Why Phi? – solar and lunar rotation

Posted: October 28, 2017 by oldbrew in Maths, moon, Phi, solar system dynamics
Tags: , ,


This started as a search for a period when the Sun and the Moon would both complete a whole number of rotations.
The result was:
Solar: 25.38 days * 197 = 4999.860 d
Lunar: 27.321662 * 183 = 4999.864 d
(data sources: see reference notes at end)

Taking these as equivalent, we have 197-183 = 14 ‘beats’.
197 = 14*14, +1
183 = 13*14, +1
4999.864 / 14 = 357.13314 days
357.13314 days * 45/44 = 365.2498 days
45 * 14 (630) beats = 44 * 14 (616) calendar years, difference = 0.022 day

So the beat period of the two rotations is 44/45ths of a year, i.e. the difference in number of rotations is exactly 1 in that length of time.
630 beats = 616 years (630 – 616 = 14)
616/45 = 13.68888 calendar years = 4999.8663 days
184 lunar sidereal months (rotations) = 4999.864 days

Then something else popped up…

The Phi factor:
‘We recover a 22.14-year cycle of the solar dynamo.’ (2016 paper)
See: Why Phi? – modelling the solar cycle

Solar Hale cycle = ~22.14 years (est. mean)
13.68888 * Phi = 22.149~ years
22.14 / 13.68888 = 1.61737 (99.96% of Phi)
(55/34 = 1.617647)

From the same post:
Jupiter-Saturn axial period (J+S) is 8.456146 years.
That’s when the sum of J and S orbital movement in the conjunction period = 1

13.68888 / 8.456146 = 1.618808
Phi = 1.618034

Conclusion:
This cycle of solar and lunar sidereal rotation (SRC) sits at the mid-point of the Phi²:1 ratio between the J+S axial period and the mean solar Hale cycle, i.e. with a Phi ratio to one and inverse Phi to the other.
SRC = (J+S) * Phi
SRC = Hale / Phi
SRC = Hale – (J+S)
(Mean Hale value is assumed)

In a period of 616 years there are 45 SRC.
The period is 44 * 14 years = 45 SRC = 45 * 14 beats.
SRC * (45/44) = 14 years.

Cross-checks:
Carrington rotations per 616 y = 8249
8249 CR / 45 = 4999.865 days

Synodic months per 616 y = 7619
7619 SM / 45 = 4999.856 days
8249 – 7619 = 630 = 45 * 14

45*183 sidereal months = 8235
8235 – 7619 = 616
8249 CR – 8235 Sid.M = 14
Beat period of CR and Sid.M = 616/14 = 44 years = 45 * (13.6888 / 14)
Every 44 years there will be exactly one less lunar rotation (sidereal month) than the number of Carrington rotations.

8249 CR – 7619 synodic months = 630 = 45 * 14
630 – 616 = 14
– – –
The anomalistic year

The beat period of the tropical month and solar sidereal rotation * 45/44 = the anomalistic year.
(27.321582 * 25.38) / (27.321582 – 25.38) = 357.14265 days
45 * 357.14265 = 16071.419 days
44 * 365.259636 = 16071.423 days

The anomalistic year is the time taken for the Earth to complete one revolution with respect to its apsides. The orbit of the Earth is elliptical; the extreme points, called apsides, are the perihelion, where the Earth is closest to the Sun (January 3 in 2011), and the aphelion, where the Earth is farthest from the Sun (July 4 in 2011). The anomalistic year is usually defined as the time between perihelion passages. Its average duration is 365.259636 days (365 d 6 h 13 min 52.6 s) (at the epoch J2011.0).
http://en.wikipedia.org/wiki/Year#Sidereal.2C_tropical.2C_and_anomalistic_years
– – –
Data sources

— Carrington Solar Coordinates:
Richard C. Carrington determined the solar rotation rate by watching low-latitude sunspots in the 1850s. He defined a fixed solar coordinate system that rotates in a sidereal frame exactly once every 25.38 days (Carrington, Observations of the Spots on the Sun, 1863, p 221, 244). The synodic rotation rate varies a little during the year because of the eccentricity of the Earth’s orbit; the mean synodic value is about 27.2753 days.
http://wso.stanford.edu/words/Coordinates.html

