THE SHEARING HYPOTHESIS AND THE ALLAIS ECLIPSE EFFECT

Thomas J. Goodey

Just Two Data Points

Eclipse of 30 June 1954 Eclipse of 2 October 1959

In the 1950s Professor Maurice Allais undertook several marathon experimental series in Paris which involved repeated determinations of the rate of precession of a paraconical pendulum (which he had invented). He detected various periodic anomalies in the motion of this pendulum by using elaborate statistical analysis. However he also serendipitously observed a quite large scale effect which was absolutely unexpected. During two of these experimental series, solar eclipses partial at Paris occurred on 30 June 1954 and 2 October 1959. In both cases a well-defined anomaly was detected in the motion of the paraconical pendulum: its plane of oscillation shifted abruptly. Currently accepted physical theory offers no explanation whatsoever for this phenomenon. It is the only gross anomaly outstanding in the current scheme of physical knowledge.

Allais later shifted his personal emphasis from the field of physics to economic theory, and in 1988 he was awarded the Nobel prize in economics. However, physics remained his first love. He has always maintained that his unexplained pendulum results – both the periodic anomalies and the Eclipse Effect - were genuine and valid. He attributes them both to the anisotropy of inertial space - the title of his recent book on the subject. In this paper I shall confine myself to discussion of the Allais Eclipse Effect.

Attempts to confirm Allais's observations upon the behavior of a pendulum during a solar eclipse have met with varied results: some trials have confirmed the presence of anomalies, while some yielded ambiguous results, and others detected nothing unusual. However none of these experiments used a paraconical pendulum according to Allais's design; nor did the experimenters follow Allais's operational procedures or ask his advice on design of the experiments. Nor – as I shall show – has there ever been any idea of contriving a geometrico-astronomical layout, similar to the layout during the crucial observations of 30 June 1954 and 2 October 1959.

I believe that not enough consideration has been given to the fundamentals: what sort of unusual happening can be hypothesized to have actually taken place during the two eclipses in question? Such considerations suggest important new avenues for exploration. It might well be the case that the Allais Eclipse Effect does not manifest itself at every location during a solar eclipse, or indeed during every solar eclipse; various types of special condition (upon the geometry of the eclipse and upon the position of the observer, for example) might be prerequisites. Such conditions presumably also regulate the intensity of the Effect, and perhaps also determine others of its parameters. By investigating such dependencies we may be able to get a handle on this apparently incomprehensible phenomenon.

The effect Allais observed during the 1954 eclipse was very marked – it has even been described as "brutal". However during the 1959 eclipse the effect was manifested to a much lesser degree. So we have two data points to reason from. The only previous attempt at analysis of the geometry of these eclipses has been Allais's comment that "in 1959 the amount of the solar surface eclipsed (at Paris) was only 36.8% of the surface eclipsed in 1954". It is obviously desirable to go into more detail.

Here are magnified views (taken from Fred Espenak's superb website) of the most relevant portions of the eclipse tracks in 1954 and 1959. The green crosses show Paris, while the stars show the eclipse points at the moment of greatest eclipse and the corresponding sub-solar points (for definitions, see later), and the arrows show the directions in which those points were moving, relative to the Earth's surface.

30 June 1954 2 October 1959
gamma=0.61 gamma=0.42
G.E.: 60°28' N, 04°10' E G.E.: 20°25' N, 01°26' W
S.S.: 23°12' N, 8°02' W S.S.: 03°24' S, 06°36' W

(Paris is at 48°48' N, 2°20' E.)

(gamma is the minimum distance of the Sun-Moon line from the Earth-Moon line
during an eclipse (either solar or lunar) measured in units of Earth radii.)

In fact in 1954 the distance between Paris, the experimental location, and the point of greatest eclipse was 1300 km, while in 1959 it was 3180 km. Moreover, in 1954 the closest the umbral path came to Paris was about the same, 1300 km; in other words, the eclipse was maximum at Paris at about the worldwide moment of greatest eclipse. But in 1959 the path of totality came closest to Paris substantially before the moment of greatest eclipse - roughly half an hour before - at a distance of 2790 km. Anyway basically Paris was much further from the action in 1959 than in 1954, so a priori it's no wonder that the eclipse effect was weaker.

