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Unformatted text preview: INERTIAL AND GRAVITATIONAL MASS 443 ANNALS OF PHYSICS: 26, 442-517 (1964) The Equivalence of Inertial and Passive Gravitational P. G. Palrner Physical ROLL, Mass * R. KROTKOV, t AND R. H. DICKE Laboratory, Princeton University, Pl'inceton, New Jersey Torsion balance measurements of the difference in ratios of gravitational to inertial mass for different materials have been carried out, confirming to higher precision the null results obtained 60 years ago by Eotvos and assumed by Einstein as the Principle of Equivalence upon which the General Theory of Relativity is founded. If the parameter 'leA, B) is defined as '1(A, B) = [(M/rn)A - (M/rnhl/!-.z[M/m).{ + (M/rn)B], where M and rn represent the passive gravitational and inertial masses respectively of materials A and B, then the results from the most sensitive torsion balance used enable us to conclude with 95% confidence that I '1(Au, AI) I < 3 X 10-11 Stated more exactly, the various measurements of '1, obtained from the gravitational acceleration toward the sun, gave a substantially Gaussian distribution with mean value 'l(Au, AI) = (1.3 x 1.0) X lO-lI. The probable error quoted for the mean is based upon the observed scatter in results from individual data runs, assuming a Gaussian distribution. The importance of the Eotvos experiment to contemporary gravitational theories is discussed, and the earlier measurements of Eotvos and J. Renner are examined critically. The torsion balance and associated equipment used in the present experiment are described in detail, along with the considerations involved in their design. Methods of data analysis are also discussed extensively and tables of individual results are presented. I. INTRODUCTION It is possible to ascribe three types of mass to a body (1); inertial, passive gravitational, and active gravitational. By active gravitational mass one means a measure of the source strength of a body for generating a gravitational field, and by the passive gravitational mass one means a measure of the gravitational force acting on a body in a given gravitational field. If the inertial and passive masses are always proportional, independent of composition; the gravitational accelerations will be composition-independent. The first crude experiments that demonstrated the uniqueness of the gravitational acceleration are probably lost in antiquity. While Galileo may not have * This work has been supported by research contracts with the Office of Naval Research, the U. S. }.It.tomic Energy Commission, and the ~!&tional Science Foundation. t Present address: Sloane Physics Laboratory, Yale University, New Haven, Connecticut. 442 .... ~ I L~ ~'. ,~ y~ ~ :-1: "f dropped iron and wooden balls from the leaning tower of Pisa, he was certainly aware that the gravitational acceleration of a body is substantially independent of the material of which the body is composed. Later, a pendulum experiment which showed the equivalence of inertial and passive gravitational mass with an accuracy of a part in 103 was performed by Newton. Bessel (2) in 1827 improved this null result to an accuracy of 2 X 10-5 H. H. Potter (3) in 1923, also using the pendulum technique, was able to im6 prove the precision to 3 X 10- The most accJrate of the classic experiments were those of Baron Roland v. Eatvas, perfornwd with a torsion balance. His first series (4), published in 1890, established the constancy of the gravitational acceleration with a precision of .5 X 10-8. Later, with D. Pekar and E. Fekete 9 (5), he improved this to 3 X 10- Although this work first appeared in 1909 as an essay which won the Benecke Trust Fund award of the University of Gattingen, it was not published until 1922, three years after the death of Eatvas. These later remarkable experiments, using only classical techniques, showed that a wide variety of substances, including such exotic materials as "Schlangenholz" and magnalium, fell with the same acceleration. It is of interest to note that as recently as 1935 one of Eatvas' torsion balances was used with little modification by J. Renner (6) to repeat Eatvas' old experiment, substituting for snake wood the equally exotic material "Batavian glass drops." Perhaps the first question to occur to a physicist is, "Why is there any necessity of increasing the precision of the Eatvas experiment? Is not a precision of 9 3 X 10- enough?" What this question usually implies is the feeling that this experiment and others establish the ~quivalence principle, and that Einstein's "general relativity," being based on this principle, is almost certainly correct. If general relativity is beyond question, the experiment is without merit, for the null result of the Eatvas experiment is a result which can be derived from "general relativity." It is well known that Einstein's principle of equivalence, the "strong principle" (7), is such a rigid constraint on gra vitatibnal theory that "general relativity" is almost uniquely determined by this principle, together with the principle of general covariance. Far from being the most general of relativity theories, it is actually a strongly restricted special case. According to Synge (8), the presence of the word relativity in the narhe is something of a misnomer. The theory is more properly described as a. theory of the dynamies of a Riemannian space-time, rather than a relativistic theory in the sense of the concept of space introduced by G. Berkeley and E. Mach. As will be discussed below, it is a mistake to regard Einstein's equivalence principle as directly established by the Eatvas experiment. Actually, what is directly supported is a wea.ker forIn of the principle, This '(,veak equivalence principle" can be used as a basis for theories more general than "general relativity." Perhaps the best way to answer the question posed above, as to the impor. .. 444 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 445 tance of the Eotvos experiment, is to exhibit its unique role as a basis for rela- constants are fixed. It should be noted, however, that the Eotv6s experiment tivistic theories. This has been discussed previously in a number of publications. and the improved version to be reported here do not set useful limits on the Here we simply summarize several important points: variability of the weak coupling "constant" or the gravitational coupling "constant." 1. It is well known that Einstein's "general relativity" is based on his assumption, sometimes called the "strong equivalence principle" (7). This is the as4. The strong equivalence principle requires that the gravitational interaction sumption that in an electromagnetically shielded laboratory, freely falling and between two small bodies, closely spaced, should be independent of the distribunonrotating, the laws of physics, including their numerical content, are inde- tion of distant matter surrounding the bodies. This may not be consistent with pendent of the location of the laboratory. In such a laboratory all particles, free the requirements of Mach's principle, according to which the gravitational of nongravitational forces, move without acceleration. Hence, more generally, acceleration depends upon the distribution of the total mass of the universe in any arbitrary coordinate system all particles move with the same acceleration. _about the point in question. These two basic principles are not necessarily inIt should be remarked that the uniqueness of the trajectory of a particle compatible. For example, it is possible that, with the application of suitable moving gravitationally is essential to the geometrical interpretation of the space- boundary conditions, only certain distributions of distant'matter may be pertime orbit as a geodesic of the geometry. Thus, the fact that the gravitational mitted. However, it is also possible that only the weak form of the equivalence field tensor is interpreted as the metric tensor of the geometry, rather than as a principle is valid. Unfortunately, the Eotvos experiment is of no assistance iii. prosaic force field, is due in part to the unique gravitational acceleration. deciding between these two alternatives. It must be noted that it is here assumed that the "partiele" is so small and has 5. Lee and Yang have suggested the existence of a gauge-invariant vector so little spin angular momentum that the "gravitational tidal forces" or "gravi- field, similar to electromagnetism, associated with baryon conservation (9). tational gradient" effects on the particle are negligible. These forces may be They have also shown that the E6tvos experiment sets drastic limits to the defined as those associated with nonzero second derivatives of metric tensor strength of an interaction with such a field. components in a locally Minkowskian coordinate system. It is an experimental 6. It has frequently been suggested that anti-matter may fall up, not down. problem associated with the Eotvos experiment to measure accelerations under L. Schiff (10) has made use of the null result of the E6tvos experiment in an ingenious argument to suggest that it falls down, not up. conditions for which such forces are negligible. 2. A weaker form of the equivalence principle has been stated (7). It requires It is clear from the above that if one believes that general relativity is estabthe uniqueness of the gravitational acceleration, at least to the accuracy of the lished beyond question by its elegance, beauty, and the three famous experiEotvos experilnent, while permitting the numerical content of physical laws to mental checks, then the Eotv6s experiment has no point, for its null result can vary from point to point. It provides a broader base upon which gravitational be derived from the theory. However, if gravitational theory is to be based Oil theories can be constructed. experiment, the null result of the Eotvos experiment is probably the single most It may be noted that a massless scalar field has no place in "general relativity" important experimental result available to us. As such, it is important that its under the "strong equivalence principle," for the value of the scalar varying froIn accuracy be improved, if possible. point to point would contribute a variable' element as part of the numerical content of physical laws. In similar fashion a seQ<?ndlong range tensor field. II. EOT\'OS' APPARATUS would be impossible, for it would generate a scalar by contraction. However, the It IS a good general experimental rule that the most precise experiments are existence of both of these fields might be possible under the "weak equivalence null experiments, and the E6tv6s experiment is no exception. With E6tv6s' principle." a~para.tus, consisting of a Cavendish balance (more exactly Boy's modification) 3. Under the "weak ~quivalence princi.ple" particle masses and coupling Withdlffere~It materials at opposite ends of the beam, the centrifugal forces on constants co~ld vary, b~mg pe~~aps functIOns .of a 9 scalar fiel~. :r:o~ever, the the two weI~hts d~e t~ the earth's rotation are balanced against a component Eotvos experIment and Its preCISIOnof 3 parts m 10 sets drastIC limIts to such ~ft~e earth s gravitatIOnal field. If the ratio of passive gravitational mass to possible variations (7). These limit~ are improv~~ su~stantially by the experi- Illert.mlmass ~hould diffe.r from one weight to the other, there would be a torque ment r~ported here. Thus, to a consIderabl~ preCISIOn,It may be ~oncluded that ten~mg to tWiSt the torSIOn balance. A rotation of the whole apparatus through the ratIOS of the masses of elementary partICles to each other are mdependent of 180 would then reverse the sense of this twist. position in 'pace.thoe, and also that the ,trongand eleotmmagoetie eoupling At a latitude of 45, the ho","ntal eomponent of the ea<th'e een"if"" l a,. I 446 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 447 celeration is only 1.6 cm/sec2 If there were an anomaly of 1 part in 1011 in the inertial-gravitational mass ratio, there would appear an anomalous turning force of at most 1.6 X 10-11 dyn/gm on one of the weights. For purposes of design considerations we shall have in mind a standard gravitational anomaly of 1 11 part in lOll; namely, an added anomalous gravitational force, 10- of the normal gravitational force, acting on one of the bodies. In a Galileo free-fall experiment, the anomalous force would be much greater, 10-8 dyn/gm, but it would now be necessary to measure the acceleration to l' part in 1011. In a more sophisticated experiment, two 'weights might be dropped together and their relative accelerations compared. (The extraneous effect of gas drag could be eliminated by dropping them inside a freely falling container.) In a drop of 5 meters taking 1 sec, an anomalous acceleration of one of the' 9 weights of 10-8 cm/sec2 would lead to a relative displacement of only 5 X' 10- c between the two weights. A great advantage of the E6tv6s technique is that it: permits an anomalous force, though much smaller, to act for a far longer periodi of time and accumulate a bigger displacement. An interesting experimental possibility would be to put an apparatus in a artificial satellite. In this case, the advantage of the large force anomaly coul be combined with that of a long observation time. It is interesting to calculate the maximum rotation to be obtained in Eotvosl apparatus from an anomaly of 10-11 His apparatus employed weights of 30 gml had a beam length of 40 em, and the suspension wire had a torsion constant 0 11 0.5 (dyn cm)-l. The resulting maximum rotation from an anomaly of 10- 1.9 X 10-8 rad. E6tv6s observed rotations with the traditional instrument 0 that day, a telescope and scale. A rotation of this size represents 1/20,000 0 ERTICAL ROTATIONAL AIR SPAf:;E ADJUSTMENT ADJUSTMENT SUSPENSION WIRE MIRROR TELESCO~E WEIGHT I c:=:::> c=== \!I ; .. WEIGHT IIj r ~ n "J ~ FIG; 1. Reproduction of a drawing of a single torsion balance used by Eatvas for some of the smallest division on his scale. 's measurements (11). The scale below the drawing is one meter in length. Perhaps the most serious objection which can be raised to the E6tv6s exper.i ment is the lack of a suitable control. There is no way of turning off the centrifui gal force field of the earth. Hence, there is no over-all zero check upon the pell' . formance of the torsion balance with some specific set of weights in place. T While E6tv6s made use of ac~eleration toward the sun for some of his meas avoid this difficulty, it is possible to use the a~celeration of the a~paratus towar ~nts,. these results we~e of mferior accuracy and his experiment depen~:~ the sun. At 6 a.m. and 6 p.m., and for a torslOn balance beam m a north-sout rll1~anly upon the centnfugal force field (5). His principal apparatus is "h . direction, an anomalous gravita~ional pull upon one .of the .weightsat an e~d ~ FIg. 1, reproduced from an old article (11). It may be noted that one ~~e~~~~ the beam would produce a turnmg force. The resultmg tWIst would be penodi as suspend~d lower than the other. The resulting elimination of the tw f Id with a 24-hr period. While the horizontal component of acceleration towar ~mmetryId aXISd' the apparatus increased the number of basic types., of gl'aVI,aof .at 2 lonal fi. ( . the sun is at most only 0.6 em/sec, about three-eighths of the rotational ce e gra I~nts tIdal effects) to which the apparatus was sensitive a d trifug~l a~celeration, it occurs wi~h a 24-hr period.',and hence the me~~ureme. d no worthwh~le purpose. It .is difficult to see why this was done, as it' OI~~V contaIns Its own zero check. ThIS type of expenment has the addltlOnal a ade the. expenment more dIfficult, requiring twice as many observ t' . 0 vantage that a rotation of the apparatus is unnecessary, eliminating the resultin observatIOns 90 apa:t) as would have been required by an apparatus \~i~,~ns 0 disturbance of the delic.ate balance, for the rotation is generated automaticallf ... ~Of?ld symmetry aXIs (180 rota~ions). The only reason that we can see f. ~ o and gently by the rotatmg earth. IS IS that the apparatus was deSIgned for making geophysical measurements 448 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 449 of gravitational gradients, and was used only incidentally for this imp9rtant basic experiment. . In the apparatus shown in Fig. 1, a platinum weight was inserted into one end of the torsion beam, while the weight suspended from the opposite end could be changed. The observing telescope was supported by a bracket at the right of the apparatus and carried a short scale on top of it. The telescope had a 90 bend in its optical axis, and the figure shows the open end of the eyepiece. This arrangement required the observer to place himself approximately a meter from the torsion fiber, on a line making an angle of about 45 with the torsion beam. The distance from the torsion fiber to the scale was 62 em, and the variable weight was suspended 21.2 em below the torsion beam, 40 em long. In order to make an observation, the whole apparatus was rotated about its vertical axis and then left undisturbed for one hour or more until air damped the oscillations, after which the scale was read. Many of the measurements of E6tv6s, Pekar, and Feteke (5), and all of those of Renner (6), were made with a so-called "double gravity variometer." 