PHYSICAL REVIEW D 67, 022002 ~2003!

Allais gravity and pendulum effects during solar eclipses explained

T. Van Flandern*Meta Research, 6327 Western Avenue, NW, Washington, DC 20015-2456

X. S. Yang†

Faculty of Engineering, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, United Kingdom~Received 30 July 2002; published 27 January 2003!

Gravitational and other anomalies seen repeatedly in connection with solar eclipses have led to speculationabout a possible gravitational shielding effect as the cause. Here we show that an unusual phenomenon thatoccurs only during solar eclipses, rapid air mass movement for the bulk of the atmosphere above normal cloudlevels, appears to be a sufficient explanation for both the magnitude and behavior of the anomaly previouslyreported in these pages.

DOI: 10.1103/PhysRevD.67.022002 PACS number~s!: 04.80.Cc, 95.10.Gi

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I. INTRODUCTION

A very accurate Foucault-type pendulum slightly icreases its period of oscillation and/or changes its planswing ~by up to 13.5o) at sites experiencing a partial eclipsof the Sun, as compared with any other time. This effect wfirst noticed by Allais over 40 years ago@1#, and both it andrelated phenomena are now named after him. Some seffect has been seen at several eclipses since then, bunot seen at other eclipses. In recent years, an anomaeclipse effect on gravimeters has become well-establiseven under controlled environmental conditions~especiallypressure! @2#, which some of the pendulum experiments dnot have. Several exotic explanations have been propothe most interesting of which is a possible shielding effecthe Sun’s gravity while the Moon is partly in front of thSun. The size of the effect as measured with a gravimduring the 1997 eclipse was roughly (5 –7)31029g% ,whereg% is the acceleration of gravity at the Earth’s surfa~about 1000 cm/s2). So the acceleration of gravity from thEarth seems todecreaseduring solar eclipses by abou(5 –7)31026 cm/s2 @3,4#. ~See Fig. 1.!

However, the upper limit on any gravitational shieldingnow set by Lageos satellites, which suffer an anomalousceleration of only about 3310210 cm/s2 during ‘‘seasons’’where the satellite experiences eclipses of the Sun byEarth @5#. This seems to rule out the possibility that theclipse effect on gravimeters might be gravitational ininteresting sense, such as a shielding effect. But interpreany manifestation of the Allais pendulum effect as gravitional shielding was always problematic because it failedexplain why the effect set in well before the partial stagean eclipse began and lasted at least half an hour afteended; why it had two large excursions in local gravity, onear the beginning and one near the end of the partial phwith little effect near mid-eclipse; and why the effect wouact as adecreasein the Earth’s gravity instead of an appareincrease. Because the Sun’s gravity opposes the Earth

*Email address: tomvf@metaresearch.org†Email address: x.s.yang@swansea.ac.uk

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the Moon shielded part of the Sun’s gravity, this shouldcrease the acceleration of nearby objects toward the Eartdecreasing the component toward the Sun.

II. ANALYTICAL MODEL

As veterans of many eclipse expeditions, we were keeaware of the sharp temperature drop that accompanieseclipses, with onset soon after the partial phase begins.temperature drop creates air-mass movement into and othe eclipse zone. This is analogous to what happens fometeorological front with a large temperature differentialeither side in the absence of rapid air circulation to stabilthe two regions of unequal-pressure air. However, in normmeteorological conditions, only the lower atmosphere in abelow the cloud decks is strongly affected by large blocksair with a pressure differential. But over 90% of the atmsphere’s mass is above the cloud levels. This high atmsphere never experiences a comparable sudden temperdrop in a limited geographic region except at times of soeclipses because it is normally always in continuous, unscured sunlight by day. Eclipses partially shut off a hesource for the atmosphere above the clouds withinMoon’s penumbral~partial eclipse! shadow, and that losheat creates a relatively steep temperature gradient attudes where such a phenomenon normally never exists.can then be forced to move in total volumes much larger tare otherwise possible.

