Role of EarthMoon rotational dynamics in the shaping of the surface of our planet
* published posthumously

Published:May 03, 2022
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Marcus P. Borom*, 2022. "Role of EarthMoon rotational dynamics in the shaping of the surface of our planet", In the Footsteps of Warren B. Hamilton: New Ideas in Earth Science, Gillian R. Foulger, Lawrence C. Hamilton, Donna M. Jurdy, Carol A. Stein, Keith A. Howard, Seth Stein
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ABSTRACT
The age of the Moon (1.55–1.78 b.y. old) as calculated from its regression as a function of geological time is much younger than the currently accepted age (ca. 4.52 Ga) determined by radiometric dating of lunar samples collected by Apollo astronauts. This discrepancy has posed a serious challenge for planetary scientists to account satisfactorily for the formation and subsequent breakup of Pangea. Conventional orbital models of the EarthMoon system cannot explain why Pangea formed on only one hemisphere of Earth, whereas this study’s proposed twostage rotation model can provide a plausible explanation. Calculations and a plot of the EarthMoon separation distance against geologic age suggest that, during their first ~3.0 b.y., Earth and the Moon were mutually tidally locked, rotating as an integrated unit about a barycenter (designated as stage I rotation). Beginning 1.55 Ga, however, Earth disengaged from its tidal lock with the Moon and entered its current orbital mode (designated as stage II rotation). The dynamics associated with the two rotational modes of the EarthMoon system throughout Earth’s history are hypothesized to constitute the driving forces for the migration and coalescence of landmasses during stage I rotation to create Pangea, and its ultimate breakup and drifting during stage II rotation.
INTRODUCTION
When Alfred Lothar Wegener formulated the “theory of continental drift” early in the twentieth century (Wegener, 1920), he stunned the scientific community by proposing that all Earth’s continents were derived from the breakup of a single supercontinent, which he named “Pangea.” His theory and ideas were initially rejected and then ignored for decades. Beginning in the mid1950s, however, abundant new data and findings from oceanfloor mapping (Hess, 1954, 1955, 1960) and other geoscientific studies revived Wegener’s theory, ultimately leading to the development of the “theory of plate tectonics.” While plate tectonics is now widely accepted as the basis for continental drift (Kious and Tilling, 1996), some key questions remain either unaddressed or poorly answered, such as: (1) Why did Pangea form on only one hemisphere? (2) What caused Pangea to fracture only 200 m.y. ago? (3) What caused the fractured, continental components to drift apart? (4) What were the driving forces for each of these processes, and were they different?
By 1956, Earth had already been determined to be ~4.6 b.y. old (Patterson, 1956), but the age of the Moon was still a mystery. Geophysicist Gordon MacDonald (1964) sought to determine the age of the Moon by developing an expression to describe the evolution of the lunar orbit by calculating the effects of tidal friction on Earth’s rotation. As tidal friction slows Earth’s rotation, conservation of angular momentum requires that the loss in Earth’s rotational angular momentum must be compensated by an increase in the orbital angular momentum of the Moon. Both the orbital period of the Moon and the EarthMoon separation distance, therefore, increase as Earth’s rotational angular momentum diminishes. MacDonald (1964) described the effects of this momentum transfer through a complicated derivation that produced the following simple expression,
where r is the mean EarthMoon distance, t is time, and k is a constant dependent on Earth and Moon parameters and forces of tidal friction, where
in which m is Moon’s mass, M is Earth’s mass, R is Earth’s radius, l is an effective Earth + ocean, tidal Love number, Q is a dissipation equality factor, and
where δ is the tidal lag angle, and µ = G (M + m) is the combined gravitational acceleration of the EarthMoon system. Note: The use of Equations 1–3 is restricted to circular orbits and an invariant k. All associated calculations in this manuscript follow the orbit simplification provided by MacDonald (1964).
