A geophysical experiment on Newton’s inverse-square law
V. ACHILLI(1), P. BALDI(2), G. CASULA(3), M. ERRANI(4) S. FOCARDI(5), F. PALMONARI(5) and F. PEDRIELLI(6) (1) Osservatorio Vesuviano - Napoli, Italy
DISTART, Università di Bologna - Bologna, Italy
(2) Dipartimento di Fisica, Università di Bologna - Bologna, Italy
(3) Istituto Nazionale di Geofisica - Roma, Italy
(4) INFN, Sezione di Bologna - Bologna, Italy
(5) Dipartimento di Fisica, Università di Bologna and INFN - Bologna, Italy
(6) Dipartimento di Fisica, Università di Ferrara - Ferrara, Italy
(ricevuto il 27 Agosto 1996; approvato il 30 Ottobre 1996)
Summary. — A geophysical experiment consisting of the measurement of the
gravitational effect produced by a large water mass was performed in order to verify Newton’s law. The use of a superconducting gravimeter, the detailed analysis of the local tidal perturbation, the precise topographic and geological surveys lead to a precision of about 0.1% in the final result. The ratio between the measured and the expected gravitational effect differs from 1 by more than 9 standard deviations. This may be explained by adding to the Newtonian potential a Yukawa repulsive term. The experimental result leads to constraints for the relationship between the relative magnitude (a) of the new term and the range (l) of the interaction. In the region 20 m ElE500 m, a ranges from 2.6% to 1.3%.
PACS 04.80 – Experimental studies of gravity.
1. – Introduction
One of the most ambitious projects in physics is the description of all fundamental interactions by a single force. The recent success of the electroweak unification in the Weinberg-Salam model [1] is a strong support of the opinion that further unifications of all the actual fundamental interactions might be possible. On the other hand, great difficulties must be overridden in the attempt to unify the gravitational force with the other ones. The main difficulty is that gravitation is presently described within the framework of general-relativity theory which gives essentially a geometric interpret-ation for this interaction, while the other interactions are treated in terms of virtual particle exchange. Classical gravitation requires definite information on position and momentum, in contrast with quantization rules. Until today gravitation has resisted incorporation into a renormalizable quantum field theory. It appears that these difficulties can be avoided in superstring theories. In fact, in contrast with point-field
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theories, superstrings are believed to be a finite theory which avoids completely the necessity of the renormalization prescription.
Within the framework of supergravity unification, Scherk [2] suggested the existence of a possible force due to a spin-1 particle (graviphoton). Moody and Wilczek [3] and Kim [4] advanced the hypothesis of a pseudoscalar massive particle (graviscalar). Both particles, graviphoton and graviscalar, would acquire non-zero masses because of symmetry breaking.
More generally, supergravity and superstring theories can provide candidates of vector and scalar fields [5]. These new particles, generally, are massive and their interaction ranges are finite [6]. The simplest modification of the Newtonian potential energy, between pointlike masses m1 and m2, taking into account a new particle with finite-range interaction, can be written in the form
U 42 G m1m2
r ( 1 1ae
2rOl) . (1)
The coefficient a sets the magnitude of the new term relative to the Newtonian one and
l determines the range of the interaction. The same expression was proposed by
Fischbach et al. [7] as possible explanation of their re-analysis of the classical Eötwös experiment [8]. The simplest consequence of such a hypothesis leads, obviously, to a modification of the inverse-square law
F 4Gm1m2
r2 (2)
given by Newton [9]. The validity of this law has been carefully tested in the laboratory experiments at distances less than 1 m. Above 10 km, the law has been verified by comparing the G-values obtained by the Moon and LAGEOS satellite motions. For intermediate distances, Cook [10] and De Rujula [11] report experimental limits on a of the order of 1022 (for a D0) and 231022 (for a E0) for l in the region 1–1000 m. Recently, two experiments [12, 13] quote better limits of the order of 1023; however, as pointed out also by Fischbach et al. [14], geophysical measurements for this range of distances cannot be easily modelled.
Here we describe an experiment to verify Newton’s law for this range of distances. Taking advantage of the most accurate instrumentation for the measure of artificially induced small variations of gravity, the experiment was designed to minimize any model dependence.
2. – Overview of the experiment
The fundamental characteristic of our geophysical experiment is the independence of the measurements from the terrestrial gravitational field. We searched for a situation in which a gravitational signal can be obtained as response to the displacement of a large mass. Such a condition can be found near a lake used as power storage, where the variations of the water level induce gravitational changes. Brasimone-Suviana [15, 16] turned out to be the most interesting system among all possibilities existing in Italy. The lake Brasimone (845 m above sea level, approximately midway between Bologna and Florence), together with the Suviana lake, constitutes a
power storage used by ENEL (the Italian national electric power company). During the night, water is pumped up from Suviana to Brasimone and the inverse process is used to produce electric power at hours of maximum need. On the Brasimone lake shore, in a research centre of ENEA (the Italian national institute for energy development), 30 m below the ground level, a 100 m long tunnel extends towards the centre of the lake (fig. 1). We placed a gravimeter at the end of the tunnel in order to measure gravity variations induced by the moving water mass. The resulting gravitational signal differs only by 8 % from that produced by an infinite sheet, and does not depend very critically on the shore shape.
We used a superconducting gravimeter which is capable of recording gravity variations of the order of 10 nGal (1 Gal 4 1 cmOs2). Such a figure must be compared with the B280 mGal total gravity change due to the water level variation. Because of tidal effects, the gravitational acceleration varies continuously in time giving a contribution to the recorded signal in the 100–250 mGal range. The need to subtract the tidal effect requires its precise knowledge. The tidal variation cannot be exactly determined from the Moon and the Sun positions only since a part of it depends on the local geophysical characteristics. The tide at Brasimone was measured by running the gravimeter for five months in an ENEA laboratory 400 m far from the lake. All gravimeter calibrations were executed in the same laboratory.
