 13, 2014 at 7:27 pm)
    “It does offer the option of step sizes in minutes, which can get you pretty close.”

    Upon closer inspection of the menu there’s also “equal intervals (unitless)”. Next chance I get I’ll explore these options as potential work-arounds. Thanks for the note.

  121. Ian Wilson says:

    Paul,

    A preliminary comment first, you have 1.00000 sidereal year = 1.0000174 sidereal year. This is minor error.

    1.0 sidereal year = 365.2536360(1) days = 1.0000174(2) astronomical years

    where

    1.0 astronomical years = 365.25 days

  122. Ian Wilson says:

    Paul Vaughan said:
    December 14, 2014 at 4:22 am

    J2000 1800AD-2050AD http://ssd.jpl.nasa.gov/txt/p_elem_t1.txt
    241.1339144 sidereal years

    J2000 3000BC-3000AD http://ssd.jpl.nasa.gov/txt/p_elem_t2.txt
    222.4944428 sidereal years

    Comments Ian?
    Do these models all fit historical observations or not?

    Response:

    We need clarify a few things here. Are you proposing that orbital elements, averaged over
    periods of 250 years and 6000 years be used rather instantaneous values?

    In my study, I used the actual Proleptic Gregorian date/time (or its Julian day) of the event (e.g. the date at which Venus crossed the ecliptic at the time of inferior conjunction with Earth) obtained from the JPL (Horizon) ephemeris.

    A long term average of an orbital element of a planet contains important information but it strictly only applies to the period over which the averaging is done. In addition, it also smooths out the short term changes which might be important.

    I would be good to know your opinion about the long-term mean versus instantaneous value problem.

  123. Paul Vaughan says:

    Ian, is the convention that unless otherwise stated “year” means
    astronomical year = Julian year = 365.25 days = 31557600 seconds ?

    1.00000261 http://ssd.jpl.nasa.gov/txt/p_elem_t1.txt
    1.00000018 http://ssd.jpl.nasa.gov/txt/p_elem_t2.txt
    1.0000174 http://ssd.jpl.nasa.gov/?planet_phys_par

    Are these all expressed in units of
    astronomical year = Julian year = 365.25 days = 31557600 seconds ?

  124. tallbloke says:

    Hopefully not off-topic but feel free to leave it hanging for now if it interrupts the flow:

    Given that the period of the precession of the Earth-Venus conjunction cycle by 1/5 was calculated by Ian some time ago to be 239.8 years, and this period falls between the two periods Paul calculated for Venus transits, is it possible/likely/almost certain that that the E-V cycle precession and the transit period are physically linked, notwithstanding long term cyclic nutations which cause the two periods to diverge by a few years inter-alia?

    If so, does this tell us anything about what causes the relative inclination between orbits and the rate those orbital planes precess at?

  125. Ian Wilson says:

    Paul,

    The reason that I use the instantaneous method in my paper is that it makes no assumptions about which long-term average value to choose. In addition, the results apply over the time interval of the study. The down-side is that the results may not necessarily apply outside the time limits of the study.

    The limitations of my study were not the orbital elements of Venus and the Earth, which can be reasonably estimated over much longer times scales than 3000 years. The main limitation(s) is the difficulty of projecting the properties of the lunar orbit back and forward in time.

    Ignoring this more general problem, if we restrict ourselves to the difficulties associated with the orbital elements of Venus and the Earth we need to consider the following:

    a) the ~ 239 year period that you get from the harmonic analysis is the 239.8 year = 150 VE required for Venus to re-synchronize with the pentagonal pattern of the Earth-Venus alignments. The are an additional two VE alignments for Venus to appear at the same point with respect the stars (i.e. 152 VE = 243.0 years). This comes about because the whole pentagonal pattern drifts ~ 72 degrees in a retrograde direction with respect the stars.

    b) Of course, the length of the ~ 243 year VE transit cycle slowly drifts with time. Although, figure 2 in my paper shows that there is little drift from this value over the 700 year period 1600 and 2300 A.D. Figure shows that the ~ 243 year average period for the spacing between every second transit of Venus across the Sun pretty much applies between -100 B.C. and 3100 A.D.

    Note: My paper uses the Proleptic Gregorian date/time (or its Julian day) for each of the Venus transit events

  126. Paul Vaughan says:

    TB: What I’ve calculated could certainly be described as a VE conjunction precession cycle. When it didn’t quite line up with the annually aliased 243.1074578 tropical year lunisolar envelope, I started having questions which I haven’t finished answering. We appear to be having similar questions.

