De Rop’s long-term lunar cycle

Posted: January 5, 2016 by oldbrew in moon, solar system dynamics, Tides
Tags:

Perihelion precession by season [credit: Wikipedia]

Perihelion precession by season [credit: Wikipedia]


Willy de Rop of the Royal Observatory of Belgium wrote a paper entitled ‘A tidal period of 1800 years’ in 1971 about tides and the motion of the Moon. It generated some interest and was referred to in at least one other paper, but on closer consideration leads to some ideas we can put forward here.

The opening paragraph states:
‘The Swedish oceanographer O. Pettersson
has presented evidence indicating that the last
maximum of oceanic tides occurred about 1433.
He pointed out that there is a coincidence
between a tidal period of 1800 years and climatic
changes of the same period. We think we
can explain this period as follows.’


De Rop’s basic premise is that there’s a correlation between the so-called ‘lunar wobble’ period and the anomalistic year.
His paper contains a geometric proof, and the final numbers are:
300 lunar wobbles in 1799 anomalistic* years (the lunar wobble is known to repeat in just under 6 years).
Update: *this should be tropical years – see numbers chart below re. the eventual one year difference between tropical and anomalistic years.

What do we make of this? Several things, all interlinked.

Great Year

A few months ago the Talkshop featured a thesis by Ralph Ellis based on his concept of the Great Year, namely the perihelion precession cycle.
Ellis said: ‘The Seasonal Great Year of 21,700 years in length is the reason for the variability in Interglacial spacings.’
—————————————————————
Update 6/1/16: New science paper from Ralph Ellis:
Modulation of Ice Ages via Precession and Dust-Albedo Feedbacks
—————————————————————
Several sources quote a period of 21,600 years including this one which says:
‘For the Earth we have an approximately 41,000-year axis tilt period plus the 100,000-year and 400,000-year cycles of variation in the Earth’s orbit shape. Add in the 21,600-year cycle of precession of the Earth’s perihelion date, you will find the net effect to be quite complex to predict.’

Wikipedia is less precise, saying: ‘it takes about 21000 years for the ellipse to revolve once relative to the vernal equinox’

If we apply the concept of the Great Year to de Rop’s lunar cycle, we find his period looking a lot like a ‘Great Month’ (noting that the word ‘month’ itself derives from ‘Moon’).

A difference of 1 between tropical and anomalistic years (5397 tropical years = 1799 x 3)

A difference of 1 between tropical and anomalistic years
(5397 tropical years = 1799 x 3)


In numbers: 1799 anomalistic years(AY) x 12 = 21588 AY
Correction: we find 1799 tropical years(TY) x 12 = 21588 TY = ~21587 AY (see chart).
Obviously this fits closely with the estimated 21,600 year period quoted fairly widely on websites, and close to the 21,700 years used by Ralph Ellis.

If we have a Great Month and a Great Year, is there anything else? In a word: yes.

It turns out that 1799 AY TY x 3 is equivalent to whole numbers of lunar apsidal cycles (610) and lunar nodal cycles (290), so that would be the Great Season (3 Great Months x 4 = 12 = Great Year).

That would be 900 lunar wobbles (300 x 3, also 610 + 290), so in a Great Year there would be 3600 wobbles. Since perihelion precession means a full revolution of the perihelion precession date (currently early January), there are 10 wobbles per degree of movement of the perihelion (10 x 360 = 3600).

After the Great Month, Season and Year, the other period we can look for is the Great Day. If we say there are 30 Great Days in a Great Month of 300 wobbles, there will be 10 wobbles in a Great Day. The period will be 1799 AY TY / 30 = 59.9666~ AY TY, which looks a lot like the 60-year climate period often referred to in science papers and elsewhere, e.g.:

‘Given that records of solar activity are accurate, solar activity may have contributed to part of the modern warming that peaked in the 1930s, in addition to the 60-year temperature cycles that result in roughly 0.5 °C of warming during the increasing temperature phase.’ – Wikipedia

Also: The Sixty-Year Climate Cycle

So de Rop may have provided a foundation for the period of the perihelion precession and the (controversial in some quarters) 60-year climate period.