In order to compare the theoretical Newtonian effect with the measured one, the lake shore shape must be known as accurately as possible. For that reason an aero-photogrammetric survey was carried out when the lake was at the minimum level.
The water level was measured and recorded with a 0.1 mm sensitivity.
Samples of water were taken from the lake, in different times, in various positions and at different depths. Their density was measured and no significant differences were observed.
The water temperature in the lake near the gravimeter location was recorded at 21 m and 25 m below water level.
Air temperature, relative humidity and atmospheric pressure were also continuously recorded, in order to deduce the air density which was needed to subtract the air mass from the displaced water mass.
Other less important effects were also taken into account. The water load induces elastic deformations of the site. For such a reason very accurate measurements of this deformation were performed.
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The behaviour of the water table and its possible relationships with water level in the lake were also studied.
In the next sections all the aspects of the experiment here summarized will be discussed in detail.
3. – The superconducting gravimeter
The superconducting gravimeter [17] differs from traditional mechanical spring instruments because it uses the magnetic levitation of a superconducting spherical mass through the field gradient generated by the persistent currents of two superconducting coils. The inherent stability of supercurrents allows for a gravimeter of high stability and precision. The theoretical instrumental sensitivity is of the order of 1 nGal; geophysical and environmental noise reduce sensibly the effective accuracy (0.01 mGal or better).
Figure 2 shows a view of the cross-section of the instrument core, which is immersed in a liquid-He bath inside a dewar. Our instrument, which is operated in a very restricted space inside a tunnel, was built with a specially small dewar of 50 cm diameter and a 60 l liquid-He capacity. A cold-head refrigeration system allows the gravimeter to run for about four months without interruptions for He refilling.
The instrument, in order to measure the vertical component of local gravity, is provided with a very accurate feedback system controlling and adjusting its vertical setting. This ensures also a total insensitivity to deformations of the gravimeter basement. For a complete description of the instrument we refer to the literature [18]; here we recall briefly its operating principle.
The gravimeter signal corresponds to a feedback force used to hold the levitating sphere in a fixed position when the gravity force changes. The position of the sphere is sensed by the imbalance of a capacitance bridge formed by plates above and below the spherical mass and a ring around its center. The feedback force is provided by a small
Fig. 3. – The calibration apparatus: an annular mass is placed around the gravimeter and moved up and down. A high-precision (0.1 mm) electronic digitizer is used to sense the mass position.
coil underneath the sphere which generates a variable magnetic field; the voltage applied to the feedback coil depends linearly on the variations of gravity. In order to obtain such relationship a calibration of the gravimeter must be performed. The scale factor depends on the superconducting coils current setting, which determines the levitation magnetic field gradient.
Until now the usual approach for the calibration consisted in comparing the superconducting gravimetric tidal signal with the corresponding one of an absolute gravimeter; in this way a calibration at 1 % level is obtained [19].
To reach a higher precision we adopted a method [20] consisting of moving a mass which produces a known change of the gravitional field.
The calibration apparatus, shown in fig. 3, is described in detail by Achilli et al. [21]. The choice of an annular mass (273.40 6 0.01 kg) placed around the meter, reduces systematic errors (e.g. due to an erroneous determination of the mass position) well below the statistical accuracy of the method, which turns out to be 0.1%.
The vertical component of the gravitational force exerted by this mass is calculable exactly for the simple geometry adopted; it can be expressed as a function of the vertical distance h between the mass center and the sphere center, by the following
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Fig. 4. – Gravity variation induced by the mass movement.
expression: Fz(h) 4maz(h) 42pGm M p(R222 R12) H f(h) , (3)
where G is the gravitational constant, m and M the mass of the sphere and of the calibration ring, R1 and R2 the internal and external radii, H the thickness of the annular mass, and f(h) a smooth function of h. Figure 4 shows the excitation signal
a(h) best fitted to the experimental gravimeter data (tidal signal subtracted) produced
by the cylindrical mass movement. Three parameters enter the fit: a constant voltage offset, the gravimeter sphere position and the calibration scale factor (referred as CSS in table I).
Another approach (PSC method) consists in the direct measurement of the total peak-to-peak effect, whose calculated value, including the small correction for the mobile mass-supporting parts, corresponds to a gravity change of
Dgcalc4 6.731( 1 ) mGal (4)
with a total relative uncertainty of about 0.01%.
In fig. 5 a typical series of peak-to-peak measurements is shown. The transitions between the two peaks were made at time intervals longer than 3 minutes, in order to take into account the gravimeter response delay time. Calibration runs were made at different times, during the experiment.
Our gravimeter is equipped with an auxiliary tool, the electrostatic calibrator, allowing the measure of the feedback voltage response to a constant electrostatic force applied to the sphere. This electrostatic calibration allows the monitoring of the time stability of the scale factor to better than 1 part per ten thousand.
In table I the calibration constants obtained with both methods at different times, each of them related to different coil settings, are listed together with the corres-ponding electrostatic-calibration values.
TABLE I. – Calibration constants and electrostatic calibration values for different coil settings. Period (year) Method Calibration factor Electrostatic calibration (mGalOvolt) (volts) 1991 CSS 66.09 6 0.31 5.1875 6 0.0009 1991 PSC 65.91 6 0.16 5.1875 6 0.0009 1992 CSS 65.54 6 0.27 5.2198 6 0.0003 1992 PSC 65.71 6 0.17 5.2198 6 0.0003 1993 CSS 65.14 6 0.22 5.2715 6 0.0003 1993 PSC 65.32 6 0.10 5.2715 6 0.0003 1994 PSC 65.50 6 0.10 5.2722 6 0.0003 1995 PSC 64.95 6 0.10 5.3650 6 0.0003 1995 PSC 64.88 6 0.33 5.3691 6 0.0005
determine the correct scale factor to be used for the measurements in the tunnel under the lake, where only electrostatic calibrations can be done. In table II the electrostatic-calibration values measured during the two runs performed in the tunnel in 1992 and 1993 are reported. In the third column the corresponding calibration constants used in the analysis are given. Their quoted errors take into account both the interpolation error and the scale factor global error.