    I’m not satisfied with my current level of understanding of V & E orbit coupling — for example as addressed in links I gave in my first comment above, which included a link to the Cuk seminar video.

    This page on Milankovitch cycles makes it appear that in some expert communities this stuff is just widely accepted common knowledge: http://www.eoearth.org/view/article/154612/ . That reminded me of Linda Hinnov’s papers. I’ll need to explore further to develop better cross-disciplinary grounding.

  127. Paul Vaughan says:

    Ian Wilson (December 14, 2014 at 10:06 am) wrote:
    “I would be good to know your opinion about the long-term mean versus instantaneous value problem.”

    If I can manage to develop a little more background knowledge I should be able to easily reconcile the two.

  128. Paul Vaughan says:

    Ian Wilson (December 14, 2014 at 11:01 am) wrote:
    “[…] for Venus to appear at the same point with respect the stars”

    has to be at a node — of course — so:
    The calculation I gave above needs a further level of generalization. I’ll give the matter further attention whenever the natural spark happens. (Right now other responsibilities call…)

  129. Paul Vaughan says:

    I’m becoming increasingly convinced that 243 / 34 is the root of WWI & WWII excursions from RI & SCD. This is passing the diagnostics I’m throwing at it.

    During 2030-2070 (a) 34 aligns differently with 243, (b) the polarity reversal is in the opposite direction, and (c) RI & SCD will be different, so without good historical proxies, we have no analogs. Since there’s no available avenue by which to cheat with analog odds, the only available option is deeper exploration of fundamentals (via iterative refinement) to the point of geometric lockdown. This appears feasible.

    However, there are ethical issues. If we could know, would we want to?

    I would definitely not trust politicians, the government, and the general public to understand (without potentially severe misinterpretation) and respond appropriately. There would just be floods of misunderstanding & propagandistic distortion artistry, especially if financial & military (& perhaps healthcare) strategists perceived a security risk.

    We have to be satisfied with exploring for entertainment purposes & no credit in such a context. I absolutely would not trust politicians, the government, and the general public to understand (without potentially severe misinterpretation whether deliberate or accidental) and respond appropriately even if a perfect forecast was given. The darker side of human nature won’t be overwritten by any forecast no matter how perfect.

    On the brighter side: Nature’s still beautiful to simply explore.

    [Moderation note] Discussion of this comment will take place on the open thread.

  130. Paul Vaughan says:

    “[Moderation note] Discussion of this comment will take place on the open thread.”

    I’m not going to volunteer anything about paragraphs 1 & 2 on the other thread. (I can also live with not discussing them further here.)

    As for discussing the remaining paragraphs: That’s unnecessary. I’ll instead spend my time exploring what I think may be a combined JVS evection signal. It’s a busy paid-work week so there may be a long delay before I comment substantively on this.

  131. Paul Vaughan says:

    Ian, could you please clarify whether I now have the conventions right?

  132. Ian Wilson says:

    Paul,

    Yes you have the conventions right. My concern is with this post you made above:

    (1.0000174)*(0.61519726) / (1.0000174 – 0.61519726) = 1.598689623 sidereal years

    If the 1.0000174 in this equation is in Julian (astronomical) years then the answer must be
    astronomical years, not sidereal years. It’s only a minor hiccup but important in the long run.

  133. Paul Vaughan says:

    Ian: I understood your concern. Thanks for clarifying.

  134. tallbloke says:

    Paul: I’ll instead spend my time exploring what I think may be a combined JVS evection signal.

    This may help

  135. Paul Vaughan says:

    tallbloke (October 19, 2013 at 1:32 am) wrote:
    “lunar evection-in-latitude oscillation period […] 32.3”
    “lunar evection oscillation period […] 31.8”

    terminology
    of course I have these calculated (to higher precision) on file
    — but I didn’t know they went by these names
    Ian, are these the standard terms?
    And do these phenomena go by any other names in the literature?

  136. tallbloke says:

    Paul,
    Be careful, there’s more than one type of ‘evection’.

    ‘Evection in latitude’ is a different beast to the more commonly referred to ‘lunar evection’.

    No-one really knows how ‘Evection in latitude’ works, until I came along with my WAG, so no-one studies it much.

