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Pattern Recogn. Phys., 1, 199–202, 2013

www.pattern-recogn-phys.net/1/199/2013/

doi:10.5194/prp-1-199-2013

©Author(s) 2013. CC Attribution 3.0 License.

Open Access

Pattern Recognition

in Physics

Apparent relations between planetary spin, orbit,

and solar differential rotation

R. Tattersall

University of Leeds, Leeds, UK

Correspondence to: R. Tattersall (rog@tallbloke.net)

Received: 6 October 2013 – Revised: 26 November 2013 – Accepted: 28 November 2013 – Published: 16 December 2013

Abstract. A correlation is found between changes in Earth’s length of day [LOD] and the spatio–temporal

disposition of the planetary masses in the solar system, characterised by the zaxis displacement of the centre

of mass of the solar system [CMSS] with respect to the solar equatorial plane smoothed over a bi-decadal

period. To test whether this apparent relation is coincidental, other planetary axial rotation rates and orbital

periods are compared, and spin–orbit relations are found. Earth’s axial angular momentum moment of inertia,

and internal dynamics are considered in relation to the temporal displacement between the potential stimulus

and the terrestrial response. The diﬀerential rotation rate of the Sun is considered in relation to the rotational

and orbital periods of the Earth–Moon system and Venus and Mercury, and harmonic ratios are found. These

suggest a physical coupling between the bodies of an as yet undetermined nature. Additional evidence for

a resonant coupling is found in the relation of total solar irradiance (TSI) and galactic cosmic ray (GCR)

measurements to the resonant harmonic periods discovered.

1 Introduction

Earth’s length of day [LOD] varies cyclically at various

timescales. These small variations in the order of a millisec-

ond are believed to be related to exchanges of angular mo-

mentum between the atmosphere and Earth, the displacement

of oceans away from and toward the equator (Axel-Mörner,

2013), and the changing Earth–Moon distance. On longer

timescales, the variation is considerably larger, on the order

of several milliseconds, and these variations take place over

several decades or more. It is thought by Gross (2007) that

the cause of the longer-term variation is due to shifts in the

circulation of convecting molten ﬂuid in Earth’s ﬂuid outer

core. If this is the case, it begs the question: what is the cause

of those shifts?

2 Data and method

LOD Data from (Gross, 2007) is plotted against the zaxis

motion of the centre of mass of the solar system [CMSS] with

respect to the solar equatorial plane using the NASA/JPL

DE14 ephemeris. This curve is smoothed at around the pe-

riod of two Jupiter orbits (24yr) in order to mimic the damp-

ing eﬀect of the changes of motion in a viscous ﬂuid (like

that in Earth’s interior). The curve is shifted temporally to

obtain the best ﬁt to the LOD curve, and the period of the lag

is found to be 30yr (Fig. 1).

3 Result

The result is suggestive of a dynamic coupling between

changes in the disposition of solar system masses, predomi-

nantly the gas giant planets. These planets possess an over-

whelming percentage of the mass in the solar system out-

side the Sun, and also possess a high proportion of the entire

system’s angular momentum. Resonant coupling between

Jupiter–Saturn and the inner planets in the early history of

the solar system had signiﬁcant impact on the planets’ even-

tual orbits (Agnor and Lin, 2011).

If the planets are able to transfer orbital angular momen-

tum to the axial angular momentum of neighbour planets, we

might expect to see evidence of this in the axial rotation peri-

ods of smaller planets relative to the orbital periods of larger

neighbours. To investigate this possibility, the rotation rates

Published by Copernicus Publications.

200 R. Tattersall: Planetary spin–orbit coupling and solar differential rotation

2

3. Result

Figure 1: z-axis motion of the CMSS relative to the solar equatorial plane plotted against LOD (Gross 2010) 1840-2005

The result is suggestive of a dynamic coupling between changes in the disposition of solar

system masses, predominantly the Gas Giant planets. These planets possess an

overwhelming percentage of the mass in the solar system outside the Sun, and also possess

a high proportion of the entire system’s angular momentum. In the early history of the solar

system, resonant coupling between Jupiter-Saturn and the inner planets had significant

impact on their eventual orbits. (Agnor & Lin 2011)

If the planets are able to transfer orbital angular momentum to the axial angular

momentum of neighbour planets, we might expect to see evidence of this in the axial

rotation periods of smaller planets relative to the orbital periods of larger neighbours. To

investigate this possibility, the rotation rates and orbital periods of several planets are

compared with the rotation rate and orbital period of Jupiter.