In May 1994 a comparison between an absolute and our superconducting gravity meter was performed [22]. The instrument of the IMGC (Italian Institute of Metrology G. Colonnetti, Turin), which adopts a symmetrical rise-and-fall principle, was used;
Fig. 5. – An example of repeated gravity measurements moving the mass between the positions of maximum effect.
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Fig. 6. – Calibration factor vs. electrostatic calibration.
TABLE II. – Electrostatic calibration values of the two runs and corresponding calibration factors. Period (year) Electrostatic calibration (volts) Normalized calibration factor ( mGalOvolt) 1992 5.2712 6 0.0001 65.42 6 0.05 1993 5.2354 6 0.0006 65.61 6 0.07
166 drops distributed in three periods, which include maximum and minimum values of the tidal effect, were made.
To get the calibration constant we compared our gravimeter tidal signal (in volts) with the corresponding one of the absolute gravimeter. The calibration factors, calculated both using all data from different runs, and a set of 65 drops collected in one single run in correspondence with the best tidal signal, are reported in table III. In the same table the mass calibration constant calculated from the
TABLE III. – Calibration factors obtained by comparing our gravimeter with an absolute one.
Date Absolute gravimeter Mass calibration
number calibration electrostatic calibration
of drops factor calibration factor
1994-5-9,10,11 166 64.4 6 0.5 5.2687 6 0.0003 65.44 6 0.06
electrostatic calibration is also reported in the last two columns. At the level of the absolute-gravimeter precision (about 1%), the comparison is satisfactory.
The absolute gravity value at the laboratory was measured as
g 4980.3065357(8) cm s22.
(5)
4. – The local tidal signal
The tidal signal depends both on the direct Moon and Sun effects and on the elastic response of the Earth. Furthermore, oceanic load, barometric pressure, local anomalous elastic response of the Earth, topographic and geological features, induce deviations from a global Earth tides model which are not determinable without a direct measurement. For this reason we recorded for five months the tidal signal at the experimental site. The instantaneous gravimeter voltage signal, synchronized to better than 60.01 s to the universal time signal UTC received by an antenna on-site, was recorded at a rate of 0.5 s. A 20 s rate signal was generated filtering the original one by means of a low-pass electronic Butterworth filter which eliminates high-frequency noise. A further standard format modification with detection and flagging of interruptions was applied to obtain 1 min sampled data.
As a successive step we computed hourly data by adjusting a polynomial on a 23 min time interval, centered on the hour.
The tidal analysis itself [23] consisted of two steps according to the technique developed by Venedikov [24, 25], starting from the harmonic formulation which describes the n hourly readings li at the epoch ti:
li4
!
jHj cos [vj(ti2 t0) 1fj] 1C1D ,
(6)
where Hj is the observed amplitude of the wave of frequency vj, fj is the observed
phase of the same wave at a conventional fixed time t0, C is a constant depending upon the choice of the reference and D represents the instrumental drift [26].
According to Venedikov’s approach, the residues of a best-fit solution to the above equation were analysed separately for the estimation of the long-term instrumental drift and for the computation of the influence of atmospheric-pressure variations. These are the major cause of random fluctuations of gravity, due to the gravitational attraction by the air and to the distortion of the Earth’s surface resulting from pressure changes [27]. The strong correlation between residues and barometric pressure is shown in fig. 7. The computed mean pressure admittance is 2 0.201 6 0.007 mGalOmbar and the instrumental drift turns out to be linear (2.709 6 0.001 mGalOday).
The data corrected for the pressure and drift effects were used as input for the computation of a final least-square solution. The r.m.s. value of the final residues is 0.2 mGal. The results are listed in table IV: the main tidal wave types [28] (column 1), their amplitudes (column 2) and the phase differences Df (column 3) with respect to the reference model, based on the Cartwright-Tayler-Edden development [26].
The high quality of the results is due both to the high stability of the super-conducting gravimeter and to the good quality of the site, characterized by a very low environmental noise level.
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Fig. 7. – Tidal signal, residues after removal of tide and drift, atmospheric pressure.
Our tidal model was checked by diurnal sequences of gravimeter data taken at various times. The r.m.s. residues found are consistent with those of the fit. However, on longer data sequences, long-term tidal modes become important and the residue distribution broadens considerably. The knowledge of such terms is in any case not relevant in our method of analysis (which will be discussed in sect. 8) which uses time intervals of the order of ten minutes.
TABLEIV. – Results of the tidal analysis at the Brasimone station.
Wave H Df
Estimated amplitude Phase difference
( mGal) (degrees) Q1 6.91 6 0.03 20.18 6 0.26 O1 35.99 6 0.03 20.05 6 0.05 NO1 2.82 6 0.02 0.3860.46 P1 16.83 6 0.03 0.10 6 0.09 S1K1 50.03 6 0.03 0.07 6 0.03 J1 2.83 6 0.03 20.41 6 0.58 OO1 1.50 6 0.03 1.17 6 0.99 2N2 1.37 6 0.01 0.48 6 0.50 N2 8.71 6 0.01 1.48 6 0.09 M2 46.00 6 0.01 0.94 6 0.02 L2 1.29 6 0.02 2.26 6 0.94 S2 21.44 6 0.01 20.01 6 0.04 K2 5.82 6 0.01 20.436 6 0.14 M3 0.59 6 0.01 2 1.63 6 0.80
5. – Lake survey and modelling
In order to calculate the gravity variations induced by the moving water masses, the shape of the lake and the position of the gravimeter in the same reference system must be known.
A preliminary study was performed by using a topographic map in order to evaluate the gravitational effect as a function of the lake shore profile. In fig. 8 the ratio between this effect and that due to a water infinite layer is plotted as a function of the azimuthal angle f, where f is measured in a polar reference frame centered on the gravimeter station. As one can see, the lake shore correction is quite small (B5%) almost everywhere, apart from a sector of B 607 corresponding to the nearest shore. Therefore, both a general topographic survey and a more accurate one in this small region were performed.