    Bottom of this page may help
    http://farside.ph.utexas.edu/teaching/336k/Newtonhtml/node133.html

  137. Paul Vaughan says:

    Reading the comments in the parallel thread, I have to stop and wonder if commentators aren’t aware of one of the first things any newcomer would discover about lunar orbits:

    (27.55455)*(27.212221) / (27.55455 – 27.212221) = 2190.350523 days
    (2190.350523) / 365.242189 = 5.996981151 years

    It’s a trivial exercise to prove algebraically that the following is equivalent:

    (32.28077687)*(31.81194133) / (32.28077687 – 31.81194133) = 2190.350523 days
    (2190.350523) / 365.242189 = 5.996981151 years

  138. tallbloke says:

    So that’s:
    anomalistic month 27.55 days (perigee to perigee)
    sidereal month 27.21 days (Ea-Moon alignment to fixed stars)
    Long period tide Msm ?? 31.81 days
    32.28 days (can’t find a reference for this)

    Please tell us what the latter two periods are.

  139. oldbrew says:

    TB: that can also be done using whole numbers – see the lunar chart.

    1973 tropical years / 329 = 5.9969604

    RE: ‘tell us what’ [etc]…
    See here: PV @ December 15, 2014 at 2:24 pm

  140. Paul Vaughan says:

    Good eye OB. Good to see you back here. I enjoy your positive attitude towards exploration.
    TB: I’ll detail the calculations for you probably later today.

  141. Ian Wilson says:

    Sorry, I have had to re-post with a different symbol for the mean as it is already used by hypertext.

    Paul, Tallbloke and Oldbrew,

    If you want to see how sausages are made then go to:

    http://farside.ph.utexas.edu/teaching/336k/Newtonhtml/node133.html

    It is highly recommended that those with a weak stomach jump to END1*** and END2***

    START***

    There they explain that [note [ ] symbols is used to designate the mean of a variable]:

    Here is a summary of this page:

    The mean moon is an imaginary body which orbits the Earth, in the ecliptic plane, at a steady angular velocity that is equal to the Moon’s mean orbital angular velocity, n. Likewise, the mean sun is a second imaginary body which orbits the Earth, in the ecliptic plane, at a steady angular velocity that is equal to the Sun’s mean orbital angular velocity, n’. Hence, the ecliptic longitudes of the mean moon and the mean sun are

    [theta] = n * t
    [theta]’ = n’ * t

    1. ECLIPTIC LONGITUDE OF THE MOON

    Thus true ecliptic longitude of the Moon is: theta = [theta] + lambda,

    where

    lambda = 2e*sin M + 5/4*{e^2}*sin(2M) – 1/4*{iota^2}*sin(2F) + 11/8*{m^2}*sin(2D)+ 15/4*m*e*sin(2D-M)

    and

    M = ([theta] – alpha) is the lunar mean anomaly, which is defined as the angular distance (in longitude) between the mean Moon and the perigee.

    F = ([theta] – gamma) is the lunar mean argument of latitude, which is defined as the angular distance (in longitude) between the mean Moon and the ascending node.

    D = ([theta] – [theta]’) is the mean elongation of the Moon which is defined as the difference between the longitudes of the mean Moon and the mean Sun. (i.e. the Moon’s phase)

    e = eccentricity of the lunar orbit
    iota = the inclination of the lunar orbit
    m = n / n’
    alpha = the mean ecliptic longitude of the perigee
    gamma = the mean ecliptic longitude of the ascending node

    the first three terms on the right-hand side of the above expression are Keplerian (i.e., they are independent of the perturbing action of the Sun). In fact, the first is due to the eccentricity of the lunar orbit (i.e., the fact that the geometric center of the orbit is slightly shifted from the Earth), the second is due to the ellipticity of the orbit (i.e., the fact that the orbit is slightly non-circular), and the third is due to the slight inclination of the orbit to the ecliptic plane. However, the final two terms are caused by the perturbing action of the Sun. In fact, the fourth term corresponds to variation, whilst the fifth corresponds to evection.

    NOTE : The equation above must be corrected for eccentricity of the Earth’s orbit. This give rise to a small addition term -3me’ * sin(M’) on the right-hand side of the above equation, where M’ is the Sun’s mean anomaly. This term, which is known as the annual equation, oscillates with a period of a solar year.

    END***

    Evection can be thought of as causing a slight reduction in the eccentricity of the lunar orbit around the times of the new moon and the full moon (i.e. D=0 degrees and D=180 degrees), and causing a corresponding slight increase in the eccentricity around the times of the first and last quarter moons (i.e. D=90 degrees and D=270 degrees).

    “The variation and evection terms oscillate sinusoidally with periods of half a synodic month, or 14.8 days, and 31.8 days, respectively. These periods are in good agreement with observations.

