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KRONOS Vol VII, No. 2
SENMUT'S CEILING AND THE EARTH AS TIPPE TOP
To the Editor of KRONOS:
The Senmut ceiling may be subject to various interpretations, but Velikovsky and others have interpreted it as suggesting that Orion once followed Sirius across the night sky, whereas at the time of Senmut, Sirius followed Orion across the sky, just as it does today.
The change from Orion-following-Sirius to Sirius-following-Orion could have been accomplished either (1) by a reversal of the direction of Earth's rotation, from left-handed to right-handed, with the pole of rotation continuing to point in approximately the same direction, or (2) by a movement of Earth with its axis of rotation through approximately one hundred eighty degrees, with the rotation remaining right-handed. It could not have been accomplished by a tippe top movement of Earth, with the axis of rotation retaining its orientation in space, but with Earth itself moving through one hundred eighty degrees on the axis, so that each of Earth's geographical poles was now at the opposite end of the axis of rotation.
The reports that the Sun once rose in the west and set in the east could be explained either (1) by a reversal of the direction of Earth's rotation, or (2) by a tippe top movement of Earth. They cannot be explained by a movement of Earth with its axis of rotation through about one hundred eighty degrees, with the rotation remaining right-handed.
Thus my position is that the tippe top movement of Earth merits further study, but that it is neither necessary nor sufficient for Velikovsky's scenario: it is not sufficient for explaining how Orion could follow Sirius across the night sky, and it is not necessary for explaining how the Sun could rise in the west and set in the east.
Lynn E. Rose
C. Leroy Ellenberger Responds:
Prof. Rose's remarks, which were prompted by my comments on a criticism of Warlow (see below) at the Princeton Seminar in September 1981, are not only well-taken, but provide a timely opportunity to up-date the status of the "Earth as tippe top" hypothesis. As Rose correctly points out, an axial tilt of 180° will not change the direction in which the Sun rises. In 1978, Peter Warlow advanced the idea that geomagnetic reversals and the reversing of the Sun's apparent motion across the sky might be produced by the Earth flipping 180° on its spin axis through a fixed magnetic field(1) in a manner analogous to the flipping of a tippe top.(2) This insight was an outstanding example of novelty serving utility. The motion of a spinning body flipping over while its spin axis remains fixed is seldom appreciated until it is seen. This idea initially looked very worthwhile because it was thought that a tippe top-like inversion of the Earth could be accomplished much more easily than reversing the Earth's rotation. The tippe top motion, as Prof. Rose's second paragraph indicates, is not the same as an axial tilt which has traditionally been contrasted with spin reversal when discussing either the Sun standing still or anomalous periods of day and/or night.(3)
Warlow suggested that such a geographical inversion could be accomplished in 24 hours by the gravitational torque on the Earth's equatorial bulge produced by a body of 10 lunar masses passing Earth at a distance of about 5.17 Earth radii, or an Earth mass passing at about 10.35 Earth radii. In addition, drawing heavily on Velikovsky's work, Warlow collated an impressive body of evidence strongly suggesting that such inversions have occurred several times within the last 13 ,000 years.(4)
Subsequently, Eric Crew, in reporting on a Russian's calculations which suggested "that the molten core of the Earth is spinning about 17 times faster than its surface",(5) proposed that a tippe top-like inversion would occur even more easily if the Earth's crust, as a thin shell, could slip around the mantle. However, in using the terms crust and core, Crew was semantically imprecise because his usage did not distinguish core from mantle. Later, he presented some preliminary calculations supporting his shell-slipping model.(6)
The apparent ease with which a tippe top and other small scale models flip over is deceptive in considering the Earth executing the same maneuver. The advantage afforded by the Earth's slower spin, about six orders of magnitude less, is overwhelmed by its greater mass, over 24 orders of magnitude larger. A recent article by Dr. Victor J. Slabinski(7) poses a very serious challenge to all efforts to explain how the Earth could undergo a tippe top inversion. Although Warlow has known about Slabinski's criticism since March 1980, to date he has not answered it satisfactorily.(8) In fact, in his continuing posturing he has not even acknowledged Slabinski's criticism, either specifically or in general.