A reference plano-altimetric network, based on 17 primary and 5 secondary stations, was installed along the border of the lake (fig. 9); the vertices are located on pillars based directly on bedrock. High-precision geometric leveling, electromagnetic distance measurements and angular measurements, were performed; also a GPS (Global Positioning System) static observation campaign was done in order to improve the rigidity of some portions of the network. The 62 planimetric connections allowed us to obtain the planimetric coordinates of the primary vertices with an accuracy of the order of 1 cm (3 s). This error, together with the systematic scale error associated with the electromagnetic distance measurements (G1025), gives a negligible contribution to the final computation.
The vertical coordinates were measured with an accuracy of 1 mm or less in a geopotential reference system. The corrections for the mean curvature of the equipotential surface, needed to obtain Cartesian coordinates, reduce the accuracy to about 1 cm. An aerophotogrammetric survey was successively performed, using the network for the definition of the reference frame; a flight altitude of 500 m above the surface allowed us to obtain photograms with a scale of 1:7500; the expected errors in the coordinates derived from the restitution are estimated in 20 cm. An independent
Fig. 8. – Influence of the shore geometry: Newtonian effect (relative to that of an infinite layer) vs. the azimuthal angle.
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Fig. 9. – Map of the Brasimone area, showing the gravimeter position, the topographical network and the alignment of the clinometric surveys.
check of this accuracy was performed by means of a GPS kinematic survey. Altimetric profiles were obtained using two receivers, initially located at two stations of known relative position, and moving one antenna to the points whose position were to be determined. It is well known that this method provides centimetric accuracy on small areas [29]. We controlled the vertical coordinates of about 200 points distributed in two different sectors of the shore; the comparison gives a mean systematic difference of 2 and 5 cm, respectively, for the two areas, with a root mean-square scatter of about 10 cm. A more detailed terrestrial photogrammetric survey was done in order to define with better accuracy the profile of the part of the shore close to the gravimetric station; in this case, starting from photograms with a scale 1:1500, we reduced the errors to about 5 cm.
The digital model of the shore obtained by the above surveys is based on the coordinates of 50 000 points distributed along 21 contour lines covering 10 m of water level variation.
The position of the gravimeter in the tunnel was obtained using the following approach: from the plano-altimetric network a top point was determined above the access pit, then vertically transferred to the tunnel level with plumb-line and distance measurement; finally, linking measurements to the gravimeter station were performed involving distance, leveling and orientation by a gyroscope. The final
Fig. 10. – Calculated Newtonian effect of 10 cm water level vs. the vertical distance from the gravimeter.
errors are of the order of 6 cm in horizontal coordinates, and of 1 cm in the vertical one.
The calculation of the gravitational effect of the water was done by numerical integration of the Newtonian force on parallelepipedal volume elements.
Let x * , y * , z * be the integration variables, and x , y , z the gravimeter coordinates, the vertical component of the gravitational acceleration by a right parallelepiped mass is given by g(x , y , z) 4Gr
x1 x2 y1 y2 z1 z2 z * 2z [ (x * 2x)2 1 (y * 2 y)21 (z * 2 z)2]3 O2dx * dy * dz * . (7) By integration (8) g(x , y , z) 4 4 2 GrNNN
k
(x * 2x) ln[
y * 2y1 [ (x *2x)2 1 (y * 2 y)21 (z * 2 z)2]1 O2]
1 1(y * 2 y) ln[
x * 2x1 [ (x *2x)2 1 (y * 2 y)21 (z * 2 z)2]1 O2]
1 1(z * 2 z) tan21 (z * 2z)[ (x *2x) 2 1 (y * 2 y)21 (z * 2 z)2]1 O2 (x * 2x)(y *2y)z
N
z2 z1N
y2 y1N
x2 x1 .Starting from the contour lines with elevation differences of 0.5 m, representing the measured model of the shore, a new digital model, computed by interpolating on a regular planimetric grid (0 .5 3 0.5 m2) the profile of the topography, was used for the numerical integration. The above calculation was checked with a totally different integration method based on polar coordinates around the gravimeter and no significant differences (at the 1025 level) were found.
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function of their vertical distance above the gravimeter. The overall uncertainty was evaluated taking into account three different sources of errors:
– the uncertainty in the gravimeter position;
– a random measurement error, normally distributed, affecting all points of the contour lines used for the digital model;
– the systematic effect of water absorption in the lake shore.
The gravitational effect of the displaced water mass in the Suviana lake is negligible. The numerical integration was repeated changing the nominal gravimeter position. Taking into account the accuracy of the gravimeter position in the plano-altimetric reference network, and combining errors in quadrature, a total relative error of 1.431024 can be obtained.
The second error, due to the uncertainty on the coordinates of the contour points, was numerically evaluated assuming a normal distribution of the errors with a variance of 25 cm; it appears to be at least an order of magnitude lower than the preceding one (i.e. G231025).
The last source of error causes a systematic increase of the Newtonian effect, corresponding to an additional water mass on the lake shore. We calculated the effect taking into account the expected water absorption into the clays (B5%) and a maximum diffusion length of the order of 1 m in 10 minutes. This systematic effect, corresponding to a total volume of B17 m3of water, was evaluated to be smaller than 5 31025 of the gravitational signal.
Using our experimental data we could verify the correctness of the above figures. In between periods of exploitation of the power plant, the lake level should be constant, apart from small known contribution of water from rivers or rain, and losses due to evaporation; one unknown contribution is due to ground diffusion. We analysed the lake level time dependence just at the end of many pumping or emptying cycles, and measured the level slope for periods of about 15 min. The measured effects correspond to volumes of water which are always a factor two smaller than 17 m3, the figure given above for the lake shore diffusion.
In conclusion, the relative error on the theoretical calculation of the Newtonian effect is 1.4 31024.
6. – Water density and level monitoring
For the purpose of the experiment we needed to measure the lake level variations with an integral accuracy of the order of 1 mm over the useful range of 7.4 m and to have a 0.1 mm sensitivity in order to be able to monitor short-term variations of the water level during operation of the electric power station when a level change rate of 0.2 mmOs is reached at pumping rates of 100 m3Os.