'This device was nothing more than two torsion balances of the type shown in -Fig. 1, mounted i~ the same housing with torsion beams approximately parallel and the suspended variable weights at opposite ends. Table I summarizes the results reported by Eotv6s, Pekar, and Feteke (5), and by Renner (6). Since E6tvos and his colleagues did not publish their data 'and details of their calculations, we are not certain how to evaluate the standard deviations which accompany their measurements (see the third column of Table I). However, assuming that the measurements they made were independent and normally distributed, it is possible to estimate the implied accuracy of a single position measurement on the torsion balance scale from the number of measurements made and the quoted standard deviation of the final result. Such estimates have been made and are all consistent with the reading _ accuracy claimed by E6tv6s et al., 7'20 of the smallest division on the scale used, provided this was the only significant source of error. Similar estimates of the scale reading accuracies for the measurements of Renner (6), however, are not consistent with his claimed 7'20 scale division reading accuracy. This inconsistency has been at least partially resolved through the courtesy of Dr. Renner himself, who recently made available to us copies of his original data and calculations for the measurements on brass and paraffin. A careful examination of these records has revealed that his method of calculating differences in position readings of the torsion balance yielded sets of values which were not statistically independent. For instance, his method utilized each of 24 independent north and 24 independent south position readings a number of times, and in various combinations, to obtain 90 values of t.he north-south difference. The rms fluctuation of these differences about their MEASUREMENTS IN. PASSIVE BY EOTVOS TABLE I et al. (5) AND RENNER (6) MATERIALS OF THE DIFFERENCE FOR VARIOUS GRAVITATIONAL TO INERTIAL MASS RATIOS Materials A B TJ (A,B) ::I: standard deviation of the mean Magnalium Snakewood Copper Ag2SO. + FeSO. glass and brass vials + Water brass vial Cu'SO. crystals + brass vial CuSO. solution + brass vial Asbestos + brass vial Talc brass + + Platinum Batavian glass drops Ground Batavian glass drops Paraffin brass vial NH.F brass vial Copper Copper Bismuth Eotvos, Pekar, and Feteke Platinum Platinum Platinum Reacted Ag2SO. + FeSO. + glass and brass vials Copper Copper Copper Copper Copper Renner" Brass Brass Brass Brass Brass Manganese alloy Manganese alloy Brass (4 (-1 (4 (0 ::I: 1) ::I: 2) ::I: 2) ::I: 1) X X X X 10-' 10-' 10-' 10-' (-5 ::I: 1) (-3::1: 1) (-4 ::I: 1) (-2 ::I: 1) (-3 ::I: 1) X X X X X 10-9 10-' 10-' 10-' 10-' (0.45 ::I: 0.65) X 10-' (-0.06 ::I: 0.67) X 10-' (0.21 ::I: 0.65) X 10-' (0.24 ::I: 0.26) X 10-' (0.06 ::I: 0.25) X 10-' -0.08 ::I: 0.20) X 10-' -0.12 ::I: 0.22) X 10-' -0.14::1: 0.74) X 10-' + + a As discussed in the text, the standard deviations given in this table for Renner's results should be increased by a factor of three. r" I;" I 1: Co '; mean was then computed, and divided by V90 to obtain a standard deviation for the mean. When the north -south differences are reduced to a set of 24 or fewer statistically independent values and the calculations carried through in the proper way, the resulting standard deviation of the final result is about three times larger than that quoted by Renner. The similarity in the descriptions of how Eotv6s et al. (5) and Renner (6) analyzed their data suggests that the results quoted by the former (Table I) could have suffered from the same faulty statistical treatment. Since we do not have access to the original data and calculations which would answer this question, and since E6tvos' standard deviations are indeed consi"tent with his scale reading accuracy, this suspicion is without any substantial foundation. Another peculiarity is evident in the results of Renner. Referring to Table . , < '. ;,~; 450 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 451 I, it is seen that his eight mean values are all less than his quoted standard deviations of the means. When the standard deviations are corrected by multiplying by a factor of 3, with two exceptions the means are all less than 0.2 of their respective standard deviations. If Renner's measurements are distributed normally, as he assumes, the probability of obtaining by chance 8 means as small as those observed is less than 10-7 The comparison between the means and standard deviations given by E6tv6s is much more reasonable, however. If anything, E6tv6s' mean values tend to exceed his standard deviations by more than would be expected, but this could represent systematic disturbances of the balance. -In view of the above considerations, it seems reasonable to ascribe a crude 9 "average probable error" of about 1.4 X 10- to the results of Renner (6), and about 3 X 10-9 to the results of E6tv6s et al. (5). (The crude "probable error" for the results of E6tv6s is obtained from his quoted mean values, rather than the smaller standard deviations.) If we interpret these "probable errors" in the same way that we have interpreted the results of our own experiment (see Section VI), E6tvos and Renner have shown with 95 % confidence that E6tvos (5): 11/1 < 9 X 10- 9 (95 % confidence limit) Renner (6): 11/ [ < 4.2 X 10- 9 for the materials which they investigated. III. THE DESIGN OF AN E()TV()S EXPERIMENT We early decided to avoid the basic flaw in the Eotvos experiment, the lack of control, by using the acceleration toward the sun rather than the rotational acceleration of the earth. We believe this decision was correct, for the highly sensitive balance we employed would have suffered serious irreversible changes under a gross rotation of the apparatus. As it was, the apparatus was used in one position for several weeks and then rotated to a new position for a new series of measurements. No attempt was made to preserve the zero of the apparatus from one position to the next. If one thinks in terms of an apparatus capable of exhibiting a gravitational force anomaly of 1 part in 1011, the turning force on a 30 gm weight, the largest 10 used in these experiments, is at most 1.8 X 10- dyn. It is interesting to note 2 12 that a body starting frOln rest and accelerating at the rate of 6 X 10- cm/sec , 11 representing an anomalous acceleration toward the sun of one part in 10 , 4 would reach the. magnificent velocity of 2 X 10- em/sec after a whole year. At this velocity it would move 7 mm in one hour. It is evident that great care is required to avoid extraneous forces greater than this. The following is a list of effects which were considered in the design of the experiment: 1. Disturbances due to variation in torqlles caused by gravitational field gradients. 2. Variable torque from a varying magnetic field acting upon magnetic contaminants. In particular, the diurnal variation of the earth's magnetic field might be a source of trouble. 3. Variable electrostatic forces associated with varying contact differences of potential between the torsion balance and surfaces "seen" by the balance. 4. Disturbances due to gross gas pressure effects. 5. Brownian motion effects. 6. Disturbances due to the rotation detection system. 7. Extraction of signal from noise. 8. Temperature variation effects. 9. Ground vibration disturbances. Each of these effects will be discussed in turn. 1. Field gradient effects. It was already mentioned that Eotv6s' apparatus was sensitive to gravitational field gradients. It is easily computed that a 100 kg man would need to be at least 30 meters away if the resulting torque on his balance were to be definitely less than that of a gravitational anomaly of 1O-11 Such a torque clearly depends on the location bf the man relative to the torsion beam. As previously mentioned, the position of -an observer reading the scale on Eotvos' apparatus was necessarily along a line making an angle of about 45 with the torsion beam. It is interesting to note that this is the worst possible position for an observer, as far as the effect of his gravitational gradient on the balance is concerned. To reduce difficulties of this type, we did three things. First, the torsion balance was made small, with a moment arm of only 3.3 Cm. Second, the balance was given an approximate threefold symmetry axis, with the weights hung from the corners of an equilateral triangle constructed from fused silica. Two of the weights were of one material and the third of another. Third, the instrument was operated remotely. With this balance a 100 kgman could be as close as 4 meters from the balance before a torque equivalent to a 10-11 anomaly would occur. In order to reduce to small values the time varying disturbances due to gradients in the gravitational field, the torsion balance in our experiment was placed in an instrument well 12 ft deep and 100 ft from the nearest building, 150 ft from the nearest road (a lightly-traveled service road), and 1200 ft from the nearest highway. See Fig. 2 for a diagram showing the location of the well. Inasmuch as the ground drops away from a point 50 ft from one side of the instrument well, and is wooded on this slope, it is important to be sure that a 452 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 453 ilTHlETIC FIELDS ~~~~T ~ "'" '\ ~~ '",~ - .~-~--~~~~= ~ ..~-=-~ ~~~ 0.0:::;::~ LA . ~ K E CARNEGIE 40' 80' 120' o I I I I SCALE FIG. 2. Location of the instrument well and control shack on the Princeton University campus. daily period in rainfall, snowfall, or simply morning dew does not introduce a spurious 24-hr period through the influence of gravitational gradients. A rough estimate of this effect can be made by assuming a rain or snowfall which loads the open field on one side of the instrument well uniformly and the wooded area on the opposite side not at all, with a strip 20 meters wide, in which the density of water varies linearly from one extreme to the other, separating the two areas. 2 Such a calculation shows that an accumulation of 4 gm/cm on the open field would be required to,produce an effect as great as the standard gravitational anomaly of 10-11 Another question which can be raised is whether the varying gravitational gradients due to atmospheric highs and lows could cause noticable disturbances (time varying). Assuming a rather extreme condition, an atmospheric low of 4 em Hg located at a distance of 100 km and having an area of 50 X 50 km\ i gives a gravitational inhomogeneity producing a torque only lO- of that associated with our standard gravitational anomaly. The above estimates of gravitational gradient effects have all been based on the assumption of perfect threefold symmetry of the triangular torsion balance, which leads to an octupole interaction for the torque. It would appear that the departure from perfection may be sufficiently great that an interaction with the quadrupole moment is greater than that with the octupole moment. Assuming one moment arm to be 1 % longer than the others, one finds that a 100 kg man would need to be at least 6 meters away, and rain or snowfall could load the open field by no more than 1 gm/cm2 before producing a torque greater than the standard gravitational anomaly of 10-11 Since the quadrupole interaction becomes relatively more important at large distances, the atmospheric low pressure area mentioned above would produce a qu~drupole torque of 104 times the octupole torque. However, all of these gradient effects are still small enough to be negleeted, particularly because they must occur with 24-hr periods to be significant. One mystery connected with E6tv6s' original experiment is how he could 9 achieve a precision of 3 X 10- if the effect of his own'body on his apparatus (through gravitational field gradients) was as much as 80 times this (as may be easily calculated) . The answer probably is that he left the balance undisturbed until it came to rest, and then he observed for only a short period of time before the instrument would respond to his own gravitational field. Also, he ;-./ may have observed in such a standardized fashion that this effect may have always been nearly the same, tending to cancel out in the data reduction. One may easily calculate that with E6tv6s' apparatus the observation time (total time sitting at apparatus) would need to be as short as 30 sec if errors from the ~. I!' field gradient effect were not to exceed 3 X 10-9 ~r 2. Magnetic contaminants. One problem connected with our measurement, a difficulty which apparently also bothered E6tvos, was the torque caused by magnetic contaminants. While only time-varying magnetic disturbances were of importance to us, in particular only the 24-hr period, the basic torque associated with a gravitational force anomaly of 10-11 was, in our experiment, only a little greater than 0.1 % of the torque corresponding to the precision attained by E6tvos. Thus, the magnetic contaminant problem was perhaps even more severe than Eotvos'. In order to indicate the magnitude of this problem which we faced, we may note that a single filament of iron 10-3 X 10-3 X 10-2 cm3 with a magnetization of 100 gauss can introduce a torque from the earth's magnetic field of 10-8 dyn-cm. This is about 20 times the torque corresponding to a 10-11 anomaly. Each of the three balances we built was freer of magnetic contaminants than the preceeding one. The last one, which used a quartz suspension fiber instead of the tungsten employed in the others, seems to have been substantially free of contamination. All the materials were chosen to be free of contamination or else were treated to eliminate contaminants. The details of construction and magnetic measurements will be discussed in Section IV, 1. 3. Electrostatic effects. The various disturbances of the torsion balance can be 454 ROLL, KROTKOV, AND DICKE 2 INERTIAL AND GRAVITATIONAL MASS 455 divided into two classes; those resulting from external sources, such as ground vibration and magnetic field variation, and those of internal origin. The electrostatic effects are of the latter type. The electric fields present in the vacuum ehamber housing the torsion balance exert forces, and hence torques, on the balance. In order to see the order of magnitude of the forces involved, we note that the weights have an area of the order of magnitude of 10 cm2 and are spaced from neighboring surfaces by the order of 1 em. Contact differences of potential of the order of 0..5 volts between dissimilar materials might be expected. Hence, electrostatic forces are of the 10 order of 1O-6'dyn. This is to be compared with a force of about 10- dyn asso11 ciated with our standard gravitational anomally of 10- A disturbing force this large is not a priori a catastrophe, as it is the time variation in the force that matters, not the force itself. The order of magnitude of variation (24-hr period) of contact difference of potential which can be tolerated is 10-5 volts. (This will be made more exact in Section IV, G, 3.) It is well known that the work function of a surface depends iIi a rather sensitive way upon absorbed gas, which in turn depends upon temperature. For purposes of making a very rough estimate, it may be assumed that absorbed gas on the surface changes the work function by roughly 0.5 volts and that the gas absorption activation energy is 0.1 volt per molecule, requiring a temperature stability of the order of 1O-3oC if electrostatic forces are to be sufficiently constant. This is only one of many temperature-dependent effects which require fairly good temperature regulation. The means by which this is accomplished are discussed in Section IV, G. 4. Gas pressure effects. Perhaps the most severe requirements on the apparatus 10 are those imposed by gas pressure effects. With an anomalous force of 10- dyn 2 and a cross-sectional area of 10 cm , it is necessary to reduce variation in pres~ 17 sure differences on two sides of a weight to 10- atm. Hence, an extraordinary degree of elimination or control of these forces is required. EotvOs did not employ a vacuum, but instead used air at 1atm pressure in a close-fitting jacket. For temperature uniformity, the jacket had triple-layered metal walls with air spaces between layers (see Fig. 1). In such a system temperature gradients must be very small, for convection must be substantially eliminated. The viscous force on a cylinder may be crudely estimated by Stoke's Law. To reduce 7 this force to 10-10 dyn requires the air flow velocity to be less than 10- cm/sec. It seems certain that a better technique is to put the torsion balance in a vacuum. However, the vacuum must be good and temperature gradients must 6 9 still be small. Note for example, at a pressure of 10- mm or 10- atm, the temperature difference of gas streaming in two opposite directions in the chamber 10 must be less than 3 X 1O-6oC if the pressure difference is to be less than 10- dyn/cm Thus, considerable care in multiple jacketing and insulation is necessary to achieve the required temperature uniformity. Also, it is clear that a 6 vacuum better than 10- mm is desirable, suggesting that ultrahigh vacuum techniques and a bakeable system be employed. This requirement in turn imposes severe limits on the types of materials used in the torsion balance. These will be discussed in Sections IV, A and IV, C. .5. Brownian motion effects. One added, but relatively unimportant, advantage of going to a high vacuum (of, say, 10-8 mm) is that pressure fluctuation disturbances of molecular origin are reduced to a qegligibly low value, substantially eliminating Brownian motion disturbances of the balance. This statement is easily misunderstood, as it is remembered that the Brownian motion kinetic energy of the rotational mode of the balance is kT /2, independent of the gas pressure. The lower the gas pressure, the smaller are the pressure fluctuations acting on the weights. But the viscous damping of the balance is also smaller, leading to the same average fluctuation energy. The fluctuation in the gas pressure is given by the momentum transferred per molecular collision, multiplied by the square root of the number of collisions. For a total transverse area of the three weights of 15 cm2, this f1uetuation 8 force at a pressure of 10- mm is 3 X 10-11 dyn for a 1 sec average, small com10 pared with the force of 10- dyn associated with the standard gravitational 11 anomaly of 10- The fluctuation force is proportional to the square root of the pressure (for a mean free path greater than 1 cm) and varies inversely as the square toot of the averaging time. While this driving force of thermal origin is small, it is sufficient, because of the accompanying low viscosity, to maintain the necessary kT as the fluctuation energy of the balance. The mean potential energy of kT /2 leads to an uncertaint yin rotation angle of f:,.