User:Walter Baltensperger/Sandbox

Motivation
During the Last Glacial Maximum, about 20'000 years ago, arctic East Siberia was not ice covered, while in the west the ice sheet reached the region of New York. This asymmetry suggests that the North Pole was situated somewhere in Greenland. If this was the case, the North Pole must have moved geographically to its present position in the Arctic Sea at the end of the Pleistocene, about 13'000 years ago. This problem motivated theoretical investigations since the end of the 19th century with negative result. However, a recent study by W. Woelfli et al. claims that a scenario for a rapid geographic polar shift compatible with laws of nature and observations exists. An example of a proposed polar shift is shown in Fig. 1.



Earth's shape
If the Earth were a perfect sphere, an arbitrarily small deformation, e.g. a slight stretching in a direction at a finite angle to the rotation axis, would lead to a geographic wandering of the poles around that direction. In space the polar axis would stay fixed. The geographic motion of the poles would be produced by rotations of the globe. This follows from the law of free motion of a rigid body.

Earth's radius at the equator is larger by 21 km than at the poles. The excess mass at the equator stabilizes the rotation around the axis perpendicular to the plane of the equator.

For a significant wandering of the poles, a deformation of the Earth is required, which can compete with the stabilizing exess mass around the equator. For Fig. 1, a stretching of the globe in a direction 30° away from the polar direction was assumed, such that the radius in that direction was increased by 6.5 km. A rapid deformation of the globe, compatible with the continuation of life on Earth, could be due to a close passage of a planet sized object. The Earth is stretched by the tidal force in the direction of the object. This situation was considered in, where the authors estimate that a Mars sized mass passing at about 15'000 km from Earth's center with a relative speed of about 40 km/s would produce the deformation mentioned above. While this estimate is not detailed, the narrow passage lasts about 10 minutes only, so that inertial considerations may cover dominant parts. The rapid movement of Earth's surface is truly cataclysmic and could represent  the quaternary extinction event, which killed many species of large animals.

The following two sections resume the physics which leads to the path in Fig.1.

Inertial tensor and geographic movement of the poles
The mass distribution of a solid body enters into its law of motion through the inertial tensor   $$ \Xi$$. Its components are determined from the mass density $$\rho (\vec r) $$ by the integral

$$\Xi_{jk} = \int d^3r\,\rho (\vec r) ( r_j^2 \delta_{ jk} - r_j r_k)  $$

where $$\delta$$ is the unit tensor. There exists a cartesian coordinate system fixed to the solid body in which $$ \Xi$$ is diagonal. In this coordinate system $$ \Xi $$  is represented by lengths along the three axes. In the case of the unperturbed Earth, two axes have equal length, $$ a=b $$, and they lie in the plane of the equator, while the third, $$ c $$, in the polar direction, is longer.

In a coordinate system fixed to the solid, the motion of the angular rotation vector $$\vec\omega (t)$$ is determined by Euler's equation , a vector equation, in which the square braket is a cross product:

$$\frac{d \vec{\,\Xi \omega}}{ dt} = [\vec{\Xi \omega},\vec \omega] $$

When the rotation axis coincides with the axis of largest inertial momentum $$c$$, the motion is stable. However, when the direction of the instantaneous rotation $$\vec\omega (t)$$ makes an angle with this axis, $$\vec\omega (t)$$ will precess around it forming a cone. The period of this precession is larger than the rotation period (= 1 day) by the factor $$a/(c-a) $$. For the actual Earth this precession period is about 300 days.

Deformation and relaxation
A deformation of the Earth modifies its inertial tensor. Since the Earth has its equatorial bulge, a sizable deformation in a oblique direction is required to obtain a notable change of the direction of the large axis. As an example, it has been estimated that a one per mil stretching in a direction 30 degrees from the previous large axis would modify the inertial tensor in such a way that the direction of the new largest axis had moved about 17 degrees.