Confirming that air-mass movement in fact happehourly data recorded on the day of the 1999 eclipse fromsite in the partial eclipse zone~north latitude 38.1°, east lon-gitude 20.6°, maximum eclipse magnitude 80%) shows thatmospheric pressure increased sharply by at least 0.6%ing the eclipse~from 979 to 985 g/cm2), returning to pre-eclipse levels afterward@6#. ~See Fig. 2.! Unfortunately, thehourly data does not have very good time resolution, anfor a location roughly 700 km from the path of totality. Butgives important clues about the magnitude of the air mmovement.

From the kinetic theory of gases, for any given rms speof air moleculesv̄ and air densityr, pressurep is given byp5 1

3 r v̄2. ~This is because pressure changes affect all th

©2003 The American Physical Society02-1

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T. VAN FLANDERN AND X. S. YANG PHYSICAL REVIEW D 67, 022002 ~2003!

FIG. 1. Observed effect of solar eclipse olocal acceleration of gravity@4#.

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dimensions, so the pressure change in one dimension,as downward, is reduced by a factor of three@7#.! If wedifferentiate this pressure formula, divide by the original fomula, and rearrange terms, we get a relation betweencentage changes:dr/r5dp/p22d v̄/ v̄ . We do not havemeasurements ofv̄ during the eclipse, but it varies with thsquare root of absolute temperature, which obviously gdown during the eclipse. If we assumev̄ drops by the same0.6% change as the pressure rises, this would imply achange in air density of 1.8%. Other physically reasonaassumptions can lead to percentage changes in air massing from 1.2% to 2.4%; but we will use this median value1.8% for our further analysis.

Simple reasoning suggests that the cooler air insideeclipse zone will decrease in volume~increasing in density!in accord with Boyle’s law as its temperature drops, creata ‘‘low’’ pressure region with the unusual character thatwould extend to great altitudes. This leaves room for warmair from outside the eclipse zone on all sides of the advaing shadow of the Moon to flow rapidly into the eclipse zoand fill the volume emptied by the cooler, denser air theThis is what happens on a smaller scale across meteorocal fronts. When ‘‘highs’’ and ‘‘lows’’ collide, winds are created that attempt to equalize those discordant pressures.that for eclipses, the redistribution of air mass would affbroad areas well outside the eclipse zone through thiscess because those areas are the reservoir from whichextra air mass would be drawn.

If the shadow were static or slowly moving, air wouflow deep into the eclipse zone until pressures equaliagain for the higher density of air mass present. The greadensity would be found in the center of the shadow whthe eclipse is total. However, reality is far from a static siation. The speed of sound is 330 m/s at sea level, andmally changes little with altitude. But the speed of the Morelative to the Earth averages close to 1000 m/s. The sha

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moves at the same speed as the Moon when projectedpendicular to the surface, or faster when projected obliquFrom that speed we must subtract the speed of the obseon the rotating Earth, which is generally in the same dirtion, but is never faster than 500 m/s at the equator. Sobalance, we see that the Moon’s shadow is always movinsupersonic speeds relative to ground observers.

The result is that warmer air from outside the eclipse zois continually trying to rush toward the cooler regions juinside the shadow, increasing the total mass of air overground below. But that air never gets a chance to penetvery deeply before the shadow has rushed onward, carrthe high-altitude ‘‘front’’ with it faster than air can travelHence, the ground barometric pressure is seen to rise duthe eclipse, but the amount will be a complex function of teclipse geometry. Of course, the shadow cools a much lavolume of air than can be above the observer’s horizon.the production of gravity anomalies at the observer willdependent on what the upper atmosphere is doing locallthe shadow approaches, covers, and recedes. But we ca

FIG. 2. Hourly barometric pressure measures. Bars markcontact, middle of eclipse, and last contact for the 1999 Augussolar eclipse, as measured at a site in Kefallonia, Greece.