The required sensitivity was achieved thanks to the existence of a large well (5 m diameter, 30 m deep) communicating with the lake through a 50 cm diameter, 150 m long pipe extending horizontally towards the center of the lake (see fig. 1), thus sensing the water level near the gravimeter underground station. This well, being free from water surface atmospheric disturbances, turned out to be ideal for monitoring the lake level, on also taking into account the stability of the temperature which shows only slow seasonal variations from about 6 to 16 7C. In any case free oscillation modes of the lake itself (160 s fundamental mode period, few mm in amplitude) eventually excited by
wind and observable in the well, were also eliminated by smoothing the measured level. Small corrections to the level readings due to temperature differences between the well and lake waters were also applied.
The lake level was monitored by measuring the length of a stainless-steel wire fixed to a buoy, which was suitably stabilized and vertically guided. The wire was maintained at constant tension and wound on a precision aluminum pulley wheel. The angular position of the pulley was monitored by an absolute digitizer having a sensitivity of 0.1 mmOdigit. The digitizer readings were sent via a data acquisition card to a PC and, usually every 20 seconds, the mean value of five successive readings were recorded in a disk file.
The correspondence between digitizer readings and the lake level in the local coordinate reference system was established to better than 1 cm by measuring the lake level with a stage standing on a submerged point whose elevation was known.
The calibration of the digitized level-meter was obtained by comparison of the digitizer readings with the measurements of an electromagnetic distance-measuring instrument [30] having a sensitivity of 0.1 mm and a calibrated accuracy of 1 mm. The laser beam pointed vertically down the well onto a reflecting-mirror system floating on the water surface. The calibration procedure was repeated at various lake levels, both in static conditions and during rapid variations of the water level.
In fig. 11a) the measurements performed from December ’93 to April ’94, in order to study the long-term reproducibility of the absolute digitizer readings, are shown. They span a 7.8 m range of water level variations corresponding to the digitizer readings from 33 500 to 111 000. A linear fit superimposed to the data gave the residues shown in fig. 11b): they lie within 25 digits, and a non-linear effect is visible, amounting to about 2 mm over the entire range. This non-linearity is compatible with a thermal effect due to the air temperature gradient in the well between the water surface and the digitizer wheel many meters above. In table V we report the results of a global fit of all measurements, showing the differences between a linear and a quadratic fit.
In summary, in the gravimeter data analysis the following errors were used: – Water elevation in the local reference system 61 cm (relative error 831025), – Integral error over an 8 m change in the lake level 615 digits (relative error equal to 2 31024).
The level measurements performed in the well must be corrected for the water thermal differences between the well and the lake [31]. For this reason three platinum thermometers [32] with 0.1 7C sensitivity, were placed at different depths in the well. The deepest one was in front of the horizontal pipe communicating with the lake, and the other two, respectively, 5 m and 10 m above. Their temperatures were recorded at time intervals of 5 minutes. At the same time the temperatures at a depth of 1 m and 5 m were measured in the lake along the vertical above the gravimeter station. The temperatures measured in the well and in the lake, for the period January-August 1993, are shown in fig. 12. In the lake we assumed as reference only the temperature at 2 5 m; in fact, as reported in reference [33] and verified by us, the maximum temperature difference, measured at any depth in various places and in different seasons, never exceeded 60.8 7C. The ratio R between the water density in the lake and in the well is shown in fig. 13 as a function of the water temperature in the lake; the best fit of the data obtained with a fourth-order polynomial, is also shown. The effective water level variations of the lake can be obtained dividing by R those measured in the
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Fig. 11. – Calibration of the digitized level-meter: a) level-meter digits vs. distance measurements, b) residues of a linear fit.
well. As shown in the figure the maximum correction is less than 4 31024and the error can reach in the worst conditions 1024.
Water samples were drawn from the lake in order to measure the water density. The presence of suspended particles (typically SiO2) and phytoplankton was disregarded because their relative contribution to the density is about 5 31026[34]. TABLEV. – Results of the digitizer calibration (z given in meters).
No. of Type of Chisquare r.m.s. Slope
points fit value (digits) (digitsOm)
205 linear 537 16.2 10032.8 6 0.4
Fig. 12. – Temperatures in the well and in the lake (2 5 m) showing the seasonal effect in the period January-August 1993.
Fig. 13. – Ratio R between the water density in the lake and in the well vs. the lake temperature.
Using an absolute densitometer with a 1026gOcm3 sensitivity the mean density measured at 21 7C was
( 998.146 60.016) kgOm3. (9)
This value was extrapolated to the different lake temperatures by using the known density-temperature relations [35]. Finally, we can estimate the water specific mass with a relative error of 2.6 31025.
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displaced air mass. The air density was computed by the relation [35]
r 41.29293
g
273.3T
h
3g
P 20.37833E
760
h
, (10)where T is the ambient temperature in kelvin degrees, P the atmospheric pressure in Torr and E the water vapor pressure in the same units. The relative uncertainty of this correction can be estimated to be 1 31025.
7. – Geological effects
This experiment, as anticipated in sect. 2, is model independent. Such statement is obviously not rigorously true because the water basin is not perfectly stable. In this section two main geological effects that must be taken into account will be discussed. The first is the ground subsidence due to the water load, the second is related to the water table and underground absorbed waters.
Let us consider the subsidence and uplift produced, on the gravimeter station at the end of the tunnel under the lake, by the load variations. The total vertical displacement
V can be split into two parts,
V 4V11 V2, (11)
where V1is the displacement between the gravimeter and the well (used to correct the lake level changes starting from the measurements in the well) and V2is the residual displacement. For the correction of gravimeter readings the total vertical movement in the terrestrial gravity field is needed.