By integrating Equation 1 and using reasonable variations in the constant k, MacDonald (1964) was able to plot lunar separation distance versus geological time and thereby bracket the age of the Moon between 1.55 Ga and 1.78 Ga. A curve similar to one obtained by MacDonald is shown in Figure 1 as stage II rotation. MacDonald’s prediction of lunar age was not initially questioned simply because there were no other data for comparison. However, using radiometric dating of lunar samples collected by Apollo missions 11–17 (1969–1972), the age of Moon was determined to be much older (ca. 4.52 Ga). This major finding spurred numerous attempts to modify Equation 1 to produce an EarthMoon coincidence consistent with a lunar origin ca. 4 Ga (e.g., Bills and Ray, 1999; Wisdom, 2006; Čuk and Stewart, 2012). The results of these attempts were mixed, but they all assumed that, throughout its orbital history, the EarthMoon system has functioned only with the rotational mode of the present, tidally locked Moon orbiting an independently rotating Earth. The basic assumption of a single rotational mode is challenged in this study.
In contrast to the single MoonEarth rotational mode approach referenced above, a model similar to the one described in this chapter was proposed earlier by Grjebine (1980). However, Grjebine’s model (1980) predated acceptance of the “giant impact” origin for the formation of the Moon, and it is both mechanistically different from and has a mutual tidal lock covering a shorter time span than that proposed in this work.
RESULTS AND DISCUSSION
Beginning Point: The Formation of the Moon—The Giant Impact
At present, the most widely accepted explanation for the formation of the Moon is the “giant impact” model of Hartmann and Davis (1975). In this hypothesis, an Earth orbit–crossing planetary body, approximately the size of Mars (given the name Theia), impacted Earth during the early stage of formation of our solar system ~4.56 b.y. ago. Though not immediately embraced by the astrophysical or geological communities, the giant impact model became more widely accepted almost three decades later following the computer model of Canup and Asphaug (2001), which provided a stepbystep explanation of the impact processes summarized below:
Molten material, having about the same bulk density of the impactor, was ejected into an equatorial disk extending beyond the Roche radius (limit).
The impact imparted a 5 h spin on the rotating Earthdebrisdisk system.
The Moon formed by agglomeration of the molten particles in the debris disk, just beyond the Roche limit at a distance of 22,000 km from Earth’s center.
During this initial period, both Earth and its debris disk were rotating symmetrically around Earth’s geometric axis. As Earth gradually reacquired debris from within the Roche radius, and the debris beyond the Roche radius agglomerated to form the lunar body, the EarthMoon system began to rotate about its barycenter, regardless of whether tidal locking occurred or not. Such reconfiguration contributed to the slowing of the rotation rate of Earth as well as the orbital rate of the Moon. Neither the original model of Canup and Asphaug (2001) nor the more detailed simulation of Canup (2004a, 2004b) directly addressed the rotational mode of the EarthMoon pair. Finch (1982), however, had previously suggested that, during their early history, the EarthMoon system likely rotated in lock step around the system’s center of mass, i.e., they were mutually tidally locked in a geosynchronous orbit.
During this planetsatellite configuration, both Earth and the Moon remained in a molten state for around 200 m.y., during the remainder of the Hadean Eon. At a separation distance of around 22,000 km, the Moon would have imposed a gravitational acceleration on Earth greater than that imposed by Earth on the currently tidally locked Moon. Such a force would have been sufficiently large enough to mutually tidally lock the EarthMoon system. These forces would have contributed to a slowing of Earth’s rotation, and ultimately to achieving a mutual tidal locking as discussed by Richardson and Walsh (2006), and as observed for the PlutoCharon system (Stern and Mitton, 2005).
With a mutually tidally locked system, the starting point in the rotational evolution of the EarthMoon system necessarily would have been quite different from the current configuration. It should be noted that the current “conventional wisdom” is that the rotational mode of the MoonEarth system has undergone no change during its entire history. Unfortunately, a single, unchanging rotational mode provides no explanation for the formation, fragmentation, and drifting of the supercontinent Pangea within a platetectonics context. To remedy this shortcoming, I propose a hypothesis that challenges “conventional wisdom,” namely, that the orbit of the EarthMoon system has experienced two different rotational stages: Stage I was a mutually tidally locked EarthMoon system rotating about its barycenter; and stage II was (and is) a tidally locked Moon orbiting an independently rotating Earth.