WARLOW'S THREE ERRORS
Slabinski identifies three errors indicating that Warlow had significantly underestimated the torque required to invert the Earth. The torque about the equatorial axis, [tau]3, that Warlow calculated, is lower than the correct value by a factor of 200 because the importance of the angular acceleration 3 on the left side of his equation 8 was overlooked. More seriously, the torque required for a 24 hour inversion was underestimated because Warlow assumed that the torques about the other two axes, [tau]1 and [tau]2, were negligible with respect to [tau]3 when, in fact, they are of the same order and cannot be neglected. Further, "Warlow's appeal to 'precessional momentum' to carry the inversion through to completion if the torque applied to the Earth falls below the required magnitude" is erroneous. Newton's laws of motion require that the precession stops when the torque stops.
Slabinski then shows that the maximum torque necessary for a postulated 24 hour inversion would require a cosmic body with a mass of 417 Earths passing at two Earth radii. This exceeds the mass of Jupiter which is 318 Earths. This result applies to a torque about the spin axis, [tau]1, which was chosen because it is more difficult to attain than [tau]3 which Warlow calculated. However, Slabinski points out: "Even if there were such a compact cosmic body that just missed hitting the Earth, its motion past the Earth . . . would not keep it within [2 radii] of the Earth's center for an hour, let alone the 24 h that the torque must act to invert the Earth on its spin axis." A passage farther away would require a more massive body to maintain the torque. Nothing in Worlds in Collision indicates that Earth was menaced by such a massive object, but the ability of an Earth sized object to invert the Earth appears to be severely undermined.
Finally, Slabinski shows "that the torque impulse required by any inversion scheme is the same as the torque impulse required for a spin reversal, that is to just stop the Earth's spin about its axis of figure (the xl axis) and then restore its spin to the same magnitude, but with opposite sense". Thus, a tippe top inversion of the Earth offers no advantage over stopping and restarting the Earth's rotation in order to explain a 180° change in the Sun's rising and setting positions. However, as Slabinski pointed out in a conversation with this writer, inspection of the relevant integrals reveals that these two mechanisms, geographical inversion and spin reversal, require a torque impulse 2/ or about 64% of that required to invert both the Earth and its spin axis, i.e., a 180° axial tilt.
SHELL-SLIPPING NO SAVIOR
Producing an inversion by slipping the Earth's crust over the mantle would appear to offer advantages compared to inverting the entire Earth since the moment of inertia of a spherical shell with a thickness equal to 1% of the Earth's radius is about 1/30 the moment of inertia of the entire Earth. Simply scaling the 417 Earth mass object by the ratio of the moments of inertia implies that a 14 Earth mass object passing at two Earth radii would be required to invert a thin crustal shell, neglecting friction.
A "shell-slipping" model, however, offers no real advantage as Slabinski showed in a private communication.(9) The "catch" is that the 24 hour duration assumed by Warlow is fictitious. In other words, a hypothetical inversion lasting 24 hours implies a certain maximum torque at the mid-point. However, as Slabinski indicated in his article, an actual cosmic body capable of inverting the Earth would pass Earth in much less than a day. The torque calculated by Warlow and corrected by Slabinski does not correspond to a realistic physical event. With a shorter period of action, the torque, and hence the mass of the cosmic body, would need to be increased in compensation. The actual requirement is truly astonishing. Slabinski's analysis shows that the gravitational torque impulse (i.e., the integral of the torque over time from minus infinity to plus infinity) required to invert Earth's crust would take a foreign body with a mass of 68 Jupiters passing at two Earth radii, neglecting friction and assuming a parabolic trajectory past Earth.
