Ian Wilson: Tidal-Torque model of Solar-Planetary Interaction

Posted: November 25, 2012 by Rog Tallbloke in Astronomy, Astrophysics, Cycles, Solar physics, solar system dynamics, Tides
Tags: , , , ,

My thanks to Ian Wilson for an update on his tidal-torquing model, which relates the motion of Venus, Earth and Jupiter to changes in sunspot numbers and the flows observed on the Solar surface. This elegant solution looks very promising in terms of forecasting solar variation, as well as offering a hypothesis explaining a mechanism underlying the strong correlations between solar variation and planetary motion. The following article is reposted from Ian’s excellent blog.

THE UPDATED V-E-J TIDAL TORQUING MODEL
Ian Wilson : November 2012

The problem with the collective blog postings about the
Spin-Orbit Coupling or Tidal-Torquing Model that are described
at the end of this post is that they only look at the tidal-torquing
(i.e. the pushing and pulling of Jupiter upon the Venus-Earth
tidal bulge in the Solar convective zone) when Venus and Earth
are inferior conjunction (i.e. when Venus and Earth are on the
same side of the Sun). However, a tidal bulge is also produced
when Venus and the Earth align on opposites sides of the Sun,
as well (i.e at superior conjunction).

This means that in the real world, tidal bulges are induced in
the convective layer of the Sun once every 0.8 years rather
than every 1.6 years, as assumed in the original basic model.
This is achieved by a sequence of alternating conjunctions
of Venus and the Earth:

IC –> SC –> IC –> SC –> IC –> etc..

[where IC = Inferior conjunction & SC = Superior conjunction]

Unfortunately, logic tells you the gravitational pushing/pulling
of Jupiter on the V-E tidal bulge at a given inferior conjunction
will be roughly equal opposite to the pushing/pulling that occurs
at the next superior conjunction. At first glance, this would seem
to destroy any chance for the gravitational force of Jupiter (acting
on the V-E tidal bulge) to produce any nett spin in the outer
convective layers of the Sun. However, it turns out that the
gravitational tugging of Jupiter at inferior V-E conjunction is
not completely cancelled by the tugging at the next superior
V-E conjunction. This lack of cancellation is primarily related
to the changing orientation and tilts of the respective orbits
of Venus and Jupiter.

The diagram immediately below shows sunspot number
(SSN) for solar cycles 0 through to 9. Plotted below the
sunspot number curve in this figure is the net tangential torque
of Jupiter acting the V-E tidal bulge, where Jupiter’s tangential
torque at one V-E inferior conjunction is added to Jupiter’s
tangential torque at the next V-E superior conjunction to
get the nett tangential torque. In this diagram, a positive
nett torque means that the rotational speed of the Sun’s
equatorial convective layer is sped-up and a negative
nett torque means that the equatorial convective layer
is slowed-down.

[N.B. The nett torque curve has been smoothed with a
5th and 7th order binomial filter to isolate low frequency
changes]

Some important things to note are:

a) The nett torque of Jupiter acting on the V-E tidal bulge
has a natural 22 year peridocity which matches the 22
year hale (magnetic) cycle of solar activity.

b) the equatorial convective layers of the Sun are sped-up
during ODD solar cycles and slowed-down during EVEN
solar cycles.

These two points provide a logical explanation for the
Gnevyshev−Ohl (G−O) Rule for the Sun.

This rule states that if you sum up the mean annual Wolf
sunspot number over an 11 year solar cycle, you find
that the sum for a given even numbered sunspot cycle is
usually less than that for the following odd numbered
sunspot cycle (Gnevyshev and Ohl 1948). The physical
significance of the G−O rule is that the fundamental activity
cycle of the Sun is the 22 year magnetic Hale cycle, which
consists of two 11 year Schwabe cycles, the first of
which is an even number cycle (Obridko 1995). While
this empirical rule generally holds, there are occasional
exceptions such as cycle 23 which was noticeably weaker
than cycle 22.

These two points are also in agreement with the results of
Wilson et al. 2008.

Does a Spin–Orbit Coupling Between the Sun and the Jovian Planets Govern the Solar Cycle?

