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WHY ARE THE ECLIPSES DANCING THROUGH THE CALENDAR?Roland Szostak
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WHY ARE THE ECLIPSES DANCING THROUGH THE CALENDAR?Roland Szostak
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 | Ancient cultures knew already counting rules for predicting the dates of eclipses. This worked fine for lunar eclipses and but not so reliably for solar eclipses. Nevertheless they had a good experience, how the eclipses dance through the calendar. Our students start on a different level. They learn to understand the shadowing geometry for the eclipse from the heliocentric view, as it is caused by the orbital motions of Earth and Moon. But they don't understand this dance.
Quite separately they are taught the dynamics of a gyroscope. The somewhat mysterious behaviour of the precession of a rotating wheel is described by a mathematical formalism, which unfortunately leaves the impression that this is a complicated matter. As a practical consequence the precession of the terrestrial axis is described, which is very slow and needs 26000 years. But there is a much langer effect with the quick precession of a period of 19 years. It is the gyration of the lunar orbital plane, which makes the eclipses dance through the calendar. It is worthwhile to discuss this effect: The practical results can be checked easily in a calendar by the students. And the system Earth-Moon can be treated simply as a system of two mass points.
But the main advantage of this lecture will be, that it intends to explain every detail of the curious gyrating motion in a visible way on the level of very simple Newtonian mechanics, so that the intimidating mathematical formalism can be suspended. |
From our understanding since Newton it is evident, that the eclipses occur when the moon is in a position near to the line where the lunar orbital plane crosses the terrestrial orbital plane. As long as these two planes are constant in space, one expects the eclipses to happen at two positions which are stable with the year, only hopping a bit to the next neighbouring coincidence with full Moon or new Moon. This hopping in the calendar should be only +- 2 weeks. The fact that this is not the case, is due to the gyroscopic motion of the lunar orbital plane. By this the crossing line with the terrestrial orbit generates a precession, which has an angular velocity of a period of 18,61 years.
We all have learned the phenomenon of the precession in a different context. The textbooks refer to the gyroscopic motion of the terrestrial axis which rotates on a cone with an angle of 23,50 with a period of 26000 years. By this the polar star is only accidentally near to the north pole in our millennium. And by this there is a slow precession of the crossing line of the equatorial plane with the orbital plane of the Earth, which is called the precession of the vernal equinox. This effect generates a small difference between the length of the sidereal and tropical year.
For the didactical consideration it would be better to teach the precession of the lunar orbit than to teach the precession of the equatorial plane: The period of 26000 years is 50 long that it cannot be experienced within own life. Meanwhile the period of 19 years provides data for the shift of the eclipses through the calendar in a way which can be checked easily by ourselves.
- There is also an advantage on the level of theoretical mechanics: It is possible to treat the system Earth-Moon as a system of two mass points and to apply the simple laws of mechanics for mass points. Analysing the behaviour of the rotating earth by considering the forces acting on the equatorial bulge is much more complex. The explanation of the gyroscopic motion of the rotating earth affords the formalism of the gyroscopic dynamics, which works fine on an abstract mathematical level but does not offer a simple visualizable access on the level of point mechanics.
First I will explain the gyroscopic effect in the workshop in traditional terms. After that I will explain the dance of eclipses without the tool of gyroscopic mechanics. I will give here a short outline of this strategy: Let us start by considering the circular motion of the moon around the earth without precession. The earth is assumed to be at rest. The circular motion can be regarded as composed by two oscillating components, which are perpendicular to each other in x and y. These components are equivalently two sinusoidal motions with a phase shift of 900.
Let us see next how this undistorted lunar orbit will be affected: There is a periodic synchronous force which results from the inclination of the lunar orbital plane (50) against the plane of the ecliptic plane. This force is acting in a way, that the moon is always pushed towards the ecliptic plane.
In order to estimate in which way the lunar orbit is affected by this force, let us consider for a moment the behaviour of an oscillator which is driven by an external force of the same frequency: The oscillator will follow the external force retarded by an angle of 900. This can be seen in the case of an oscillating mass, which is coupled to the external force by a string. The external force may come, for instance, from a pendulum with the same frequency.
Let us extend this a little step further: Usually the external force will be regarded as acting parallel to the motion of the oscillator along the x-direction. But now imagine that the external force acts (as a z-component) perpendicularly to the motion of the oscillator. What happens? The oscillator will start to build Up a motion in this perpendicular direction. As this component is delayed by 900, the resulting motion will be an ellipse in the xz-plane.
Let us go back now to the moon which was orbiting in the xy-plane. A motion builds up in the z-direction, which is by 900 retarded to the motion of the x-component and in phase with the y-component. Then you see that the orbiting plane is inclined in the way as described by full gyroscopic dynamics.
Finally we have to consider a resume by comparing this new method with the standard method: We did not apply gyroscopic dynamics. So we did not need to care for terms connected with the momentum of inertia and could stay with the simple terms of dynamics of mass points. By this we escape the abstract level of mathematical solutions and gain more insight on a visualized and illustrative level. But this method cannot be well applied to the precession of the equatorial plane, because there we have to consider the momentum acting on the bulge of the earth and its influence on the momentum of inertia. So the method which uses only mass point mechanics, is bound to the precession of the lunar orbit.
If we feel that it is an advantage to treat the problem on this easier theoretical level, it becomes preferable on the didactical level to talk about the dance of the eclipses through the calendar before we talk about the shift of the vernal equinox. The period of 19 years of the eclipses provides data in calendars, which can also be better practically examined by the students themselves than the long period of 26000 years of the vernal equinox. So there is some advantage in treating the dance of the eclipses through the calendar first.
In the background there is still a point, which I have cut away silently in describing the new method and which is not solved perfectly on the didactical level: The amplitude of an oscillator as a function of frequency of the external force shows the curve of resonance. This curve is sensitive to the damping in the vicinity of resonance. Now the problem: We are acting in resonance, and there is no damping in our system. Nevertheless there is no contradiction. The resonance curve describes the equilibrium amplitude of the oscillator in the stationary mode. But we do not refer to this. We consider only the change per cycle.
In addition to the resonance curve of the amplitude there is an assigned phase shift as a function of frequency. In the case of no damping this curve jumps at resonance from f = 0° to f = 180°. So the phase angle of 90° is applicable only by a nontrivial approximation to resonance.