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ALLAIS GRAVITY AND PENDULUM EFFECTS DURING . . . PHYSICAL REVIEW D67, 022002 ~2003!

confident that the air mass movement usually makesgravity anomaly an upward-pointing force relative to normair ~i.e., a weaker downward acceleration of Earth’s gravi!because air drawn from huge volumes well outsideeclipse zone is concentrated into a smaller volume onperiphery of that zone. Correspondingly, the air deficit ovewider region outside the eclipse zone should produce a srise in the local gravitational acceleration—an effect sugested by the data in Fig. 1 for about an hour before thetrough and about an hour after the second trough.

From these considerations, we see that air mass flowing eclipses might have the right qualitative behavior toplain the observed Allais gravity anomaly because it occmainly near the periphery of the eclipse zone and is inright direction. The following question now arises: Canexcess air mass of order 1.8% during eclipses be enougproduce a gravitational force of the observed magnituThis question has not previously come up becausegravimeter was usually protected from any direct effectstemperature and barometric pressure changes. Howevesee here that air mass movement toward an eclipse zonein principle produce a gravitational effect from which thgravimeter cannot be shielded.

To understand the geometry intuitively, we note thatnormal net force of the entire atmosphere on an observezero because, to good approximation, the atmosphere isof spherical shells each of uniform density; and the net foon any point inside any uniform spherical shell is zero. Tnet force of the air above the observer’s horizon is obviouupward. The net force of the rest of the air around the globelow the observer’s horizon exactly cancels the upwforce. Although that air is much farther away, its massmuch greater than the air mass above the observerchanges in this distant air mass below the observer’s horcannot be neglected in calculating the gravitational effectair on the observer.

In Fig. 3, let P be any given surface point~e.g., the ob-server!, O be the origin at the center of the Earth, andQ bea point in the atmosphere at a heightz above the ground. Fothe moment, we will approximate the atmosphere as iwere a spherical shell of uniform density with infinitesim

FIG. 3. Geometry of Earth’s atmosphere near a particular plaO5 Earth’s center;P5 observation place;Q5 point in atmo-sphere;r 5 Earth’s radius;z5 height of atmosphere pointQ aboveground;q5 distance fromP to Q. a andb are angles of the triangleat O andP, respectively.

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thickness at average heightz. In a later step, we will integrateover many such spherical shells with exponentially decreing density from the ground up. Ifr is the radius of the Earthandq is the distance from pointP to point Q, then the dis-tance fromO to Q is r 1z. We will also need the two angleQ-O-P5a andQ-P-O5b. An element of atmosphere withmassdm at pointQ will exert an acceleration on pointP ofmagnitudeda5Gdm/q2, directed toward the mass elemenwhereG is the universal gravitational constant. We cansolve this into horizontal and vertical components and ingrate over all mass elements in the whole atmosphere orsubset of them to get the total acceleration caused by themass considered.

First let us examine the forces for normal air symmecally placed around an observer at pointP. By symmetry, thehorizontal components~in the x-coordinate andy-coordinatedirections perpendicular to thez-coordinate direction! mustaverage to zero. So we can limit our consideration to justvertical component of the acceleration in thez-coordinatedirection ~positive upward from the ground!, for which daz52Gdmcosb/q2. Then the mass element isdm52pr 2r sinadadz, wherer is the mean density of air in theshell,dz is the vertical thickness of the mass element inz-coordinate,rda is the width of a ring of mass elements oradius r centered on theOP axis, and 2pr sina is the cir-cumference of the same infinitesimal ring of atmospheWith this setup, we need only integrate overa from 0 top tosum all mass elements at a given heightz, and then integrateover all heightsz from 0 to`. So the total vertical acceleration az at pointP caused by the atmosphere is

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Working with the triangle in the figure, we can eliminateqandb for the independent variablea. The relations we needare

q25~r 1z!21r 222r ~r 1z!cosa

5z212r ~r 1z!~12cosa!, ~2!

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@Equation~4! would be needed for the horizontal forcwhich we will not develop further here.# To simplify ourintegrand, we will substitute a new independent variablej512cosa. Thendj5sinada, and the range of integrationmust proceed fromj50 to j52. Making all these substitutions, Eq.~1! becomes

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T. VAN FLANDERN AND X. S. YANG PHYSICAL REVIEW D 67, 022002 ~2003!