(j = (kT/K/12, (j(j (1) due to the stand- where K is the torsion constant of the balance. The rotation ard 1O-11 anomaly is 08 = 1O-Umgsl/ K, (2) where l is the monient arm and gs is the transverse component of the acceleration toward the sun. Inserting the characteristic values m = 30 g, g., = 0.6 2 cm/sec , l = 3 cm and K = 0.2 dyn cm/rad, we find that f:,.(j is a remarkably large 4..5 X 10-7 rad, and that f:,.(j/08 ,......, 200. (;3 ) This fluctuation would represent a serious limit to the precision of the experiment if, as inEotvos' experiment, a measurement consisted of a position observa- 456 ROLL; KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 457 tion of short duration. However, a measurement in the form of an average over a time interval of one or more natural periods of the torsion balance leads to a thermal fluctuation error of negligible proportions. This type of time-average measurement is easily carried out .. However, as will be discussed in detail later, it is more convenient in practice to dampen the oscillation with a feedback amplifier system. This effectively introduces an artificial viscosity without the accompanying thermal fluctuations. As a result of this dampening, the thermal fluctuation energy of the balance is reduced to much less than kT. The energy necessary to drive this refrigeration system is of course derived from the feedback amplifier. It will be clear from the subsequent discussion that there are important reasons conne,cted with the signal to noise problem for introducing this feedback damping. 11 6. Rotation detection system. The standard gravitational anomaly of 109 implies a rotation of our torsion balance of about 3 X 10- rad and a resulting displacement of the weights of about 10-8 cm. It would not be sufficient to detect the displacement of one of the weights to infer the rotation angle, as displace6 ments due to the swinging of the balance (pendulum mode) could be 10 times as great. The displacement due to tidal-induced tip of the apparatus also would be much greater. While a displacement measurement system, such as a condenser or inductance bridge, could be used if it were employed on all three masses in a balanced bridge system responsive to rotation only, the balence would be very critical and the resulting electrically-induced torque on the balance would be serious. Therefore, the best angle measurement technique appears to be based on the use of the old-fashioned optical lever. This method has two advantages; it gives an indication of rotation only, and the resulting disturbance of the balance is minor in a carefully designed system. The details of the rotation detection system will be discussed in Section IV, D. We note here, however, that with a torsion balance a few centimeters on a side. 5 the diffraction width of a light beam would be of the order of 10- rad. In order to determine the center of such a diffraction pattern with an accuracy of 3 X 10-9 rad, 107 photons are required. It might be asked whether the disturbance due to the momentum transferred by 107 photons is negligible. Even if the total light pressure interaction occurred transversely on one weight, a ,light flux of 107 photons per second would produce a force 5 orders of magnitude too small to be significant. In similar fashion, the heat transferred to the mirror through absorption of a few per cent of these photons causes a negligible force through the radiometer effect, at a gas pressure of 10- mm. While these disturbances are negligible, reasonable care must be taken to devise a rotation detector capable of giving the necessary precision with a very 4 weak light. Furthermore, to detect a rotation as small as 3 X 10- of the diffrac8 ~F ~(;"l,' ~ tion width requires a drift-free detection system. For example, the standard dc balanced double photomultiplier system would not have sufficient long term stability. (Remember that stability over a 24-hr period is required.) The method used here will be discussed later. An important part of the solution to the problem is maintaining sufficient temperature stability to assure the position stability of the mechanical parts. 7. The noise problem. Some of the various aspects of the noise problem have been discussed above. Although the noise problem will be treated in detail in Section V, C, some of the results of that later discussion are summarized here in general terms. The apparatus is a substantially linear device generating an output voltage (measured as a trace on a chart recorder) in response to a possible gravitational anomaly and in response to various noiselike disturbances. As there are so many disturbances of the balance itself, disturbances beyond the control of the experimenter, it is desirable that unessential noise, such as that generated in an amplifier of a poorly designed rotation detection system, should be substantially eliminated. After this the only noise requiring consideration is that associated with the pendulum itself. Noise disturbances may be conveniently sepltrated into two classes; Gaussian and nonGaussian. When a large number of small disturbances, randomly dis-. tributed in time, act on the balance, the Fourier transform of the disturbance is characterized by a Gaussian probability distribution function and, most important, a lack of correlation between individual Fourier components. The best that can be done to discriminate against noise of this type is to filter, passing the signal plus noise through a filter matched to the spectrum of the information pulse (a 24-hr period signal limited to an observation period of a few days). There are various ways of performing this noise filtering and they are all equivalent. The type of non Gaussian noise of importance to the subject experiment is the noise pulse. This is a large disturbance of relatively short duration, a disturbance which is clearly much larger than the Gaussian noise. Such disturbances may excite various high frequency modes of the torsion balance and appear as rotational noise of low frequency only through nonlinear effects. On the other hand, the disturbance may excite the rotational mode directly. While it is possible to treat the non Gaussian noise by filtering as in the case of the Gaussian noise, to do so may lead to a poor system, and it is often possible to do very much better. A large noise disturbance of short duration carries with it enough information to allow its identification and removal as a source of noise. However, a simple narrow-band filter would pass the 24-hr eomponent of the noise pulse without removing it. The technique for eliminating this type of nonGaussian disturbance is to 458 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 459 employ a double filter. The information plus noise is first passed through a filter with a wide spectral band pass, and then through the narrow Gaussian-noise filter. After the first filter, because of its wide spectral band pass, the noise pulse appears substantially undistorted as a large disturbance of short duration. It can be eliminated by simply deleting this part of the recorder trace and interpolating smoothly over the short gap. The time series, including interpolated gaps, is then passed through the final narrow filter with a band pass at a frequency of >14 cycle per hour. This narrow-band filtering is actually accomplished by numerical calculations on a digital computer, as is explained in Section IV, D. 'While the judgment as to noise pulses to delete and the interpolation could be carried out automatically and electronically, it is easy to do this manually, and this was the procedure employed. When relatively long periods of time are marred by large noise pulses, it may not be possible to interpolate reliably. In this case, these long defective periods may be deleted and the remaining data fitted by least squares to the appropriate assumed information time series. This technique will be discussed later. At a gas pressure as low as 10-8 mm, and with a quartz suspension, the Q of the torsion balance may exceed 105 Since the corresponding time constant is the order of a year, the torsion balance has a long memory, continuing to respond this long to large noise disturbances. The balance is basically a very narrow-band filter tuned to its characteristic oscillation frequency. In order to eliminate this extremely narrow band-pass filter and replace it by a broad-band low-pass filter, the oscillation frequency is increased somewhat and the Q is reduced to about unity by means of a feedback amplifier. The details will be discussed in Section IV, D. 8. Temperature effects. As was discussed previously, the torsion balance would be expected to be quite sensitive to temperature variations. The nlOst critical effect is that of differential gas pressures associated with temperature gradients acting on the balance. However, temperature gradients could have other effects. For example, a temperature gradient across the telescope support tube would cause it to bend. There could also be direct temperature effects on the deflection of the balance. For example, the twist of the torsion fiber might be weakly temperature sensitive. Also, as was discussed previously, the contact difference of potential, particularly between the gold weight and the deflection electrodes, could depend upon absorbed gas and hence be temperature dependent. It seems hopeless to try to understand in detail and quantitatively all of these temperature variation effects. Fortunately, this is not necessary. It is only necessary to study statistically the correlation between the temperature, its time derivatives, and the balance signal, assuming statistically significant cor. relations occur. Once the correlation coefficients are known, the torsion balance signal can be corrected for the temperature effects by making use of the measured temperature as a function of time. The details of this procedure are discussed in Section V, D. 9. Ground vibration effects. It would be expect~d that rotational accelerations of the ground with a'24.hr period would be far too small to be important. To be significant, an acceleration of 10-12 radjsec2 would be required. This would lead to a 24-hr period amplitude of 2 X 10-4 rad, far too large to be considered. High frequency ground vibrations can be a serious difficulty if of sufficient amplitude. With a completely linear system they would be unimportant, for they would occur at the wrong frequency and would be filtered out. However, the torsion balance was not a completely linear device. Also it was capable of oscillating in a large number of vibrational modes, such as the pendulum swinging mode, the various rocking modes of the balance, and the vertical fiber stretching mode. These, and many others, have high Q's and are easily excited with. an amplitude sufficiently great that the torsion zero is affected. Fortunately, the Q's of these modes were sufficiently low, and their frequencies sufficiently high, that vibrations of this type would die out in a reasonably short time after their excitation by a strong ground disturbance. The two pendulum ;;winging modes had the longest life, and a special provision was made in the apparatus for damping them if they were strongly excited by an unusually violent seismic disturbance. IV.APPARATUS The equipment which most successfully overcame the difficulties c\i;;cussed above represents the culmination of an evolutionary process extending over several years. Various features of the last and best version of the instruments will be described next. Since significant results were also obtained from the previous generation of instruments, reference will be made at the end of this section to significant differences between the last two versions of the apparatus. A. PENDULUM AND SUSPENSION Basically, the torsion balance consists of a triangular quartz (fused silica) frame with a weight suspended from each corner, the frame being suspended by a quartz torsion fiber. Figure 3 shows the mechanical details of this arrangement. In the final version of the apparatus, the two identical weights were right circular cylinders of pure aluminum, 3.2 em long X 2.1 em diam, and weighing 30 gm each. The third weight was a 3.2 em X 0.78 cm right circular cylinder of high-purity gold. Small aluminum tubes were fastened to the top of each weight and crimped around the suspension wires. These thin copper wires were lashed to the notched corners of the quartz ftame with finer copper wire, and I I' ,t 460 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 461 QUARTZ ROD DETAIL "A,' SUSPENSI~ PLATE SUSPENSION FITTING PLATE ALUMINUM WEIGHT----- ALUMINUM WEIGHT GOLD WEIGHT I I INCHES I 012 FIG. 3. The torsign balance suspension. The construction of both upper and lower fiber fittings is illustrated in Detail "A." then crimped into copper tubes held by a suspensIon plate. The triangular quartz frame itself was fabricated from quartz rod, and had an aluminized optical fiat on the side between the two identical weights. Its weight was 3.4 gm. The techniques for fabricating and mounting the quartz torsion fiber were devised by Dr. Barry Block, who also made the fiber used. A 1 nun quartz rod served as the source of the torsion fiber. After inserting the rod into aluminum fittings (Fig. 3), small balls were formed on its ends, and wedges ground on the balls. The rod and fittings were then mounted in a spring-loaded "gun." By heating the center of the rod in a fine oxyhydrogen cross-fire and firing the gun when the quartz reached the proper temperature, it was possible to pull a fiber whose length and. diameter were controlled by the throw of the gun and the strength of the spring used, respectively. The quartz fiber employed in the experiment was approximately 10.8 in. long, 0.001 in. in diameter, had a torsion .constant of 0.24 dyn cm rad-\ and a breaking strength of the order of 200 gm. As shown in Fig. 3, the top of the fiber was attached to its support, and the bottom to the suspension plate fittings, by means of aluminum screws which threaded into the fiber fittings. The wedges ground onto the quartz halls dug into the aluminum screw when it was tightened to prevent any twist tipping of the shanks of the quartz rods in the fittings. The coupling of the lower fitting to the suspension plate is shown in Fig. 3. In the same figure may be noted a copper disk fastened to a long copper rod, which is in turn secured to the bottom of the suspension plate fitting. When in place in the vacuum chamber this disk hangs just above a soft ironslug, which may be magnetized, if desired, by a movable pe-r:manent magnet outside of and beneath the chamber. Such an arrangement was occasionally required to provide eddy-current damping of the swinging modes of the torsion balance. The magnet is normally retracted into the interior of a soft iron shield, but may be slowly ejected pneumatically by remote control. To prevent the accumulation of electrical ch~rge, such as might be produced by cosmic radiation, it was necessary that the entire suspension posses;;;a none zero conductivity and be connected to ground. The only nonmetallic material used in the balance was quartz (fused silica) for the triangular frame and torsion fiber. Before assembly and installation, a thin layer of aluminum was evaporated onto the torsion fiber, giving its entire 1O.8-in. length a resistance of some 3000 ohms. After assembling the lower part of the sJspension, the triangular quartz frame was coated with silver, which made good electrical contact with the copper wires lashed to the corners. The optically flat si~e of the triangle was, of c~urse, aluminized to serve as the mjrror for the optical lever system, and the silver made contact with but did not cover this layer of aluminum. As was learned by bitter experience, the elimination of significant magnetic contamination required the most scrupulous care in selecting materials and fabricating and assembling the balance. The four materials used in the suspension were aluminum, copper, gold, and a silver "paint." All of the tools used in cutting the first three of these were made of ordinary high-speed steel, but were first washed in an ultransonic cleaner. Pure 2S aluminum and electrolytically refined copper were used in all cases. In addition, the copper was heat-treated by holding at about 900C for several hours. Such treatment destroys the ferromagnetism of any magnetic contaminants by putting them into solid solution in the copper. In fabricating the gold weight, the principal source of contamination was not or 462 ROLL, KROTKOV, AND DICKE I2\"ERTIAL AND GRAVITATIONAL MASS 463 the gold itself, which had a total impurity concentration of less than 10 ppm, but the crucible in which the weight was cast. This crucible, made of spectroscopic grade graphite, was machined with a new tool cleaned in the manner described above. Subsequently, the crucible was treated with acid to remove possible surface contaminants. A gold brick was then melted into the crucible under vacuum in an induction furnace. (The procedure here was essentially that used in preparing high purity semiconducting elements for use in transistor