DETERMINATION OF THE GRAVITATIONAL CONSTANT G USING A FABRY-P EROT PENDULUM RESONATOR

Transcription

1 DETERMINATION OF THE GRAVITATIONAL CONSTANT G USING A FABRY-P EROT PENDULUM RESONATOR A. SCHU:ACHER, H. SCHUTT, H. WALESCH, H. :EYER Bergische Universitiit Gesamthochschu/e Wupperta/, Fachbereich Physik Gauftstrafte 20, D Wuppertal, Germany We present the status of the Wuppertal gravitational experiment. Our unique method to determine G is described, folloed by a discussion of recently discovered systematic errors and the means to minimize them in the near future. Introduction The gravitational constant G is the least ell knon fundamental constant in physics. Different experiments, all claiming relative accuracies near 0-4, disagree in their results on a level of up to 74 io-4. This is a hint of huge unknon systematic errors of probably all experiments. It can be said that G is knon ith an accuracy of only about 0-3 so far. Our method of measuring G is based on a different principle than the commonly used torsion balance experiments. This is of particular interest because certain systematic errors cannot occur due to constructional reasons, see for example Kurodct. Furthermore e are able to test Neton's inverse square la ith the same set of data. 2 Experimental Setup Our setup consists of a pair of pendula the distance of hich depends on the gravitational force of to field masses acting on them, see Fig.. The four cylindrical bodies are aligned along one common axis of symmetry. The CM-distance beteen the fieldmasses and the pendula can be changed ithin a range of 0.7m to 2.2m hich leads to a change of pendula distance of up to 20nm. The pendula form an open Fabry-Perot resonator ith the resonance frequency a function of the pendula distance. The resonance frequency can be measured ith high precision, so that 343

2 pendulum ire 50 cm eddy-current brake guide rail resonator stepping motor fieldmass Figure : Schematic vie of the Fabry-Perot gravimeter. changes of the pendula distance as small as a fe picometers can be detected. 3 Data Acquisition A typical set of data is shon in fig. 2a, here the measured resonance frequency change is plotted against the time. At t = 0 the fieldmasses are in a position far aay from the resonator, at t = lomin they are moved to a position near the resonator. The fieldmasses rest in each position for 5min and change to the other one periodically. The resulting change in resonance frequency b.. f is used to determine the gravitational constant G. As can be seen in fig 2b, hich shos the to day behaviour of the same set of data, there is a longterm drift in the resonance frequency that is not correlated to the movement of the fieldmasses. The drift has to be corrected for, e. g. by means of a polynominal fit. The obtained values of b.. f are distributed gaussian like ith a typical idth of corresponding lopm. This means e have to run 00 cycles to achieve a resolution of lpm. If e do this ithout moving the fieldmasses e obtain a mean value hose deviation from zero is less than lpm. In other ords the longterm drift leads to no systematic shift in G. The drift is mainly due to the thermal expansion of a quartzsheet used to precisely fix the distance of the pendulum ires near the suspension points. Fig. 2c shos the corresponding temperature of the quartzsheet. Loer order origins of the frequency drift are temperature variations of the resonator and small variations in the local gravity field caused by moving masses like the moon or people moving near the experiment. As long as these disturbances are not modulated ith the same period our fieldmasses are moved ith they do not lead to a systematic shift in G, but may iden the statistical distribution. Once e have replaced the quartzsheet by one made of Zerodut' e ill be able to study this loer order effects in detail. 3. Numerical Analysis To compute a value of G e can use equations and 2: M [( ) - (- - b..b -- G- - -r;. (rref 5 r2 (r b)2 Kr + ith a thermal expansion coefficient a factor + b)2 40 ) K t] loer than that of the quartz in use 344 re ()

3 o m time[ min) time [days] i ':: WED TUE time [days] Figure 2: Typical set of data. a(top): the modulation of the resonance frequency resulting from moving the fieldmasses. b(center): longtime behaviour of (a). c(bottom) : temperature of the quartzsheet. Ab = {3 6./, (2 ) here Ab denotes the change of pendula distance resulting from a displacement of one 6.eldmass from a reference position rref to the ne position r. the fieldmass and the near pendulum. The mass M r denotes the CM distance beteen of the 6.eldmass is precisely measured at the PTB in Braunscheig, the eigenfrequency o of the penduhi' and the distances r have to be measured. The CM distance b beteen the pendula and the conversion factor {3 are derived from the frequency spektrum of the resonator and calculated from cavity theory. The factors K in Eq. correct for the fact that the interacting masses are not pointlike. They are calculated by a standard gaussian integration over the volumes of the bodies. Since this method only gives approximate values a sophisticated interval arithmetic as carried out in collaboration ith the Dep. of Num. Math.2 giving guaranteed bounds for the effect of rounding errors and geometrical tolerances. As a result e could confirm the magnitudes of the standard gaussian integration. In Tab. some possible error sources are listed. The specified tolerances ould produce a relative error of G of io- 5 each. Note that by using a symmetrical setup ith to 6.eldmasses e can afford relative high tolerances in the distance beteen the 6.eldmasses and the pendula. If only one 6.eldmass is in use e have to kno the distance to the pendula in the vacuum chamber ith an accuracy of some µm, hich is unachievable. Not taken into account in Eq. are the effects caused by gravitational forces of the pendulum G of 4 io- 4 and an increase of ires and the spindle nuts hich lead to a relative decrease in 7 io- 4 respectively. the simplified Eq. is valid only if the eigenfrequencies of both pendula are identical 345