EFFECTIVE TIME OF ACTION
In following Warlow's presentation, the torque from the passage of the 417 Earth mass body was not coupled with a trajectory and its associated time of flight. The table below presents the "effective time of action" for three cosmic bodies passing Earth on a parabolic trajectory with a distance of closest approach of two Earth radii. The effective times are expressed as the duration over which the torque equals or exceeds I/n of its maximum, for n = 1000, 300, 100, 30, 10. These five thresholds were selected in the absence of specific knowledge when the effect of the torque would perceptibly affect Earth.
the Torque Is At Least 1/n of Maximum
Warlow was remiss in not tying his torque estimate and 24 hour duration to a physical trajectory. The results in the table feature the modelling of a near collision as a two body problem using a parabolic trajectory to approximate the severe perturbation produced during the event. For the close approaches examined by Slabinski, this procedure is more realistic than conceiving the near collision as occurring along the near tangent orbits of two undisturbed bodies. In order to deal meaningfully with the components of Velikovsky's cosmic scenario, physical parameters need to be handled in a consistent manner and as rigorously as practicable.
For the entire Earth or its crust to flip over solely under the gravitational influence of a passing cosmic body of any size seems impossible. The tremendous inadequacy of the models that have been considered does not appear to be surmountable by either the addition of electromagnetic effects or the expansion to higher orders of the dynamic analysis or the Earth's shape. The convection of electric charge developed by Ralph Juergens(10) neither stopped nor reversed Earth's rotation, but merely slowed it down and speeded it up with the direction of rotation unchanged. Thus, it had no bearing on the problem of how the Sun changed direction. Slabinski closed his article in J. Phys. with the following challenge: "Those who would now appeal to electromagnetic forces from a cosmic body to invert the Earth have the burden of demonstrating quantitatively that (i) an electromagnetic force can produce torques of sufficient magnitude, and that (ii) the torque components along the x1 x2 x3 axes have the proper time dependence to produce an inversion." The analyses of geographical inversions have not taken account of the fact that the moment of inertia over continents exceeds that over oceans by about one part in a thousand. However, such refinement would not be expected to affect the analyses materially.
THE DOUBLE-TOP MODEL
The prospect for explaining the ancient reports of reversed solar movement with a feasible physical mechanism would be bleak indeed were it not for a paper presented in December 1980 in Stockholm, Sweden.(11) Stig Flodmark, of the Institute of Theoretical Physics, University of Stockholm, described the dynamics of a double-top model of the Earth in which the core and mantle, as two separate solid bodies, are capable of differential movement. The model indicates that, at present, the core is rotating with approximately the same period, one day, as the shell. This geometry is supported by the discontinuous fluctuations in mass density at the core-mantle boundary and the several experimental indications that the layer surrounding the solid core is viscous.(12) The flip-flop behavior of the double-top model is caused by instabilities arising when the core's ellipsoid of inertia is prolate and influenced by the effects of fluid friction and tidal effects in the viscous layer.(13)
This model can explain the observed "annual" (360 day) and Chandler (432 day) periods for the relative motion of the geographic north pole and the motion of the magnetic north pole. The "annual" and Chandler periods cannot be explained together in terms of a solid body Earth.(14) Magnetic reversals are explained as core-mantle reversals producing a tippe top-like movement of the mantle over the core. After even tens of thousands of years of quiescent wobbling, small amplitude motions can increase exponentially to a critical level. The transition can be "due to extremely small changes with time for the elements of the secular equation".
Although a near-collision with a cosmic body may precipitate an inversion, such encounters are not necessary for inversions to take place. In the event of a near-collision, Flodmark is non-committal on whether the inversion would occur immediately (on the order of a day) or over a long period (years) marked by an accelerating build-up to the inversion. Flodmark states that such a reversal cannot occur for a solid body. His computer modelling shows "that a core-mantle reversal can take place mainly within a period of one single day".(14a) By this is meant that, after a long period of slow acceleration, once a critical level is reached, the reversal proceeds rapidly to completion.
