3. The forces associated with Earth’s rotational velocity
3.1 Rotational behaviour of the Earth
Despite the apparent paucity of published research work on the influence of the rotation of the Earth on tectonic activity, there have been notable contributions on the rotational behavioural of the rotating planet. Waller & Home51 considered the rotating Earth as a non-homogenous shell that comprises an inner mantle which in turn surrounds a semi molten outer core, and a solid inner core. They further considered the core as being subject to dynamic heated convection currents as well as having a different rotational velocity to the upper layers. Sager & Koppers42 described the movement of the Earth’s spin-axis from as far back as the late Cretaceous. The movement of the Earth’s spin-axis referred to by the authors as an ‘apparent polar wander path’ (APWP), is of the order of 3°-10° per million years. Sager & Koppers42, Kearney and Vine22 as well as Courtillot and Besse6 suggested that this phenomenon might be the result of changes in the Earth’s principal axis of inertia caused by the redistribution of mass in the mantle. The literature survey did not uncover viable agreed explanations regarding both the origin of the variable tilt angle of the Earth’s axis (22.1°-24.5°) as well as the reasons for the Milankovitch precession movement cycles (Figs 7 & 8).
Laskar et al.24 suggested that:
The gravitational pull of the moon on the Earth has stabilised the tilt deviation to the order of 1.3° and
The absence in the case of Mars of a stabilising gravitational force by a relatively large moon has allowed its axial tilt to vary from 10° to 60° in a manner over tens of millions of years.
Following the observations of the uplifted sedimentary sequences (Fig 1a & 1b) in the Andes it became apparent that the forces associated with the continuous unidirectional northward movement of Pangea from the Permian to the Jurassic, followed by the westward movement of the American plates and the north-east movement of the Indian and Australian plates (over 275 Ma, would have to be constant over this large geological time span.
It was this observation that prompted the investigation of the forces associated with the constant rotational velocity of the Earth. The most notable observations were the Milankovitch cycles30,31 (Fig 7 & 8) which display cycles in:
The variation in the eccentricity of the Earth’s orbit30 (over 100,000 years)
Oscillations in its degree of axial tilt between 21.5° and 24.5° (over 41,000 years) and
The precession (‘wobble’) of its axis as it changes from pointing towards Polaris (the North Star) to Vega then back to Polaris (over 23,000 years).
Taken together with the Chandler and other minor cyclical ‘wobbles’ the rotating Earth displays very similar characteristics to the mechanical behaviour of a rotating shaft with an unbalanced load2,24. The ‘Chandler Wobble’ (3-15 metres at the North Pole) which is superimposed on the other wobbling motions has a rotation period of 433 days. The wobble is not unlike that of a spinning toy top. The following simplified diagrams are given to demonstrate the similarity between the end motions of the unbalanced shaft and the Milankovitch cycles.
Fig 9 shows the damaging effect on journal bearings accommodating an unbalanced tilted shaft rotating around its mass centre rather than the designed geometrical centre line. Fig 10a shows the similarity the unbalanced rotating toy top and the unbalanced rotating earth accommodating an offset centre of gravity. Fig 10b shows the end view of the elliptical path taken by an unbalanced rotating circular body and Fig 10c the instrumentation surrounding it, to compute the position and magnitude of the counter-balance weight needed to affect balance and remove the inclined tilt. This motion plotted in Fig 10d shows a similarity to the Milankovitch precession cycles in depicting the elliptical movement of an unbalanced rotating shaft whose COM is offset from the centre of rotation. Fig 10d, also, shows a typical plot of the vibrational movement along the length of the unbalanced rotating shaft and Fig 10e will show the animated vibrational motions in PowerPoint45.
An everyday example is the balancing of a motor vehicle wheel from the measurements taken at the test positions (Fig 10b) to ensure a smooth ride when a new tyre is fitted. There are International Standards such as ISO 1940-1:2003 Mechanical Vibration, relating to the equations and methods adopted to dynamically balance rotating machinery such as flywheels, ship’s propellers, motor armatures, etc. the equations are also well documented in almost every textbook on applied mechanics26,33,41.
3.2 Determination of position of offset Centre of Mass (COM)
In order to try and determine a possible source or cause responsible for the planet behaving like an unbalanced rotating body, some principal features of global tectonic activity need to be considered. As the ratio of the mass of the crust to the body mass of the Earth is small, the crust’s surface position will have a negligible impact on the Earth’s Moment of Inertia. It thus seemed sensible to try and determine the COM of the earth and use that value to estimate the ‘differential circumferential stress forces’ (DCSF) created in the Earth’s lithosphere. With reference to Fig 11 the following observations are noted: (a) the geologically quiescent African Plate shows the characteristics of being in tension in that while there no evidence of subduction (except in the north), splitting of the plate is taking place at the Rift Valley. In contradiction, (b) the Pacific Basin with its deep peripheral trenches, crumpled topography (west of the Hawaiian chain) and subducted areas of the lithosphere (e.g. the Nazca plate under the South American plate) have all the appearances of being in compression.