I. R. G. Wilson, B. D. Carter, and I. A. Waite
Publications of the Astronomical Society of
Australia, 2008, 25, 85–93.

Figure 8 from Wilson et al. 2008 (above) shows the moment arm
of the torque for the quadrature Jupiter and Saturn nearest the
maximum for a given solar cycle, plotted against the change in the
average equatorial (spin) angular velocity of the Sun since the previous
solar cycle (measured in μrad s−1). The equatorial (≤±15 deg)
angular velocities published by Javaraiah (2003) for cycles
12 to 23 have been used to determine the changes in the
Sun’s angular velocity (since the previous cycle) for cycles
13 to 23.

What this graph clearly shows is that the Sun’s equatorial
angular velocity increases in ODD solar cycles and decreases
in EVEN solar cycles, in agreement with the V-E-J
Tidal-Torquing model.

c) The 11 year solar sunspot cycle cycle constantly tries to
synchronize itself with the Jupiter’s nett tidal torque.

The original figure plotted at the top of this blog post is
reproduced here with superimposed blue and red vertical
lines showing the times where the Jupiter’s nett torque
(acting on the V-E tidal bulge) changes sign (i.e. direction
with respect the axis of the Sun’s rotation). The points
of sign change in Jupter’s nett torque that occur just
before solar sunspot minimum are marked by blue lines
while the points that occur after solar minimum are
marked by red lines. The figure below shows that:

i) Normally their is a phase-lock between the time of sign
change in Jupiter’s torque and solar minimum.

ii) As soon as this phase-lock is broken (i.e. around about 1777)
22 years later (i.e. one Hale cycle) after the loss of lock, there is
a collapse in the strength of the solar sunspot cycle.

References:

1. Gnevyshev, M. N. and Ohl, A. I., 1948, Astron. Zh., 25, 18
2. Obridko, V.N., 1995, Solar Phys., 156, 179
3. Javaraiah, J., 2003, SoPh, 212, 23
4. Wilson I.R.G, Carter B.D, and Waite I.A., 2008,

     Publications of the Astronomical Society of
     Australia, 25, 85–93.

5. Wilson, I. R. G., 2010, General Science Journal
http://www.gsjournal.net/h/papers_view.php?id=3812

Essential background reading if you knowledge of
the evolution of this model is not up-to-date :

Does a Spin–Orbit Coupling Between the Sun and the Jovian Planets Govern the Solar Cycle?

I. R. G. Wilson, B. D. Carter, and I. A. Waite
Publications of the Astronomical Society of
Australia, 2008, 25, 85–93.

Do Periodic Peaks in the Planetary Tidal Forces Acting Upon the Sun Influence the Sunspot Cycle?

I.R.G. Wilson
A free download of the paper is available in the General 
Science Journal where it was published in 2010 
 
Two blog entries that give a basic explanation of the 
V-E-J Tidal-Torquing Model
 
Six blog entries that investigate the properties of the  
V-E-J Tidal-Torquing Model
 
Comments
  1. Paul Vaughan says:

    As some readers here may recall, I carefully audited claims of VEJ synchronicity with the solar cycle a few years ago. My conclusion: The claims are rock solid.

    I will take this opportunity to make an announcement: I have found a clear Jupiter signal on Earth in one of the major ocean basins. I made the finding probably 3 years ago, but at the time I had not yet pieced together enough big-picture awareness of terrestrial heat & water circulatory morphology to overcome my skepticism about the period. I now realize that the terrestrial Jupiter signal is doubly indirect (via moon & sun). It seems unlikely that I’ll ever have time, resources, & inclination to explain formally, but it’s probably feasible to find time to share a single illustration sometime during the next few years.

    I sincerely hope Ian Wilson will be awarded the longterm secure support he needs to take his work as far as he possibly can within his lifetime.

  2. Ninderthana says:

    Thanks Paul for your kind words – I have long been inspired by your ability to be open to new ideas and your dogged determination to let the evidence guide your excellent research.

    Congratulations on your discovery about the Jupiter Signal – I know how difficult it is to explain the full details of what you are doing to others but I look forward to seeing graph on your work. I am sure that it will get others thinking about the true nature of the planetary interactions.

  3. J Martin says:

    Ian, Your graphs in this post seem to focus upon the Dalton minimum and following one of your links I found this;
    “1996.5 First minimum of cycle 23 marking the start of the next “Dalton-like” Minimum.

    I was wondering if your torque graph allowed you to project the next 100 years, and if that showed the nature of the pending (Landscheidt) minimum, and how it compared to the graphs you have shown in this post.