Stage I: Initial Rotational Configuration of the EarthMoon System: Geostationary, Mutually Tidally Locked, Barycentric Rotation
From a simple calculation, based only on the radii of Earth and the Moon and their mass ratio of ~81/1, and the EarthMoon separation distance proposed by Canup and Asphaug (2001) of 22,000 km, the calculated barycentric, rotational axis of the pair would have initially lain ~270 km from the center of Earth toward the Moon along the line drawn through the centers of mass of the two bodies. As discussed later, this seemingly minor offset of the rotational axis would have been sufficient to generate a migration of surface material to the far side of Earth. As presented by Richardson and Walsh (2006), Newton’s form of Kepler’s third law can be used to determine the initial rotational period of the Moon in a geosynchronous Earth orbit. When either the separation distance (orbital semimajor axis), r, or the orbital period of the satellite, P_{geo}, is known, the other can be calculated using:
where G is the gravitational constant, and M is the combined mass M_{Earth} + M_{Moon}.
With this equation and using the initial r value of 22,000 km specified by Canup and Asphaug (2001), an initial synchronous period P_{geo} is calculated to be 9.091 h. This is in contrast to the proposed initial 5 h spin of the equatorial debris disk. The process of agglomeration of disk debris into an orbiting, spherical satellite along with the forces required to slow down the orbital rate of the satellite to match the requirements of Kepler’s third law, as expressed in Equation 4, could also slow down the rotational rate of the planet so that the two would match.
During a geosynchronous, mutually tidally locked rotation, both bodies would orbit their barycenter about a common axis and maintain the same surfaces facing one another. This mode will be referred to as stage I rotation. When Earth disengages from the gravitational acceleration forces of the Moon and begins to rotate independent of the Moon, that rotational mode will be referred to as stage II rotation.
To calculate the orbital evolution of the Moon over its entire history, one must recognize that Equation 1 for the orbital evolution of the Moon remains applicable for both proposed stages I and II, since the orbital evolution for each stage is driven by tidal friction, but with extremely different values of dissipation equality factor (Q).
The current recession rate of the Moon, as determined by laserranging measurements on cubecorner reflectors placed on the Moon by astronauts, was reported by Dickey et al. (1994) to be (3.82 ± 0.07) × 10^{−2} m/yr. Using this value of the recession rate and Equation 1, the constant for the independent rotation (denoted as subscript “ir”) of Earth, k_{ir}, is determined to be k_{ir} = 6.286 × 10^{45} m^{6.5}/yr.
To calculate the recession rate constant for a geosynchronous Earth orbit, k_{geo}, one must backcalculate to the time at which mutually tidally locked barycentric rotation ceased by integrating Equation 1 to yield Equation 5:
where t_{1} is the time at maximum separation, r_{1}, and t_{0} is the time at minimum separation, r_{0}.
By using Equation 5 and the beginning and ending parameters for independent rotation, one can calculate the age of Earth at the beginning of stage II rotation to be:
Once again using Equation 5 and the values for the beginning and ending parameters for stage I, one can calculate:
Equation 5 with the new values for k is a valid expression for mutually tidally locked barycentric rotation because tidal friction is still the factor driving the reduction in rotational speed. There is now sufficient information to calculate the MoonEarth separation distance as a function of geologic time. Recession of the Moon versus geologic time is calculated using Equation 5 to cover the entire geological history of the Moon using different constants for each section. The combined plot for both stage I and stage II rotation is shown in Figure 1. Selected portions of the data for each rotational stage are given in Tables 1 and 2.
Tables 1 and 2 were generated by first manually selecting EarthMoon separation distances covering the entire orbital evolution. These separation distances then were used to calculate the orbital periods using the appropriate equations shown at the bottom of each table. The geologic times were calculated from the separation distances using Equation 5 and the value of either k_{ir} or k_{geo} appropriate to the rotational stage being calculated.
During stage I, since Earth was not rotating on its axis with respect to the Moon, there were no lunar tides. During this mutual tidal lock, however, solar tidal forces were still at play. Because the EarthSun orbital configuration has not changed appreciably during the evolution of the solar system, the solar tides in the past would have had essentially the same minormagnitude values as are felt on Earth today. Even though these solar tidal forces are small, they nonetheless would have contributed sufficient tidal friction to gradually slow the spin of the binary pair during stage I, with the result that both the separation distance and the orbital period would have, as shown in Figure 1, slowly increased to conserve angular momentum.