A deformed globe with rotation vector $$ \vec \omega (t) $$ tends to relax into a shape with the equilibrium equator bulge in the plane perpendicular to $$\vec \omega (t) $$. Let the corresponding inertial tensor be $$ \Xi_0 [\vec\omega (t)] $$. What returns to equilibrium is the Earth with all its layers, and with variing intensity this process covers a range of times. The authors of use a single relaxation time  $$ \tau $$ and apply it directly to the inertial tensor:

$$\frac{d\,\Xi}{dt} = -\frac{\Xi (t)-\Xi_0[\vec\omega (t)]}{\tau } $$

Initially, the bulge at the old equator is still the dominant deformation, while the stretching in an oblique direction moved the largest axis of $$  \Xi $$   away from the initial $$\vec\omega (t)$$. Therefore the motion of $$\vec \omega (t) $$  begins as a geographic precession around the new major axis of  $$  \Xi $$   with a period between 300 and 400 days. If $$ \tau $$ were much smaller than this period, the movement would not go far enough. This would be the case for a liquid globe, since the distances of the deformation would be covered by liquid flow in days at most. A sufficiently large $$ \tau $$ may exist, if the streching broke solid structures, which thereafter plastically deformed into the new equilibrium shape . The authors of  solved the two differential equations numerically to obtain the geographic path of the North Pole  $$\vec \omega (t)  $$. For $$\tau=1000 $$   days, the path is shown in Fig.1. The result is a geographic shift of the North pole from the center of the observed largest glaciation in Greenland to its present position in the Arctic Sea.

Postulated scenario
In the scientific community a rapid polar shift, consistent with the laws of nature and the known facts, is considered impossible. The obvious difficulty with the cited model is, that the Mars sized object does no longer exist. The authors of therefore postulate a special scenario before the pole shift, which, so they claim, can make the object disappear afterwards, within the last 10'000 years. They assume that the massive object was in an extremely eccentric orbit. In each passage near the Sun it was heated by tidal work and solar radiation. It was hot inside, liquid and evaporating. It is essential that during the close passage near the Earth it disintegrated into several fractions. Careful modeling by Erik Asphaug and Willy Benz, undertaken after the disintegration of the Levy-Schumaker comet passing near Jupiter, shows  in Fig. 13 of that a Mars sized pile of stones (and therefore probably also a fluid sphere) passing near the Earth would not disintegrate into several fractions. Perhaps this could happen to the hot planet, if the preheating of its interior brought it close to instability, so that a slight deformation from its spherical form resulted in an expansion. The assumed fractions have a reduced escape velocity, which determines the evaporation from a heavy object. Therefore, the evaporation rate of the fractions is increased sufficiently to make the disappearance of the hot planet within 10'000 years plausible. In this model the Ice Ages have ended.

Consequences of this scenario
Before the pole shift, the hot planet in its eccentric orbit evaporated, so that a disk shaped cloud of ions, which moved around the Sun, was formed. This reduced the solar irradiation on Earth, when its orbit was within the cloud. R.A. Muller and G.J. MacDonald have pointed out, that the angle of Earth's orbit with the invariant plane, i.e. the inclination, varies with a period of about 100'000 years. This is the dominant period of the Ice Age over the last 800'000 years. Therefore, Muller et al. postulated the existence of such a cloud.

The cloud itself has dynamics. The evaporation from the hot planet increases the cloud's density steadily until inelastic scatterings between ions lead to the cloud's collapse. The corresponding sequence of a gradual reduction of the global temperature followed by a rapid increase can be identified with  a Dansgaard-Oeschger event.

The dust inclusions in the ice of Greenland and of Antarctis  are vastly denser during cold periods, when Earth's orbit is in the disk shaped cloud. This refers to small grains with diameters less than 4 microns. They have a consistent size distribution and may have formed in Earth's atmosphere. The larger grains have been carefully examined and were found to be terrestrial. .

After the pole shift, the fractured objects evaporate or even erupt more material than before, however, in an orbital plane, which can make an angle of several degrees with the Ecliptic. Apart from sporadic effects, the mean influence of this material on Earth is therefore small.

Conclusion
A claim exists that a rapid geographic pole shift is possible. However, the conditions, which it imposes, allow for a narrow possibility only. Improved simulations are necessary. If the crude estimates were optimistic, the narrow window may close.

The model postulates an object of a type, which in the whole lifespan of the Earth was probably unique, considering that an actual collision with Earth would have stopped life. The model is therefore not backed by similar observations. On the other hand, the conditions that it requires before the pole shift lead to external influences on Earth, which reproduce basic features of the Pleistocene Ice Ages. The model has not been discussed in publications of the scientific community.