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