With passage of geologic time, the EarthMoon separation distance would have increased, and the barycentric axis for the geostationary EarthMoon system would have moved farther from Earth’s center toward the Moon, reaching an offset of as much as 427 km at the end of stage I rotation (Table 1).
Stage II: Decoupling of Earth from Its Tidally Locked Relationship with the Moon
As EarthMoon separation increased during stage I, the gravitational acceleration of the Moon ultimately would have become too weak to impose its slowing rotation rate on that of Earth, and Earth’s rotation would have unlocked from that of the Moon. After unlocking, Earth would have begun to rotate on its own axis, independent of the previously cojoined rotation of the pair. The system then entered the current rotational mode designated as stage II rotation.
I am unaware of any published study that defines the parameters required for determining the separation distance at which a mutually tidally locked planet and satellite would unlock. In the absence of such information, I have conservatively chosen a separation distance of 35,000 km for the unlocking point, which yields a rotation rate determined from Equation 4 for the binary planet of 18 h. A separation distance of 35,000 km establishes the transition from stage I, with no lunar tides, to stage II, with lunar tides. Beyond this transition point, the Moon continues to be tidally locked to Earth and continues to rotate only once per lunar orbit.
When the rotation of Earth decouples from that of the Moon, Earth continues to experience constantly moderate solar tides, but it begins, for the first time, also to experience lunar tides. The initial lunar tides, due to the small separation distance, would have had a much greater amplitude, but lower frequency than the tides of today (see Table 2). The frequency of the tides would have increased rapidly over the first 200 m.y. of stage II as the difference between the orbital period of the Moon increased relative to the rotational rate of Earth, but the magnitudes of the tides would have dramatically decreased.
Stage I: Centrifugal Migration of Continental Material to the Far Side of Earth
During the indicated ~3 b.y. of stage I, Earth underwent eccentric rotation about a barycenter, which, by 1.55 Ga, had become offset from Earth’s axis by as much as 427 km (see Fig. 1; Table 2). As the surface cooled, the uppermost layers of the lithosphere began to solidify, separating into hot, isolated masses (“islands”) of solidified, continentbuilding, lowdensity, felsic minerals. These isolated, felsic islands floated on a molten, viscous underlayer of more fusible, higherdensity, mafic composition and migrated under the influence of centrifugal forces associated with increases in the barycentric radii around Earth’s circumference (see Fig. 2). Over geologic time, these migrating islands coalesced to form protocontinents that ultimately accumulated on Earth’s far side to form the supercontinent Pangea.
The force responsible for the migration of continentbuilding materials along latitudinal lines was the tangential component of the barycentric, centrifugal force arising from the EarthMoon system’s rotation about the barycenter (see Fig. 2). The migration of the protocontinental islands left behind thinner layers of denser materials that formed when the ambient temperature reduced sufficiently to promote solidification. As Earth’s surface temperature continued to lower with time, the formation, aggregation, and migration of continentbuilding masses by centrifugal forces became more sluggish, but this process was neither improbable nor impossible. Material transport could have continued through the migration of tectonic plates driven by the centrifugal gradient. Over time, these protocontinents could have been driven together ultimately to coalesce to form the supercontinent of Pangea. According to this model, migration of continentbuilding materials would have continued throughout the entire period of stage I rotation.
Stage I: Validation of Centrifugal Migration of ContinentBuilding Material
Earth experiences semidiurnal tides, with two high tides per day, but the reason for Earth’s tides is complex and controversial. Some investigators (e.g., NOAA, 1998), including the author, consider that the tides on the side of Earth away from the Moon are generated by centrifugal forces derived from rotation of the EarthMoon system about its barycenter, and that tides on the side of Earth facing the Moon are due to lunar gravitational acceleration. The current barycenter lies ~4688 km from Earth’s center of mass toward the Moon and on the line through the centers of mass of the two bodies (see Fig. 2). However, other investigators attribute both of Earth’s tidal bulges to the first differential of lunar gravitational acceleration (i.e., that the acceleration is inversely proportional to the cube of the separation distance). This mathematical construct is simply an attempt to minimize the effect of solar gravitational acceleration. If the undifferentiated equation for tidal forces is used (i.e., that the tidal force is inversely proportional to the square of the separation distance), the contribution of the solar component overwhelms the lunar component.