Flodmark's work suggests that the Earth may presently be in a mode that is susceptible to a sudden core-mantle reversal. However, the input data "are not sufficiently accurate to fix the time coordinate". To do so would require more accurate measurements of the Earth's rotation and magnetic field variations plus refinements to the model such as introducing non-linear coupling terms for the mutual friction between the core and mantle.
The double-top model is supported qualitatively by the work of the Formosan oceanographer Ting Ying H. Ma, who was cited by Velikovsky in The Velikovsky Affair, p. 243, and in "The Ocean" [KRONOS V:4 (Summer 1980), pp. 24-25]. From his study of fossil corals, Ma deduced that the position of the equator with respect to the crust had changed from one geological age to another in a manner that could not be explained by continental drift alone. According to Hapgood, by 1949, Ma was forced by the accumulated evidence "to adopt a theory of total displacements of all the outer shells of the earth over the liquid core" down to a depth of several hundred miles.(15) Later, Ma hypothesized that Globigerina ooze alternating with red clay results from the repeated sudden displacements of the solid Earth. "When the solid earth shell is displaced the depth of the ocean floor shifted from lower to higher latitudes would be suddenly reduced so that a region for red clay rises into the level for accumulation of Globigerina ooze and a lamina of Globigerina ooze is laid on until it gradually sinks back to below 4,000 meters depth with resumption to normal depth and the deposition of red clay continues."(16)
Ma, it should be noted, dealt with crustal displacements of less than 180° and did not offer an explanation for how the movements inferred from his data were accomplished. He deduced that four sudden total displacements of the solid Earth shell, all in the same direction, have occurred in the last million years; that the crust takes 55,000 to 70,000 years to readjust to normality after such a displacement; and that the last readjustment was completed from about 65,000 to 2,800 years ago based on Pacific cores. Atlantic cores date the last adjustment period from 61,900 to 2,600 years ago. The relation between this terminal date and the last cosmic catastrophe in Worlds in Collision would appear to be coincidental, but further investigation is needed.
Tracing the paths of crustal displacements, however, does not appear to yield unequivocal results. Without necessarily discrediting Ma's deductions, Hapgood's results are not concurrent. Within the past 80,000 years, Hapgood believes that the location of the north pole has changed three times in shifts lasting 5,000 years. The north pole, according to Hapgood, has been in its present location for 12,000 years.(17) These shifts did not all occur in the same direction as Ma's did. None are shown along the preferred longitude of 60° W identified by Warlow and independently by others.(18) Thus Earth's shifting crust, if such it is, appears to leave contradictory clues. Naturally, radioisotope datings, upon which both Ma and Hapgood relied, are subject to reservations within a catastrophist, Velikovskian framework.
In light of the unsolved and apparently unsolvable problems attending the solid body tippe top model, Flodmark's double-top model, at this stage, seems a viable replacement.(19) Considering the experience with the tippe top model, however, a high priority should be placed on validating the double-top model. While near-collisions are compatible with the double-top model, Flodmark's paper does not reflect any calculations with them. Therefore, much needs to be done to examine the actual compatibility between it and near-collisions. Of primary concern is the determination of 1) the minimum torque impulse required to initiate an inversion and 2) the perigees of Mars and Venus-sized bodies to produce that torque impulse.
A tippe top-like reversal of the shell would appear to be more economical than a spin reversal of the entire Earth or the shell since the former is consistent with the double top's normal motion, whereas the latter is not. Curiously, all recent theoretical work has been concerned with explaining tippe top-like movements of the Earth's surface without having to invoke electromagnetic effects. Since motions occur along the path of least resistance, the possibility that a spin reversal has occurred would appear to be greatly reduced and the interpretation of Senmut's ceiling may be in need of a raison d'être other than evidencing a spin reversal. If a spin reversal is a viable alternative, where are there discussions and quantifications of its mechanism?