We used a hydrostatic clinometer [36], consisting essentially of a horizontal water pipe equipped at both ends with high-precision capacitive level sensors. A 93.3 m long water pipe was placed in the tunnel, horizontally aligned to better than 1 cm. The instrument can measure a 1 mm vertical displacement over the tunnel length, thanks to a thermal correction of the level readings and to the thermal stability inside the tunnel.
An example of the correlation between the lake level and the clinometer response is shown in fig. 14.
In fig. 15a) and b) the subsidence and, respectively, the uplift vs. the lake level, are shown. Both figures refer to 10 cm water level variations. As one can see, the phenomenon is not reversible but a hysteresis occurs, presumably due to the clays that constitute the ground basin. Two separate linear fits to the data give the following results:
– lake moving up dh 417.2(2)23(2)31022Q H mmOdm, – lake moving down dh 424.6(2)29.5(2)31021Q H mmOdm where H is the water level.
Finally, we estimated V1 extrapolating the measured value dh for the total distance between the gravimeter and the well, obtaining a mean value
aV1b 419.3 mmOdm. (12)
Note that the gravity equipotential surface correction due to 10 cm of water is only 0.2 mm, comparable to the maximum error in the fitted vertical displacement
Fig. 14. – A five-day sample of lake level (continuous line) given in a local reference system and clinometric signal vs. time.
dh. Such an error may give only a 4 31026 relative contribution to the final
computation of the gravitational effect.
In order to evaluate the total displacement V, we first simulated the load variation effect starting from a finite-element approach. To figure the elastic properties of the terrain we used both a regional geological profile for the deep rocks and the results of a local seismic survey for the material of the first 70 m thick ground layer [37]. The models predict a vertical displacement in the range 0.25–0.50 mm for every meter of water at the gravimeter station. They do not show any appreciable difference between the deformations at ground surface and 30 m depth (the gravimetric station). Furthermore, the deformation reduces to 10% of its maximum value 1.5 km away from the lake centre. The theoretical models were improved with many field measurements. The differential deformation between surface and gravimetric station level was monitored by means of a digitized level meter confirming the finite-element model results. High-precision spirit leveling surveys with empty and full lake were repeatedly performed from the well to the building B and to a benchmark 1.5 km away from the lake centre (see fig. 9). Furthermore, four clinometric surveys, besides that in the tunnel, were performed inside building B, along the lines shown in the same figure. The total vertical displacement obtained combining all the measurements is 0.52 6 0.1 mm for 1 m of water level variation. This value should be increased by 0.05 mm in order to take into account the residual displacement from the last benchmark to infinity (sic!). In conclusion we used for the correction of the gravimetric data the value
aV b 40.57 6 0.11 mm for 1 m of water . (13)
To evaluate the effects of the water table variations, the results of the existing studies of the local hydrology [38] were used. Most of the lake shore is characterised by rocks with a very low porosity (clays); the water table variations are only of a few centimeters and their delay in response to a single filling of the lake is in the range of 2–10 days. Our differential analysis method, based on measurements in time intervals of about 10 minutes, excludes any contribution derived from these delayed effects. In a small area,
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Fig. 15. – Clinometric signal for a 10 cm water level variation showing a different behaviour for water moving up a) and moving down b). The lake level is given in a local reference system.
where a geological survey identified fractured rocks (Massiccio del Brasimone), on the contrary, some piezometric measurements showed [39] an instantaneous response of the water table to the lake level variations. For this area, evaluating the total volume involved in immediate water table variations and assuming a maximum porosity of 33%, we estimate a gravitational effect of 0.15% relative to the lake signal. This correlation vanishes near the border of this structure, as shown by the continuous monitoring of the water table performed during this experiment [40].
8. – Data collection and analysis
The gravimetric effect of the moving mass was monitored during two different periods, in 1992 from 4 March to 13 July and in 1993 from 9 January to 15 August, for a total of 327 days. Figure 16 shows an example of the recorded gravimetric signal
compared with the theoretical tide and the lake level. During the two runs several electrostatic calibrations were also performed to calculate the conversion coefficient from volts to mGals (see table II). For each day, a file containing the most important information was prepared. Every 20 s the following data were recorded: UTC time, TIDE-signal (in volts), HR-signal (in volts), mean square root of HR-signal (in volts), lake level (in digits), atmospheric pressure (in mbars). The TIDE-signal is the gravimeter output filtered through a low-pass filter and recorded after analogic digital (AD) conversion every 20 seconds. The HR-signal is another gravimeter output signal recorded every 0.5 s, passed through a wider band filter and treated with a higher-resolution AD converter. Each mean HR-value and its mean-square root was calculated starting from a sequence of 40 consecutive HR-data. The value of the mean-square root is very sensitive to short-term perturbations like earthquakes and to possible errors of the AD converter. Data with a standard deviation higher than a predetermined threshold value were excluded in the analysis.
In order to calculate the water and the air density, a general file was also set up: for each day we were interested in, the 2 1 m and 2 5 m lake mean hourly temperatures were there recorded.
A visual inspection of the graphical representation of the diurnal files permitted us to select time periods for which the lake level was changing. In the same way every period characterized by some malfunction, earthquakes, or run interruptions due to electrostatic calibrations was excluded. For each useful period the day and the initial
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and final times were recorded in a main file. Such a file was used as the starting point for the subsequent analysis. The analysis programs can operate on TIDE or HR-data. By using the amplitudes and the phases obtained from the tidal measurements (table IV), the tidal value is calculated in mGals for every recorded time. Every gravitational measured value (TIDE-signal or HR-signal) is first converted from volts to mGals by using the conversion coefficient determined by electrostatic calibrations. Afterwards, such a value is corrected for the following effects: gravitational variations due to pressure changes and gravimeter drift. Successively, the tidal contribution is also subtracted. The residual signal so obtained represents the gravitational effect associated with the lake level change. Two different data analysis, integral and differential were performed. In both the techniques the following effects were taken into account:
a) The conversion from digits to centimeters was performed by using the
Distomat calibrations, described in sect. 6;
b) The effective level variation of the lake was calculated from that measured in
the well, by taking into account the differential vertical displacement between the gravimeter and the well. The mean value of this correction is 19.3 mm for every 10 cm (see sect. 7);
c) The level variation of the lake was corrected for water density differences
between lake and well due to the different thermal regimes (see sect. 6).