The interior integral here leads to three terms, all proptional to 21/r 2. When that constant is factored out, the rmaining integral is then a dimensionless quantityg(0,g<1). The value ofg is 11 when evaluated at eitherj50 orj52, showing that the whole atmosphere gives zeroforce. g has a value nearer to zero in between. To goapproximation, thez dependence of this interior integral cabe neglected for just the portion of the atmosphere abovnear the observer’s horizon. For that air, the exterior integbecomes simply

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whereh is the scale height of the atmosphere andr0 is thedensity at sea level. This gives the solution we seek,az52pGgr0h.

For computation, we adopt these numerical values:G56.67231028 cm3/g s2 ~universal gravitational constant!,r 56.373108cm ~radius of the Earth!, r051.2931023

g/cm3 ~mean sea level density of dry air at standard tempeture and pressure!, h58.53105cm ~scale height at a typicasurface temperature of 16 °C)@8#. g511.0 ~its maximumvalue!. From these, we derive the valueaz54.631024 cm/s2, which is the upward-directed gravitational aceleration of the observer due to the atmosphere abovehorizon.

III. NUMERICAL MODEL

Now the effect of the atmosphere might easily have btoo weak by many orders of magnitude to be of furtherterest here. But instead we see that this numerical valualready nearly 100 times bigger than the size of the eclieffect we are seeking, verifying that we are dealing wquantities of a sufficient order of magnitude to producegravitational anomaly seen. However, this coefficient istotal upward acceleration for a non-eclipse situation. Tstandard atmospheric pressure at sea level is 1035 g/2.The increase of barometric pressure by 0.6% signifiesmore air mass has moved into the eclipse zone, increasindensity by roughly 1.8%. So the upward acceleration michange by a comparably small percentage, i.e., about31026 cm/s2). And this is indeed close to the magnitudethe observed changes, (5 –7)31026 cm/s2. We have there-fore verified that air mass movement is quantitatively aqualitatively of the correct size and character to explaingravitational anomalies seen at the times of eclipses.

Unfortunately, computing the details of how air massmove during an eclipse would be a non-trivial meteorolocal problem, one we will not attempt here. What we canthat is less demanding is to set up a numerical integratiothe effect of air mass changes meeting the constraintsscribed above to determine their calculated effect ongravimeter, for comparison with the observed effect. Sowill assumethat the air mass flow at subsonic speeds crea region of higher air mass near the shadow edges withexponential drop-off to either side, and with the magnituof this excess air mass scaled approximately by the meas

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pressure changes in the 1999 eclipse.The equation to be numerically integrated will be simil

to Eq.~1! except that we cannot assume circular symmetrythe atmosphere around the observer. Moreover, we havinformation about how air mass flow may change with heigabove ground, so we will assume that all change occurthe scale height, about 8.5 km up. With those alterations,equation we will integrate becomes

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Two of these quantities need further definition. First, tfactor of 2p in the y-coordinate integral shown earlie2pr sina, is here replaced with the interior integral in E~7!, where h is defined in Eq.~8!. This integral must beintegrated numerically because of the variations in the atmspheric densityr. The r sina part of this factor is alreadyincorporated in the exterior coefficient and the exterior ingral. Second, the density of an element of atmospherer canvary at every point as a function of projected distance ofelement from the Moon’s shadow axis,r s . To compute thislatter distance, we project the observer, the atmospherement, and the shadow axis onto a plane passing throughcenter of the Earth perpendicular to the line from Eartcenter to the observer, and examine the geometry in Fig

Viewed from space at any given moment, the Moonpenumbral shadow~wherein observers see at least part of tSun’s disk obscured! encloses a certain volume of atmosphere which then cools, compresses, and draws more afrom the surrounding regions. The radius of this conical pumbral shadow near the Earth’s surface is typically ab3.53108 cm. The shadow usually encompasses an elarger area on the ground because the Earth’s surface ismally inclined at some arbitrary angleu ~up to 90°) to theshadow cone, making the projected shadow ellipticalshape. So for any given projected distance of the obsefrom the shadow axis,X ~which is the distancePS in thefigure!, we can calculate thex,y coordinates of the distancSQ. This would immediately be the distancer s we seek ifthe projected shadow were circular. But it is elliptical a

FIG. 4. Projection of observerP, atmosphere elementQ, and theMoon’s shadow axisS onto a plane through the center of the Earperpendicular to a line fromP.