For a geometric description of the lunar wobble, see de Rop’s paper (shorter version – only 2 pages) and in particular Figure 3, showing the interaction of the apsidal and nodal cycles in a ‘half-wobble’ or 180 degrees of movement (=~3 years). Once the meaning of Figure 3 is clear, everything should start falling into place (hopefully).

Note: Keeling and Whorf also found the same tidal cycle in their paper entitled:
‘The 1,800-year oceanic tidal cycle: A possible cause of rapid climate change’

Comments
  1. oldbrew says:

    Keeling & Whorf’s paper says (referring to a graph – Figure 1):
    ‘The maxima, labeled A, B, C, D, of the most prominent sequences, all at full moon, are spaced about 180 years apart. The maxima, labeled a, b, c, of the next most prominent sequences, all at new moon, are also spaced about 180 years apart. The two sets of maxima together produce strong tidal forcing at approximately 90-year intervals.’

    In de Rop’s theory ‘about 180 years’ would be 30 lunar wobbles = 179.9 Anomalistic Years (1/10th of 1799)

    Or 3 Great Days.

  2. oldbrew says:

    There’s a typo in De Rop’s paper:
    ‘A period of 300 p corresponds to 17.990002050 anomalistic years, hence, to exactly an entire number of anomalistic years’ – either ‘17.990002050 anomalistic years’ should say ‘17.990002050 anomalistic centuries’ or the decimal point needs to move.

  3. tallbloke says:

    All the numbers are boggling my poor old battered brain. Can we lay this out in a table of some kind, starting with short periods and working up to the long ones?

  4. oldbrew says:

    TB: possibly, but let’s just focus on the wobble period.

    10 = the Great Day (the ’60 year’ climate cycle)
    30 = ~180 years (Keeling & Whorf’s maxima – see comment above)
    300 = the Great Month (de Rop’s 1799 year lunar tidal period) or 30 x 10 Great Days
    900 = the Great Season (3 Great Months) or 610 lunar apsidal cycles
    3600 = the Great Year (300 x 12 = 3600 wobbles) or the perihelion precession of 21588 anom. years

  5. oldbrew says:

    In this video the display shows one month of perihelion movement, right at the start.

    January is crossed out and replaced by February, the blue blobs are Earth in Jan. and Feb. precession positions (upper left) i.e. a month apart, and the note says ‘~1800 years later‘ i.e. the time taken for Earth to move its perihelion point by 1/12th of a full revolution (12 x 1800 = 21600y)

    So that’s your Great Month🙂

  6. oldbrew says:

    Keeling & Whorf also say:

    ‘A fourth condition necessary for determining maximal tidal forcing is the closeness of the earth to perihelion, the point on the earth’s elliptical orbit closest to the sun, occupied every anomalistic year of 365.2596 days. When an analysis is made to find the times when all four conditions are most closely met, the 1,800-year cycle becomes apparent as a slow progression of solar-lunar declinational differences that coincide with progressive weakening and then strengthening of successive centennial maxima in tidal forcing (Fig. 2). The 1,800-year cycle thus represents the time for the recurrence of perigean eclipses closely matched to the time of perihelion. Progressively less close matching of perigee, node, and perihelion with syzygy occur, on average, at intervals of 360, 180, 90, 18, and 9 years.’ [bold added]

    Same point as de Rop.

    Update – K & W say:
    ‘The timing of dust layers, from lake sediments, and of the Little Ice Age, from historic data, correspond closely with peaks of the 1,800-year tidal cycle.’

    Ralph Ellis (see post above) bases his ice ages theory on perihelion precession and dust cycles.

  7. tallbloke says:

    K&W should have cited Ropp if they were aware of his paper.
    So, we have the Earth-Moon system oscillating at the same periodicities as the (thought to be solar) Gleissberg cycle of 90yr, and at the (thought to be solar) Jose cycle of ~180 year, which also roughly coincides with Maunder>Dalton>Landscheidt solar grand minima periodicity and the approx realignment of the gas giant planets.

    Nice evidence that both solar and lunar-terrestrial timings are linked to solar system motion periods.
    Not forgetting the beach ridges which evidence climate-storminess cycles at all the same periodicities.