But there is another very important difference between the situations in 1954 and 1959: IMHO, not sufficient attention has been paid to the sub-solar point, which is the intersection of the Sun-Earth line with the surface of the Earth. It might well be the case that gravitational effects along this Sun-Earth line interact in combination with gravitational effects along the Sun-Moon line to result in the Allais effect. In 1954 the observer at Paris was positioned between the path of totality (the path of the eclipse point) and the path of the sub-solar point, whereas in 1959 the path of the eclipse point passed between the observer at Paris and the path of the sub-solar point. Now in general, during a total solar eclipse, with respect to the Earth's surface, the eclipse point moves eastwards along its path at about 1/2 km/sec while the sub-solar point moves westwards along its parallel of latitude at about the same speed (however, the exact speeds vary for each individual case). Thus, if one visualizes the Earth-Sun line as one blade of a scissors and the Moon-Sun line as the other blade, these lines move towards, transversely past, and away from one another at a relative speed of about 1 km/sec while remaining substantially parallel with one another, rather as scissor blades shear past one another. In 1954 the observer (Allais in Paris) was between these two notional scissor blades around the time of their closest mutual approach, whereas in 1959 he was not. I surmise that this may be the reason why the Eclipse Effect was so much greater in 1954.

The Shearing Hypothesis

Therefore I have formulated the "Shearing Hypothesis". This postulates that the Eclipse Effect is somehow due to the Sun-Moon line and the Sun-Earth line momentarily getting close to one another as they shear past one another at the relative speed of about 1 km/sec, and that the Eclipse Effect occurs primarily in the region between these lines at the time of their closest mutual approach.

There is a often-deployed counter-argument against the existence of the Allais Eclipse Effect as follows: If such an effect really existed, and if it appeared close to the Sun-Moon axis at all times, then it would be manifested during the normal course of planetary motion, thus stultifying conventional orbital dynamics. It would also exert an effect upon the orbital movements of satellites. Now, the orbits of the GPS satellites (in particular) are never disturbed in this way; so such an eclipse effect can't exist.

But if the Shearing Hypothesis is valid, this counter-argument loses its force. To repeat this Hypothesis, it postulates that the disturbance of pendulum motions, and presumably of other dynamic gravito-inertial processes, is a very transient effect which only occurs in the spatial volume generally between the Sun-Moon line and the Sun-Earth line as they shear past one another at the high relative speed of about 1 km/sec. It only occurs over the period of an hour or so in a restricted cylindrical space whose cross section extends a few thousand kilometers (although its longitudinal dimension is likely very great). It would be reasonable that no significant effect would be exerted upon the orbits of satellites or planetary bodies by such a short-lived effect; it would be unlikely for an orbiting body ever to run into the effect, and certainly the resulting force could never be accumulated in the unique way that a pendulum accumulates small forces over periods of hours.

According to this Shearing Hypothesis, therefore, for each solar eclipse, the area where it would be best to locate an experiment for observing the Allais Eclipse Effect is quite restricted: the ideal position (on the Earth's surface) is somewhere on or near the middle portion of the line joining the point of greatest eclipse to the corresponding sub-solar point. And during the 1954 eclipse Paris was - quite fortuitously – a very suitable place for observation according to this criterion. However in the 1959 eclipse Paris was far from being so suitable, so that the effect was less outstanding. Actually the gamma in 1954 was not particularly low (0.61), but nevertheless a remarkably pronounced Eclipse Effect was observed.

Moreover, a matter which has never been considered is the question of the anti-solar-eclipse. If the Shearing Hypothesis is valid, the Allais Effect may well extend right through the Earth to the other (night) side, along the prolongation of the Sun-Moon line. This should be tested – presumably when the eclipse itself is inaccessible, so that a direct experiment for the eclipse itself in the location specified above is in any case difficult or impossible.