manufacture.) Thereafter, the only cutting process necessary on the gold weight was threading a hole for the tubular aluminum fitting which was crimped around the support wire. The tap used for this purpose was cleaned as described earlier. -(Fortunately it was not even necessary to drill a hole for the tap'. The casting process left a hole of the right size in just the right location.) Before assembly, all parts of the suspension except the torsion fiber itself were cleaned in the ultrasonic cleaner, organic solvent, acid, and distilled Water, successively. At no time during or after this cleaning were any of the parts allowed to come in contact with metals other than 2S aluminum, or with other substances which could leave significant amounts of magnetic contamination, dirt, or grease on them. The assembly was performed on clean glass plates, using rubber gloves and special aluminum-jawed tools. The gloved hands, tools, glass assembly plates, and glass storage jars were all cleaned in the same manner ~s the balance parts themselves. As mentioned previously, the quartz triangle was coated with a suspension of high purity silver particles in banana oil. Again, all objects with which this silver "paint" came into contact had undergone essentially the same cleaning procedure outlined above. After drying for a day or two in a closed tube containing desiccant, the aluminized torsion fiber with its fittings was connected to the suspension, and the entire arrangement installed in the vacuum chamber. B. :MECHANISM FOR ADJUSTING SION BALANCE HEIGHT AND ANGULAR POSITION OF THE TOR- FORK VACUUM CHAMBER HAT SOFT IRON SLUG MOTION) (VERTICAL PHOSPHOR-BRONZE SPRING VACUUM CHAMBER STEM FI BER SUPPORT 't:ii;,' ~ t ~ In order to position the torsion balance for the optical lever system, it was necessary to be able both to raise and lower the suspension and to rotate its equilibrium position. FurtherIl1ore, these adjustments had to be made from outside the chamber, after the chamber had been baked under vacuum and sealed off. The required degrees of freedom were afforded by the mechanism illustrated in Fig. 4. The two soft iron slugs shown could be rotated independently by means of a small magnetron magnet just outside the stainless steel wall of the vacuum chamber. Rotating the top iron slug causes the entire mechanism below the support plate to move vertically without rotating, over a span of about one-half inch. The lower iron slug, on the other hand, is coupled to a differential gear arrangement which causes the fiber support to rotate without any substantial FIG. 4. Detail of the mechanism for independently raising and lowering, and rotat.ing the torsion balance inside the vacuum chamber. Turning the lower soft iron slug with a magnet outside the vacuum chamber stem causes the torsion fiber to rotate slowly. The difIerential gears provide a 20: 1 reduction ratio. Using a magnet outside the vacuum chamber hat to rotate the upper iron slug causes the torsion balance to move vertically over a span of about )i in. One revolution of the upper slug causes a vertical motion of ~'2"o in. vertical motion. Twenty full turns of the lower slug cause one full turn of the fiber support. Since the torques which can be exerted on the iron slugs by external magnets are necessarily limited, the mechanism had to be constructed so as to operate freely after the baking. Except where otherwise noted, the mechanism in Fig. 4 was constructed of stainless steel, with carefully fitted gears. Molybdenum disulphide powder was used as lubricant, since it could survive the baking and it retained good lubricating properties under vacuum. 464 ROLL, KROTKOV, AND DICKE IXERTIAL AND GRAVITATIONAL MASS 46.5 BAYARD-ALPERT GAUGE ALUMINUM BALANCE CAN BASE DAMPING DISK IRON SLUG~ CUP PLATE ~ ~ELECTRODES FIG. 5. Cutaway drawing of the vacuum chamber showing the torsion balance in place. Flanges are sealed with gold "0" rings. C. VACUUM CHAMBER When mounted in its vacuum chamber, the torsion balance appears as shown in the cutaway drawing, Fig. 5. The stainless steel vacuum chamber itself consists of four parts: base plate, balance can, stem, and hat. Flanges and glass-tometal Kovar seals are heliarc welded to the balance can and stem, while gold "0" rings provide the vacuum seal between the four parts. On the base plate are mounted two electrodes which straddle the gold torsion balance weight, and the soft iron slug used in eddy-current damping of swinging modes of the balance. As will be discussed in the next section, the two electrodes are used to provide electrostatic torque feedback. J\lechanically, they consist of copper plates mounted on the stems of two Kovar glass-to-metal feed-through seals which have been silver soldered to the base plate. These form a parallel plate condenser, between the plates of which is i:)uspended the gold weight. The soft iron slug is a cylinder threaded into a blind hole at the center of the base plate and surmounted by a shallow aluminum cup. In operation, the damping disk on the torsion balance is suspended in this cup but does not touch it. Since they can easily come in contact with parts of the torsion balance, the base plate and all items mounted on it went through the same cleaning process as the parts of the balance suspension itself. The sole appendage on the balance can is the optical lever telescope window, an optical-quality window fused to a glass-to-metal seal. This window provides an inlet and outlet for the incident and reflected light beam of the optical lever. The aluminized optical flat (mirror) on the quartz triangle is located directly behind the window and normal to the axis of the optical lever telescope. It should be noted that this window and the electrodes are the only asymmetries in the vacuum chamber near the torsion balance itself. Since the window and balance are both symmetric. with respect to the telescope axis, any nonuniformities produced by the former (e.g., a temperature difference between the window and balance can) to a first approximation should produce no torque on the balance. (This was not true of the older torsion balance described in Section IV, J. It had a lead chloride weight in a glass bottle to the right as seen looking in the window.) A Bayard-Alpert type ionization vacuum gauge was mounted on a glass-tometal seal welded to the lower flange of the stem, and could be used to measure gas pressure in the vacuum chamber. The side arm from the pyrex stem of this gauge was connected to the vacuum system used in evacuating the chamber during baking, and was subsequently sealed off. To remove absorbed gas given off by the materials in the chamber, a small 0.2 liter/sec Vac-Ion pump was welded to the top of the hat. 3000 volts were applied to this pump continuously during the accumulation of data, from a remotely located rf high voltage supply. The 50 J.1.A panel meter available for monitoring pump current drain never showed a discernable deflection, indicating that the power dissipation of the pump was less thana few tenths of a milliwatt. As monitored by several thermocouples, the vacuum chamber was held at a temperature of about 370C for a period of 70 hI' during the baking procedure. A mercury diffusion pump with liquid nitrogen cold trap maintained vacuum inside the chamber and reached a pressure of about 10-6 mm Hg during the latter part of the baking. After sealing off and removing the vacuum chamber from the oven, and pumping for a few days with the Vac.lon pump, the pressure as reo 466 ROLL, KROTKOV, AND DICKF. INERTIAL AND GRAVITATIONAL MASS IRON POLE PIECES MAGNET O~CILLATING WIRE MAGNET 467 4V 0.5",! (e) + 1.34V MOTOR---':'~ DRIVEN IK LIGHT PIPE 200K (a) (d) DIFFRACTION PATTERN OSCILLATING WIRE "~S;RING PRISM 50JL ~L1P MAGNET . IRON POLE PIECES (b) -50 ('lI I ! -25 0 25 50J-L FIG. 6. Block diagram of the optical lever detection system corded by the ionization gauge remained reasonably constant at about 1 X 10-8 mm. When the apparatus was removed from service 15 months later, the pressure, as carefully measured with the Bayard-Alpert gauge at redueed emis8 sion current, was still 10- mm Hg. D. OPTICAL LEVER DETECTION SYSTEM At the heart of the experiment is the instrumentation for measuring very small rotations of the torsion balance. This equipment is shown schematically in Fig. 6. The source of light for the optical lever is a 6-volt flashlight bulb, operated at 5 volts from a regulated power supply to prolong the life of the bulb indefinitely. The light is focused through a 25JL slit, reflected from the aluminized flat on the quartz frame of the torsion balance, deflected off the telescope axis by a small prism, and the image of the slit focused on a 25JL-diamtungsten wire. By locating this wire in the field of a small magnet and connecting it in a balanced bridge oscillator circuit, it was made to oscillate at its mechanical resonance frequency of about 3000 cps and with an amplitude of 25 to 50JL. Mechanical and electronic details of the oscillating wire arrangement are shown in Fig. 7. When the diffraction pattern of the 25JL slit produced by the 40 mm diameter telescope lens is centered exactly on the equilibrium position of the oscillating wire (Fig. 7( d)), the photomultiplier will detect only the even harmonics of the FIG. 7. Details of the oscillating wire light modulator. (a) Top view of the oscillating; wire device, showing the magnet and pole piece assembly, prism, and light pipe. (b) Side view, showing the method of mounting the oscillating wire between the pole pieces. (c) Block diagram of the balanced bridge oscillator which drives the oscillating wire. (d) Sketch of the diffraction image of the slit focused and centered on the equilibrium position of the oscillating wire. As the wire oscillates about the position illustrated, the light received by the photomultiplier is modulated at the second harmonic of the wire frequency. Only when the diffraction image shifts off-center from the equilibrium position of the wire is the fundamental wire frequency detected by the photomultiplier. (e) Calculated fractional light intensity received by the photomultiplier as a function of displacement of the diffraction image of the slit from the center of the wire. 3000 cps fundamental frequency. As the torsion balance rotates slightly and shifts the diffraction pattern off center, the fundamental frequency will begin to appear in the photomultiplier output. The phase of the fundamental (0 or 180 relative to the oscillator signal driving the wire) indicates the direction of rotation of the pendulum, and its amplitude is proportional to the magnitude of the rotation for sufficiently small angular displacements. The calculated fractional light intensity received by the photomultiplier is sketched in Fig. 7(e) as a function of displacement of the center of the diffraction pattern from the center of the wire. The full width at half maximum of this curve (the "line width;' which must be split by the detection apparatus) is about 30JL or 3 X 10-5 rad. Next, the photomultiplier output is increased by a preamplifier and an ampli- 0 468 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 469 fier tuned to the fundamental frequency, and phase detected by mixing with the wire oscillator signal. The output of the phase detector is pulsating direct current which is filtered and further amplified. As indicated in Fig. 6, these processes are' all performed by a lock-in amplifier. The output of the lock-in amplifier, then, is proportional in sign and magnitude to the angular displacement of the torsion balance from the position at which the slit image is centered on the wire. This output goes to a fjlter circuit, whence part of it drives a chart recorder and part of it is fed back to the torsion balance. High frequency noise, including the 0.82 cps signal which appears as a result of the balance swinging in a plane ingluding the telescope axis, is effectively removed from the chart recorder signal by the two 48 sec time constants in the recorder filter. The two RC = 0.8 sec sections in the feedback filter serve to remove high frequencies from the feedback electrodes. Following these two "integrating" filter sections is a "differentiating" circuit with a time constant of 3.6 sec. This provides the velocity feedback or damping which is necessary to prevent oscillation due to a Nyquist instability. Finally, the 51 kn potentiometer serves as a feedback gain and torque sensitivity control. The time response of the "servo system" illustrated in Fig. 6 is determined by the feedback filter, and predominantly by the differentiating element in that circuit. Given an initial displacement, the fed-back torsion balance will oscillate with a complex frequency w = (wr iWi) which is a function of the open-loop feedback gain A. The cubic equation for w(A), resulting from the feedback filter used, was solved in a straightforward but tedious manner to obtain the results shown in Fig. 8. Also indicated on this graph is the range of operating gain in which data was accumulated. As is desirable to minimize the effects of transient disturbances, the operating range was in or very near to the region of critical damping (vanishing real frequency). The effects of the magnitude and stability of A on the sensitivity of the torsion balance will be discussed in the , u w rJ) 3 w u z w ::> rJ) 1.0 OPERATING~ GAIN RANGE o w a:: lJ.. a:: <l: ...J -Wi ::> <l: o z >a:: <l: <l: :twr~ ---l~ -COMPLEX ROOTS -+wr ~ , I z o I I I I :2: o Z <l: ...J <l: -3 I --- REAL ROOTS ROOTS I I __ I I I I 1.0 10 IMAGINARY + ~ 0.1 DC OPEN 10.0 100.0 LOOP FEEDBACK GAIN A FIG. 8. Resona~t angular frequencies of the torsion balance system as a function of the dc open loop feedback gain A. The dashed curves represent real parts of the resonance frequencies, while the solid curves represent imaginary parts which lead to damping of torsional oscillations. Also shown as solid curves are purely imaginary roots. The curves were calculated from the transfer characteristic of the feedback filter circuit shown in Fig. 6. next section. Ignoring for the moment the drift compensator, a constant potential difference V B of 4 volts is applied across the two feedback electrodes. However, the potential of each electrode relative to ground (the gold weight) is determined by the position of the movable contact on the 5 kn electrode bias potentiometer, and by the potential V applied to this movable contact from the feedback filter circuit. If the gold weight is approximately centered between the electrodes, it can be shown (see Appendix A) that the electrostatic torque on it is Le = electrode bias potentiometer (actually a 10 kn' four-decade preClSlOn voltage divider in parallel with a precision 10 kn resistor) provides a fine adjustment of the constant torque on the torsion balance, and therefore of its angular position. As indicated in Fig. 6, VB is the potential supplied to the feedback electrodes by the bias battery. The actual potential VB * between the surfaces of these electrodes, however, is the sum of VB and the difference ~ Ys in the contact potentials between the surfaces: VB * = cVB*[(l - 25)VB + 2V], (4) VB + ~Vs . where c is a constant involving the geometry of the arrangement, and 5 is the electrode bias potentiometer setting (0 ~ 5 ~ 1.0, and 5 = 0 corresponds to the potentiometer slider connected to the negative battery terminal). Hence, the ~ Vs could amount to a volt or so, while Y B was 4 volts under operating conditions. In similar fashion, V should not be interpreted as the potential applied to the electrode bias potentiometer wiper, but the contact difference of potential 470 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 471 between the gold weight and the copper electrodes should be added. However, as long as this contact potential difference does not vary, it is without effect on the experiment, and for simplicity it is ignored in the remainder of the discussion. There is apparently more than one source of steady drift in the equilibrium angular position of the torsion balance. One well-established source of drift is the time rate of change of temperature. Since the apparatus is adequately shielded from 24-hr fluctuations in temperature, as will be discussed later, slow temperature drifts do not affect the experiment seriously. In addition to temperaturerelated drifts, there seems to be a small, steady drift which decreases with a time constant of months. Although the origin of this drift is not completely undersfood, it again does not affect the measurements of a 24-hr period, and may possibly be attributed to a slow relaxation of strains in the torsion fiber or other mechanical parts of the suspension. A third source of drift arises from the potential 17B of the batteries across the electrodes. These consist of three carefully selected RM 42-R mercury cells in series, enclosed in a temperature-regulated oven to be described later. As the batteries discharge through the ;) kn potentiometer, the total potential across the electrodes decreases at a rate of 0.5 mV/day or less, but in a linear fashion with no sign of 24-hr periodicities. In order to keep the chart recorder on scale when operating at high sensitivities, it was necessary to remove as much of this drift as possible. Hence, a drift compensator was used to insert a voltage which increased linearly with time between the output of the feedback filter circuit and the electrode bias potentiometer. This device consisted of a 1 kn, 0.025 % linearity, 25-turn Helipot driven by a synchronous motor at a rate of 6rev /day; and in series with an Rl\I-42R mercury cell and a large variable resistor. By adjusting the variable resistor to give 10 to 30 mV across the Helipot (generating a linear drift of 2.5 to 7.5 mV per day), depending on the weather (i.e., on the long-term behavior of the outdoor temperature), it wa.s possible to keep the chart recorder on scale for reasonable periods of time. Nevertheless, it was still occasionally necessary to adjust the electrode bias potentiometer to discontinuously reposition the recorder trace on the chart. Since the torque sensitivity is independent of this potentiometer, the resulting discontinuity in the chart recorder trace could easily be removed by measuring and subtracting it from all subsequent data. The drift rate of the drift compensator was only infrequently changed, never during a run, so it could not introduce a spurious 24-hr period. The potential of the drift compensator mercury cell was monitored With a potentiometer for one or two month periods a few times during the accumulation of data, and was found to fluctuate typically by less than 10 or 20 }J.V (close to the resolution of the potentiometer) from day to day, in a nonperiodic way. Since only 1 to 3 % of the total cell voltage was applied to the bias potentiomet.er, such fluctuations were entirely negligible as far as the torsion balance output was concerned. E. TORQUE CALIBRATION AND SENSITIVITY OF THE SYSTEM In order to obtain a quantitative measure of the external torque acting on the torsion balance, it was necessary to calibrate the apparatus. This was accomplished by using a micrometer head to rotate the telescope of the optical level through a known angle relative to the balance. Such a rotation is fully equivalent to applying an external torque which would rotate the balance through the same angle in the opposite direction. The angle of rotation, together with the known torsion constant K of the fiber, gives the effective torque applied to the balance, which is then registered as a displacement of the trace on the chart recorder. (The torsion constant was obtained from the torsional oscillation period, knowing the moment of inertia of the torsion balance.) Figure 9 illustrates the mechanical arrangement used in making this rotation. The clamping screw opposite the micrometer stem was loosened and the telescope gently pushed toward it with the micrometer, generally 1 or 2 X 10-;; in. at a time. The set of three clamping screws nearest the vacuum chamber served as the fulcrum for the rotation, and the weight of the telescope at the micrometer end was borne by the third clamping screw unde~neath the telescope (not visible LEVELING SCREW CAN TO SUPPORT AND SHIELD TELESCOPE RADIATION t.: ~ ~ \% ~; o FIG. I I ! 