  8. oldbrew says:

    TB: yes, the 1800 year cycle is also 2/3rds of 2400 years (J-S, solar) and half of 3600 years (21 U-N) – periods you pinpointed in the PRP paper and a recent post:
    https://tallbloke.wordpress.com/2015/10/22/why-phi-simplified-a-brief-fibonacci-tour-of-the-solar-system/
    (see under: ‘The long term cycles’)

    Another paper: Is there a 60-year oscillation in global mean sea level? (2012)
    – Don P. Chambers, Mark A. Merrifield and R. Steven Nerem
    http://onlinelibrary.wiley.com/doi/10.1029/2012GL052885/pdf

    Discussion: ‘We have demonstrated that a quasi 60-year oscillation
    exists in long tide gauge records around the world and that
    the gauges in the North Atlantic, western North Pacific, and
    Indian Ocean have similar phase (within 10 years, or 55)
    and amplitude when regionally averaged.’

    They later seem to water this down a bit – maybe for ‘political’ reasons?

  9. oldbrew says:

    I need to re-visit this. It turns out the 1799 years are tropical years not anomalistic, despite what Willy de Rop says.

    The ‘trick’ is that the 21588 TY (1799 x 12) equals 21587 anomalistic years i.e. one less.
    I’ll put up a numbers chart to explain. The number of full moon cycles gave the game away.

    Update: post revised, numbers chart added.

    The point is: over 3 x 1799 tropical years (TY), the difference between the number of anomalistic and synodic months is an exact number (of full moon cycles), and also an exact number of TY.

  10. Chaeremon says:

    @oldbrew (January 5, 2016 at 10:17 pm) Re. Update

    Have you compared to my 300 years post, just 1/6th of the here 1800 years, where all is still integer and only ratios are fractions,
    https://tallbloke.wordpress.com/2015/11/09/why-phi-some-moon-earth-interactions/#comment-110977

    I think that 1799 anomalistic in the paper means perihelion, no?

  11. oldbrew says:

    C: ‘1799 anomalistic in the paper means perihelion’

    To get a whole number of wobble periods we need a whole number of apsidal cycles.
    In 3 x 1799 tropical years the criteria are met (610 isn’t divisible by 3, so 1799 years won’t do it).

    The numbers chart I added meets this rule:
    ‘apsidal precession completes one rotation in the same time as the number of sidereal months exceeds the number of anomalistic months by exactly one’
    http://en.wikipedia.org/wiki/Lunar_precession
    [Note: I use tropical not sidereal months]

    The anomalistic year should match perihelion at the end of the full period (21588 TY) because one year – or nearly one – is ‘dropped’ compared to tropical years.

    ‘Have you compared to my 300 years post’

    Yes, the thing is the wobble period is known to be slightly less than 6 years.
    De Rop is saying one year is lost every 1800 so it’s 6 x 300, minus one.

    In the added chart, the number of days of the full moon cycles and the tropical years is a 100% match.

    Also: ‘the full moon cycle [which] is the beat period of the synodic and anomalistic month, and also the period after which the apsides point to the Sun again
    http://en.wikipedia.org/wiki/Lunar_month#Anomalistic_month

  12. oldbrew says:

    Dodwell’s theory, based on his studies of ancient observations, reckons it took 4194 years for the obliquity to re-align (2345 BC – 1850 AD) after a major ‘event’.

    ‘The analogy with the earth’s return to a steady point in 1850 A.D., during the period of 4194 years after its axial disturbance in 2345 B.C., is evident. The oscillations, thus disclosed by the harmonic analysis of the observations, are a striking confirmation of the disturbance itself, and of the course of the earth’s recovery in the subsequent ages.’
    https://tallbloke.wordpress.com/2014/12/24/dodwells-surprising-study-of-the-obliquity-of-the-ecliptic/

    Dodwell breaks it down to 7 x 599 years as the obliquity curve progresses:
    ‘Seven semi-oscillations are indicated between 2345 B.C. and 1850 A.D., the exact semi-oscillation period being 599 years.’

    What follows here is speculative.
    100 lunar wobbles = 599.667 years
    700 LW = 4197.667 years = just half a wobble more than the Dodwell period.

    Dodwell: ‘when limiting lines are drawn through the maxima, and through the minima, they converge towards the zero point at 1850 A.D., where the principal or Mean Curve also is finished.’

  13. Chaeremon says:

    @oldbrew (January 6, 2016 at 10:03 am) Re. De Rop is saying one year is lost every 1800 so it’s 6 x 300, minus one.