Finally, suppose that the portion of the Sun-Moon line which intersects the Earth's surface is its portion between the Moon and the Sun (rather than its portion on the side of the Moon remote from the Sun). In this case a lunar eclipse occurs. Perhaps the Allais Effect will be manifested near the points of intersection in these cases as well.

Nomenclature

We will consider three straight lines, each of which passes through the centers of two astronomical bodies: the Sun-Moon line ("SML"); the Sun-Earth line ("SEL"); and the Moon-Earth line ("MEL"). At any moment, the point upon the Earth's surface at which the Sun is at the zenith, i.e. one of the points of intersection of the SEL with the Earth's surface, is termed the "Sub-Solar" (abbreviated as "SS"); and the other such point of intersection, at which the Sun is at the nadir, is herein termed the "Anti-Sub-Solar" (abbreviated as "ASS). Here is an illustrative figure:

Similarly the point upon the Earth's surface where the Moon is at the zenith, i.e. one of the points of intersection of the MEL with the Earth's surface, is herein termed the "Sub-Lunar" (abbreviated as "SL"); and the other such point of intersection, where the Moon is at the nadir, is herein termed the "Anti-Sub-Lunar" (abbreviated as "ASL").

On Solar Eclipses

An observer located upon the sunlit side of the Earth experiences a total solar eclipse, when (referred to the Sun as stationary) the motions of the Earth and the Moon conspire to bring the Moon momentarily directly in front of the Sun from the point of view of the observer on the Earth's surface, so that the center of the Sun, the center of the Moon, and the observer are momentarily collinear in that order. (This configuration can be abbreviated as C_S-C_M-O along the SML.)

In other words, the SML is intersecting the surface of the Earth, which is unusual, and the observer is positioned at that one of the intersection points which faces towards the Sun. This point is termed the Eclipse Point ("EP") at that instant; and the other of the intersection points is herein termed the Anti-Eclipse Point ("AEP"). And, at the moment that the SML passes closest to the center of the Earth (so that the distance between them is equal to gamma, and the eclipse is the greatest), the current positions of these points EP and AEP along their tracks upon the Earth's surface are herein termed the points of Greatest Eclipse and Anti-Greatest-Eclipse — GE and AGE.

By the way, during a solar eclipse it is virtually never the case, that the center of the Earth also lies upon the SML; that would require the total eclipse to occur at the observer's local noon, and simultaneously his latitude to be equal to the Sun's current declination. In other words, we can almost forget about the theoretical possibility of all the three celestial bodies being arranged in a straight line; this is illustrated here:

Such circumstances hardly ever come to pass.

However, they will almost come to pass during the eclipse of 22 July 2009, when the center of the Earth comes within less than 450 km of the Sun-Moon line. This matter, which is of historic importance, is discussed later.

Digression: it is an odd fact that the angular diameters of the Sun and the Moon, as seen from the surface of the Earth, are almost the same, so that, depending upon the exact distance between the Moon and the Earth at the time of the eclipse (the Moon's orbit around the Earth is not perfectly circular), either the Moon may actually cover the Sun (the eclipse is total), or a ring at the extreme edge of the Sun may remain uncovered (the eclipse is annular). We will assume that this coincidence of angular diameters is just that: a strange coincidence. Any other hypothesis leads us into wild realms of thought which can scarcely be said to be scientific according to any currently imaginable paradigm.

On Lunar Eclipses

In the complementary case that the SML intersects the surface of the Earth, but with the Moon on the opposite side of the Earth from the Sun, then the Moon will be located within the Earth's penumbra at least, if not its umbra, so that a lunar eclipse is taking place. In this case, an observer can see the eclipsed Moon provided that he is positioned anywhere upon the dark side of the Earth. (In this respect, lunar eclipses are quite different from solar eclipses, during which a good view of the eclipse is only available from a very restricted set of locations.)