    A number of people, Geoff Sharp amongst others think it will be a Daltonesque minimum, whereas, Landscheidt, Abdussamatov, and others are expecting a more Maunderesque style minimum.

  4. Sparks says:

    Your doing it wrong! It’s very good, but it’s wrong. sorry :)

    [Reply] 1) State who you are replying to. 2) don’t argue by assertion, back up your statements with reasoned argument. Cheers – TB

  5. Sparks says:

    The planetary influence on the Sun is there, I apologize, please have a look at this, it’s my first attempt at reconstructing it.

    http://thetempestspark.files.wordpress.com/2012/11/ujs.jpg

    [Reply] Thanks, now you just need to tell us what the variously coloured sinusoids represent with some sort of legend or list.

  6. Ninderthana says:

    Sparks,

    The physics involved in the model that I present is very straight forward. A temporary tidally induced distortion appears in some of the convective layers of the Sun along an axis that is co-linear with the (temporary) alignment axis of Venus and the Earth. The level of distortion is maximized by the reinforcement of the induced tidal distortions of both the Earth and Venus.

    The simple model that I present only deals with the pushing and pulling of Jupiter’s gravity upon the temporarily reinforced tidal bulge. This, of course, is only an approximation to reality.
    Clearly, both the Earth and Venus continuously produce tidal distortions on the Sun, upon which the gravitational force of Jupiter acts. However, when the tidal distortions of these two planets are located at different solar longitudes, it is likely that long term net pushing/pulling forces of Jupiter on one will effectively cancel out the other. This means that we can ignore the effects of Jupiter’s tidal torquing at times other than those near the alignments of Earth and Venus. It is only when the two tidally induced distortions combine that we get a net tidal-torquing because of changes in the distance of Jupiter from the Sun and the changes in relative orientation of the orbits of Venus and Jupiter.

  7. janbenestad says:

    about tides:
    The strong tide is at Venus-Jupiter conjunct/opposite, about every 118 days
    Every 12 years these are stronger because of the Jupiter perihelion
    But at the Jupiter perihelion there is a dampening effect from Mercury, so the overall Jupiter-Mercury tide shows a double top (about two years before/after Jupiter perihelion).
    And about every 11 years we have favorable allignments with Earth.

    So picture shows a rigid double top (12years) and a moving 11-year top – three tops.
    These tops have different qualities. The two 12-year tops can start a solar cycle, but the 11 year-top cannot. And all three tops can prolong a cycle.

    Finally there is the 10-year cycle. It determins the sunspot cycle strength. And at a certain configuration of the three tidal tops, it can start a sunspot cycle.

    This is what you need to understand the sunspot cycle – now we just need the physics to explain :)
    http://www.sibet.org/solar

  8. Ninderthana says:

    Janbenestad,

    A viable solar cycle model must be based physical principles. The VEJ Tidal-Torquing model is based upon the idea that you need a net torque acting upon the outer layers of the Sun to change the rotation rate of layers in the outer convective layer of the Sun. This can only be achieved by the largest planetary gravitational force (i.e. Jupiter’s) acting upon the largest tidal asymmetry that is produced by the remaining planets (i.e. the tidal asymmetry that is produced by alignments of Venus and the Earth).

    There is little point producing a tidal bulge if there are no significant gravitational forces that are acting upon it.

  9. Ninderthana says:

    Jan Benestad,

    Please look at my 2008 paper:

    Publications of the Astronomical Society of Australia, 2008,
    25, 85–93.

    Does a Spin–Orbit Coupling Between the Sun and
    the Jovian Planets Govern the Solar Cycle?

    http://www.publish.csiro.au/?act=view_file&file_id=AS06018.pdf

  10. Ninderthana says:

    J. Martin,

    Good question! Will we have a Dalton or a Maunder-like minimum in this century?

    Unfortunately, the VEJ Tidal-Torquing Model is a model indicates that the Sun’s Babcock-Leighton dynamo is a phase-locked resonance with the dominate cycles in the gravitational/tidal forces of the planets.

    This means that natural resonances within the Sun’s Babcock-Leighton dynamo, which are mostly linked to long term changes in Sun’s differential rotation rate, are only modulated by the tiny changes in the planetary torques applied to the outer layers of the Sun.