NOTES AND REFERENCES:1. Peter Warlow, "Geomagnetic Reversals?" Journal of Physics A, 11 (October 1978), pp. 2107-2130 (Page 2112 appears between pp. 2011 and 2013). Reprinted in SIS Review III:4 (Spring 1979), pp. 100-112. Eleven of 63 references are to the Velikovskian literature: KRONOS (7), Pensée (2), Velikovsky (1). See also, colloquium in SISR IV:2/3 (Winter 1979/80), pp. 62-67; this writer's contribution was written in July 1979. Unfortunately, none of the criticisms raised by Crew and Ellenberger which Warlow skillfully answered, addressed the real problems in the tippe top model. Brief discussions appear in KRONOS V:4 (Summer 1980), p. 68 and KRONOS VI:4 (Summer 1981), p. 51. A summary appears in John White, Pole Shift (New York: Doubleday, 1980; Berkley Books, 1982).
2. Scientific American (October 1979), pp. 1834; Scientific American (March 1981), p. 192; The Physics Teacher (May 1978), p. 322; American Journal of Physics 45:1 (January 1977), pp. 12-17; John Perry, Spinning Tops and Gyroscopic Motion (New York: Dover, 1957). Tippe tops have been used as premiums by Cracker Jack and Burger King.
3. Earl R. Milton, "As Worlds Collide", KRONOS II:3 (February 1977), pp. 3-11; M. G. Reade, "Poles Uprooted?" SISR I:1 (January 1976),pp. 18-19.
4. The following is a sampling of the evidence. Herodotus reported that Egyptian priests told him that the Sun had changed its usual position four times within their recorded history, twice having risen where it normally sets and twice setting where it normally rises. The priests assured him that Egypt was quite unaffected by this. In answer to Vitaliano's criticism in Legends of the Earth, this would be the case if the inversion occurred about a fixed equatorial axis at 30° E and 150° W; i.e., flooding from tidal waves would be minimized in Africa. Some centuries later, Pomponius Mela, from written Egyptian annals, corroborated Herodotus. The Papyrus Harris, kept at Leiden, mentions the Earth turned over at the time of a cosmic upheaval involving fire and water. Inversions can be inferred from Hopi, Chinese and Maori legends. Folgheraiter in 1896 and 1899, confirmed by Mercanton in 1907, measured reversed polarity in 8th century B.C. Greek and Etruscan pottery. Over the last 10,000 years, there have been two periods when British weather came from the east and two when it came from the west which has been inferred from the changing pollen record. Although this alternation of the direction of the prevailing winds and currents is acknowledged by mainstream scientists, its cause is unexplained. An inversion would produce just such behavior.
5. Eric Crew, "Slipping Shell and Tippe-Tops", SIS Workshop III:2 (October 1980), pp. 39-41. Crew originally advanced the slipping shell idea in the Colloquium in SISR [see Note 1], but disavowed it following Warlow's remarks.
6. Eric Crew, "Further Comments on Shell Slipping", SIS Workshop III:3 (January 1981), pp. 35-37. Corrected in SISW III:4 (April 1981), p. 25.
7. Victor J. Slabinski, "A dynamical objection to the inversion of the Earth on its spin axis", J. Phys. A, 14 (September 1981), pp. 2503-2507. Dr. Slabinski is an astronomer for COMSAT in the Astrodynamics Dept. No reply from Warlow appeared. See also "Fatal flaw in pole-Ripping theory", New Scientist (12 November 1981), p. 433, a follow up to their original coverage of Warlow's paper in New Scientist (9 November 1978), p. 436.