10 INCHES 20 9. f:lchematic top view of the torsion balanc'e mounted in the instrument well 472 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 473 (Xl TABLE II MEASURED TORQUE SENSITIVITY OF THE GOLD-ALUMINUM ST TORSION BALANCE ~ (J) .70 60 10.0 <l <l l') Dates of measurements Feb. 24 and 26, 1962 April 19, 1963 (1 + A)/A" (mV/dyn em) >:!= e_ 1_: ,_ . " . ..,.. ---.. ..,., ~ . , ., 5. 0 ~ 2.0 ~ .~50 .,. 2.17 X 108 Mean value 2.1 X 108 6% 2.04 X 108 :!= 4% 36 ~ 2:40 <030 o 0.5 u 1.0;i!i 0 t:J lJ. ~ w Cf) 4 ~ " The errors quoted are probable errors of the mean of five measurements February 1962 and April 1963 calibrations. each for the w 2 ::l > " ",20 10 JUL. AUG. SEPT OCT. NOV. DEC. JAN. FEB. MAR. APR. <I o ~ 0 a:: 0 1962 1963 0.. 0 0 10-1 .J 10-2 Z 0.2 ~ ~ 0 in Fig. 9). Sensitivity calibration measurements of this sort were carried out in February 1962 and April 1963. The higher the gain A of the feedback loop which stabilizes the torsion balance, the less the sensitivity, as seen at the recorder, to extremely small external torques. In calibrating the apparatus, the disturbances caused by mechanical rotation of the telescope, and indeed by the mere presence of the person performing this operation, required the use of a much higher feedback gain and lower sensitivity than was used during actual data accumulation. Hence, the results of calibrations had to be corrected to the standard operating conditions. The torque sensitivity of the torsion pendulum may be defined as the change ~ V R in voltage applied to the chart recorder divided by the externally applied torque ~L which causes that change: ~VR(mV) (5) ST = ----~L(dyne/cm)' . Simple manipulation (see Appendix A) of the static equations for the feedback loop show that 10. Measured values of the ratio R of a change in recorder output voltage 6.17" to the causative change ill) in electrode bias potentiometer setting. The points representing measurements were taken over the lO-month period in which data was obtained. By means of Eq. (8) and the measured values of torque sensitivity (Table II), the scale in It was transformed into a scale in torque sensitivity. shown on the far left. Similarly, Eq. (7) was used to calculate the right-hand scale in dc open-loop feedback gain A.. The straight line fit to the measured points was used to obtain values of torque sensitivity corresponding to the time of each of the 39 data runs. FIG. where fi V R is the change in recorder output voltage caused by a change do in the electrode bias potentiometer setting. In fact, it can easily be shown (see Appendix A) that R = (YB/,lg)[A/(l + A)J, (7) where f is the ratio of voltage at the top of the gain potentiometer to voltage output at the recorder (see Fig. 6) : jg = V j V R . Combining Eqs. (6) and (7), STCX RjVBVB*. (8) 8T = (cd g VB *)[A / (1 + A)]. (6) Here Cl is a constant, g is the setting of the feedback gain potentiometer shown in Fig. 6 (0 ~ g ~ 1.0, withg = 0 when the potentiometer wiper is connected to ground) and A is the (dc) open-loop feedback gain. The results of the two measurements of torque sensitivity are given in Table II, reduced via Eq. (6) to operating values of g = 0.0147 and VB * = 4 volts, and multiplied by (1 A) / A to make them independent of the feedback gain during calibration. (Since A > 35 or so during calibration, this latter correction is not very significant.) A parameter which is proportional to ST and much more easily measured than the open-loop gain A is the ratio + R = ~VRjfio, Because of the. frequent changes fio in electrode bias potentiometer settings required to keep the torque chart recorder on scale, the ratio R was measured at least every day or two during data runs. These values of R are plotted in Fig. 10 as a function of time over a ten-month period, and indicate the type of fluctuations and slow changes to which the torque sensitivity was subject. A portion of the fluctuation was due to ground noise, which sometimes made it difficult to measure ~ V R with very great precision. Superimposed on the fluctuations, however, is an unmistakable decrease in torque sensitivity of a little less than 20O,{ over the ten-month period. The torque sensitivity and open-loop feedback gain scales in Fig. 10 were computed from the results in Table II, together with Eqs. (8) and (7). For use in analyzing the data, the mean torque sensitivity during a given run was taken from the straight line fit to the points in Fig. 10. 474 INERTIAL AND GRAVITATIONAL MASS ROLL, KROTKOV, AND DICKE 475 On first glance, the feedback gain A used during operation appears remarkably low. However, the characteristics of the feedback filter are such as to place this low operating gain range very close to the desirable critical dampillg region (Fig. 8). In fact, the feedback gain was originally adjusted to minimize t.he noise appearing on the torque chart recorder. Although the feedback filter used was certainly not an optimum design, it did function most satisfactorily at low gain. Another parameter relevant to the torque sensitivity is the photoelectron flux from the cathode of the photomultiplier detector. Fluctuations in this flux will eventually limit the angle and torque sensitivity of the optical lever. detector. Measurement of the de output voltage from the emitter follower stage driven by the photomultiplier showed a signal of 32 mV when the balance was locked in and the lock-in amplifier output approximately null. Of this, about 0.5 mV was dark current. Since this voltage is applied across a 1 Mn impedance and the current gain of the transistor used in the emitter follower is about 100 (order of magnitude), a current of about 3 X 10-6 amp emanates from the anode. Using the manufacturer's specification of 70,000 for the current gain of a IP21 photo8 tube operated at 625 volts, this implies a photoelectron flux of about 3 X 10 per sec. As pointed out in Section III, 6, such a photoelectron flux (photon flux times 10 photocathode efficiency) permits the detection of a rotation as small as oX 10rad in 1 sec. Thus, the photon fluctuation noise does not represent an important limit to the performance of the system. angular position was required. This was obtained by gearing a multiturn potentiometer to the turntable and connecting it as one arm of a Wheatstone bridge, with a similar potentiometer in the control shack as the balancing control. Calibration of this balance control then permitted the angular position to be measured easily to :f:3 or better. G. TEMPERATURE CONTROL AND :MEASUREMENT The importance of maintaining the environment of the torsion balance at constant temperature has already been discussed. To accomplish this, and also to ROOF (ROLLED BACK) F. ROTATION OF THE TORSION BALANCE If there were to be a real difference in the ratios of passive gravitational mass to inertial mass for the materials of the torsion balance, the phase of the resulting 24-hr period in the torque produced by the sun would depend on the orientation of the balance. This dependence upon orientation provides a convenient means of verifying any suspected positive result. Should a 24-hr period appear in the recorded tor"qu~, itsph3:~e can be required to have the predicted value, and to vary with orientation of the balance in the proper way (see Appendix B for a derivation of the phase relations). Hence, the vacuum chamber and telescope for the optical lever were mounted on a turntable, which allowed the entire arrangement to be rotated through 3600 in the horizontal plane. A shaded pole motor located two feet from the mounting pedestal (Fig. 9) drove the turntable through a set of 1500: 1 reduction gears, a long shaft with universal joint and shear pin couplings, and a set of worm and bevel gears. The speed and direction of this motor could be controlled remotely by varying the current (magnitude and sign) through the shading coils. To minimize disturbances caused by the rotation, the high gear reduction ratio was used. Hence, a 1800 rotation required about 172 hr at the maximum motor speed. \ In addition to a means of rotating the balance, some remote indication of its SCALES : ~ 3 I 4fl I INSTRUMENT PIT FIG. 11. Schematic cutaway drawing of the instrument well with the torsion balance installed. TCI, TC2, and TC3 are thermocouples used to measure temperature gradients between various points in the well and the insulation plug. TI, T2. T3, T5, and Tn are thermistor temperature sensors which were used in Wheatstone bridge circuits to monitor very small (10-3C) temperature changes near the torsion balance. 476 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 477 isolate the device from other disturbances such as building vibrations, the vacuum chamber and optical lever were mounted in an instrument pit 12 ft deep by 8 ft square, and constructed of 8 in. cinder block with a poured concrete floor resting on rock (see Fig. 11). Appropriate measures were taken to prevent water from draining into the well during wet weather. The top was protected by a wooden cover sheathed with asphalt roofing material, and could be raised and rolled back on rails. Situated on the Princeton University campus, it was located near the edge of a wooded hillside overlooking Lake Carnegie (see Fig. 2). When originally constructed, the pit was reasonably remote from sources of microseismic noise such as traffic and construction activity. Unfortunately, how.ever, during accumulation of the most sensitive data, the area within a few hundred yards on one side was the site of construction of a new field house and a road. The problems introduced by the related micro- (or more precisely macro- ) seismic noise will be described in Section V. Most of the electronic equipment associated with the experiment was located in a control shack about 160 ft away and connected to the apparatus in the pit by underground cables (see Fig. 2). In addition to maintaining the vacuum chamber at constant temperature, it was also considered desirable to regulate the temperature of the control shack to within a few degrees, in order to avoid the possibility of temperature-related drift in the electronic system. The torque on the torsion balance was especially. sensitive to the potential on the electrodes, and although the control shack was temperature regulated to ::f::2For so, it was necessary to enclose the electrode bias cells in a specially-designed oven which controlled their temperature to better than 1O-3oC. The various methods used to regulate and measure temperatures in the pit, the control shack, and the battery oven will be described in turn. 1. Temperature Regulation of the Instrument Pit There were two separate aspects to the problem of temperature stabilization of the torsion-balance itself. As already explained, the balance could be easily deflected by gas pressure gradients resulting from temperature gradients across the apparatus, These gradients were minimized by surrounding the vacuum chamber with concentric, thermally-conducting radiation shields separated by glass wool insulation. In particular, the lower part of the vacuum chamber, up to the top of the Bayard-Alpert ionization gauge, was enclosed in a magnetic shield can which was lightly packed with glass wool. Concentric with this was a sheet-aluminum radiation shield extending almost to the top of the Vac-Ion pump. The one-inch gap between the magnetic and radiation shields, as well as the space at the top between the stem, the hat, and the radiation shield, was similarly packed with glass wool. The shields were mounted on the .turntable with the vacuum chamber and telescope. An equilateral triangular. plate 24 in. ;wi i ;'1', .~;' .~: ~" on a side and 1 in. thick served as a base for the turntable, and was mounted on three concrete block piers about 2 ft above the floor of the instrument pit, as shown in Figs. 9 and 11. There were also several elements in the apparatus that were temperature sensitive, rather than gradient sensitive, and it was necessary to stabilize the temperature itself. As the insulation about the vacuum chamber was not adequate, and could not easily be made adequate to eliminate temperature fluctuations with a 24-hr period, no attempt was made to approach the problem in this way. Rather, the temperature in the whole instrument well was stabilized. This had the additional advantage that other temperature sensitive elements, such as the telescope tube, could be controlled. In early preliminary experiments, attempts were made to lise servo systems to stabilize the temperature; however, because of the poor thermal conductivity of air and the necessity for convective transport of heat, this was not very successful. It was decided instead to resort to an essentially passive system employing massive thermal insulation. Whereas the sides of. the pit were well insulated from external temperature fluctuations by the high heat capacity and low thermal conductivity of the surrounding soil, insulation of the top relied on the low thermal conductivity of a plug of insulating materials about 4 ft thick. The exact composition of this plug is illustrated in Fig. 11. To prevent the corvection of air through cracks in the insulation plug into the interior of the pit, polyethylene sheets were taped to the walls at the top, and just above the lower plywood battens. Air could circulate freely in the spaces just below and above these two sheets to form a layer of reasonably constant temperature. In order to prevent circulation of air in the interior of the pit, it was necessary to maintain the top warmer than the bottom. This condition was realized automatically between May and October. During the cold months, however, it was necessary to warm the insulation plug with the two pairs of conventional electric blankets shown in Fig. 11. As long as the mean daily temperature did not drop below 20F or so, it was possible during the cold months to actually maintain the temperature inside the pit constant to within several millidegrees by manually adjusting the voltage on the lower pair of electric blankets. A third pair of electric blankets was also installed on two opposite walls of the pit for use in controlled investigation of the effects of temperature variations on the torsion balance. To obtain information on the behavior of the temperature and temperature gradients in the instrument pit, a number of sensing devices were installed. Five Wheatstone bridges containing thermistors in one arm were mounted at locations illustrated in Fig. 11. These bridges were each powered by single mercury cells located in the control shack. lO-turn helipots, also located in the control shack, served as fine balance controls, and unbalance voltage was measured with dc 478 ROLL, KROTKOV, AKD DICKE INERTIAL AND GRAVITATIONAL MASS 479 amplifiers. The sensitivity of each bridge to temperature changes was calibrated and, by connecting the dc amplifier output to a recording galvanometer, a continuous record of temperature could be obtained for each thermistor. Thermistors TI, T3, and T6 were mounted in or on small aluminum .Miniboxes. T5 was thermally connected to the heat capacity of the aluminum base triangle by means of silicone grease and tape. T2 was .similarly fastened to a 1 in. X % in. X 3 in. copper block, which was surrounded by 1 in. of styrofoam insulation. The thermal time constants associated with the heat capacity and thermal insulation of T2 and T5 both amounted to two or three hours, and were designed to simulate the time lag in torsion balance response to temperature changes caused by the thermal insulation around and heat capacity of the vacuum chamber. In addition to the thermistors, three thermocouples were installed, as illustrated in Fig. 11, to monitor temperature gradients. During the actual accumulation of data, the outputs of thermistor bridges T2 and T5 were recorded continuously, with a sensitivity of about 0.05C full scale on a 4yz in. Esterine-Angus recorder chart. The thermocouple outputs were measured frequently and, when the weather was such as to make possible large changes in the vertical temperature gradient in the pit (particularly an approach to the adiabatic gradient), the output of TC3 was amplified and recorded. With the instrument pit insulated as described above, the amplitude of the daily period in temperature, as measured by thermistor bridges T2 and T5, typically amounted to 5 X 10-4 C or so. Temperature fluctuations due to gross changes in weather, and corresponding to periods of several days, would occur with amplitudes of 10 to 50 times this amount. Such changes would be superimposed during the warm months on a steady drift upward in temperature, which might amount to as much as 0.05C/day. During the winter, of course, the corresponding temperature decrease was compensated approximately by heat from the electric blankets in the insulating plug. The temperature gradient near ..