The integral analysis was done by looking at the gravity value g at the gravimeter station g 4g(H), where H is the lake level.
This type of analysis is obviously affected by many long-term effects, like water table variations, long-term tidal components and deep-water temperature changes. It could however take advantage of the following fact: the same water level is reached by the lake many times during each week, so that some unwanted long-term effects can be mediated on long periods and hopefully smeared out.
Considering in fact the value of g as measured by the gravimeter at time t as a function of the lake level H, of time t, and of the value of g at some time t 2Dt, we can write
g(t , H) 4f(H)1T(t)1F1(t 2Dt1) 1F2(t 2Dt2) 1R ,
(14)
where f (H) is the lake effect, T(t) is the local tide and the other terms represent all possible effects with typical time delays Dt1, Dt2, etc.
While the term T(t) can be subtracted, once the local tide is known, the others are difficult to evaluate. In this first analysis we assume that for a given level Hi, measured n times in different epochs,
mean [Fj] 40
(15)
for n sufficiently large.
The full data sample, covering two years, and winter, spring, summer seasons, was grouped in ten shorter periods, five in 1992 and five in 1993. Each of these periods represents an uninterrupted run of at least half a month.
In table VI the main characteristics of the 10 periods in which data have been grouped are summarized. Column 4 shows the mean number of points for the 251 lake levels used. Column 5 gives the mean water temperature and the last two columns show
TABLEVI. – Characteristics of the ten periods of the integral analysis. Period Date of start Number of days aNo .b points per level Water atemp.b (7C) Measured Dg ( mGal) Stat. error ( mGal) 92a 5O3O92 26 67 5.81 193.19 0.29 92b 1O4O92 22 39 7.52 193.40 0.57 92c 1O5O92 14 20 11.33 193.88 0.36 92d 20O5O92 24 37 14.18 192.57 0.61 92e 16O6O92 21 44 16.49 193.16 0.76 93a 17O1O93 13 38 4.61 193.45 0.61 93b 1O2O93 39 93 4.10 194.96 0.51 93d 26O3O93 29 70 6.81 192.83 1.05 93e 2O5O93 21 47 11.27 193.75 0.25 93f 25O6O93 17 32 18.00 191.86 0.35
the total gravitational effect relative to 5 m of lake level variation (obtained by difference) and its statistical error. Those values are plotted in fig. 17 as a function of the mean water temperature of each period. The dependence of the measured effect on the temperature is apparent and even larger than expected from the water density effects discussed in sect. 6.
Making the hypothesis that g(Hi) must be constant in time for each lake level Hi if
the water table does not change, we studied the lake effect as a function of the absolute difference in the water table level from the first to the last day of each period. In table VII the density corrections for temperature for each period, the corresponding corrected values and the water table variations are reported. Data are plotted in fig. 18 showing the correlation between the measured lake effect and the absolute values of the water table variation.
The correlation was best fitted by a linear function obtaining Dg 4 (192.8160.27)1 (0.189 6 0.046) NdwN mGal . (16)
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TABLEVII. – Relationship between the corrected lake effects and the water table variations.
Period Temperature Corrected Water table
correction Dg variations ( mGal) ( mGal) dw (m) 92a 20.04 193.15 23.02 92b 20.00 193.40 28.90 92c 0.09 193.97 25.79 92d 0.16 192.73 0.41 92e 0.23 193.39 23.99 93a 20.05 193.40 23.57 93b 20.05 194.91 5.57 93d 20.02 192.81 0.12 93e 0.08 193.83 28.30 92a 0.28 192.14 21.47
The water table effect is sizeable, and in any case it is more important than the temperature effect of the lake water.
Extrapolating to NdwN40, we obtain
Dg 4 (192.8160.27) mGal . (17)
Taking into account the uncertainty in the gravimeter calibration (see table II), the experimental value becomes
Dgexp4 ( 192.81 6 0.31) mGal . (18)
Such a value must be compared with the theoretical one Dgth4 ( 192.19 6 0.16) mGal , (19)
TABLEVIII. – Summary of the dominant errors.
Source Relative error 31024
Experimental effect 3.8
Theoretical effect 1.4
Gravimeter calibration 9.2 Water level (digitizer) 2.2
Vertical deformation 8.0
Water level (density correction) 1.0
Water specific mass 0.3
Air mass correction 0.1
where the real displaced mass (water minus air) and the subsidence effect are considered. The dominant contribution to the error arises from the vertical deformation effect (see table VIII).
To conclude, the analysis of the integral effect shows that important systematic effects are present in the gravimetric measurements under the lake, due to both the basin movements and to seasonal changes in the ground and underground water circu-lation. Therefore, the only valid data seem to be those taken on a seasonal time base or on a very short time base in order to smear out and correct environmental effects. The differential analysis was performed by developing a program which operates on the files above described. At the beginning it reads in the main file the day and the time interval. Afterwards it opens the diurnal file of interest picking up from it all data in the selected time interval and working on TIDE or HR-data. We verified that the analysis results are independent of the signal used.
Starting from the original data, a third-order polynomial fit of the lake level vs. the time is performed in order to cut down the effect of the lake oscillations. The fitted values are then used instead of the measured ones. For every 10 000 digits interval of level change (B10 cm) a linear fit is performed relating the gravity signal and the lake level. The calculated slope gives the corresponding gravitational change. Because this level change occurs in a time of about 10-12 minutes, the method allows one to eliminate two noise effects: temperature variations of the water bulk and changes in the water table. Every slope value is successively corrected for the effects above reported
(
see a), b) and c) on p. 796)
. Furthermore:d) Starting from the measured values of the temperature and density in the lake,
the effective water density is calculated for every 10 cm tract. For the same tract, the air density is evaluated from air temperature, atmospheric pressure and relative humidity. The difference between water and air densities gives the density of the displaced mass. This allows one to normalize the gravitational effect to the density value, 1000 kgOm3, used in the theoretical model;
e) The gravitational effect is corrected for the vertical displacement of the
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Fig. 19. – Experimental and theoretical (continuous curve) gravitational effect for 10 cm of water level variation vs. the vertical distance from the gravimeter.
which takes into account the free air gradient (0.309 mGalOmm), the Bouguer correction and the density variation of the surrounding medium. The uncertainty arises from the comparison between different deformation models. It corresponds to a relative error for the gravitational signal of 8 31024.