    2.. ECLIPTIC LATITUDE OF THE MOON

    The ecliptic latitude of the Moon is given by:

    beta = iota*sin(F+ lambda) + 3/8*m*iota*sin(2D-F)

    END2***

    The first term on the right-hand side of this expression is Keplerian (i.e., it is independent of the perturbing influence of the Sun). However, the final term, which is known as evection in latitude, is due to the Sun’s action. Evection in latitude can be thought of as causing a slight increase in the inclination of the lunar orbit to the ecliptic at the times of the first and last quarter moons, and a slight decrease at the times of the new moon and the full moon. The evection in latitude term oscillates sinusoidally with a period of 32.3 days.

  142. Ian Wilson says:

    The above puts the following post by Paul in a new light:

    (27.55455)*(14.7652945) / (27.55455 – 14.7652945) = 31.81194133 days

    which is the beat between the half synodic month and the anomalistic month.
    The question is why does it equal the evection oscillation period in ecliptic longitude.

  143. wayne says:

    I recognize that: ((27.55455 * 14.7652945) / (27.55455 – 14.7652945))² = 1012
    or that is (27.55455 * 14.7652945) / (27.55455 – 14.7652945) = √1012.
    It does, lands right on it since your max precision in the data is seven digits.😉
    Its significance if anything — you’ve got me, but dealing with periodic wave forms you might find something. Didn’t know to notify you or not.

    Ok…. I didn’t just recognize it, but I had just written a test function in my new physics calculator language, a little like ‘R’ like below to test functions without needing a ‘return’ statement and at the same time testing the ² or ‘square’ symbol of a variable and… there was 1012 bigger than Dallas.

    setprec(7)
    beat = function(hi, lo) { hi*lo/(hi-lo) };
    a = beat(27.55455, 14.7652945)

  144. tallbloke says:

    Wayne: Nice, we like whole numbers.🙂

    Ian: The question is why does it equal the evection oscillation period in ecliptic longitude.

    As Paul shows, there is a link between the evection periods and the synodic and anomalistic months.
    The 6 year period Paul derived necessarily fits with the contra-rotation of the lines of apse and nodes crossing each other every 3 years (Thanks Chaeremon).

    The nature of evection from the farside link (which I gave Paul earlier too):
    “Evection can be thought of as causing a slight reduction in the eccentricity of the lunar orbit around the times of the new moon and the full moon”
    “Evection in latitude can be thought of as causing a slight increase in the inclination of the lunar orbit to the ecliptic at the times of the first and last quarter moons, and a slight decrease at the times of the new moon and the full moon.”

    The 6 year period is three quarters of the Ea-Ve conjunction cycle, approximately half the Jupiter orbital period and approximately a third of the Jupiter-Saturn conjunction period. Coincidence? I doubt it. I think the Moon ‘fits in’ as best it can to the forces acting on it from other planets, and the evection represents the extent of its inability to balance all the forces. Something worth noting on that score is that the gravitational force felt by the Moon from Jupiter and Venus are very similar in magnitude. There has to be a reason for the squeezing and pulling that manifests itself in evection, and I suspect we might find it in JEV timings.

    Of course the Moon also affects the rate the Earth spins at, and it happens to be a fairly similar rate to the rate Mars spins at. Maybe it’s time to take a look at Mars and its moons too. Oldbrew might like to tell us about some of his discoveries on that front on the open thread.

  145. Paul Vaughan says:

    ecliptic xy plane
    Evection (in Longitude)
    (27.55455)*(14.7652945) / (27.55455 – 14.7652945) = 31.81194133 days
    1 / 31.81194133 days = 2Y-A

    ecliptic z axis
    Evection in Latitude
    (27.212221)*(14.7652945) / (27.212221 – 14.7652945) = 32.28077687 days
    1 / 32.28077687 days = 2Y-D

    (32.28077687)*(31.81194133) / (32.28077687 – 31.81194133) = 2190.350523 days
    (2190.350523) / 365.242189 = 5.996981151 years
    1 / 5.996981151 years = (2Y-A) – (2Y-D) = D-A

    (27.55455)*(27.212221) / (27.55455 – 27.212221) = 2190.350523 days
    (2190.350523) / 365.242189 = 5.996981151 years
    1 / 5.996981151 years = D-A

  146. tallbloke says:

    The first time these A, D and Y variables were used by Paul in this thread was here:

    Paul Vaughan says:
    December 7, 2014 at 2:53 pm
    Caution: There’s a slip cycle on longer timescales:
    1/2530.131668 years = 13A/2+5Y-148T

    No explanation was given then and I haven’t seen one since – sorry if I missed something.
    So Paul, please tell us what Y, D and A are. And T if you can spare the time.
    Thanks.