-the top of thepitenc1osure, as measured by TC3, was typically of the order of 0.3C/ft. Two or three times during the experiment, striking evidence was .obtained for the r.elation between adiabatic changes in atmospheric pressure and temperature. At the onset of a thunderstorm or similar disturbance, an atmospheric pressure decrease would be recorded on a barograph in the form of a pulse an hour or so long and 1 mm Hg or so in amplitude. With no appreciable time lag, pulses would appear in the output of thermistor bridges T2 and T5, with magnitudes of a few hundreths of a degree Centigrade. Considering the time constants connected with the thermistors, this is the order of temperature change one would expect from an adiabatic pressure change in the pit equal to that recorded on the barograph. 2. Control Shack Temperature Regulation The control shack was a conventional one-car garage built on a concrete slab. Except for the large swinging doors at one end, the interior walls and ceilings were covered with composition wallboard and insulated. During the colder part of the year, the interior was heated by three 1000 watt electric wall heaters, each with its own built-in thermostat. Except on the coldest days (below 15F with a brisk wind), and during abnormally large and sudden temperature changes, these heaters maintained the temperature within about ::l::1C of a constant value near 20C. During the summer months, a window-mounted air conditioner performed the same service. This was controlled by an ordinary home thermostat . mounted on the wall 3 ft behind the racks of electronics, with a local heater installed next to the bimetal strip in the thermostat. Passing current through the local heater when the air conditioner was off speeded up the response of the system to temperature changes and tended to smooth these out inside the control shack. It was necessary during the spring and fall to operate both the air conditioner and one or two of the wall heaters from the wall thermostat in order to adequately track the often large and unpredictable changes in outdoor temperatures. To check the operation of these temperature regulating devices, a Bristol thermo-humidigraph was used to monitor control shack temperature during most of the time data was being accumulated. In addition, the barograph previously mentioned and a second thermograph, which recorded outdoor temperature, were kept in continuous operation. These instruments were sometimes useful in anticipating changes required in the control shack thermostat settings and/or the voltage applied to the electric blankets in the insulation plug of the instrument pit. . 3. Battery Oven Small changes in the electrode bias potential VB can result in uncomfortably large spurious torque signals. The magnitude of these spurious torques can be estimated from Eq. (4) by asking for the real torque (to be applied to the torsion balance), such that a change in battery voltage ~ VB = ~ F B * would result in no . deflection of the balance (~V = 0). From Eq. (4), :~ ~L = 2c(l - 25)VB~VB (9) t ~ (assuming that V ~ 0). Under operating conditions, /1 - 25 I < 0.5, FB = 4 volts, and c = 1 X 10-5 dyn cm/(volt)2 (see Appendix A), so that I ~L/ ~ VB/ < 3 X 10-5 dyn cm/volt. Since the maximum torque of the sun on one of the torsion balance weights is iT 480 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVI'l'ATIONAL MASS 481 3 MERCURY IN SERIES CELLS COPPER BOX (1/32") THERMOSTAT IN ALUMINUM BLOCK WOODEN ( 3/e") BOX ALUMINUM NICHROME ~ HEATING WIRES SLEEVED IN ASBESTOS " 'Z <: \: ;( <:" '1 "" " "t:l'r\ltl BOX (SIS") SCALE: 05' I I I I I I ! I I I Iflnch was adjusted with a Variac to equalize the on and off times at approximately 2 min. The low thermal conductivity of the 2-in. styrofoam layer between the copper and aluminum boxes, in combination with the large heat capacity of the latter, provided a thermal time constant of approximately 12 hr connecting the two boxes. Hence, the temperature fluctuations due to the 2-min, O.03C thermostat cycling were effectively reduced to 10-4 DCor so. Temperature changes occurring outside of the oven caused appropriate changes in the cycling times of the heater, and were thereby prevented from propagating in to the mercury cells. A carefully-handled standard cell was also installed inside the oven to be used as a comparison voltage standard for the electrode bias potential. For a period of several days after the oven reached thermal equilibrium, the standard cell and an appropriate fraction of the electrode bias potential were compared by means of a microvoltmeter which drove a chart recorder. This test showed slow, continuous drifts amounting to some tens of microvolts per day, but no sign of a 24-hr periodicity (to within a few microvolts). Subsequent monitoring of the bias potential for periods of several weeks during the accumulation of data confirmed this behavior. H. PENDULUM OSCILLATIONS AND THE DAMPING l\1AGNET OVEN FIG. 12. Schematic cutltway drawing .of the battery .oven. Temperature the mercury cells were reduced ta 10-4C .or sa inside the even. BATTERY fluctuatians .of about 60 dyn cm, an accuracy of a part in lOll for the experiment requires a stability of better than 20J.lV in VB' The typical temperature coefficient of a mercury cell is 40J.lV JOC (12), or 120J.lV;oC for three of them in series. Hence, any attempt to obtain precisions of a part in lOll or better requires that daily fluctuations in electrode bias potential VB be less'than a few microvolts, and therefore that such fluctuations in temperature of the mercury cells be less than ~afew milliaegrees-:-To eliminate these fluctuations, the three mercury cells were enclosed in the oven illustrated in Fig. 12. A glass mercury capillary thermostat with a dead band of 0.03C regulated the copper box at about 30C. To ensure thermal contact with the copper wall, this thermostat was mounted in a small hole in an aluminum block which was tightly bolted to the wall. A slurry consisting of copper powder slightly wetted with diffusion pump oil was packed into the space between the thermostat stem and the sides of the hole in the' aluminum block. Since the thermostat could pass only a small amount of current, it was used to control a sensitive relay, which in turn cycled current through the nichrome heating wires wound around the copper box. The on-off cycling of the heater was monitored continuously by a chart recorder, and the current through the heater ',~ .: ;,... ~.,.. i~ ~ I. .1" j~ As mentioned previously, one of the principal objectionable results of seismic disturbances is the stimulation of swinging modes or "pendulum oscillations" of the torsion balance. The swinging of the balance in a plane parallel to the telescope axis could couple to the detection system, because it involves a tilting of the mirror on the quartz triangle. Swinging in the plane normal to this axis, on the other hand, would not affect the detection system, since the mirror remains in the same plane. Neither direction of swing would couple. if the telescope slit were accurately vertical (an<;lsufficiently long). The adjustment of the vertical was facilitated through the use of a bubble level fastened to the telescope tube and adjusted by setting the slit image parallel to a plumb line. Although this adjustment was probably accurate to a few minutes of are, there was still a substantial residual coupling of this swinging mode to the rotation detection system. It is apparent from the arrangement of weights at the three corners of the equilateral triangle that the two normal swinging modes of the balance should be very nearly perpendicular and parallel to the mirror. As discussed below, the record of the amplitude of this period, detected by the rotation detector, shows a weak beat at a period of 16.. min, suggesting that the two normal modes were 5 not accurately oriented in these two directions, and that the orthogonal mode was also capable of coupling weakly to the detection system. The fact that the pendulum vibration modes could be detected in this way was of no particular significance. However, the torque recorder traces also show that ~82 L ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 483 deflection of the balance occurred at this beat frequency. The dominant mechalism for producing this deflection is believed to be one suggested by S. Liebes 3). If the pendulum is instantaneously swinging in a plane not the symmetry )lane, or perpendicular to it, there is a nonzero average torque tending to align ,he symmetry plane either parallel or perpendicular to the direction of swing. fhis can be understood most easily by considering the extreme case of a torsion :>alancein the form of a dumb-bell. If such a torsion balance were to swing in a e>lane 45 with respect to the axis of the dumb-bell, it is evident that the mean at value of the centripetal force on the weights results in a nonzero average torque lpplied to the weights. This torque has a mean value proportional to the square )f the amplitude of swing (13). Fortunately this coupling was very small, being troublesome only when the lmplitude of swing was abnormally large. If the threefold symmetry axis of the ;alance had been perfect, this effect would not have occurred. However, this ,ymmetry was not perfect because of the smaller diameter of the gold weight. The equipment mentioned above for measuring the amplitude of the pendulum )scillation consisted of an amplifier tuned to 0.82 cps and connected to the output of the lock-in amplifier in Fig. 6. The swinging mode of this frequency, which ~ouples to the optical lever system, is thus selectively amplified and displayed e>na chart recorder. When the amplitude of the pendulum oscillation is large, results such as those shown in Fig. 13 are obtained. The above mentioned beat period of 16.5 min between the two normal swinging modes is evident in both the pendulum oscillation amplitude and torque outputs shown in this figure. :J f :tt . ~ ~: t:,;:~ Ii ~ 'iJ; Contributions to the torque as large as this would not be serious if one could be sure that they had zero mean over long time averages. Indications were, however, that this was not true. Hence, a means for damping the pendulum oscillation was necessary. In the torsion balance being described, a mechanism was apparently present which would provide the necessary damping within a large fraction of a day after removal of the driving disturbance. Although the exact nature of this damping mechanism is not known with certainty, we suspect that it arises from the bending of the copper wires supporting the weights as the torsion balance swings.! When more rapid damping was desired, such as after a rotation or other severe mechanical disturbance, a magnet could be raised directly underneath the vacuum chamber to magnetize the soft iron siug inside and produce eddy currents in the damping plate attached to the torsion balance. The magnet fit snugly into a magnetically-shielded cylinder and rested several inches below the bottom of the vacuum chamber when not in use. A compressed air line from the control shack to this cylinder in the instrument pit could be pressurized at will to raise the magnet in the cylinder, through holes in the center of the aluminum base triangle and turntable, to just underneath the stainless steel base plate of the vacuum chamber. Use of the damping magnet in this way would ordinarily reduce the most severe pendulum oscillations to a negligible amplitude in a few hours. The effects of this magnet on the torsion balance were sufficient to offset its equilibrium position by a considerable amount, however, and data could not be accumulated when the magnet was in use. L MAGNETIC EFFECTS AND INSTRUMENTATION The precautions taken .to keep magnetic contamination out of the torsion balance have already been described in detail (Section IV, A), and the use of a magnetic shield can around the vacuum chamber has been mentioned. This shield can was constructed of two >t2-in. laminations of shielding material and was in the form of a right circular cylinder 6.75 in. in diameter and 13 in. high. Other than the open top, the only openings in the can were a 2 X 2.75 in. oval hole on one side to accommodate the window for the optical lever telescope, a 2 in. 1 In the most sensitive balance used, the weights had large moments of inertia about their centers of mass (27 gm cm' and 34 gm cm' for the gold arid aluminum weights, respectively). When the balance was set into swinging motion, relatively large torques would be exerted on the weights by the support wires, in order to twist them into following the circular arc of pendulum motion. Such torques could result in energy losses and damping through the bending of the support wires. In the previous version of the apparatus, on the other hand, the weights could be more rigidly supported by the wires, since their moments of inertia were much smaller (3.7 gm cm' and 6.6 gm cm' for the copper and PbCl, weights, respectively). In fact, the pendulum oscillations would not damp out sufficiently without feeding back an electrostatic force on this earlier balance (see Section n-, J). FIG. 13. Chart records of torque on the torsion balance in solar torque units (1 stu = 58.6 dyn cm), and relative pendulum oscillation amplitude (arbitrary units) during stormy weather. The envelope of the chart labeled relative pendulum oscillation amplitude is the selectively amplified amplitude of the 0.82 cps component of lock-in amplifier output. A 16.5min beat period between the two normal modes of pendulum oscillation is prominent on both charts. 484 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS FEEDBACK ELECTRODES 485 hole centered on the bottom for the damping magnet system, and twelve H in. holes in the bottom for leveling and clamping screws. After installation in the instrument pit, the sensitivity of the apparatus to changes in the horizontal component of magnetic field was measured using the dipole field of a, magnetron magnet. The magnetic field at a given distance from this magnet was determined by using a compass to find the distance at which 3 it was equal to the known field of the earth, and then applying the 1/r law for a dipole field. To measure the magnetic sensitivity of the torsion balance, the magnet was placed at a known distance from and orientation relative to the quartz triangle. The deflection of the torque chart recorder caused by rotating the magnet and reversing the direction of its field then provided the necessary information. Such measurements were carried out both before and after the 10month period of data accumulation, and in each case a number of measurements involving reversal of magnetic fields in two. orthogonal horizontal directions were made. The results indicate that the sensitivity to magnetic field changes was 5 less than 1.3 X 10-13 solar torque units (stu) per')' (1')' = 10- gauss). (A solar torque unit is 58.6 dyn cm, the maximum torque of the sun on one of the torsion balance weights. The size of an anomalous gravitational torque in stu is approximately equal numerically to the anomalous difference in ratios of passive gravitational to inertial mass to which it would correspond.) Since typical daily periods in the earth's magnetic field have an amplitude of 10')' or less, magnetic effects begin to become important only for sensitivities sufficient to detect differences 12 in the passive gravitational to inertial mass ratio of less than one part in 10 Had the magnetic sensitivity of the torsion balance been greater, or had the torque sensitivity been higher, two magnetometers were available to measure the two horizontal components of the earth's magnetic field and record them on a chart with a full scale sensitivity of about 100')'. The information on these charts could then be multiplied by the appropriate measured coefficients ~ VR/ ~H (m V h), and subtracted from the output of the torque recorder to remove a'significant fraction of the effects of a varying magnetic field. t \ / "\ ~M ) COPPER PbCR 2 !