But if the observer is positioned anywhere upon the sunlit side of the Earth, then he is unable to see the eclipsed Moon:

This may be termed an "anti-lunar-eclipse" situation.

In analogy to the nomenclature for a solar eclipse, that one of the two points at which the SML intersects the Earth's surface during a lunar eclipse, from which the eclipse (the Moon) is visible, will herein be termed the Eclipse Point ("EP") [although it has no intrinsic right to this designation]; and the other one of the intersection points (from which the Sun is visible) will herein be termed the Anti-Eclipse Point ("AEP"). And, as before, at the moment that the SML passes closest to the center of the Earth (so that the eclipse is greatest), the current positions of these points EP and AEP upon their tracks will be termed the points of Greatest Eclipse and Anti-Greatest-Eclipse — abbreviated as "GE" and "AGE" [these terms are actually only meaningful in terms of the solar eclipse analogy].

(During a lunar eclipse, no particularly outstanding phenomenon is apparent to an observer upon the track of the eclipse point EP, or at the point GE; the eclipse looks much the same from any point. This is quite different from the case of a solar eclipse.)

-o0o-

The remainder of this paper is an attempt to analyze upcoming solar and lunar eclipses over the next few years from the point of view of the Allais Eclipse Effect, and to develop recommendations for experimental disposition in each case. It should be noted that these recommendations can never actually lead the experimenter seriously astray, even if the Shearing Hypothesis is fundamentally incorrect. This is because, for each eclipse, the area recommended for experiments will naturally fall quite near to the path of the eclipse point EP, as was the case for the 1954 eclipse in which a pronounced Eclipse Effect was actually observed. However, the basic recommendation of the Shearing Hypothesis is not to position the experimental pendulum(s) actually in the EP track, but rather to the side of it towards the sub-solar point. However I consider that, in a suitable case where observations can be freely set up in any desired position, (i.e. where the EP track crosses land), as a cross-check, it would be advisable also to establish an independent pendulum observation directly upon the EP track.

Relevant solar eclipses

It seems fairly obvious that the smaller is the gamma of a solar (or a lunar) eclipse, the stronger will the associated Eclipse Effect be. Accordingly there is no real imperative to take partial eclipses (where gamma > 1) into account. However it is considered a matter of course that total, annular and hybrid solar eclipses are all on a par as far as the Eclipse Effect is concerned. (These are collectively termed 'central eclipses').

The following central solar eclipses should be considered: 8 April 2005; 3 October 2005; 29 March 2006; 22 September 2006; 7 February 2008; 1 August 2008; 26 January 2009; 22 July 2009; 15 January 2010; 11 July 2010; 20 May 2012; 13 November 2012; 10 May 2013; and 3 November 2013. (No central solar eclipses occur in 2004, 2007, and 2011.)

Hybrid solar eclipse of 8 April 2005
(gamma=0.33)

Greatest Eclipse:
0°0 N, 0°0 E

Sub-Solar (at G.E.):
0°0 S, 0°0 E

. <DATA LATER>

This eclipse itself cannot be well accessed from land (it runs largely through an empty part of the Pacific), but the anti-eclipse is accessible. The southern end of Madagascar is an ideal location. Note that usually, for an anti-eclipse, again with respect to the Earth's surface, the sub-solar moves westward at about 1/2 km/sec as before, but the umbra point now also moves westward, at about 3/2 km/sec. This means that the observer will be moving quite fast with respect to both the Sun-Earth and the Sun-Moon lines; how this will affect the Allais Eclipse Effect, if at all, I cannot guess. (It may prove to be very significant.) The geometry during the anti-eclipse is as shown here:

Annular solar eclipse of 3 October 2005
(gamma=0.33)

Greatest Eclipse:
12°52 N, 28°44 E

Sub-Solar (at G.E.):
4°05' S, 22°05' E

Total solar eclipse of
29 March 2006
(gamma=0.38)