  147. Paul Vaughan says:

    The current discussion is a continuation of an earlier discussion.
    The variables were defined here:
    https://tallbloke.wordpress.com/2014/11/29/paul-pukite-sloshing-model-for-enso/comment-page-1/#comment-94021

    But note well that even without that link the definitions can be inferred unambiguously from context above (by recognizing lunar anomalistic, draconic, & synodic month periods).

    Also note that Ian’s link has the same algebra (as it should). It only looks a little different because the same cycles are defined using different symbols.

  148. tallbloke says:

    Thanks Paul, good to get confirmation from the originator rather than guessing. The thing confusing me now is that when I offered the inverse of a period as a period I was told it wasn’t. So what’s going on with your 1/period lines in your comment? Are they frequencies equal to the algebra following? If the symbols represent periods, how does this work?
    Thanks, and again, sorry if I’m missing something obvious.

  149. Ian Wilson says:

    Paul,

    I am sure that Ulrich, Rog, oldbrew and yourself have noted this at one time or other but I feel that it may be important:

    243.107457826 year x 19 = 4619.0416987 years = 4627.0 years – 7.9583013 years

    the last number is very close to 1 VE cycle (difference ~ 12.8 days).

    Also, there are 20 Venus transits is a transit cycle separated by:

    19 x 243.107457826 years.

    After 20 Venus transits, that particular cycle slips off the edge of the Sun, and a new Venus transit cycle begins.

  150. Paul Vaughan says:

    TB:

    The symbols represent frequency.

    frequency = 1 / period

    Do that for every term before doing any subtraction/addition.

    T = 1 / 365.242189 days = 0.002737909 / day
    D = 1 / 27.212221 days = 0.036748195 / day
    A = 1 / 27.55455 days = 0.036291647 / day
    Y = 1 / 29.530589 days = 0.033863192 / day

    Then do the subtraction & addition.
    You end up with a frequency.
    Then recall that:
    period = 1 / frequency

    Example #1: Evection (in Longitude)

    2Y – A
    = 2*(1/29.530589) – (1/27.55455)
    = 2*(0.033863192) – (0.036291647)
    = 0.067726384 – 0.036291647
    = 0.031434737 / day

    1 / 0.031434737 = 31.81194133 days

    As you can see, that’s mathematically equivalent to this:
    (27.55455)*(14.7652945) / (27.55455 – 14.7652945) = 31.81194133 days

    Example #2: Evection in Latitude

    2Y – D
    = 2*(1/29.530589) – (1/27.212221)
    = 2*(0.033863192) – (0.036748195)
    = 0.067726384 – 0.036748195
    = 0.030978189 / day

    1 / 0.030978189 = 32.28077687 days

    which is mathematically equivalent to:
    (27.212221)*(14.7652945) / (27.212221 – 14.7652945) = 32.28077687 days

  151. Paul Vaughan says:

    TB: Try 3A-2Y-15T to see how frequency algebra shortcuts you past hierarchical application of beat period calculations.

  152. wayne says:

    “… nest the ‘pre’ tag inside the ‘code’ tag …”

    A really great Christmas surpise Rog, thank you for finally cracking that nut open! Like a picture is worth a thousand words sometimes a clear table is worth more than a hundred comments.😎

  153. tallbloke says:

    Wayne: I think I alerted you to this months ago on the page you were submitting ‘test’ latex comments to. Anyway, glad you’re happy🙂

  154. wayne says:

    Rog, I see what has happened. You said on that latex page to wrap the code tag around and it would fix it and I had (mistakenly) assumed you meant to use the code instead of pre and I even tried that on a past thread and that too failed. Argg. But looking at the source code of the latex test page you had used ‘pre’ outside with ‘code’ inside and I see now that both will work, code inside pre or pre inside code, but you know, that wordpress linked page never clearly spelled that out… I just read it yet again. So it IS a great Christmas present for me anyway, I’ve fought over that problem for years on this particular wordpress style that you use! The others wordpress styles don’t seem to have any problem when just using pre by itself. So… sincerely, thanks again.

  155. tallbloke says:

    Wayne: Thanks for you usual thoroughness, if only my disorganised and busy mind could emulate it.