+PYREX o I 123 I I ! eM OPTICAL LEVER FIG. 14. Location of the weights in the copper-PbCh torsion balance. The copper weights each weighed 6.5 gm, while the pyrex flask of carefully recrystallized PbCh weighed 6.6 gm. Of this latter weight, 3.5 gm was pyrex and 3.1 gmPbCh . J. THE COPPER-PbCh TORSION BALANCE During the period between fall 1960 and spring 1961, a torsion balance of lower sensitivity was operated and useful data obtained. Although similar or identical in most respects to the balance which has just been described, there were a few significant diffeences which should be noted. The most important of these concerns the weights suspended from the quartz triangle. As shown in Fig. 14, two of these were electrolytic, heat-treated copper, while the third consisted of an evacuated pyrex flask containing PbCl2 which had been purified by careful recrystallization. The copper weights measured about 2.5 cm X 0.6 cm and weighed 6.5 gm each, while the flask of PbCh was 3.3 cm X 1.2 em and weighed 6.6 gm. To prevent charge accumulation, the pyrex flask was coated with silver, along with the quartz triangle. Instead of a quartz torsion fiber, this balance used a 0.001 x 0.0005 in. tungsten ribbon with a torsion constant of 0.07 dyn cm/radian. It was fastened to the top and bottom fiber supports by twisting tightly around a yoke at each end. The vacuum chamber was identical with that used for the gold-aluminum balance, except for the lack of a Vac-Ion pump and the use of an ordinary ionization gauge rather than one of the Bayard-Alpert type. To remove residual gas after baking and sealing off the chamber, a getter tube \vas mounted in parallel with the ionization gauge. However, itsJ1eater was inadvertently burned out, and in practice the ionization gauge itself was used occasionally to pump the system. When removed from service in June 1961, the pressure in the vacuum chamber was measured to be about 10-6 mm Hg. With the exception of the electrode bias battery, and a few resistance and capacitance values in the filter circuitry, the optical lever detection system and associated electronics were identical for the two torsion balances. The small changes in the filter circuit were inconsequential, but the use of a lead-acid automibile storage battery instead of mercury cells is noteworthy. This battery was enclosed in a somewhat less heavily insulated oven than were the mercury ~ 486 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 487 TABLE III SENSITIVITY OF COPPER-PbCh TORSION BALANCE TO TORQUES AND MAGNETIC FIELD CHANGES. Torque sensitivity 4.3 X 10'mV/dynem or 5.6 X 109 mY/stu For this balance, 1 stu (solar torque unit) = Magnetic field sensitivity 2 X 10-3 mV/y or. 5 X 10-12 stu/y 13.1 dyn em. cells. After completion of the measurements with the copper-PbClz torsion balance, it was found that the storage battery was significantly inferior to mercury cells as far as drift and temperature coefficients were concerned. The feedback electrodes were operated with 2 volt bias from one cell of the storage battery. The temperature coefficient of one cell is about 400 !J.V;oC (14), as opposed to 120 !J.V;oC for three mercury cells in series. Because of a higher noise level observed with this older torsion balance, however, battery fluctuations still could not affect the results significantly. This high noise level required that the torsion balance be operated at considerably lower sensitivity than the gold-almninum balance. Torque and magnetic sensitivities were measured for the copper-PbClz balance in the manner described above, and are shown in Table III. In terms of solar torque units (1 stu = 13.1 dyn cm for this torsion balance), the magnetic sensitivity was considerably higher than for the gold-almninum balance. Taking into account 10 the lower sensitivity of the older balance, which was just able to detect 10- stu, the effect of a 10,.. daily period in magnetic field would not quite be detectable. Because of the higher pressure in the vacuum chamber and the lack of symmetry of the torsion balance itself about the telescope axis (the PbC12 flask was --considerably larger in size than the copper weights; see Fig. 14), it was somewhat more sensitive to temperature fluctuations than the gold-almninum balance. It was possible to remove a considerable fraction of the temperature dependence by statistical regression analyses of the thennistor bridge and torque outputs. The methods used to do this will be described in detail in Section V, D. As was mentioned above, the behavior of the pendulum oscillations of this torsion balance was different from that of the gold-almninum balance. The period of the beat frequency between the two normal modes of swinging was about U~ hr ratherthan 16.5 min. Since there was no natural mechanism which significantly damped the pendulum oscillation,2 it was necessary to raise the damping magnet when it became very large. During the winter months when microseismic dis2 i~ rif: ';:: turbances due to weather were large, a feedback mechanism was used to keep the amplitude of this oscillation down. The 0.82 cps signal from the lock-in amplifier was selectively amplified and applied to the feedback electrodes. The phase of this signal. fortuitously was proper for reducing the amplitude of the swinging modes. A timing switch was used to increase the feedback gain and apply this pendulum signal to the electrodes for 40 or 45 min each hour. For the remaining 15 or 20 min, the torsion balance operated at standard sensitivity (Table III), and torque data was taken from these intervals on the chart recorder. Apparently because of the different torsion fiber and lower sensitivity, the copper-PbCl2 torsion balance did not require the use of a drift compensator . .In addition, microseismic noise from construction activity and traffic near the instrument pit was much less in 1960--61 than during 1962-63. These two facts made it possible to obtain a continuous record of torque on the balance for much longer periods during 1960--61 than during 1962-63. Runs of two weeks duration were completed with the copper-PbClz balance, while it was difficult to obtain a continuous run more than 3 or 4 days long with the gold-aluminum apparatus. This particular difference in the data necessitated different methods of analysis, which will be described in the next section. V. RESULTS A. PROCEDURE FOLLOWED IN OBTAINING DATA The ultimate object of the experiment is to determine a value or an upper limit for the parameter TJ(A, B). In modern terms this parameter may be defined as the difference in ratios of passive gravitational mass J1 to interial mass m for materials A and B: TJ(A, B) = [(M/m)A - (M/mhJ/3/z[(M/m)A + (Jf/mh]. (lOa) This definition is fully equivalent to that used by Eotvos (5), \\'ho in effect expressed the results of his experiments in terms of the fractional difference in the "universal" constant of gravitation G for materials A and B: TJ(A, B) = (GA - GB)/Ge, (lOb) where Ge is the universal constant of gravitation averaged over the earth. The relation between "I and the counterclockwise torque L (in solar torque units) on the torsion balance is derived in Appendix B; L (stu) = "I (A, B)[ ( cos \0 cos D) sin t - (sin \0 sin A cos D) cos t + (11 ) (sin \0 cos A sin D) ], See footnote 1. where the radius vector from the torsion fiber axis to the single weight of material A makes a counterclockwise angle \0 with the geographical north di- 488 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 489 rection. The other two weights are made of material B, while D is the declination of the sun, A the latitude of the torsion balance; and t the true local solar time in radians, with noon corresponding to t = O. Analysis of the data, then, amounts to finding the coefficients of sin t and cos t in the measured torque on the torsion balance, and using them to compute values or upper limits for "I from Eq. (11). During data runs five different quantities were recorded continuously OIl chart recorders: (1) the torque L OIl the torsion balance, (2) the output T2 of thermistor bridge No.2, (3) the output T5 of thermistor bridge No.5, (4) the relative amplitude of the pendulum oscillation mode which couples to the optical lever, and (5) the times of on-off cycling of the battery oven. In addition, the voltages of the electrode bias cells and the drift compensator battery, as well as the three thermocouple outputs, were checked frequently enough to ensure their proper behavior. Coptinuous records of outdoor temperature, control shack temperature, and barometric pressure were also kept, but these did not enter into the data analysis. The record of battery oven cycling served merely as a confirmation that this device was operating properly. Similarly, the pendulum oscillation amplitude record was not used in analysis of data, but simply provided a measure of some of the seismic disturbances to which the torsion balance was subject, and an explanation of the 16.5 min modulation of L when it appeared. The records of T2 and T5 did playa significant part in analysis of the torque data, however. As will be described in detail below, the statistical correlation and linear regression between T2, T5, their time derivatives, and L was analyzed with the hope of subtracting any temperature effects from the torque data. A "run" of data is defined as an uninterrupted record of the torque on the torsion balance greater than 36 hr in length. Because of the methods of analysis used, it was not feasible to include data records shorter than 36 hr. A run was terminated when the torque recorder ran off scale, so that a change in the electrode, bias potentiometer to bring it back would produce an equivalent torque r--change wliich could not actually be measured on the chart. Occasionally, the torsion balance would be disturbed enough by earth shocks from construction activity to lead to suspicion that its equilibrium position had been changed. In such case, a run was also terminated. Because of large scale construction activity within a few hundred yards of the instrument pit during the summer and fall of 1962, including construction of a field house and a road, accompanied by rock blasting, it was rarely possible to obtain a run longer than three days. Although the advent of winter reduced this type of activity, the more severe microseisms produced by the weather and the large temperature fluctuations with periods of several days (including cold spells which could not be controlled by the electric blankets) made it almost as difficult to obtain longer runs. Hence, the three runs which were much longer than 3 days were broken up into shorter ~( ~) " runs and all data was analyzed in runs of length between :38 and 86 hr. The results reported here are derived from 39 such runs. To eliminate some of the terms in Eq. (1l), all data were taken with the telescope pointed either south or north (cos ~ = ~ 1, since the gold weight is located directly opposite the telescope). The immediately obvious reason for eliminating the second term in cos t rather than the first in sin t is to avoid the reduction in sensitivity occasioned by the latitude-dependent factor sin A in the former term. Just as important, however, is the fact that many effects which are capable of introducing spurious 24-hr periods in torque are related to the daynight change, and therefore depend on cos t rather than sin t. Such effects as daily periods in temperature or magnetic field will be approximately out of phase with the sin t dependence of the signal from any gravitational anomaly, and their influence on the results of the experiment will accordingly be minimized. Equation (11) in this case reduces to L (stu) = "I 'II (Au, AI) cos 'P cos D sin t. (12) )I} l f I~/ A few runs were taken in one orientation, the balance was rotated through 180, a few runs obtained in the new position, and the process repeated. About half of the runs correspond to each of the two orientations. In this way, any spurious 24-hr periods which do not reverse sign when the apparatus is rotated should cancel out in the average. Between every few runs, the operation of the oscillator driving the oscillating wire was checked; the lock-in amplifier was retuned, and various voltages measured. At no time was there a significant change in any parameter which could have affected the torque output by a large amount. As mentioned in Section IV, E, a continuous check of the torque sensitivity was available in the form of recorder displacements caused by necessary changes in the electrode bias potentiometer (see Eq. (8) and Fig. 10). B. TYPICAL DATA Some typical data charts are shown in Fig. 15. The most striking feature of the torque chart in Fig. 15(a) is the extremely low quality of data during the working day. As previously mentioned, this is the result of nearby construction activity. In the particular case illustrated, there is no evidence that these disturbances affect the equilibrium position of the torsion balance by a significant amount. The relatively quiet section during the lunch hour fits reasonably well on an interpolation between the quiet early morning and late afternoon periods. A peculiarity of these ground disturbances was that they almost always seemed to shift the torsion balance in one direction, corresponding to a counterclockwise external torque looking down on the balance. There could be a variety of reasons for this displacement, but a quite reasonable one is the excitation of the vertical (fiber-stretching) vibration mode of the 490 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 491 FIG. 15. Typical chart records of torque on the torsion balance (in solar torque units, with 1 stu = 58.6 dyn cm), thermistor output (in C), and relative pendulum oscillation amplitude (in arbitrary units). (a) Results from a typical run on a week day. Noise due to construction activity nearby is readily apparent during the working day. Before 0840" and after 1630\ the torque chart record is characteristic of the data which could be obtained at night and on weekends and holidays, in the absence of severe weather disturbances. (b) Chart records displaying the effects of a 5 or 6 ton dynamite charge detonated in a quarry 5 miles distant on an otherwise relatively quiet day. (c) Chart records displaying the effects on the torsion balance of an earthquake centered some 1300 miles distant in Missouri. This earthquake contained especially strong surface waves (15), which were particularly effective in exciting pendulum oscillations of the torsion balance. balance, a mode whose oscillation frequency is quite high, of the order of 6 cps. A coupling between stretch and twist was observed in several of the fibers. If, as may be presumed, this coupling was to some extent nonlinear, a twist would be excited, always in the same direction, by such a vibration. The recovery time from such a deflection (5 min) is compatible with a Q of the order oflO,OOO for this mode of oscillation. The excitation of rocking modes of the balance would induce deflections in either direction, and some of the positive deflections could be due to this cause. To make sure that construction activity really was the cause of the observed disturbances, a set of controlled tests was carried out on a day devoid of other such activity (Independence Day, July 4,1962). A post was set in the ground at various places within a few feet of the instrument pit and .given a number of blows with a heavy sledge hammer. This resulted in displacements of the torque recorder similar to those of Fig. 15(a), but considerably smaller in amplitude, and always in the same direction. Another controlled observation of the effects of construction noise is shown in Fig. 15(b), which represents the effect of 5 or 6 tons of dynamite detonated in a quarry about 5 miles from the instrument pit. Although the shock was severe and in the initial displacement was in the usual direction, the torsion balance recovered in an hour or so with no permanent aftereffects. On weekends and holidays, torque data accumulated during the daytime hours was just slightly noisier than that obtained at night. It was easy in this case to interpolate over the infrequent obvious displacements resulting from . traffic noise, and average over the less obvious noisier portions of the torque output. From Fig. 15(a) and 15(b), it is evident that traffic and construction noise is not particularly effective in exciting pendltlum oscillations. Natural microseismic noise is much more efficient in this respect, since it contains more lower frequency components near the 0.82 cps resonance of the swinging modes. Pendulum oscillations such as those shown in Fig. 13 typically occurred in conjunction with periods of stormy ''.leather and/or high winds. On a few occasions when a pendulum mode appeared during fine weather in the Princeton area, a severe storm was reported off the Atlantic coast of New Jersey. Another natural source of pendulum oscillations is illustrated in Fig. 15( c). During the accumulation of data in 1962 and 1963, such discontinuities in the amplitude of the pendulum oscillation were observed to coincide in four cases with the arrival in the New York area of strong earthquake shocks. The particular case illustrated is by far the most spectacular of the four and is the result of an earthquake centered in Missouri, about 1300 miles from the torsion balance. Most of the energy of this particular earth tremor was concentrate a in surface waves (15) 'It t which, because of their low frequency spectrum, were particularly effective in K exciting swinging modes of the torsion balance. A typical thermistor bridge output is shown in Fig. 15(a) to illustrate the ~ precision with which the temperature in the instrument pit could be monitored. ~t: Depending on the weather and season of year, the balance point of the thermistor " bridges had to be reset every day or so to keep the chart recorder on scale. Be~~ cause this did not affect the sensitivity of the bridge, the discontinuity could be 'i" removed by simply subtracting it from subsequent data. ,~: if C. SIGNAL AND NOISE As in any electronic amplifier, the information-carrying signal from the torsion balance may be easily amplified without limit before delivering it in the form of a voltage to the input of the recorder. The limits on the performance of the apparatus are not imposed by a lack of "sensitivity" but rather by "noise", unpredictable fluctuations that mask the "signal." In discussing noise fluctuations and their minimization, it is convenient to consider separately various aspects of the'problem: 1. The various nonlinear response characteristics of the apparatus will first 492 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 493 be ignored and the system will be considered as a completely linear mechanicalelectrical transducer. 