4. Inner Planet Synchrony

It is observed that the ratio of Venus and Earth’s rotation rates divided by their orbital

periods is 1.08:0.0027. This is equivalent to the ratio 400:1. During their respective synodic

periods with Jupiter, Venus completes 1.03 rotations and Earth completes 398.88. This is

close to a 400:1 ratio. Looking at the Earth and Mars’ axial rotation and orbital periods we

observe that:

Earth 1/365.256 = 0.0027

Mars 1.0275/686.98= 0.0015 The ratio of these numbers is:

0.0027: 0.0015 = 1:0.546

Earth completes 1.092 orbits between synodic conjunctions with Jupiter, while

Mars completes 1.18844. The ratio of these numbers is:

1.092: 1.18844 = 1: 1.088

The ratio of the ratios is 2:1 (99.6%)

The reason for the 2:1 ratio becomes apparent when we observe that the Mars-Jupiter

synodic conjunction period is in a 2:1 ratio with the Earth-Jupiter synodic period (97.7%)

Figure 1. z-axis motion of the CMSS relative to the solar equatorial

plane plotted against LOD (Gross, 2010) 1840–2005.

and orbital periods of several planets are compared with the

rotation rate and orbital period of Jupiter.

4 Inner planet synchrony

It is observed that the ratio of Venus and Earth’s rotation rates

divided by their orbital periods is 1.08 :0.0027. This is equiv-

alent to the ratio 400 :1. During their respective synodic pe-

riods with Jupiter, Venus completes 1.03 rotations and Earth

completes 398.88. This is close to a 400:1 ratio. Looking

at Earth and Mars’ axial rotation and orbital periods, we ob-

serve that:

–Earth 1/365.256 =0.0027.

–Mars 1.0275/686.98 =0.0015.

The ratio of these numbers is

0.0027:0.0015 =1 : 0.546.

–Earth completes 1.092 orbits between synodic

conjunctions with Jupiter, while

–Mars completes 1.18844. The ratio of these numbers is

1.092:1.18844 =1 : 1.088.

–The ratio of the ratios is 2 : 1 (99.6%).

The reason for the 2 : 1 ratio becomes apparent when we ob-

serve that the Mars–Jupiter synodic conjunction period is in

a 2 : 1 ratio with the Earth–Jupiter synodic period (97.7%).

Once again there appears to be a quantisation of spin and

orbit into simple ratios involving the largest planet in the sys-

tem, the Sun and the inner planets between them.

As a further test, it is observed that:

–The Neptune rotation rate divided by the Uranus rota-

tion rate=1.0701427.

–The Jupiter–Uranus synodic period divided by the

Jupiter–Neptune synodic period is 1.0805873.

–1.0805873/1.0701427 =1.00976 (99.03%).

These observations strongly suggest that Jupiter aﬀects the

rotation rates and orbital periods of both Earth–Venus and

Earth–Mars. In combination with the other gas giant plan-

ets, the combined eﬀect produces the curve seen in Fig. 1,

notwithstanding the much smaller contributions of the in-

ner planets. Having established that the spin and orbit of the

four inner planets relates to Jupiter’s orbital period, greater

weight can be given to the possibility that Earth’s decadal

LOD anomalies may have a celestial cause in planetary mo-

tion.

4.1 Inertia and ﬂuid damping

Earth’s high axial rotation rate, along with its density, cause

Earth to have a high angular momentum which resists

changes in angular velocity. A theory developed from the

observation of magnetic anomalies on Earth’s surface sug-

gests that columnar vortices surround Earth’s core which pro-

duce ﬂows in the viscous mantle and liquid outer core (Lister,

2008). Modelling such ﬂuid dynamics as these is beyond the

scope of this paper, but the temporal stability of these mag-

netic structures suggests that small, externally applied forces

will take a considerable period of time to produce a terres-

trial response. The eﬀect of these stabilising structures will

produce a terrestrial response which can be characterised as

a ﬂuid-damped oscillation. The signature of Jupiter’s motion

above and below the solar equatorial plane over the course of

its orbital period of around 11.86yr is not seen in LOD data.

If the correlation in Fig. 1 is indicative of a physically cou-

pled relationship, it is then evident that the damping of the

oscillation is suﬃcient to smooth out both the Jupiter orbital

period and the Jupiter–Saturn conjunction period of 19.86yr.

It is found that the best ﬁt of the celestial data to the LOD

variation magnitude is at two Jupiter orbital periods. This

matches well with the temporal lag between the celestial data

and the LOD data of around 30yr. The peak-to-peak oscil-

lation period seen in the celestial data indicates a cycle of

around 180 yr. This period was found by José (1965). The lag

of the terrestrial response appears to be at around 1/6 of this

periodic length. This is around the half period of the major

oceanic oscillations observed on Earth (Axel-Mörner, 2013).

These oceanic oscillations are in phase with the changes in

LOD and the lagged celestial data.