Equations for the calculation of the strength of tidal forces generated by both the barycentric, centrifugal and the lunar gravitational accelerations operative during stage I and stage II rotations are given below.
The equation for tidal acceleration (ga; Lowrie, 2007, p. 52), where negative values indicate an action directed away from the Moon, is:
The equation for barycentric, centrifugal acceleration (BCA), where negative values indicate an action directed away from the Moon, is:
where F is centrifugal force, V is tangential velocity, a is acceleration, t is time of tidal orbit, m is mass affected (water or crust), and R′ is rotational radius, i.e., Earth’s radius plus ∆r. Therefore:
and
where d_{bc} is the distance of the barycenter from Earth’s center.
Results of the calculations using these equations are given in Tables 1 and 2. The results in Table 1 show the lunar gravitational acceleration and the barycentric, centrifugal acceleration for both the near and far sides of Earth. The barycentric, centrifugal force can be represented by two components, one normal to Earth’s latitudinal tangent, and one parallel to Earth’s latitudinal tangent (see Fig. 2). The primary, centrifugal force vector will be at an acute angle to the tangent, in every position along latitudinal lines except two, generating a tangential component for transport of landmasses during the proposed, almost 3 b.y. period of geostationary rotation during stage I. Table 1 shows that, during stage I, lunar gravitational acceleration is less than centrifugal acceleration by more than an order of magnitude, further indicating that centrifugal acceleration is favored over lunar gravitational acceleration for transport of continentbuilding material. Significantly, the values in Table 1 also show that the barycentric, centrifugal acceleration on Earth’s early lithosphere was more than ~800–3000 times greater than that currently exerted on Earth’s surface—more than sufficient to migrate landmasses to the far side of Earth during stage I rotation. The centrifugal force during stage I was persistent, directed, and notably affected by Earth’s rotation about its barycenter.
Stage II: Earth’s Rotation Unlocked from the Moon and Initiation of Lunar Tides
During the first 200 m.y. of stage II rotation, tidal forces would have been dominant and strong enough to drive marine tides to amplitudes initially exceeding 800 m. In addition, the forces would have been great enough to produce bodily Earth tides capable of disrupting continental landmasses and generating volcanic activity (see Table 1). Consider that the magnitude of lunargenerated bodily Earth tides (Melchior, 1974) is related to the magnitude of marine tides (Agnew, 2007). Bodily Earth tides at the beginning of stage II rotation, when referenced to presentday tidal forces, would have been capable of producing solid Earth material oscillations of between 200 and 400 m. These enormous tidal oscillations, which progressed around Earth’s circumference normal to Earth’s longitude as a semidiurnal wave of solid Earth material, would have had an effect similar to earthquakes of magnitudes greater than 8 on a daily basis. Clearly, these forces would have been sufficient to deform and easily alter continentbuilding masses.
Breakup of Pangea
During the later period of stage II, as tidal forces became significantly less, rotational imbalance became the dominant force acting on Earth’s lithosphere. The imbalance occurred because the landmasses, which had all accumulated on one side of the planet through centrifugal transport, essentially behaved like misplaced weights on a tire being balanced. Misplaced weights displace the center of mass from the rotational axis and result in system vibration, wobble, and/or precession. The locations of the supercontinent Pangea and the diametrically opposed thinner, lowerdensity lithosphere covered with lowdensity ocean water on the opposite side of Earth must have changed to adjust this imbalance. In such a scenario, it is interesting to note that the plates have spread apart in the Atlantic Basin along the MidAtlantic Ridge, and they have moved, initially, in both eastward and westward directions toward the Pacific Basin. Fragmentation of Pangea and movement of the components in opposite directions were the only ways to achieve a balanced rotation of Earth. Any other movement would have enhanced and continued the rotational imbalance. As the point of dynamic balance is approached, the movement of the tectonic plates should become less focused, as they approach the locations for a final state of balance. Once rotational balance has been achieved, movement of tectonic plates will cease, and volcanism will diminish. Carbon dioxide will continue to dissolve in the oceans and be converted to insoluble carbonates. Without volcanism to recycle the carbonates, Earth will lose its CO_{2} blanket, and the oceans will freeze. Ice will sublime, and Earth will lose its water and become lifeless.