8. Letter, Peter Warlow to V. J. Slabinski, 11 March 1980 (with a copy to this writer) responding to his receipt of the February 25, 1980 draft of Slabinski's critique. Warlow closed his letter with: "I look forward to hearing more from you, especially if you can prove me wrong, for I do not like the idea of the Earth rolling all over the place.... Unfortunately, science is the search for truth, and I have the horrible feeling that I have found a lump of that truth. But nobody ever promised us that the truth would be pleasant, did they?" This unbridled combination of "objectivity" and "humility" seems ironic in view of the fact that Slabinski's March 22 reply went unanswered, as did his October 14, 1980 follow-up enclosed with the final draft of the J. Phys. article. In his March 22 letter, Slabinski pointed out: "Your appeal in your letter to the observations of our ancestors is irrelevant to the point I am discussing here. I am not questioning whether a tippe-top inversion actually occurred. I am questioning assumption [iii] of the Warlow hypothesis. I am questioning whether a close encounter by a cosmic body can indeed produce the torque required for a tippe-top inversion." Warlow's failure to answer repeated enquiries from KRONOS concerning Slabinski's criticisms adversely affected KRONOS' interest in reprinting Warlow's J. Phys. article.
9. V. J. Slabinski, March 9, 1981 calculations, 3 pp., as revised. (Also see Slabinski below.)
10. Ralph E. Juergens, "On the Convection of Electric Charge by the Rotating Earth", KRONOS II:3 (February 1977), pp. 12-30.
11. Stig Flodmark, "The Earth's Rotation", paper presented at the Institute of Mechanics, Royal Institute of Technology; December 17, 1980. Submitted to Physica Scripta for publication.
12. This is indicated "from the experimental values for the tangential components of the seismic wave velocity and the tensor of elastic constants in this region". See C. W. Allen, Astrophysical Quantities, Ch. 6.51.
13. In the double-top model, the "core" with a radius of 0.218 the Earth's radius or about 1390 km corresponds to the conventional "inner core" plus "transition zone". Thus, the core in the double-top model abuts the "outer core" which in this case is considered part of the mantle.
14. However, although according to accepted thought the cause of the Chandler wobble is still uncertain, the "annual" cycle is believed to be "the result of impulses imparted to the planet by large-scale atmospheric movements and other seasonal mass transport, such as snow accumulation and vegetation". See J. D. Mulholland, "The Chandler Wobble" Natural History (April 1980), pp. 134, 138-141; or Cheryl Simon, "Chandler Wobble" Science News 120 (October 24, 1981), pp. 268-269.
14a. Although Dr. Flodmark believes that, in principle, the shell might also be able to flip, his calculations to date have only shown flips of the core.
15. Charles H. Hapgood, The Path of the Pole (Philadelphia: Chilton, 1970), pp. 81, 26.
16. Ting Ying H. Ma, "Alteration of Sedimentary Facies on the Ocean Bottom and Shortness of the Period of Diastrophism after a Sudden Total Displacement of the Solid Earth Shell", Oceanographica Sinica, Vol. II, Fascicle I (September 1955), p. 4.
17. Hapgood, op. cit., Endplates and p. 107.
18. See, for example, Edward M. Weyer, "Pole movement and sea levels", Nature 273 (4 May 1978), pp. 18-21. Weyer shows that "seemingly incongruous shoreline samples dated 14,700 to 28,000 B.P. [suggest] rhythmic polar oscillations on a 5,600 yr cycle, synchronized with two glacial episodes". The best fit for the data occurred for a pole slippage, which was on the order of ±0.6°, along longitude 60°W and 120°E. Weyer suggests that the centrifugal force from an ice cap caused the pole to shift. "Theoreticaily (with a rigid crust), a slippage of only 1° would raise or lower sea levels relative to land as much as 373m" with the most pronounced effects "along the meridian of polar movement and in latitude 45°" Also, see John White, Pole Shift (New York, 1980), pp. 366-7.