    The bottom line, it is difficult to project the model more than one or two Schwabe cycles ahead (~ 24 years) since such projections will rely upon knowing when the planetary driving torques occur with respect to future solar minima.

    My bet is that we are about to experience an Oort-like minimum which should last about
    two Schwabe cycles (2009 to 2033).

  11. janbenestad says:

    Ninderthana
    I am a stock trader and work with cycles. I have studied the sunspot cycle, and can “explain” all the known cycles from a “technical standpoint”. Using the three cycles of 10, 11 and 12 years, I have found patterns and signals, that precisely can explain and forecast solar cycles. What remains is the physical explaination. With all the technical “hints” it should be an overcoming task.

    Here are some hints: timing is everything. Knowing when a cycle begins gives a pretty good estimate of Tmax, strength and length. And for the next cycle.

    Something has to start a sunspot cycle (start some kind of ocillation on the Sun)
    The strength of the cycle depends on the “forces” in the raising phase of the cycle
    And the tidal cycles decide when the cycle ends/dies out.
    A new sunspot cycle cannot begin before the old one is dead (fallen below a threshold)

    Normally it is the 12-year cycle that starts a sunspot cycle (strong tidal forces). This is the case when the 11-year cycle-top is lagging the 12-year cycle top.
    The strenght of the cycle is decided by the relative positions of Jupiter and Saturn.
    When the 11-year cycle leads the 12-year cycle, the 10-year cycle alone is able to start a sunspot cycle. This results is a series of short and relative strong sunspot cycles, until the 11-year cycle has come close enough (behind) the 12-year cycle

    So it is something like this: when the previos sunspot cycle is dead, a new one will be ignited. It will happen at the 12-year cycle top (double top) or at the 10-year cycle bottom.
    If it happens at the 10-year cycle bottom, we get a pretty strong sunspot cycle, but the tidal cycles are not contributing very much.
    If it happens at the 12-year cycle top, the sunspot cycle can be in or out of phase with the 10-year cycle. So either a weak sunspotcycle only driven by tidal cycle, or a strong cycle driven by both the 10-year and the tidal.

    Big question is what kind of force/cycle the 10-year cycle is? It is probably related to wobble or barycenter, and must induce some kind of deformation or flow change on the sun…

    JAn :)

  12. janbenestad says:

    Ninderthana
    here is an illustration
    http://www.sibet.org/images/2012/20121127aa.jpg
    shows rigid 12 year tidal cycle and moving 11 year tidal cycle
    And the three types of sunspot cycles

    When we get a “phase failure” with 10 year cycle it gets a bit more complicated. What happens is that the tidal cycles prolong the cycle, so that is “consumes” a second 10-year cycle. And the followng couple sunspot cycles will be out of phase.
    And at the Maunder we saw two such double cycles in short order (SC-12 and SC-10) creating a “catastrophe”

  13. tallbloke says:

    Hi Jan, and welcome.
    Timo Niroma found that cycles are rarely 11 years long. The cycle length tends to cluster around 10.38 or 11.86 years. These two periods are the average JEV cycle lengths and the Jupiter orbital period.

    The 10 year is actually 9.93 years – the Jupiter-Saturn tidal period or half the synodic period.
    The 11 year period is actually 11.08 and is a Jupiter-Neptune harmonic.

    Ninderthana (the author of this article) has done a huge amount of work on this area and I recommend his website:
    http://astroclimateconnection.blogspot.co.uk

    It’s a serious attempt to find the underlying mechanisms.

    There are also many articles on this blog by myself and others interested in this field of research, so feel free to do some browsing in the archives or use the site search facility. Google’s site specific search is handy too.

    Cheers

    TB

  14. Ninderthana says:

    Janbenestad says:
    November 27, 2012 at 2:03 pm

    Jan,

    I am astrophysicist and I try to understand the natural world. I would like to respectfully point out that cycles are best used to help guide where to look for the underlying physical principles of a phenomenon. They are not, by themselves, a physical explanation of that phenomenon.

    Scientists refer to physical principles in order to understand a phenomenon. They then propose models (based on these physical principles) which can be tested by observation.

    I am grateful that you are using your skills and expertise to try and solve this very interesting problem. Thank you for your contributions.