From the results of this study, rotational rebalancing is proposed as the root cause for the movement of the tectonic plates during stage II rotation, even though Earth currently experiences several rotational/orbital modes: Earth is orbiting the sun with one, gyroscopically maintained retrograde rotation per orbit; Earth is being orbited by the Moon around the EarthMoon barycenter; and Earth is rotating independently about its geometric axis. Rotation about its geometric axis, in competition with its rotation about the barycenter, is the source of the vibrational imbalance.
An illustrative, engineering example of why the continents are drifting can be found in commercially available dynamic balancing rings for both automotive tires and aircraft propellers. Rotational imbalance can pose a problem with automotive wheels as well as with airplane propellers. Commercial dynamic balancers are available that attach either to the rims of automotive wheels or to the hubs of aircraft propellers, normal to the axis of rotation (see, for example, the video posted on the web by Dsuban [2009] or the video supplied by Centramatic at https://vimeo.com/69169651). Applying this wheelbalancing analogy to the case of Earth’s rotation about its axis, one can view the tectonic plates themselves as the balancecompensating elements, readjusting themselves on a global scale. It is interesting to speculate that Antarctica perhaps has removed itself from the rebalancing process by becoming located directly on the polar axis.
CONCLUDING REMARKS
This study proposes that the orbital evolution of the EarthMoon system involved two distinct rotational modes: stage I, ca. 4.52 Ga to 1.55 Ga, during which the EarthMoon system rotated as a binary planet around a barycentric axis; and stage II, 1.55 Ga to present, when Earth rotated unlocked from the Moon while the Moon continued to orbit Earth as a tidally locked satellite. The discussion herein emphasizes that these two differing rotational modes generated forces that have affected the motion of Earth’s tectonic plates, with particular reference to Pangea. The rotational stresses operative in stage II of the proposed twostage rotation model serve as the primary driving forces for the movement of tectonic plates, rather than the generally invoked convective currents within the asthenosphere. Approximately 200 m.y. after the beginning of stage II, stresses associated with tidal forces had diminished to a point that stresses associated with rotational imbalances began to predominate, which initiated the breakup of Pangea.
After Pangea fragmented, the continental components began to move both to the east and to the west in an attempt to establish rotational balance. According to this model, continental drift will continue until Earth achieves—in the distant geological future—a final state of balance, at which point drifting will cease. The chronology of Earth’s crustal movements and associated phenomena during stages I and II is summarized in Table 3.
This study focused on Pangea because it is Earth’s youngest, bestdocumented, and most widely accepted, supercontinent. I recognize that a number of other supercontinents predating Pangea have been postulated (e.g., Pisarevsky et al., 2003). However, because of the dearth of pertinent geologic data for Earth’s oldest rocks, testing of this twostage rotational model for older supercontinents must await the outcome of future studies.
It is noted that the absence of lunar tides during stage I rotation provided calm, shallow zones along ocean shorelines. These calm waters would have provided favorable conditions for the proliferation of cyanobacteria (e.g., Buick, 2008; Hazen, 2012), which photosynthetically convert CO_{2} and H_{2}O into carbohydrates with the liberation of oxygen. This significant biological alteration of Earth’s atmosphere opened the gateway for the evolution of all airbreathing creatures. The presence of cyanobacteria during stage I is evidenced by the fossil record of stromatolites. The proposed dramatic tides associated with the beginning of stage II rotation offer an explanation for the rapid reduction in cyanobacteria as evidenced by the reduction of stromatolites in the fossil record.
ACKNOWLEDGMENTS
I gratefully acknowledge the following individuals: Robert I. Tilling, scientist emeritus of the Volcano Science Center, U.S. Geological Survey, for his encouragement and support in completion of this paper; my former colleagues Glauco Romeo, corrosion chemist, and Robert C. DeVries, GE Manmade Diamond Team, for their helpful critiques and advice; and my wife, Aida, for her longsuffering patience during my yearslong efforts in writing this paper. I express appreciation to the reviewers for their critical comments and to Volume Editor Donna Jurdy for her guidance and persistence during the long process in securing the success of this entire volume.