Greatest Eclipse:
23°09' N, 16°45' E

Sub-Solar (at G.E.):
3°24' N, 27°10' E

These two eclipses should be considered together, since they both are focused upon Central Africa. In fact it appears that it should be possible to set up, in a single location, a pendulum experimental station which can serve to investigate the situation during both eclipses. However this area has the great disadvantage of being the absolutely darkest part of Africa – it doesn't get any darker than this! The best location is Kisangani, but doing experimental work there is not feasible, at least for any organization without serious governmental backup. This is particularly disappointing because Kisangani is almost upon the Equator, so that the Foucault effect would not confuse the experimental results. In any case, the geometry during the eclipses is as shown here:

3 October 2005

<African chart later>

29 March 2006

<African chart later>

The corresponding anti-eclipse points are:

3 October 2005:

Anti-Greatest-Eclipse: 21°02 N, 164°34 W

Anti-Sub-Solar (at A.G.E.): 4°05' N, 157°55' W

29 March 2006:

A.G.E.: 16°21 N, 142°25 W

A.S.S.: 3°24' S, 152°50' W

(note that these positions are only approximate, but they are right within ten nautical miles)

Actually Hawaii is ideally placed for this investigation! This is extremely encouraging in view of the practical difficulties of working in the Congo. The geometries are shown below.

Anti-solar-eclipse of 3 October 2005

Anti-solar-eclipse of 29 March 2006

Annular solar eclipse of 22 September 2006
(gamma=0.41)

Greatest Eclipse:
20°40' S, 9°04' W

Sub-Solar (at G.E.):
0°16' N, 4°54' E

When one initially looks at the path of this eclipse, it seems that positioning an observation station between the point of greatest eclipse and the corresponding sub-solar is quite impossible. But no! actually the island of St. Helena is ideally situated at 15°56' S, 5°42' W. (It becomes apparent that this project necessarily entails concentration upon remote oceanic islands, since the major part of the Earth's surface is covered with water.) Moreover St. Helena is a British dependency and there will be no political or social problems to contend with, although the logistics may be rather daunting; there is no air service to St. Helena. The geometry during this eclipse is as shown here:

The situation for observing this eclipse is quite convenient. So is the anti-eclipse, which is almost exactly:

A.G.E.: 21°12 S, 161°08 W

A.S.S.: 0°16' S, 175°06' W

Samoa would be ideal. If we get positive results for the two Hawaii observations, so that the Allais Eclipse Effect has been verified and also has been shown to pass through the Earth and thus to be observable for anti-eclipses as well as for eclipses, then it would be tempting just to transport our laboratory from Hawaii to Samoa or Tonga, thus avoiding all the logistical problems of setting up on St. Helena.

Annular solar eclipse of
7 February 2008

(gamma=0.95)

Greatest Eclipse:
67°35' S, 150°28' W

Sub-Solar (at G.E.):
15°31' S, 121°15' E

This is not a very promising eclipse for testing the Allais effect, because of the large gamma. In any case the only possible position for observation would be in Southern Australia, which would be near the sub-solar point but not very near the point of greatest eclipse. Melbourne would be satisfactory.

As for the anti-eclipse:

A.G.E.: 52°40' S, 97°02' W

A.S.S.: 15°31' N, 58°45' W

The most suitable observation station would seem to be Antofagasta in Chile. Actually, to be really classy, the absolute best would be "Robinson Crusoe's Island", i.e. Alexander Selkirk in the Juan Fernandez islands, at 33°37' S, 78°50' W.

Total solar eclipse of
1 August 2008

(gamma=0.83)

Greatest Eclipse:
65°38' N, 72°16' E

Sub-Solar (at G.E.):
17°52' N, 24°44' E

Again the gamma here is quite large, so this eclipse is not very promising. However a wide range of possible sites are available in Russia upon and near the line joining the point of greatest eclipse and the sub-solar point. So there is no real point considering the anti-eclipse.

2007 and 2008 are not very good years… but we haven't considered the lunar eclipses yet!