4.2 Differential solar rotation rates relating to planetary

motion

The periods in which the Sun’s visible surface makes one

sidereal rotation vary with latitude. Near the equator the pe-

riod is near 24.47 days. This period is known to vary on a pe-

riod relating to the orbits of Jupiter, Earth and Venus (Wilson

et al., 2008). Near the poles, the period of rotation is around

35 days. These periods relate to variation in total solar irradi-

ance (TSI) (Scafetta and Willson, 2013).

Pattern Recogn. Phys., 1, 199–202, 2013 www.pattern-recogn-phys.net/1/199/2013/

R. Tattersall: Planetary spin–orbit coupling and solar differential rotation 201

Considering the relationships between the spin and orbit

of Mercury (three rotations per two orbits of the Sun) and

Venus (three rotations per two Earth orbits and two rotations

per synodic conjunction with Earth), a test is made to see if

similarly simple harmonic relations exist between these plan-

ets and the Sun’s diﬀerential rates of rotation.

Firstly, it is noted that the lengths of day of Mercury and

Venus form a ratio that is close to 2 : 3 ratio, and that the

equatorial and polar rotation rates of the solar surface also

form a ratio close to 2 : 3.

Mercury makes one sidereal rotation per 2/3 (240 degrees)

of orbit in 58.65days. A point on the Sun rotating at a rate

which brought it directly between Mercury and the solar core

in the same period would have a sidereal period of 35 days

(184 days making one full rotation plus 2/3 of a rotation,

i.e. 240 degrees). Mercury makes two sidereal rotations per

480 degrees (1, 1/3 orbits) in 117.3days. A point on the Sun

rotating at a rate which brought it directly between Mercury

and the solar core in the same period would have a sidereal

period of 27.06 days, making four sidereal rotations plus 1/3

of a rotation (120 degrees). It is noted that this is close to the

Carrington period.

In summary, it can be seen that Mercury has a 3: 2 spin-

orbit ratio which is in 3 :5 spin–spin and 2 : 5 orbit–spin ra-

tios with a solar rotation period of 35.18 days, and is in 6 : 13

spin-spin 4 :13 orbit-spin ratios with a solar rotation period

of 27.06days.

Two Mercury rotations occur in 117.3days. In a similar

period of 116.8days, Venus makes a full rotation with re-

spect to the Sun, while orbiting 180 plus 6.18 degrees (0.52

orbits) and rotating (retrograde) 180 minus 6.18 degrees in

the sidereal frame. A point near the solar equator rotating at

a rate which brought it directly between Venus and the so-

lar core in the same period after 4.52 rotations would have a

sidereal solar rotation period of 25.84days.

Points near the solar poles rotating at a rate which brought

them directly between Venus and the solar core in the same

period after 3.52 rotations would have a sidereal period of

33.2days. It is noted that the average of these two solar ro-

tation periods is 29.51days, which is close to the period of

rotation of the Earth–Moon system with respect to the Sun

(29.53days). A solar rotation period of 29.32days is found

to be in a 1 : 3 ratio with a period of 87.97days; the Mercury

orbital period, and 1 : 4 ratio with a period of 117.3days; the

period of two Mercury rotations and close to one Venusian

day.

The relationship of Venus with the Earth–Moon system is

more clearly seen by considering that the period of a Venus

rotation with respect to the Sun of 116.8days is exactly 1/5

of the Earth–Venus orbital synodic conjunction period of

1.6yr. Five synodic conjunctions occur over a period of 8

Earth years as Venus makes 13 orbits, bringing the two plan-

ets back to within two degrees of their original longitude. At

the end of this period, the various solar periods calculated in

the preceding observations make whole numbers of sidereal

6

Figure 2: Comparison of GCR measurements over Carrington rotations with planetary frequencies

The coincidence of peaks in the GCR curve with multiples of the Carrington rotation period CR

indicates a resonant effect of this frequency (27 days). Similarly, the coincidence of other peaks in

the GCR curve with multiples of the periods at which a point on the solar equator passes between

inner planets and the solar core is indicative of resonant effects. Also shown are Mercury and Venus

orbital and half orbital periods. Venus orbital periods lie close to multiples of the Venus-Solar

rotation periods near the peaks in GCR activity at 4.15 and 8.3 CR. The sharp, high amplitude peak at

4.2 CR lies between the half periods of Venus’ orbit and sidereal rotation, which are four days apart.

7. Conclusion

Harmonic ratios between the planets orbital periods are the principle cause of their

quantised semi-major axes. These ratios also affect the rate at which planets rotate, which

sets their LOD. The discovery of simple ratios of LOD between planets further underlines

the resonant nature of the effect which quantises their relations. These resonances also

affect the Sun, which has developed a differential rotation in response to the resonant

forces to which it is subjected by the planets, whose orbital elements may be modulating

the resonant periods. Cyclic variations induced in the rate of rotation of various latitudinal

plasma belts on the solar surface affect its activity cycles such as the Hale, Schwabe and

Gleissberg cycles which are found to be in synchronisation with planetary alignment cycles.