    And I would like to follow you both: what is lost in which one’s period? seems that can confuse without qualifying the body (earth or moon).

    R. H. van Gent has a good match for 1799.00~ astron. years (145 x Inex – 133 x Saros) but perigee (or apogee) is far away from syzygy. I prefer van Gent since eclipse interval lengths are empirical fact and base of ephemerides.

    B.t.w. I have “only” ephemerides for 1999bce to ce2999 for checking (can check eg. 1800y, NP), so outside of that: bon voyage old friend😉

  14. oldbrew says:

    ‘R. H. van Gent has a good match for 1799.00~ astron. years (145 x Inex – 133 x Saros) but perigee (or apogee) is far away from syzygy.’

    It will be far away, because 610 apsidal cycles isn’t divisible by 3.

    It takes 3 x 1799y, or 900 lunar wobbles, to get a whole number of apsidal (610).
    61 AC = 90 LW works, but it’s not a whole number of years or of full moon cycles, whereas 5397 TY is.

  15. ralfellis says:

    I see I have a mention here.

    Just to let you know that I have updated my ice ages paper, that mentions this. The discussion of eccentricity and obliquity in the appendix may also be of interest.

    .

    Modulation of Ice Ages via Precession and Dust-Albedo Feedbacks

    A new paper proving that CO2 is a minor player in the drama that is the Earth’s climate.

    Abstract

    We present here a simple and novel proposal for the modulation and rhythm of ice ages and interglacials during the late Pleistocene. While the standard Milankovitch-precession theory fails to explain the long intervals between interglacials, these can be accounted for by a novel forcing and feedback system involving CO2, dust and albedo. During the glacial period, the high albedo of the northern ice sheets drives down global temperatures and CO2 concentrations, despite subsequent precessional forcing maxima. Over the following millennia CO2 is sequestered in the oceans and atmospheric concentrations eventually reach a critical minima of about 200 ppm, which causes a die-back of temperate and boreal forests and grasslands, especially at high altitude. The ensuing soil erosion generates dust storms, resulting in increased dust deposition and lower albedo on the northern ice sheets. As northern hemisphere insolation increases during the next Milankovitch cycle, the dust-laden ice-sheets absorb considerably more insolation and undergo rapid melting, which forces the climate into an interglacial period. The proposed mechanism is simple, robust, and comprehensive in its scope, and its key elements are well supported by empirical evidence.

    https://www.academia.edu/20051643/Modulation_of_Ice_Ages_via_Precession_and_Dust-Albedo_Feedbacks

    Ralph Ellis

    [reply] main post updated to reference Ralph’s paper

  16. oldbrew says:

    How the Solar System Works – Other Motions of the Earth says:
    ‘The direction of perihelion moves eastward, taking 114,000 years to circle the sky’
    http://www.uwgb.edu/dutchs/AstronNotes/HowSolSysWorks.HTM

    21 x 5397 (= 63 x 1799) tropical years = 113,337 years

    In that time there are:
    21 x 610 lunar apsidal cycles [21 and 610 are Fibonacci numbers]
    21 x 290, -1 lunar nodal cycles
    21 x 900, -1 lunar wobbles
    21 x 4787 full moon cycles [4787 + 610 = 5397]

    So when one lunar nodal cycle has been ‘lost’, the cycle ends and restarts.

  17. tallbloke says:

    Good work OB! What that tells us is that Moon’s apsidal cycle is tied to Earth, but the nodal cycle is ‘relative to the fixed stars’ – hence the loss of one nodal cycle during the earth-moon system’s apsidal precession round the Sun.

    I hardly need remind you that:

    21 U-N conjunctions is ~3600 years – twice the 1799 period, and the period over which the U-N conjunction points make a full rotation around the Sun.

    Fibtastic.

  18. oldbrew says:

    Thanks TB. Looks like a result after a blur of numbers for a couple of days.

    For a bonus point: 113337y x 3, -1y = 340010y = 21 x 898 Saros

  19. tallbloke says:

    And for a double bonus: 21 x 3598 x 3/2 = 113337

  20. oldbrew says:

    That’s because 3598 = 1799 x 2.