  156. tallbloke says:

    Paul: Thank you kindly for taking the time to give a comprehensive answer with examples. A great assistance to me in understanding what you’ve achieved. Important identities!

  157. oldbrew says:

    32.28077687 days x 204 = 6585.2785 d
    31.81194133 days x 207 = 6585.071855 d
    1 Saros = 6585.3212 d

  158. tallbloke says:

    Ian says:

    I am sure that Ulrich, Rog, oldbrew and yourself have noted this at one time or other but I feel that it may be important:

    243.107457826 year x 19 = 4619.0416987 years = 4627.0 years – 7.9583013 years

    the last number is very close to 1 VE cycle (difference ~ 12.8 days).

    Yes, I noted it further up the thread.

    Also, there are 20 Venus transits is a transit cycle separated by:

    19 x 243.107457826 years.

    After 20 Venus transits, that particular cycle slips off the edge of the Sun, and a new Venus transit cycle begins.

    On the face of it, this would seem to be a random coincidence, because ‘slipping off the edge’ is a function of solar diameter, and the rate of precession of Venus’ line of nodes relative to Earth orbital tilt. But given all that Oldbrew and I have been discovering about phi relationships in orbits, spin rates, precession periods and even planetary diameters and densities, I suspect it is not a coincidence, but the inevitable outcome of the forces underlying the organisation of the solar system. 4627 years is a ‘convergent node’ of the interlocking orbital and synodic and synodic cycle precessions. Like the ‘coincidence’ that the Moon is ‘just the right diameter’ to exactly obscure the Sun at full eclipse, this is a tantalising glimpse into that underlying order that spans so much more than current theory encompasses.

  159. oldbrew says:

    ‘the last number is very close to 1 VE cycle (difference ~ 12.8 days).’

    You may be running up against the issue of which planetary dataset to use.

    If you use the one Paul Vaughan recommends there is no 4627 year period, it’s 4628y.
    That’s mostly because it gives a shorter period for Saturn than the NASA fact sheet, which in turn makes the J-S conjunction period slightly longer.

    Paul Vaughan says:
    December 14, 2014 at 4:22 am
    Seidelmann (1992) http://ssd.jpl.nasa.gov/?planet_phys_par

  160. Paul Vaughan says:

    As I’ve advised Ian privately, there are typos where Ian references me in his recent Venus-Moon paper:

    p.76 [pdf p.2] ( referring to equation (1) )
    (Scafetta 2013 and Vaughan 2013)”

    should read:

    (Scafetta 2013 and Vaughan 2009)

    and

    p.93 [pdf p.19]
    “Vaughan, P.: Private communications, 2013.”

    should read something more like:

    Vaughan, P.: Private communications, 2009
    or
    Vaughan, P.: Tallbloke’s Talkshop blog discussions, 2009
    or
    Jan. 23, 2010 Paul L. Vaughan Sunspots & JEV

    Here’s a quote from the latter:

    =
    Summarizing the above calculations algebraically:
    2(V-E)-2{2[2(E-J)]-[2(V-J)]}
    2V-2E-2{2[2E-2J]-[2V-2J]}
    2V-2E-2{4E-4J-2V+2J}
    2V-2E-2{4E-2J-2V}
    2V-2E-8E+4J+4V
    6V-10E+4J
    This is “JEV Phase”.
    =

    As I’ve mentioned previously at the Talkshop:

    Not only the frequency algebra but also the symmetric frequency algebra derivation can be generalized to JSUN (i.e. it doesn’t just apply to J).

    There has been some confusion in Talkshop discussions about why:

    JEV ~= J + N
    &
    N ~= JEV – J

    Possibly some readers haven’t yet realized that these quantities have more than one physical interpretation.

    For comparative example, above I’ve shown how the beat of lunar evection in longitude and lunar evection in latitude is the same as the beat of the new/full moon & perigee cycles.

    Similarly J+N is a condensed algebraic form that falls out of node (inclination) & perihelion (eccentricity) beats.

    By just looking at the frequency algebra terms remaining after other terms cancel out, some readers may not be appreciating how beat envelopes, slip cycles, &/or aliasing patterns are systematically derived (as in geometric proof) by hierarchical application of beat harmony calculations.

    Over time I’ll elaborate as time & audience receptivity permit. For now I’m advising the community that there have been some important misunderstandings / misinterpretations of J+N.

    Regards

  161. Ian Wilson says:

    Paul said:

    “p.76 [pdf p.2] ( referring to equation (1) )
    (Scafetta 2013 and Vaughan 2013)”

    should read:

    (Scafetta 2013 and Vaughan 2009)

    and

    p.93 [pdf p.19]
    “Vaughan, P.: Private communications, 2013.””