19. A cautionary view of this double-top model follows from the realization that the mass of the shell, with an inner radius of 1390 km, is 40 times that of the core while the moment of inertia is just over 700 times that of the core. With the shell's moment of inertia being 99.86% of the total, the problem of inverting such a shell would appear to be tantamount to inverting the entire Earth. The double-top action, if valid, would be a subtle, yet potent, difference from a solid Earth. This reservation would be nullified if the relevant core were large enough for its moment of inertia to at least equal that of the shell. For example, if the slippage occurred at the transition zone 700 km below the surface, the shell's moment of inertia would be almost 40% of the total. In contrast, if the slippage occurred at the base of the asthenosphere, the shell's moment of inertia would be only ahout 15% of the total.
20. The efforts of Anders Hagman and Jan Sammer are acknowledged for sending KRONOS copies of the Flodmark and Ma works, respectively. The assistance of Dr. Slabinski in preparing the Table is greatly appreciated.
Dr. Victor J. Slabinski Comments:
The point that Prof. Rose makes is quite instructive. To the discussion, I would like to present a more severe lower limit on the mass of the cosmic body required to invert the Earth than was found in my original critique of Warlow's hypothesis.(1)
Peter Warlow has developed the case for the Earth's having flipped over several times within the last 13,000 years.(2) However, because of several errors, his calculations drastically underestimated the necessary torque. Whereas he claimed to show that an inversion could be accomplished within a period of about a day by the gravitational torque of an Earth-mass body passing at 66,000 km,(3) the correct torque for the problem as he formulated it requires a body with a mass of 417 Earths passing at two Earth radii.(4) This distance was selected in order to minimize the mass of the foreign body. A greater distance would require a still larger body to produce the required gravitational torque.
Unfortunately, the torque Warlow calculated was not associated with a corresponding trajectory. Cosmic bodies large enough to invert the Earth act for a period much less than the 24 hours assumed by Warlow. When the calculation combines the torque with a realistic trajectory, the requirement becomes truly astronomical or, more appropriately, stellar. Not even confining the inversion to a thin crustal shell(5) yields a practical solution. As is shown in the Appendix, an inversion of the solid Earth by the gravitational torque from a cosmic body, with a parabolic trajectory and a perigee of two Earth radii, requires a body with a mass of 62 Suns. A crustal shell with a thickness of 1% the Earth's radius (about 64 km), that does not interact with the underlying layers, still requires a body with a mass of 68 Jupiters! We expect that such an interaction would be totally catastrophic.
If the Earth ever did flip over, its mechanism remains unexplained. Any appeal to electromagnetic forces that does not give a quantitative analysis of how such forces produce the required torque is equivalent to saying, with due credit to Sidney Harris, "a miracle occurs".(6)
REFERENCES:1. Victor J. Slabinski, "A dynamical objection to the inversion of the Earth on its spin Axis", J. Phys. A 14 (September 1981), pp. 2503-2507.
2. Peter Warlow, "Geomagnetic Reversals?" J. Phys. A 11 (October 1978), pp. 2107-2130 (page 2112 appears between pp. 2011 and 2013). Reprinted in SIS Review, III:4 (Spring 1979),pp. 100-112.
3. Ibid., p. 2125; p. 110.
4. Slabinski, op. cit., p. 2506.
5. Eric Crew, "Further Comments on Shell-Slipping", SIS Workshop III:3 (January 1981), pp. 35-37.
6. Sidney Harris, What's So Funny about Science? (Los Altos, CA: Wm. Kaufmann, 1977). In this collection of cartoons from American Scientist, two scientists stand before a blackboard covered with complex equations. One to the other, "I think you should be more explicit here in step two" while pointing to "Then a miracle occurs . . ." printed amidst the math. APPENDIX: TORQUE IMPULSE REQUIRED TO INVERT EARTH'S CRUST Moment of inertia (Is) of spherical shell:
Is = 8p/15.r (R2
[*!* Image] VJS 1981 March 9 (Rev. Jan '82)
Editor's Note (LMG): KRONOS regrets that Peter Warlow has ignored two invitations (September 22nd and November 12.1981) to participate in this Forum discussion