— The standard meridian on the sun is defined to be the meridian that passed through the ascending node of the sun’s equator on 1 January 1854 at 1200 UTC and is calculated for the present day by assuming a uniform sidereal period of rotation of 25.38 days (synodic rotation period of 27.2753 days, Carrington rotation).
http://jgiesen.de/sunrot/index.html

The sidereal month is the time between maximum elevations of a fixed star as seen from the Moon. In 1994-1998, it was 27.321662 days.
http://scienceworld.wolfram.com/astronomy/SiderealMonth.html

Comments
  1. oldbrew says:

    Another way of expressing it:

    45 J+S = 380.52657 years (= 45 * 8.456146)
    45 SRC = 616 years
    45 Hale = 996.3 years (est.)
    [380.52657 + 616 = 996.52657]

    1/Phi² = 0.382~
    1/Phi = 0.618~
    1 = 0.382 + 0.618
    [382 + 618 = 1000]

    Note: 84 Jupiter orbits = 996.46 years

  2. Paul Vaughan says:

    summary:

    (27.2753)*(27.3215823630557) / (27.2753 – 27.3215823630557) = 16101.2598801307 days = 44.08284704 Julian years

  3. pyromancer76 says:

    Thanks for continually providing one of the most interesting blogs in science (and politics).

    I just watched a lecture by Gurtina Besla (YouTube) from U of Arizona on “The Dynamics of the Local Group: Challenges to Convention in the Era of Precision Astrometry”

    Her measurements and math may have nothing to do with the material you present here, but I thought perhaps some of these new findings might go along with the fascinating ideas you propose.

    And these new measurements show the barycenter of the MilkyWay to be in a different place and to move depending on the movements of the Local Group.

    Anyway, just wanted to share and to say thank you again. Your blog makes each day more interesting.

  4. oldbrew says:

    (27.321662 * 27.2753) / (27.321662 – 27.2753) = 44.006486 y

    In 44 years the difference between the numbers of lunar sidereal months and solar Carrington rotations is 1.

  5. Richard111 says:

    Entropy?

  6. oldbrew says:

    NASA Goddard
    Published on Oct 27, 2011

    The number of sunspots increases and decreases over time in a regular, approximately 11-year cycle, called the sunspot cycle. The exact length of the cycle can vary. It has been as short as eight years and as long as fourteen, but the number of sunspots always increases over time, and then returns to low again. [bold added]
    – – –
    Talkshop note:
    8+14 = 22
    8.456146+13.6888 = ~22.145

  7. oldbrew says:

    Re. ‘In 44 years the difference between the numbers of lunar sidereal months and solar Carrington rotations is 1.’

    CR – Synodic Months = 45
    Sid.M. – Syn.M. = 44
    CR – Sid.M. = 1 (45 – 44)
    Solar Sidereal Months – Syn.M. = 89 (45 + 44)

    89 beats in 44 years is one per 0.494382 years, just under half a year (88/44 = 2).
    So in 616 years there would be 89 * 14 beats.
    14 beats = 6.923148 years and 89 is a Fibonacci number.

    55 * 6.923148 years = 380.67414 y
    89 * 6.923148 years = 616 y
    144 * 6.923148 years = 996.67411 y
    (55,89 and 144 are Fibonacci numbers)

    Compare with (comment: October 28, 2017 at 5:05 pm):
    45 J+S = 380.52657 years (= 45 * 8.456146)
    45 SRC = 616 years
    45 Hale = 996.3 years (est.)
    [380.52657 + 616 = 996.52657]

  8. oldbrew says:

    Solar: 25.38 days * 197 = 4999.860 d
    Lunar: 27.321662 * 183 = 4999.864 d

    197 + 183 = 380
    The axial period of these two is: 4999.86 / 380 = 13.157526 days
    (13.157526 / 25.38) + (13.157526 / 27.321662) = 1

    45 S-L axial = 13.157526 * 45 = 592.08867 = 1.62105~ years (Phi = 1.618~) = 616/380
    (616/45 = 13.68888)