Annular solar eclipse of 26 January 2009
(gamma=0.28)

Greatest Eclipse:
34°05' S, 70°17' E

Sub-Solar (at G.E.):
18°39' S, 60°22' E

The value of gamma for this eclipse is quite low, so it is a prime candidate for testing. The only possible vantage point is the island of Rodriguez, at 19.42° S, 63.24° E. The geometry during the eclipse is as shown here:

<chart later>

As for the anti-eclipse:

A.G.E.: 3°13 N, 129°34 W

A.S.S.: 18°39' N, 119°38' W

Clipperton might do; but really, at Rodriguez, the eclipse itself is bound to be easier. It appears that access to Rodriguez is not difficult.

<chart later>

Total solar eclipse of 22 July 2009
(gamma=0.069)

Greatest Eclipse:
24°12' N, 144°8' E

Sub-Solar (at G.E.):
20°16' N, 141°11' E

This eclipse is the big one!

The very small value of gamma means that the Earth-Sun line and the Earth-Moon line pass less than 450 km apart – amazingly close in astronomical terms. If this doesn't trigger the Allais effect, nothing will. Although the path of totality is passing through the western Pacific Ocean at the time, the Japanese islands of Kita-Io-Jima, Io-Jima, and Minami-Io-Jima are fairly well placed for experiments. The geometry during the eclipse is as shown here:

<chart later>

And, for the anti-eclipse:

A.G.E.: 16°20 S, 41°46 W

A.S.S.: 20°16' S, 38°49' W

This is extremely propitious – the A.G.E. is on land near Teofilo Otoni, a bit north of Belo Horizonte in a civilized part of Brazil; and the A.S.S. is just a little offshore into the Atlantic from the port of Vittoria on the Brazilian coast. There should be no difficulty in establishing pendulum observation posts in this area. Here is the geometry:

<chart later>

Annular solar eclipse of 15 January 2010
(gamma=0.40)

Greatest Eclipse:
1°37' N, 69°20' E

Sub-Solar (at G.E.):
21°08' S, 73°24' E

Diego Garcia at

6°57' S, 72°42' E

is absolutely perfect, so there is no need to consider the anti-eclipse, which is somewhere in a rather empty region of the Pacific.

<chart later>

Total solar eclipse of 11 July 2010
(gamma=0.68)

Greatest Eclipse:
19°46' S, 121°52' W

Sub-Solar (at G.E.):
22°02' N, 114°38' W

<chart later>

revilla gigedo - isla roca partida

19°00' N, 112°04' W

not very good; nor is the anti-eclipse at 62°00' S, 80°28' E.
Amsterdam island seems to be the only possibility, but it is not a really practical proposition…
 

<chart later>

Annular solar eclipse of 20 May 2012
(gamma=0.48)

Greatest Eclipse:
49°05' N, 176°19' E

Sub-Solar (at G.E.):
20°13' N, 179°50' W

Midway island at

28°13' N, 177°23' W

is ideal.

<chart later>

Total solar eclipse of 13 November 2012
(gamma=0.37)

Greatest Eclipse:
39°58' S, 161°18' W

Sub-Solar (at G.E.):
18°15' S, 153°05' W

<chart later>

SOMEWHERE IN ILES AUSTRALES – NOT MUCH GOOD

As for the anti-eclipse:

A.G.E.: 3°28 S, 35°08 E

A.S.S.: 18°15' N, 26°55' E

About the best is Fortaleza or Parnaiba or Braganza or Belem on the northwards facing coast of Brazil – not quite ideal, but not too bad…

<chart later>

Annular solar eclipse of 10 May 2013
(gamma=0.27)

Greatest Eclipse:
2°12' N, 175°30' E

Sub-Solar (at G.E.):
17°37' N, 173°43' E

The eclipse itself is not very promising. Howland, Baker, and Christmas islands seem to be in the path of totality, but not ideally positioned as far as the Shearing Hypothesis is concerned. However, for the anti-eclipse, it seems that (again) St. Helena might be a reasonable test spot.