4 z Da S amp e l o Da Sample 2 Da Sample , t... t ? t j t 3 '-'-'--'--'--'--'--'--'--'--'--'--'--'-'- -'--400 '--'--'--'--'--'600-' 300-'--'-' '--'500-'--'--'--'-.,..._, r[mm] Figure 3: Experimental results. 3.2 Experimental Results Up to no e have taken three independent data samples ith the to fieldmasses moving symmetrically to the resonator. Within each data sample different 'near' positions r ere used, i. e. G as determined ith different distances beteen the acting masses. Inbeteen the three periods of data taking the setup as partially disassembled to carry out certain changes. After reassembling the setup as realigned and the geometrical properties ere determined. Fig. 3 shos the results. The relative deviation 6.G of the measured value of the gravitational constant from the literature value is plotted against the position r. Here r = 0 denotes the smallest possible CM-distance beteen the fieldmasses and the resonator of about 920mm. Included are the statistical errors. One can see a systematic shift beteen the three different data samples and furthermore all data samples sho the same slope of /300mm on the first 300mm. The slope is not related to a deviation from Neton's inverse square la but to a systematic error in the positioning of the fieldmasses. A therupon realized examination of the positioning system of the fieldmasses revieled great discrepancies beteen the expected and the real position. The distance beteen the fieldmasses is determined in to steps: First, the fieldmasses are moved to a position near to the resonator so that the distance becomes minimal, e g. r = 0 in Fig. 3. In this position the absolute distance is measured using an optical device. A long distance microscope is mounted on a guide rail and can be moved parallel to the axis of symmetry of the fieldmasses. The position on the guide rail is determined my means of a digital lenght scale. Focusing on e. g. the edges of the fieldmasses their position can be determined as the projection on the guide rail. The resolution of this method is limited by Table l: Some possible error sources and their permissible tolerances to achieve a relative accuracy in G of kind of error geometry of the fieldmasses geometry of the pendula distance of the fieldmasses misplacement of the pendula II to the axis of symmetry misplacement of the pendula.l to the axis of symmetry 346 permissible tolerance loµm lµm 5µm lmm lmm 0-.

5 50 Figure 4: r 3 50 [rrun] The deviation of the fieldmass position from the expected position and its effect on the determination of G. that of the microscope of up to 3µm but the accuracy is affected if the microscope is not moved exactly linear. A slight curvature of the guide rail leads to a change of angle of vision of the microscope. It turned out that due to the long optical lever of almost lm the overall accuracy of the optical device is only about 200µm. This measurement as carried out only once every period of data taking inbeteen of hich even the optical device as dis- and reassembled. This explanes the systematic shift beteen the three data samples of Fig. 3. Second, during data taking the fieldmasses are moved to several positions periodically by means of a spindle and a stepping motor. The relative distance of the actual position to r = 0 is calculated by the amount of motor steps carried out and the slope of the spindle. The accuracy of the slope as specified by the manufacturer is 25µm/300mm but again the overall accuracy is influenced by mechanical properties. To check the performance e alternated the optical device by replacing the microscope by a short mechanical lever attached to the edge of a fieldmass. The resulting deviation from the expected position as a function of the expected position shos a sa 'type profile ith an amplitude of looµm and a period of locm. This arises from a tilting of the fieldmass on the guiding reels that have a distance to each other of locm. Superimposed on the first 300mm is a linear slope of about 00µm/300mm. The effect of the deviation on the determination of G shos Fig. 4, here data sample 3 of Fig. 3 is plotted together ith the calculated values that arise from the mispositioning of the fieldmasses calibrated on data sample 3 at r = 0. The measured mispositioning exactly produces the slope of the data of Fig. 3. H e correct for the slope and then calculate the mean value e obtain G = ( ± ± ) 0- m3 2, kg s - here the first specified error denotes the statistical and the second the systematic error resulting from a misdetermination of the distance of the fieldmasses of 200µm. It has to be pointed out that at the moment e cannot exclude an even higher systematic error. 4 Outlook In order to minimize the existing systematic errors e are currently building some upgrades. A ne optical device ill provide a more precise guiding of the microscope and ill be able to determine the remaining curvature of the guiding rail. We are aiming an accuracy of 3µm. A ne system ill be able to automatically determine the change of position each time a fieldmass is moved. We are aiming an accuracy of again 3µm so that the absolute distance beteen the fieldmasses ill be knon ith an accuracy of 5µm. In order to control the complete data 347

6 aquisition ith one central CPU e ill install a ne VME-bus based computer system. A loer superimposed frequency drift ill be achieved by replacing the existing quartzsheet by one made of Zerodur. A more precisely manufactured pair of pendula ill minimize errors due to deviations from perfect geometry and probably ill provide a better resonator performance. During 997 e ill carry out several measurements of G including dis- and reassembling great part of the setup to check if the reproducability is improved by means of the upgrades as expected. Special measurements ill be realized to study the effect of other possible systematic errors. Our goal is to achieve a relative accuracy of G of (5-0) 0-5 in the near future. Acknoledgments The ork on this project is supported by the Deutsche Forschungsgemeinschaft under Grant ME 577 /-4. References. K. Kuroda, Phys. Rev. Lett. 75, 2796 (995) Holzmann et al., Rel. Comp. 2.3, 229 (996). 348