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ALLAIS GRAVITY AND PENDULUM EFFECTS DURING . . . PHYSICAL REVIEW D67, 022002 ~2003!

FIG. 5. The graph shows the results of a nmerical integration of changes in the force of thatmosphere on an observer during the 19eclipse. This simulation uses plausible-buassumed air mass movements with magnituand extent chosen to give a best fit to the 199eclipse gravimeter data, but consistent with quatative reasoning and measured 1999-eclipse psure changes. Compare with Fig. 1.

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elongated along the line to the observerSP if the observer isin the path or totality. So givenr ,X,a,h,u, the relations weneed are these:

xQ2xP5r sina cosh, yQ5r sina sinh,

xS2xP5X, r s5A~xQ2xS!2cos2u1yQ2 . ~8!

We now assume our air mass model, which will consista ring of denser air near the edge of the penumbral sha~radiusr p), with a normal curve distribution having ampltude k and half-width r h . For example, a typical modewould be of this form:

r5r0~11ke2[( r s2r p)/r h] 2!. ~9!

We then adopt two sets of values fork andr h that give thebest fits to the data in Fig. 1, one set for before mid-ecliand the other for after.@The two k values differ becauseuchanges during the three hours between first and lasttacts, which slows the progress of the eclipse near the laallowing more time for air mass movement. Thek valueadopted were 0.013 and 0.018, whiler h'900 km works wellfor both contacts.# The results of the two numerical integrations with those parameter values are presented togethFig. 5. This shows that plausible air mass movement mocan reproduce the observations, especially since the lavalue from this fit to the 1997 gravimeter data~0.018! is thesame as that inferred from measured pressure changes i1999 eclipse~1.8%—see discussion of Fig. 2!. Only futuredata collection can assess whether or not the real atmospbehaves as these models do.

IV. CONCLUSION

We can therefore conclude that air mass movement dusolar eclipses is a significant effect that cannot be neglewhen trying to explain the Allais gravity anomaly, and mig

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well be the entire explanation of the effect. However, tgravitational anomaly discussed here is only a few partsbillion of Earth’s own gravitational force, so it is aboutfactor of 100 000 too small to explain the Allais excess pedulum precession or the change in pendulum swing pe~an increase of 1/3000! that sometimes shows up durineclipses. But the relatively sharp changes in barometric psure during an eclipse can certainly create local air mmovement at ground level, for example, into or out ofbuilding. So experiments that were shielded only from teperature changes but not pressure changes may have exenced an extra and unexpected driving force from localmovement perhaps responsible for these changes, wheother experiments with better controls would not have exrienced them. This is also consistent with the pendulumfect showing up most often in the early experiments, bhaving no unambiguous detections within the past 30 yewhen consciousness of the importance of controls was mwidespread.

The explanations presented here for the Allais gravity apendulum anomalies can be tested and further refined.example, we would learn more about air mass movemunique to eclipses by sending balloons to high altitude ameasuring actual changes in wind speeds and air movemat different locations and altitudes above sea level. This wpermit more realistic air movement modeling. In the meatime, the much smaller anomalous acceleration seen infrom the first two Lageos satellites still looks like a goocandidate for a possibly real gravitational shielding effe@9–11#. The Allais gravity and pendulum effects, howeveare clearly not associated with gravitational shielding.

ACKNOWLEDGMENTS

The first author~T.V.F.! thanks the Meta Research Boaand members for financial support, with a special thanksTim Seward.

ntsnes/

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//@8# Allen’s Astrophysical Quantities, 4th ed., edited by A. Cox

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