<chart later>

Hybrid solar eclipse of 3 November 2013
(gamma=0.33)

Greatest Eclipse:
3°30' N, 11°40' W

Sub-Solar (at G.E.):
15°12' S, 12°24' W

An ideal spot is Ascension Island:

7°55' S, 14°25' W

but the logistical difficulties are considerable.

<chart later>

PRELIMINARY LUNAR ECLIPSE ANALYSIS

I have not yet located good data for upcoming lunar eclipses; but here is a preliminary analysis based upon what is available at the moment.

4 May 2004 – total eclipse (gamma=0.31). The point of greatest eclipse is near the southern end of Madagascar; while the corresponding point for the anti-eclipse is Revilla Gigedo island. Fort Dauphin in Madagascar is not by any means inaccessible, but the time is relatively short.

28 October 2004 – total eclipse (gamma=0.28). The eclipse is... not really near Barbados or Dominica; this eclipse is actually inaccessible. As for the anti-eclipse…. somewhere in Irian Jaya, Indonesia, or Dili in East Timor, or even Darwin in northern Australia would do. Any of these is accessible. Even Kuching in Sarawak would not be too bad; it is close the the path of the AGE, although on the wrong side of it from the point of view of the Shearing Hypothesis. One wouldn't use Kuching if one had a free and equal choice of other points, but it might be a lot better than nothing; we have plenty of friends there anyway.

24 April 2005 – penumbral eclipse (gamma=1.09); not much good.

17 October 2005 – partial eclipse (gamma=0.98); not very good; but for the eclipse, a good location would be Kamchatka, while for the anti-eclipse, a good location would be Romania!

14 March 2006 – penumbral eclipse (gamma=1.02); not much good; but again, for the eclipse, Romania is an excellent place!

7 September 2006 – partial eclipse (gamma=0.93); not very good; eclipse = Amsterdam island; anti-eclipse = Easter Island.

3 March 2007 – total eclipse (gamma=0.32); eclipse = Lake Chad; anti-eclipse = Canton, Baker, or Howland island in the Pacific.

28 August 2007 – total eclipse (gamma=0.21); eclipse = Rarotonga or Bora-Bora in Polynesia; anti-eclipse = Luanda in Angola.

21 February 2008 – total eclipse (gamma=0.40); eclipse = one of the Guyanas; anti-eclipse = Pitcairn or Henderson island.

16 August 2008 – partial eclipse (gamma=0.56); not very good; eclipse = Cairo; anti-eclipse = maybe, Revilla Gigedo.

9 February 2009 – penumbral eclipse (gamma=1.06); not much good; eclipse = Western Australia; anti-eclipse = nowhere on land.

6 August 2009 – penumbral eclipse (gamma=1.36), not much good; eclipse = Morocco somewhere; anti-eclipse = Midway island.

31 December 2009 – partial (gamma=0.98); not very good; eclipse = Islamabad; anti-eclipse = Revilla Gigedo (again!)

26 June 2010 – partial (gamma=0.71); not very good; eclipse = Accra in Ghana; anti-eclipse = Chatham island near New Zealand.

21 December 2010 – total (gamma=0.32); eclipse = Los Angeles/San Francisco; anti-eclipse = Rodriguez island.

SUMMARY OF SOLAR ECLIPSE. POSSIBILITIES FOR EXPERIMENTATION

The following are the possibilities for solar eclipses for the next ten years.

Year	Eclipse at	Anti-Eclipse at	Gamma
2004	NONE	NONE	-
2005 Apr.	-	Madagascar
2005 Oct.	Kisangani	Hawaii	0.33
2006 Mar. 2006 Sept.	Kisangani St. Helena	Hawaii Samoa	0.38 0.41
2007	NONE	NONE	-
2008 Feb. 2008 Aug.	Australia Russia	Chile	0.95 0.83
2009 Jan. 2009 Jul.	Rodriguez Iwo-jima	- NE Brazil	0.28 0.069
2010 Jan. 2010 Jul.	Diego Garcia Revilla Gigedo	- -	0.40 0.68
2011	NONE	NONE	-
2012 May 2012 Nov.	Midway Iles Australes	- NE Brazil	0.48 0.37
2013 May 2013 Nov.	none Ascension Island	? ?	0.27 0.33