    Paul, these are not typo’s. The private communication occurred in 2013 and not 2009. I have no record of you communicating that information to me in 2009.

    The correct publishing convention on private communications is to reference the date on which the information was communicated to the author, not when the information was originally discovered.

    I was trying to ensure that you were given acknowledgement for you discovery [and not Nicola]. Unfortunately, there is no peer-reviewed journal that I can point that contains this study. So, I went back and looked for the earliest time I became aware of you algebraic expression. I far as I could determine that was in 2013.

  162. Ian Wilson says:

    Paul,

    I was doing my best to do you a favor. If I had been aware of an earlier correspondence I would have quoted it.

  163. Ian Wilson says:

    Paul said:

    “and our private correspondence is just that: private”

    Private communications have nothing to do with revealing other people’s private correspondence.

    Private Communications are a mechanism to point to information that has not appeared in a peer-reviewed journal but the author of the paper is aware of the other persons work from elsewhere [in this case I was pointing to your 2013 post on the Tallbloke’s Talkshop]. Most modern journals do not allow this form of referencing anymore. PRP was generous enough to allow the use of this convention. I used it to give you credit where credit was due.

  164. Paul Vaughan says:

    http://www.sfu.ca/~plv/SunspotsJEV.htm

    That was the link immediately above Ian’s 2010 wuwt comment
    (long since torn down by SFU as I advised it would be in that wuwt thread).

    Let’s please try to avoid more misunderstandings Ian.
    It’s no secret to this community that I inflate formality with no artificial value.
    The intention of my commentary is to criticize neither you nor your devotion to formal conventions.

    Rather I’m reminding the community that this is old news (from December 2009 & January 2010).

    Again we’re reminded:
    Cycling human awareness ensures repeated wheel reinvention.

  165. Paul Vaughan says:

    There’s potential to make formidable gains in collective exploratory efficiency.

    Collective raw exploration would go orders of magnitude faster if it wasn’t red taped to death in practice by formal conventions and associated cumulative, exponentially compounding delays.

    For example, trivial geometric proofs don’t need to be published formally in order to be true and widely recognized as such.

  166. Ian Wilson says:

    Paul Vaughan said:
    December 18, 2014 at 5:13 pm

    Your welcome.

    [Urgent] Ian, please respond to my email. Thanks. Rog

  167. Paul Vaughan says:

    If anyone deserves credit, it’s neither Nicola nor I, but rather Desmoulins.

    All I did was algebraically formulate from geometric first principles what Desmoulins’ explorations revealed.

    In comparison with the derivation I gave early in 2010, the reorganized algebraic derivation I gave in 2013 — although mathematically equivalent — is intuitively superior (for anyone trying to efficiently develop independent conceptual awarness):

    {4[(V-J)-(E-J)]}-{(V-J)+(E-J)}
    {4[V-J-E+J]}-{V-J+E-J}
    {4[V-E]}-{V+E-2J}
    {4V-4E}-{V+E-2J}
    4V-4E-V-E+2J
    3V-5E+2J

    Looking back further than Desmoulins we can of course point to an article from the 1940s and maybe it all began earlier than that in the shadow of my ignorance.

    Far more important than this issue of who deserves credit is independent wheel reinvention capacity.

    There will be a long delay before I can tidy up communication of lunisolar explorations. It will probably be some time in mid-January or later, so we’ll probably need to start a new discussion then since this thread has has fallen off everyone’s radar by now. I look forward to resuming deepening discussion when the timing’s a lot more ripe than now.

    Deep sincere thanks to Ian for supplying the Venus-moon paper, which arrived while I was looking at an envelope wondering what its period was without having yet been adequately motivated to make time to derive 243/34 from geometric first principles.

    Thanks to TB for hosting cordial discussions with superior class.

    Best Regards

  168. tallbloke says:

    Paul and Ian: Many thanks for your efforts here. The groundwork has been laid for further discussion in the New Year. Have a great Christmas.

  169. oldbrew says:

    PV says: ‘(32.28077687)*(31.81194133) / (32.28077687 – 31.81194133) = 2190.350523 days’

    32.28077687 x 34 x 6 = 1 Saros cycle (less an hour or so).