  15. janbenestad says:

    thank you for your replies :)
    My first task has been to identify the patterns in the data, and the next step is to explain them physically.
    The sunspot cycle can be explained by the 9,93 and 11,86 year cycles. Some other factors are also needed, like Mercury influencing the 11,86 year cycle, the 11,08 year Venus+Earth+Jupiter cycle, and Uranus-Neptun cycle (and the disturbances that Geoff talks about)
    Some of my “technical” discoveries are:
    -the 11-year cycle is mainly about prolonging sunspot cycles, and moderates the “war” between the 10 and 12 year cycles
    -sunspotcycles are very much about timing. And the main timing-cycle is the 12 year cycle. Solar minimum is normally a bit before or after Jupiter perihelion (Timo wrote much about this). So when the sunspot-cycle is dominated by the 12-year cycle, it is starting up new cycles before/after the perihelion.
    -when the relative position of the 11 and 12 year cycles are right, the 10-year cycle takes over and starts new sunspot cycles (where the 10-year cycle is strongest). And the 10-year cycle dominates until Jupiter/12-year cycle takes over again.

    Why isnt the sunspotcycle 9,93 years long when that cycle dominates? Because there is still needed a (relative) strong tidal-kick to start the cycle (allthough the 11 and 12 year cycles are not at the strongest). And the strong tidal-tops are spaced by ~590 days. And every ~10 years you have to add 2 or 3×118 days, so you end up with 10,3 years as one of the alternatives

    So the data gives many clues, but also many questions of how and why… :)

  16. tchannon says:

    Timo is missed.

  17. janbenestad says:

    hi Ninderthana
    would it make sense to look at the gravitational influence from Saturn on the tidal buldge?
    The strong solar tides are always orientet towards Jupiter (some degrees off depending on the position of Earth). From my data quenching it would make sense if there is some effect from Saturn, whether Saturn is near the Sun-Jupiter-axis, or perpendicular to it…

  18. Ninderthana says:

    Jan Benestad,

    Please look at the first diagram in the following blog post.

    http://astroclimateconnection.blogspot.com.au/2012/03/planetary-spin-orbit-coupling-model-for.html

    It shows that Saturn’s gravitational influence upon the Sun is very small compared to Jupiter’s
    and is so small, in fact, that it is less than that of Venus and the Earth.

    That said, there is obviously a slight reinforced of Jupiter’s influence upon the V-E
    tidal bulge when Saturn aligns with Jupiter every 19.858 years. Though, I think that
    it is very minor.

    Basically, I have concluded that the main role of the Jovian planets
    has been to create resonances in the orbital periods of the terrestrial planets – which
    believe are responsible for forming the tidal bulges upon the Sun. This makes it look like
    the motion of the Sun about the solar system barycentre (which is dependent upon
    cyclicities between Jupiter and the other Jovian planets) correlates with solar activity.
    I am now convinced that, by and large this is a false, or at minimum, an indirect
    correlation.

    Please read my 2008 paper [which has just been published in
    Russian) at:

    The Planetary-Lunar-Climate Connection
    http://www.wbabin.net/Science-Journals/Research%20Papers-Astrophysics/Download/3811

    The last part of this paper will explain the details.

  19. janbenestad says:

    Saturn tide is obviously negligable
    But the 10 year Jupiter-Saturn cycle is the major one and has to be figured out. In my work the 11 year cycle is pretty useless in explaining and forecasting cycles. But it has an important effect in the overall picture
    JAn :)

  20. tallbloke says:

    With the caveat that Ian may be right about JEV being the active kids and Jupiter-Saturn just the ushers of the active children, you might enjoy a read of this:
    http://tallbloke.wordpress.com/2011/08/05/jackpot-jupiter-and-saturn-solar-cycle-link-confirmed/

  21. janbenestad says:

    many hypothesis and playing with numbers :)
    In my opinion the 10-year cycle is the key behind sunspot-strength, and the main engine behind the variation is sunspotcycles (just check the correlation of JuSa angle and SSN)
    And the sunspot activity is related to the period between conjunctions and oppositions (which is opposite to the tidal-influence that generates sunspot AT conjunctions/oppositions).
    In the period between conj/opp of Jupiter and Saturn is the period where the solar wobble changes most, where the sun/barycenter accelerates most. It should be the natural cause, but I have no clue in what way :)
    But it is the obvious cause and should provide the answer…

    The tidal cycles of 11 and 12 years modulate and are about timing. And timing is vital in the 10-year cycle.