Preliminary information for lunar eclipses

We can assert the following definitely:

Year	Lunar Eclipse at	Anti-Eclipse at	Gamma
2004 May 4 2004 Oct. 28	Madagascar no land	no land Darwin/Timor (or Kuching)	0.31 0.28

-o0o-

Zeniths and Nadirs

However, apart from eclipses, there is another possible line to explore. Rather than taking the Sun and Moon as the massive bodies whose centers of gravity are to be in line when doing the experiment, why not substitute the Earth for one of these? In this case, when the observer, the Earth, and the other body (the Moon or the Sun) are momentarily in a straight line, that other body will be at the observer's zenith or nadir. There are four cases:

Solar zenith: the observer lies upon the line joining the centers of the Sun and the Earth, and between them. Thus the order is C_S-O-C_E. The Sun is at the observer's zenith. In other words, in the terminology introduced previously, the observer is at the sub-solar point SS:

Solar nadir: the observer lies upon the line joining the centers of the Sun and the Earth, but on the other side of the Earth from the Sun. Thus the order is C_S -C_E-O. The Sun is at the observer's nadir. In other words, the observer is at the anti-sub-solar point ASS:

Lunar zenith: the observer lies upon the line joining the centers of the Moon and the Earth, and between them. Thus the order is C_M-O-C_E. The Moon is at the observer's zenith. In other words, the observer is at the sub-lunar point SL:

Lunar nadir: the observer lies upon the line joining the centers of the Moon and the Earth, but on the other side of the Earth from the Moon. Thus the order is C_M -C_E-O. The Moon is at the observer's nadir. In other words, the observer is at the anti-sub-lunar point ASL:

Note that in the solar cases the observer must be within the tropics, i.e. between 23.5° north and 23.5° south, while in the lunar cases the observer must be within the zone of the Earth over which the Moon passes, i.e. between approximately 30° north and 30° south. I have never heard of any pendulum experiments - Foucault or otherwise - having been performed in such tropical areas. Of course, the probable reason is that the normal precession of the Foucault pendulum is much slower in low latitudes than in Europe, North America, or Japan, so nobody has thought it worthwhile to set one up.

Proposals

A first suggestion therefore is, to set up a Foucault pendulum in some location just south of the Tropic of Cancer, or just north of the Tropic of Capricorn. An ideal site would be a disused vertical mine shaft. Such a site might be found in any of the countries through which the Tropics pass: India, Mexico, Botswana, south China, or Australia might be a good bet. The famed deep sink holes in Oman lie just on or within the tropic of Cancer; they could be an exotic possibility.

The exact site chosen is not critical. In every year, any spot whatever near either tropic but towards the equator therefrom experiences each of the above conditions of solar/lunar zenith/nadir twice — a total of eight possibilities for critical observation. That's a lot better than chasing eclipses! Moreover, if the spot chosen is quite close to its tropic, the accuracy of its coincidence with the actual C_M-C_E or C_S-C_E line at the critical moment will be very high – one or two kilometers. Actually that is finer than the accuracy with which the center of gravity of the Earth is known (I suppose), so a series of observations on consecutive days/nights would be in order.

A second suggestion is to set up such a Foucault pendulum on the Equator or very near it. According to the conventional logic of the Foucault pendulum, this would be ridiculous, because there would be no, or no noticeable, Foucault effect. However, the advantage is that it would be much easier to appreciate any Allais effect which occurred, because it would not be obscured by the Foucault effect overlaying it.

Later, it would be desirable to construct Allais-type paraconical pendulums, or improvements thereof, and repeat these experiments in the above locations. However a Foucault pendulum is quite easy to make and to operate, whereas a paraconical pendulum requires very accurate machining and very painstaking operation. So first it might be an idea to work with the relatively crude Foucault pendulum and see if any interesting effects are manifested.

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