  21. tallbloke says:

    Let me put it another way:

    21U-N x 21 x 3/2 =113337y

  22. I hope you are aware of the fact that the THEORETICAL rate of precession of the perihelion of the Earth’s orbit changes with time. If you assume that the period of precession of the Earth’s axis with respect to the stars is 25,772 years, then it varies from (when rounded to the nearest decade):

    3000 B.C. – 10.85 arc seconds per year = precession of the equinoxes of ~ 20880 years.

    to

    3000 A.D. – 11.79 arc seconds per year = precession of the equinoxes of ~ 21,200 years

    (all in a pro-grade direction)

    with a current value of – 11.62 arc seconds = precession of the equinoxes of ~ 20,940 years

    Reference: Alcyone Ephemeris 4.3

    This compares to a measured rate of precession of 11.45 arc seconds per year.

  23. “The intervals between Earth’s passage through perihelion vary from 363 to 367 days. Such large variations can’t be explained by interactions with other planets or Sun. If one considers the passages of the Earth-Moon barycenter through perihelion, however, this oscillation vanishes, and therefore it is caused by Moon. (See Jean Meeus’ chapter 26, “The barycenter of the solar system” starting on page 165 and chapter 27, “On the passages of Earth in perihelion” starting on page 169 in “Mathematical Astronomy Morsels”, published in 1997 by Willmann-Bell, Richmond, Virginia, USA).

    Nevertheless, there are long-term periodic variations in the rate of perihelion advance, which can cause successive perihelion cycles to differ in duration by several millennia! Specifically, as orbital eccentricity decreases perihelion advances more quickly, and conversely as orbital eccentricity increases perihelion advances more slowly.

    At the present mean rate of about 11.6 arcseconds of advancing heliocentric longitude motion relative to the distant stars (eastward) per year, perihelion takes about 111,700 years to revolve once around Sun. At the same time, however, the northward equinox is precessing at the rate of about 50.3 arcseconds of ecliptic longitude per year (retrograde, or westward) or one cycle per 25,765 years, due to Earth wobbling on its axis like a spinning top, so the net effect (50.3+11.6 = 61.9 arcseconds per year) is that perihelion advances about 1° per 59 years and therefore takes about 360° × 59 ≈ 21,240 years to revolve once relative to the northward equinox of the date, advancing through all of the seasons in sequence.”

    Reference: http://individual.utoronto.ca/kalendis/seasons.htm

  24. oldbrew says:

    astroclimateconnection says: ‘with a current value of – 11.62 arc seconds = precession of the equinoxes of ~ 20,940 years’

    ~ 20,940 years is the time taken for anomalistic and tropical years to reach a difference of one, i.e. 1 / (AY – TY).

    ‘the northward equinox is precessing at the rate of about 50.3 arcseconds of ecliptic longitude per year (retrograde, or westward) or one cycle per 25,765 years’

    25,763 years is the time taken for tropical and sidereal years to reach a difference of one, i.e. 1 / (SY – TY).

    25,764 tropical years = 4 x 6441 TY (= 4 x 339 Metonic cycles):

    https://tallbloke.wordpress.com/2015/11/09/why-phi-some-moon-earth-interactions/

  25. tallbloke says:

    OB: That’s because 3598 = 1799 x 2.

    Looking at the relative masses of Neptune, Uranus and the Moon, I’d say it’s more likely because

    1799 = 1/2 x 3598😉

  26. oldbrew says:

    TB: whatever🙂

    Re: 21 x 290, -1 lunar nodal cycles in 113337 tropical years [comment: January 6, 2016 at 7:44 pm]

    21 x 290, -1 = 6089
    113337 TY = 119426 draconic years
    119426 – 113337 = 6089

    That’s just a cross-check to confirm the result.

  27. tallbloke says:

    Ian: Nevertheless, there are long-term periodic variations in the rate of perihelion advance, which can cause successive perihelion cycles to differ in duration by several millennia! Specifically, as orbital eccentricity decreases perihelion advances more quickly, and conversely as orbital eccentricity increases perihelion advances more slowly. At the present mean rate of about 11.6 arcseconds of advancing heliocentric longitude motion relative to the distant stars (eastward) per year, perihelion takes about 111,700 years to revolve once around Sun.

    Thanks Ian. The eccentricity variation is also on a timescale of around 100kyr, superimposed on another cycle of around 400kyr

    Useful page here:
    http://www.jgiesen.de/kepler/eccentricity1.html

    Both those cycles are likely linked to gas giant cycles, so we can have a think about that too.