The processing of all recorded data gave about 10 000 slope points at different lake levels. They were grouped in bins of equal level and the mean value and r.m.s. of each group were computed. The experimental values obtained are plotted in fig. 19 together with the standard error of their mean, and compared to the theoretical expectation (continuous curve). The experimental points reproduce quite well the shape of the expected effect on the basis of Newton’s law, but they show a significant relative upward shift of about ( 1.37 6 0.05)31022.
Equivalent results are obtained on analyzing separately the groups corresponding to lake level increasing ( 1.52 6 0.06)31022, and decreasing (1.26 6 0.06)31022, confirming the correctness of the analysis.
Starting from the differential data we computed the total effect corresponding to a water level variation of 7.376 m, discarding those values near to the minimum and maximum levels reached by the lake, where the statistics are less than 40 measurements per bin. The gravitational effect is 288.77 6 0.11 mGal, to be compared with the theoretical value of 284.715 6 0.040 mGal. The ratio R between experimental and theoretical effect is 1.0142 6 0.0004.
The quoted error must be modified in order to take into account also errors coming from the different corrections discussed above. Table VIII summarizes their relative contributions, compared with the statistical error of the experimental result and with the uncertainty of the theoretical prediction. Combining quadratically all the errors listed in table VIII, we obtain the final result
R 41.0142 6 0.0013 .
(20)
which we have estimated lower than 0.44 mGal for a 7.376 m water level variation. Such a correction reduces the R value by 0.0015.
In conclusion, we obtain for the ratio between experimental and theoretical effects the value
R 41.0127 6 0.0013 .
(21)
9. – Discussion and conclusions
The gravitational effect of the calibration mass and of the water of the lake were calculated both using the Newton law and the standard value of the gravitational constant G. We obtain, however, a ratio between experimental and theoretical effects significantly different from the expected one. This is an important characteristic of our experiment whose result is independent of G values obtained in laboratory using methods recently criticized [41].
If one accepts the validity of Newton’s law, our parameter R can also be written as
R 4 Gfar
Gnear , (22)
with the hypothesis that G is a function of the distance. This does not seem to be acceptable since it disagrees with experimental results (laboratory experiments [10], motion of celestial bodies [42]).
On the other hand, the G dependence on the distance can be derived from a modified formulation of Newton’s law. In particular, on assuming that the Newtonian potential energy is given by
U 42 G m1m2
r ( 1 1ae
2rOl) , (23)
our experimental results, compared with the theoretical ones, allow us to exclude l values smaller than 20 m. In fact for l lower than 20 m the dependence of the anomaly on the lake level should be directly observable in our experiment.
The observed effect indicates a negative a value which corresponds to a repulsive force (graviphoton). The experimental result can be explained by different pairs of a , l values. The l 2a plot shown in fig. 20 gives in the 1–500 m range the relationship between the two values in order to agree with our experimental data.
Our result disagrees with that obtained by Cornaz et al. [43] in a similar experiment where the gravitational effects produced by changes of level of a lake are measured with a differential balance. They found no violation of Newton’s law with a level of precision of 0.1% in the l range 0.01–100 m.
One of the most important differences between the two experiments consists in the fact that the effective interaction distances of the water masses are different. In fact defining reff as the weighted mean of the distances with the weights equal to the vertical gravitational force, we obtain for our geometry a value reff4 47 m, which is to be compared with the value reff4 112 m in the experiment of Cornaz
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Fig. 20. – Relationship between NaN and l in the range 20–500 m.
experiments [44-46] give G-values greater than the laboratory ones, even if their precision does not allow any conclusion.
After a critical analysis of our experiment, the following conclusions can be reached: – Assuming a Yukawa potential, the found deviation from gravitational Newton’s law can be explained for a negative by a-l couples satisfying the values shown in fig. 20.
– Our experiment is model independent and gives directly the ratio GfarOGnear, independently of the laboratory value of G.
– Almost all the measurements and also the data analysis were verified with different approaches.
As discussed in the paper, the only parameter not verified at the 0.1% level was the gravimeter calibration factor. In any case, the adopted value is in agreement with the result of the comparison with an absolute gravimeter.
* * *
This work was partly supported by the ENEA-Brasimone Laboratory. We are very grateful to the director of the Centre, Dr. GUERMANI, to Drs. CASSARINI, FILOTTO, RIGHETTI and to Mr. TAULLI and the whole technical staff for their constant support. We thank the ENEL for the collaboration in different stages of the experiment. We gratefully acknowledge the Bologna INFN director Prof. A. VITALE for the support, and the INFN technical staff, in particular Dr. GUERZONI and Mr. LOLLI for the continuous assistance. We gratefully acknowledge the important contribution, for the topographic survey, of Drs. M. ANZIDEIand F. RIGUZZI(ING-Roma), S. GANDOLFI, C. BONINI (DISTART of Bologna University) and of Mr. M. BACCHETTI and C. GUIDI (Physics Department of Bologna University). The water density measurements were performed at the Chemical Department of Modena University, and we are grateful to Prof. P. MIRONE and Dr. L. TASSI. We are indebted to Dr. B. DUCARME of the Observatoire Royal du Belgique, R. WARBURTON of GWR instruments and Prof. I.
MARSON for helpful suggestions and for the support provided. The contribution of Drs. A. AVVENUTI, M. COLUCCELLI, R. FODDIS and G. RAGUNI´ which discussed their thesis on parts of the experiment is also acknowledged.
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