    It can also be expressed as the beat period of the full moon cycle and the draconic year:
    FMC x DY / (FMC – DY) = 2190.3502 days = 5.99698 tropical years

  170. oldbrew says:

    IW says: ‘the ~ 239 year period that you get from the harmonic analysis is the 239.8 year = 150 VE required for Venus to re-synchronize with the pentagonal pattern of the Earth-Venus alignments’

    There may be an alternative or additional reason:

    99 lunar synodic months is just over 8 years.
    239 x 99 SM = 239 x 8, + 1 years (1913y)
    (99 SM – 8 years) x 239 = 1 year.

    ‘the pentagonal pattern of the Earth-Venus alignments’ = 5 V-E

    The ratio of 5 V-E: 99 SM = 761:760 (or it could be 760:759)
    760 = 152 x 5
    152 V-E = 243 year transit cycle period.

  171. oldbrew says:

    (Evection in latitude / draconic month) x 1 year looks a lot like 1/10th of the Jupiter orbit period.

    (32.28077687 d / 27.212221 d) x 1yr = 1.1862602~y = 433.28153 days (Chandler wobble?)
    JPL figure for J orbit = 4332.82 days

    Also: draconic year 346.62008 days / 32.28077687 d = 10.737662~ (evections?)
    and : draconic year 346.62008 days / 27.212221 d = 12.737662~ draconic months (DM)
    Difference = 2

    Looks like a solution needing a problem😉

    385 x 10.737662 = 4134
    385 x 12.737662 = 4904 DM (i.e. 385 DY)
    Difference = 2 x 385 = 770 (= 55 x 14)
    385 DY = 4519 synodic months (4904 – 385)
    and
    4519 SM – 4134 = 385
    (NB 4519 / 385 = 11.737662 i.e. midway between the evection and DY values)

    Also:
    5 DY = 1733.1004 days
    1.1862602 x 1461 = 1733.1261 days
    1461 days = 4 years (365.25 x 4)
    385 DY = 5 DY x 77

  172. Paul Vaughan says:

    oldbrew (December 26, 2014 at 9:44 pm) suggested:
    “Looks like a solution needing a problem”

    The problem here is being surprised to end up biting one’s own tail after unwittingly chasing it in a circle:

    4(D-Y)/5

    And it’s not a solution because if you look at the Horizons output you’ll see:

    2D-26T

    This was addressed in the previous thread.

  173. oldbrew says:

    PV: some of it was, but note the link to the evection data.

    In 385 draconic years there would be:
    4134 evection-in-latitude
    4195 evection-in-longitude (see note below)

    Also the 4134 is 385 less than the number of synodic months (4519), which in turn is 385 less than the number of draconic months (4904).

    Note: it’s not quite 4195, the nearest whole number is 113263 in 27 x 385 DY (10395) and 113263 / 27 = 4194.926

    The difference between evection-in-latitude and evection-in-longitude numbers could equal the number of ‘6-year lunar wobbles’ in the relevant period:
    385 DY = 365.37052 TY (x 27 = 9865 TY)
    4134 x 27 = 111618
    113263 – 111618 = 1645
    9865 TY / 1645 = 5.9969604 TY
    (1645 x 3 = 987 x 5 and 9865 + 5 = 987 x 10, so five ‘short’ of 1645 x 6)

    10395 DY = 8750 full moon cycles
    10395 – 8750 = 1645

  174. Paul Vaughan says:

    OB, the helmholz acoustic equation would save you a lot work.

  175. oldbrew says:

    TB says: ‘No-one really knows how ‘Evection in latitude’ works’

    It looks a lot like synodic months minus draconic years, e.g. Saros:
    223 SM = 19 DY = 204 Evection in latitude periods

    And Evection in longitude approximates to SM – FMC:
    223 SM = 16 full moon cycles (15.992~) = 207 E in L (207.007~)

  176. oldbrew says:

    Ian Wilson says:
    ‘(27.55455)*(14.7652945) / (27.55455 – 14.7652945) = 31.81194133 days

    which is the beat between the half synodic month and the anomalistic month.
    The question is why does it equal the evection oscillation period in ecliptic longitude.’

    Full moon cycle (FC) = (synodic month * anomalistic month) / (SM – AM) = 411.78443 days
    Wikipedia says: ’18×FC = 251×SM = 269×AM’

    http://en.wikipedia.org/wiki/Full_moon_cycle#Matching_synodic_and_anomalistic_months

    Add in ‘evection oscillation period in ecliptic longitude’ (EVLO):
    18×FC = 251×SM = 269×AM = 233×EVLO (233 = 251-18)

  177. tallbloke says:

    OB: To which we might add that 269=251+18