  22. tchannon says:

    I wouldn’t bet too much on Saturn being so minor if there is an effect from Jupiter.

    Reason: time constants. Saturn has 8x as long to have an effect.

  23. tallbloke says:

    Jan: bad idea to put your email address on the net. I’ll email Ian for you.

  24. Ninderthana says:

    Jan Benestad said:

    “Big question is what kind of force/cycle the 10-year cycle is? It is probably related to wobble or barycenter, and must induce some kind of deformation or flow change on the sun…”

    This is a fascinating finding Jan.

    This might be the 10.4 year cycle which seems to a sub-multiple of the solar 208 i.e.

    20 x 10.4 = 208 years.

    The average solar cycle length for cycles 15 to 22 (minimum to minimum 1913.6 to 1996.4)
    is 10.35 years.

  25. Ninderthana says:

    Tallbloke touched on the 10.4 year cycle here:

    http://tallbloke.wordpress.com/2011/08/05/jackpot-jupiter-and-saturn-solar-cycle-link-confirmed/
    August 7, 2011 at 12:27 pm

    Hi Ulric,

    The effect on the solar cycle of the other planets modulates the ‘carrier wave’ generated by Jupiter and Saturn. The two biggest planets in the solar system have the biggest effect, but the action of the other planets means that solar cycles tend to hop in length between two main periods of around 10.4 years and around 12 years. They are less freqently at or near the average length.

    The Sun wants to go with J&S but the other planets have an effect too. Saturn and Jupiter are the heavyweights which mainly account for Solar – barycentric motion and have the periods which match long term solar cycle average periods revealed in the spectragraphic analysis of the sunspot record.

    Jupiter Earth and Venus are the most tidally effective planets and the actual cycle to cycle lengths closely mirror their alignment patterns, except when the heavyweights Uranus and Neptune align so as to cause the solar barycentric motion to enter an unusual phase causing a major hiccup in the activity cycle.

    Jupiter Saturn and Earth are the most electro-dynamically effective planets and its possible that NASA’s recently discovered ‘flux tube’ reconnections between these planet’s magnetospheres and the Sun have an important role too.

    Finally, although we tend to slip into talking in terms of cause and effect, we need to bear in mind that the solar activity cycles are an outcome of solar system wide physical processes which interact to maintain the orbits of the planets and the rhythm of the solar output. This is a logical conclusion which I can expand on if you want.

  26. Ninderthana says:

    Jan,

    The again, you might be referring to the (J+S)/2 half cycle = 19.858/2= 9.929 years?

  27. janbenestad says:

    of course its 9,93 and half JuSa. I just call it the 10-year as abbrevation.
    Studying it further (like Bart Leplae) shows details and patterns in the JuSa-SC relationship that strengthens the correlation. So this cycle is probably the most important one regarding SC-strength. But strength depends on timing, and the timing-cycles are 11,08 and 11,86 years
    Therefore we are out of phase with JuSa in SC24. Because of the “mess” that the timing cycles have created :) Normally it takes about two SCs to get back in phase

  28. Paul Vaughan says:

    We are not at the stage of refinement. We are at the stage of exploration. At this stage we can learn most by looking at J & N since they set the boundaries (highest & lowest frequencies) of the framework.

    Even if we were dealing with spatiotemporal chaos in a box, wouldn’t it be useful to know the dimensions of the box?…

    We have Earth records that make the dimensions of the box clear.

    Findlater Jet / Somali Jet / Somali Current
    “During boreal summer, about half of the global low-level, cross-equatorial mass flux passes through the region associated with the Somali jet, which comprises only 10% of the Earth’s circumference (Findlater, 1969).”
    — quoted from: ftp://texmex.mit.edu/pub/emanuel/PAPERS/Boos_Emanuel_2009.pdf

    Illustration:
    http://www.ecmwf.int/research/era/ERA-40_Atlas/images/full/D4B_LL_JJA.gif

    More another day …meanwhile, here’s some essential background reading:

    Chao, B.F. (2006). Earth’s oblateness and its temporal variations. Comptes Rendus Geoscience 338, 1123-1129. doi:10.1016/j.crte.2006.09.014.
    http://www.earth.sinica.edu.tw/~bfchao/publication/eng/2006-Earth%E2%80%99s%20oblateness%20and%20its%20temporal%20variations.pdf