  28. oldbrew says:

    All the numbers in one place:

  29. tallbloke says:

    Nice work, but you might want to change the date on that.😉

    Could I also suggest you add the time periods to your key.

    [reply] done, thanks

  30. oldbrew says:

    TB: ‘The eccentricity variation is also on a timescale of around 100kyr, superimposed on another cycle of around 400kyr…Both those cycles are likely linked to gas giant cycles, so we can have a think about that too.’

    Ralph Ellis has plenty to say about the eccentricity in his new paper [ralfellis says: January 6, 2016 at 2:15 pm]

  31. tallbloke says:

    I haven’t read Ralf’s paper yet, apart from the supplementary section on eccentricity, but an inpage search on it doesn’t find any citation of Gerard Roe’s ‘In Defence of Milankovitch’ paper which found a very good correlation between 60N insolation and rate of change of N. hemisphere ice amount.

    The other thing to remember is that all these eccentricity calcs are based on standard perturbation theory, which is a mess of heuristics, not a solid ‘from first principles’ theory.

  32. oldbrew says:

    IW says: ‘At the present mean rate of about 11.6 arcseconds of advancing heliocentric longitude motion relative to the distant stars (eastward) per year, perihelion takes about 111,700 years to revolve once around Sun.’

    111,632 years is the time taken for anomalistic and sidereal years to reach a difference of one, i.e. 1 / (AY – SY) = about 11.61 arcsecs. per year.

    Comparing with precession of the equinoxes: (13 x 25763) / 3 = 111639.66

  33. tallbloke says:

    A deglaciation mystery solved in favour of orbital forcing theory:
    http://m.sciencemag.org/content/351/6269/165

  34. oldbrew says:

    Keeling and Whorf say:
    ‘A well defined 1,800-year tidal cycle is associated with gradually shifting lunar declination from one episode of maximum tidal forcing on the centennial time-scale to the next. An amplitude modulation of this cycle occurs with an average period of about 5,000 years, associated with gradually shifting separation-intervals between perihelion and syzygy at maxima of the 1,800-year cycle.’
    http://www.pnas.org/content/97/8/3814.long

    Our model shows the ‘gradually shifting lunar declination’: if one lunar nodal cycle is ‘lost’ every 113,337 years, that averages a shift of 0.06 days per cycle – hence the ‘minus one’ in 21 x 290,-1 (chart above: January 7, 2016 at 11:15 am). That can account for the ‘gradually shifting separation intervals’ described by K&W.

    The ‘1,800-year tidal cycle’ is obviously the same as de Rop’s 1799 years. The period of ‘about 5,000 years’ is described as ‘an average period of 4,650 years’ further on in the paper – close to the giant planets ‘grand synod’ of ~4628 years. 26 Jose cycles is about 4,650 years.

  35. oldbrew says:

    Keeling and Whorf say:

    ‘Variations in the anomalistic year are proportional to P/(P − 1) where P denotes the period of “climatic precession,” defined as the time-interval between two consecutive passages of the vernal equinoxial point at perihelion. The period of climatic precession has been estimated (12) to have varied between 19 and 28 kyr during the past 60 kyr, with the largest variations before 33 kyr BP.

    When a variable anomalistic year, derived from the prescribed secular variation of P , is introduced into our calculations, the average period of the 1,800-year cycle, from 33 kyr BP to the present, is only slightly altered from 1,795 to 1,805 years.’

    The reason it’s ‘only slightly altered from 1,795 to 1,805 years’ could be that, as stated before, 1799 years is not an exact number of apsidal cycles. Since there are 610 of them in 3 x 1799 years, splitting them into 3 means 203 + 203 + 204 (= 610).

    Time taken for 203 cycles = ~1796 years
    Time taken for 204 cycles = ~1805 years

    So K&W’s ideas do fit with the de Rop theory.

  36. Gail Combs says:

    ralfellis,

    I just left you a comment over at
    http://www.warwickhughes.com/blog/?p=4019

    Great paper BTW. I think you are onto something.

  37. oldbrew says:

    Link to more by Ralph Ellis on his dust theory.
    http://www.warwickhughes.com/blog/?p=4179

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