Fig 11: Convergent & Divergent boundaries
If as postulated above, the Pacific Basin is under compression whilst the African Plate is under tension, then an unbalanced rotating body model requires the COM to be positioned east of the spin axis (as we view Fig 12) but ‘west’ of the Rift Valley. This is in keeping with the mechanics of rotating unbalanced bodies (as described in Section 5 and annotated in Figs 18 & 20, in which the ‘lower mass’ side will be in compression, and the ‘higher mass’ side will be in tension. In attempting to determine the possible position of the COM, consideration was given to the physics relating to the phenomenon referred to as isostatic equilibrium. Essentially isostatic equilibrium calls for the balancing of forces (associated with different weights on different areas) acting against each other through a fluid column. A hydraulic jack is a common everyday example. In the case of Earth movement, isostatic equilibrium is associated with the balancing of forces due to different weights of landmasses in proximity. An example is the post ice age net uplift (rebound) in Fennoscandinavia which is still rising by up to 8-10 mmyr-1 following the disappearance of the Northern European ice sheet44. This reached its maximum volume c.23,000 years ago, depressing the crust/mantle under its weight. If this principle can be invoked on a global basis (Fig 13), then the ‘column’ supporting the lighter Pacific Plate will need to be longer than the opposing ‘column’ supporting the heavier African Plate with its larger mass of continental crust. In doing so, the following equation can be derived to give a simple approximation of the position of the COM by considering the difference in average elevation between the oceanic crust of the Pacific Basin and the continental crust of the African continent to be 8 km. For ease of explanation the densities of the mantle and outer core is assumed to be constant.
Taking rounded values, we have:
Average elevation difference between the Pacific Basin and African continent= 8 km
R= radius of Earth= 6400 km
ρ crust = density of crust= 2.8 kgm-3
ρ core =density of the core = 10.7 kgm-3
X-sectional area of columns= 1km2
E= distance (km) from the core centre to the balance point
Thus, the weight of the 1km2 Pacific Column to the balance point = (6400-8)x1x 2.8+Ex1x10.7= 17897.6+10.7xE
Similarly, the weight of the 1km2 African Column to the balance point = (6400)x1x2.8= 17920.
Solving for E, at the balance point we get 17897.6+10.7x E= 17920
This resolves to give the E= (17920-17897.6)/10.7= 2.09 km
For ease of calculating the circumferential forces at the Earth’s surface, the COM E is placed 1.0 km off-centre from the axis on the African plate side. Although this extremely small but feasible displacement of the COM from the centre of rotation is of the order of 0.5 to 1.0 Km, or 0.015% of the Earth’s radius, the actual magnitude of the subtended surface forces as shown by the analysis are substantial.
3.3 Analysis of an unbalanced rotating planetary body
The proposed mathematical models relate the magnitude of the circumferential forces in the outer rim to the unbalanced Earth rotating about its COM which is offset from the axis of rotation. This assumption has met with resistance on the basis that the generally held consensus is that the Earth and other planets are considered as freely rotating bodies about their COMs which is co-incident with their axis of rotation. Using these assumptions, the moment of inertia would be zero as would be any subtended forces at the surface of the planet. There is thus a notable absence of serious published study on this subject. Consideration of Kepler’s second law regarding the variable gravitational pull of the sun on the Earth (Fig 14) as it moves through a full elliptical orbit clearly demonstrates the cyclical speeding up and slowing down of the orbital velocity. This occurs both on the movement towards and away from the perihelion. Planetary movements are thus directly controlled by the mutual gravitational pull between the planets and the Sun and as such cannot be considered as freely rotating bodies. In Appendix 1 ‘Consideration of the Rotational Behaviour of the Sun and Planets’ this argument is extrapolated to suggest that a common mechanism exists which causes the planets (except Venus) to rotate in the same anti-clockwise direction as the Sun’s rotation (Fig 16). The term ‘Gravitational Connecting Crank’ (GCC) is now introduced as a possible explanation.
As it is not possible to cause rotation by any force acting solely on the dimensionless centre line of any object, rotation can only be initiated by the application of an offset torque force. This concept is illustrated in Fig 15 showing the movement of a circular object on a spindle via an offset torque force.
This concept also brings with it the exciting and unexpected conclusion that the COMs must be ‘off centre’ in order that the gravitational pull from the Sun acting on the offset COM, will in fact provide a torque moment to affect rotation and in doing so, the axis of an unbalanced rotating planet is established. This approach may also explain that the extremely low rotational velocity (58.646 Earth days = 1407.5 hours) of the planet Mercury compared to <25 hours for the other six plants except Venus is due to the possibility that the COM and the axis of rotation are almost co-incident. This is noted by its low tilt angle of 0.034° and thus a noticeable absence of an offset COM for the Sun’s gravitational pull to act on.
The gravitationally driven unbalanced rotating planets with their offset COM’s will also tilt towards the heavier side and vibrate or ‘wobble’. An everyday example is the need to re-balance a vehicle wheel after fitting a new tyre. This ‘wobbling’ action of the planet Earth will manifest itself by the precessional behaviour of its axis of rotation and is noted as one of the Milankovitch cycles. From Fig 16 shows that all the planets have a similar inclined axis of rotation to a greater or lesser degree. As their rotational velocity is driven by the Sun’s gravitational pull on their ‘offset COM’s’, they should all exhibit Milankovitch type precessional cycles.