(Wilson 2013b). Further research is required in modelling the resonant frequencies present

and studying their resultant interactions in order to better understand the magnitudes of

inertia and damping present in the oscillating subsystems which constitute the rotating solar

surface layers.

Figure 2. Comparison of GCR measurements over Carrington ro-

tations with planetary frequencies.

rotations: 113×25.84 days, 108×27.06 days, 88×33.18 days,

and 83×35.18days.

5 Discussion

A physical mechanism linking solar rotation rates with plan-

etary rotation and orbital periods may involve resonance if

the ratios are 1 : 2 or ratios such as 1 : 4, 2 : 3, 2 : 5, 1 : 3,

3 : 5, 5 : 8 etc. (Agnor and Lin, 2011). As a ﬁrst approximate

observation, the rotation rates of the solar equator and solar

poles are in a 2 : 3 ratio.

The average of the periods relating to Venus, 25.84 and

33.2days, is 29.51days. This is very close to the Earth–

Moon system rotation period relative to the Sun (29.53days).

The ratio of 25.84 to 29.51 is 7 : 8. The ratio of 29.51 to 33.2

is 8 : 9.

The ratio of the periods relating to Mercury, 27.06 and

35.184, is 20 : 26. The average of the periods is 31.12days.

The ratio of 27.06 to 31.12 is 20 : 23. The ratio of 31.12 to

35.184 is 23 :26. There are 12 Mercury orbits in 26 periods

of 27.06days each.

These observations indicate that in addition to resonance

between the orbital and rotation periods between individual

planets and the Sun, we may hypothesise that there is also

resonance between the solar rotation rates at various latitudes

reinforcing the eﬀect. If there is an eﬀect of this resonance on

solar activity levels, we would expect to see evidence of it in

accurate TSI measurement, such as the strong peaks seen at

periods around 25–27 days and 33–35days in spectrographic

analysis of TSI (Scafetta and Willson, 2013).

6 Additional analysis

Further evidence to support the hypothesis may be found in

spectrographic analysis of galactic cosmic ray incidence at

Earth, which is also indicative of solar activity levels, and is

found to be modulated at the Carrington-period length (Gil

and Alania, 2012).

A comparison of periods at which various fractional mul-

tiples of the solar equatorial rotation rate which bring a point

on the solar equator directly between the inner planets and

www.pattern-recogn-phys.net/1/199/2013/ Pattern Recogn. Phys., 1, 199–202, 2013

202 R. Tattersall: Planetary spin–orbit coupling and solar differential rotation

the solar core, and the peaks in the galactic cosmic ray mea-

surements (A27l) is made in Fig. 2.

The coincidence of peaks in the galactic cosmic ray (GCR)

curve with multiples of the Carrington Rotation (CR) pe-

riod indicates a resonant eﬀect of this frequency (27days).

Similarly, the coincidence of other peaks in the GCR curve

with multiples of the periods at which a point on the solar

equator passes between inner planets and the solar core is

indicative of resonant eﬀects. Also shown are Mercury and

Venus orbital and half-orbital periods. Venus orbital periods

lie close to multiples of the Venus–Solar rotation periods near

the peaks in GCR activity at 4.15 and 8.3CRP. The sharp,

high amplitude peak at 4.2CRP lies between the half peri-

ods of Venus’ orbit and sidereal rotation, which are four days

apart.

7 Conclusions

Harmonic ratios between the planets orbital periods are the

principle cause of their quantised semi-major axes. These ra-

tios also aﬀect the rate at which planets rotate, which in turn

sets their LOD. The discovery of simple ratios of LOD be-

tween planets further underlines the resonant nature of the

eﬀect which quantises their relations. These resonances also

aﬀect the Sun, which has developed a diﬀerential rotation in

response to the resonant forces to which it is subjected by the

planets, whose orbital elements may be modulating the reso-

nant periods. Cyclic variations induced in the rate of rotation

of various latitudinal plasma belts on the solar surface aﬀect

the Sun’s activity cycles; these include the Hale, Schwabe

and Gleissberg cycles, which are found to be in synchroni-

sation with planetary alignment cycles (Wilson, 2013). Fur-

ther research is required in modelling the resonant frequen-

cies present and studying their resultant interactions in order

to better understand the magnitudes of inertia and damping

present in the oscillating subsystems which constitute the ro-

tating solar surface layers.

Acknowledgements. The author wishes to thank the following

people for their generous assistance in the production of this

unfunded work: Stuart Graham, Ian Wilson, Roy Martin, Wayne

Jackson, Graham Stevens, Roger Andrews, and many other people

oﬀering insight and comment at “Tallbloke’s Talkshop”.

Edited by: N. A. Mörner

Reviewed by: two anonymous referees

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