2. The various types of noise disturbances of the system will first be considered to be random G~wssian noise. The design of the apparatus to minimize this noise in the linear system is discussed. 3. Next there is discussed the design of the apparatus to minimize "noise transfer" resulting from the nonlinear characteristics of the system. 4. Finally there is considered the design of the apparatus in such a way as to discriminate against the main type of non-Gaussian noise, the noise pulse. Some aspects of these problems have already been mentioned in other sections. Their repetition here is in the interest of presenting an integrated picture. 1. Gaussian Noise in a Linear Syste/u Gaussian noise may be defined as a stationary time series in the form of an arbitrarily large number of pulses occurring at random times. The Fourier representation of an ensemble of finite but very long intervals of Gaussian noise may be characterized by the statement that the various Fourier coefficients are distributed statistically as Gaussian functions, and that the various frequency components are statistically independent. Because of the statistical independence of the various Fourier components, the best that can be done to discriminate against this type of noise is to (1) design the apparatus to minimize the occurrence of noise in the signal frequency channel, and (2) eliminate bylinear filtering all frequency comopnents outside the signal channel. In connection with the above it should be remarked that a noise disturbance outside the signal frequency channel should not be of any concern, as it is eliminated by filtering. It should also be noted that the filtering need not be performed before recording the information. In fact, in this experiment it was desirable, in connection with the elimination of nonGaussian noise, to do much __ of the filtering.!?Y,:. ~mpgter after recording the data. __ For Gaussian noise, the optimum filter characteristic is one for which the power spectrum of the transfer function is equal to that of the signal divided by that of the noise. Stated more carefully, the transfer characteristic of the optimum filter is such that it responds to a delta function input pulse so as to generate the convolution of the signal function and the transform of the reciprocal square root of the noise power spectrum function. As the signal information is contained in the Fourier component at 24 hr, the optimum filter passes this single Fourier component, with a bandwidth of 0.2 to 0.5 day -1 (depending upon the length of the run). The source of noise to which a linear system responds may be separated into two classes: noise whose origin is external to the apparatus, and internal noise. The external noise is of two types; the effect of a diurnal variation in the gravita- tional gradient, and a diurnal frequency component in the angular acceleration. These effects have been discused elsewhere and are negligible in relation to other effects. To the extent that the effect of temperature on the apparatus is a well-defined linear relation, it is predictable from the past history of the temperature. Since it is predictable, it is not noise. The nature of these possible temperature-dependent effects has been discussed. It is merely noted here that a strong correlation of the output with either temperature or its time derivative did not appear (see Section V, D), and consequently these temperature-dependent effects were of no great importance. As discussed previously, one of the internal sources of noise was the Brownian fluctuation disturbance, primarily due to molecular fluctuations in gas pressure, but in part due to molecular fluctuations in the fiber torque, fluctuations associated with the mechanical hysteresis of the fiber. This was not serious; only a fraction of a second averaging time would have been necessary to determine the equality of inertial and gravitational mass with an accuracy of a part in lOll if this were the sole source of noise (see Section III, 5). Taking the internal noise in sequence, the next contribution to be considered is the fluctuation in angle measure due to a finite number of photons. The error in angle measure is roughly the diffraction limited angle divided by the square root of the number of photoelectrons. As the photoelectron rate was roughly 10' per second, only 0.3 sec of observation were required to reduce the diffraction 9 limited angle measure, 10-5 rad, to the necessary 3 X 10- rad. In addition to the photon fluctuation effect, there is connected with the problem of angle measurement a possible diurnal variation associated with slow drift of the angle sensing equipment. This could have been quite serious if a static comparison of two light intensities had been attempted. For example, a possible technique for detecting small rotations would have entailed splitting the diffraction pattern into two parts and detecting these two separate light beams with different photomultipliers. If this had been attempted, the 24-hr drifts in the sensitivities and gains of these photomultipliers could have been serious. All effects of this type are believed to have been negligible as a result of the driftfree oscillating wire technique described in Section IV, D above. As a result of the large gain of the photomultiplier, the Johnson resistor noise and tube noise in the following amplifiers was negligible. One additional source of fluctuation of some importance was the enhanced signal fluctuation due to the multiplication fluctuation in the photomultiplier. This probably increased the noise output from the photomultiplier by as much as a factor of 1.5 or 2, but this contribution was also negligible inasmuch as the photoelectron shot noise. itself was negligible. It is believed that this represents a reasonably complete list of the types of linear system responses to Gaussian noise. The noise observed was far larger 494 ROLL, KROTKOV, AND DICKE IXERTIAL AND GRAVITATIONAL MASS 495 than any of these effects and is.believed to have been largely frequency-converted noise. These various nonlinear effects were discussed previously and are here merely summarized. 2. N onlineaT Noise Response The vertical fiber stretching vibration of about 6 cps is believed to have introduced an equivalent twist in the fiber, the twist being proportional to the square of the amplitude of oscillation. Another generally less important nonlinear efi'ect was due to the various rocking modes of the torsion balance. The most important of these effects seems to have been due to the pendulum oscillation (or swinging mode) at a freq!l.ency of 0.82 cps, All of these nonlinear effects may be treated along the following lines. Consider the fiber stretching mode and assume that the fiber tries to twist as it is stretched. Let the connection between stretch and twist be All possible contributions to this beat frequency must be summed. It should be noted that the nonlinear response is essential if there is to be such a contribution to the noise. The most important single factor which made these nonlinear responses troublesome appears to have been their high Q. The mean square amplitude of the vibration mode responding to a white noise spectrum is proportional to the Q. As the frequency width of this response is inversely proportional to Q, the intensity of the diurnal torque fluctuation is proportional to Q2, an enormous 8 factor perhaps as great as 10 . Another important factor is the fact that man-made noise is apt to fall well above the diurnal frequency band, above a few cycles per second, and hence may drive these high frequency modes without providing an important direct contribution at the daily frequency. 3. N onGa'Ussian Noise e where .1.' = f(x), is the stretch of the fiber. Expanding f1(J f about the normal stretch 2 :co , =/ Ax + ~2f" ( AX ) + ... , (1::\) with Ae = e - eo and Ax = x - xo. It is evident that the first and most important nonlinear term is the second. Dropping the first term and neglecting higher terms we have f1(J ~ ~~f"( A.X ) 2 (I3a) Inasmuch as the Q of this mode is high (Q "'-' 10,000), only a narrow band of 4 frequencies Av "'-' 6 X 10- sec -1 can be excited in the vicinity of the resonance frequency Vo = 6 cps. Representing the Fourier transform of Ae by Acf> and that of Ax by AX, Eq. (I3a) is equivalent to M'(w) = (1"/411') L: L: AX(a)AX(w .~ a) da. (14) Thus, noise at tIle 24-hr period can then be written as M'(w,,) = (f" /411') AX(a)AX(w" - a) da (I4a) with Wet = (211'/24 X 3600) sec-I = 7.3 X 10-6 sec-I. As this mode contains frequencies only in a narrow band centered at 6 cps, the integral in Eq. (I4a) has nonvanishing contributions only for a falling in this narrow frequency band. Physically, the diurnal noise can be said to result froin a beating of noise frequency components near 6 cps and differing in frequency by the daily frequency. While the best that can be done to eliminate Gaussian noise is to filter with a minimum-noise filter, nonGaussian noise can be more effectively eliminated if something is known about the statistical properties of the noise. The most common example of nonGaussian noise is the noise pulse, an intense disturbance of short duration. Such a pulse can be easily identified, prior to the final narrow filtering, by its characteristic short duration. Ho"iever, after the final filtering this may not be possible. This is best illustrated with an example. A seismic disturbance due to.a blast in a New Jersey quarry might excite one or more of the various vibrational modes to produce a nonlinear noise response of the torsion balance (Fig. I5b). This might die out in an hour, in which case its effect is clearly localized in time and could be eliminated by deleting this section of the record, or alternately by extrapolating through the hiatus produced by this disturbance. This clearly could not be done after the final filtering which introduces an information passband with a frequency width only 0.2 to 0.5 day-I. Hence, it is essential that the instrumental bandwidth be kept reasonably wide until this noise is removed. In this connection, the torsion servo system played an important role, for without artificial damping of the torsion mode, this resonant response to a noise disturbance would require about 70 days to decrease by a factor of 1/ e, assuming a Q of 10". The servo response characteristics were so chosen, and the recorder filter so designed, that the instrumental band pass was reasonably flat from 0 to 2 X 1O~:l cps (-3 db). This made possible a recorder chart speed as slovv as 1.5 in./hr without noise broadening of the trace. Such a band pass was about optimum, in the sense that the bandwidth was sufficient for a very short pulse (duration of only a minute or two) to appear on the recorder. The filter cricuit did the 496 ROLL, KROTKOV, AND DICKE I",ERTIAL AND GRAVITATIONAL MASS 497 . lion's share of the filtering, reducing the band width from 100 cps at the output of the lock-in amplifier to about 2 X 10-3 cps at the recorder. Mter the manual elimination of noise pulses, using a definite set of rules to be discussed in the next section, the final filtering of each run was accomplished on 6 the computer, reducing the band pass to between 3 and 7 X 10- cps. D. ANALYSIS OF DATA FROM THE GOLD-ALUMINUM TORSION BALANCE In the actual analysis of data to determine a value for the parameter TI, three outputs from the experiment were used: the torque L on the torsion balance, and relative temperatures T2 and T5 as measured by their respective thermistor bridges. Each of these was a function of local solar time t, and was subjected to the following procedures to arrive at a value of TI. 1. First, the data L, T2, and T5 were read from the recorder charts as functions of t, in the form of one-hour averages. 2. A low order polynomial was fit to L, T2, and TS by the method of least squares and subtracted to remove long period drifts. All further calculations were performed with the results L*(t), T2*(t), and TS*(t) of this step. 3. Next, the relationships between the two temperatures and the torque L * were investigated statistically and experimentally, with the object of arriving at linear regression coefficients B which could be used to remove temperature effects from L*. 4. The torque L* was fit by least squares to a function of the form (S sin t C cos t K), both with and without subtraction of the regressions on temperature found in the previous step. From the resulting value of S, and the known solar declination D and orientation cos 'P = ::I:: 1 of the balance, a value of TJ could be computed from Eq. (12) for each run. S. Finally, various schemes of weighting the results of individual runs were investigated to arrive at a final mean value of TI for all runs together. of a small division on the chart. As has already been mentioned, discontinuities in the chart record, caused by adjustment of the electrode bias potentiometer or the thermistor bridge balance, were measured on the chart and subtracted out. This procedure was quite straightforward for the temperature charts. With the torque charts, however, special treatment was necessary for those portions which were disturbed by microseismic noise. In the case of serious disturbances which lasted for more than an hour, the noisy portion of data was simply omitted from the analysis. Examplse of such cases are the hours between 0900h and 160011 in Fig. 15a. Obvious nonGaussian noise pulses shorter than one hour in length, such as those at 1900\ 2000\ and 2040h in Fig. 15a and at noon in Fig. 15b, were interpolated over and not averaged into the data. I, 2. Removal of Low-Order Polynomial Because noise in the torque data may have a 24-hr period, and some data during the working day may have been discarded altogether, subtraction of a straightforward least squares fit of a polynomial to all the data could conceivably bias any true 24-hr period. Hence, only torque data obtained during the quietest 12 hI' or so of each run was used to compute the least squares polYlloInialfit. In each run, torque values corresponding to the consistently quietest 8 to 14 hI' period during the run were averaged, resulting in a set of average values spaced in time by 24 hr. For example, consider a 64-hr run beginning at 180011, with the data between 1900h and 0700h inclusive on each day relatively free of seismic disturbances. The data between these hours would then be averaged to give a single point corresponding to a time coordinate of 010011 for each of the three days. The daily points so obtained were then fit to a first, second, or third order polynoInial pet) and this polynomial subtracted from the original torque data: L*(t) + + "'ft Iii:' = L(t) - pet). At the conclusion of-the first step of this process, values of L(t), T2(t), and T5(t) were punched on IBM cards, and all further numerical work performed with the 7090 and 1620 data processing systems of the Princeton University Computer Center. Since none of the five steps mentioned above is trivial, each of them will be described in detail. 1. Extraction of Data fr01n Recorder Charts ~i'; ,}~ The general method used for reading numerical values of both torque and temperature from the charts was to average the recorder traces over one-hour intervals and assign the average value to the center of that interval. With the aid of fine lines ruled on a small piece of lucite, this averaging could be done repeatedly and consistently by two or three persons to an accuracy of a few tenths Since the period of the averaging procedure is 24 hI', such a method cannot bias any real daily period present in the original data. The order of polynomial subtracted from L was determined by whether there were two, three or four points to fit, being one less than the number of points to fit. In the case of the temperature T2 and TS, the same polynoInial subtraction procedurewas followed for the sake of consistency, even though the nonGaussian noise problems were not present. Figure 16 shows typical results; the torque and thermistor bridge outputs of a 63-hr run after subtracting polynomials. 3. Temperature Regressions The, object of analyzing the relation between temperature T2* and T5* and the torque output L* was to deterInine whether there existed a significant relationship, and if so, to establish the magnitude of a linear regression coefficient INERTIAL AND GRAVITATIONAL MASS 499 498 6 ROLL, KROTKOV, AND DICKE tions of the form (S sin t L**(t) + C cos t + K): T*(t) (ST = L*(t) - (SL sin t = > E f- T**(t) 4 + C cos t + Kd sin t + CT cos t + KT) L :::l ll. 2 The regression coefficients were then calculated using L ** and T**. For theith run, Bi(T) = [(Li*Ti*)av -(Li*)av(Ti*)avJl[(Ti*21av - (Ti*l;v] (16) Histograms showing the distribution 15, f- o lJJ :::l of numerical values obtained from these :::l o a:: f- o (0) t ~ ~ ~ a:: '" '0 lJJ CD 'or 5 O. T2 T5 o~ " ~ Z ::l C\I d a:: (f) z -2 -I a 1 2 -2 -I 0 COE FFICIENTS 8 ~ f- -5 TEMPERATURE a:: -10 ~ 0000 1200 MAR.24,1963 0000 REGRESSION B(mV/I0-3OCl 1200 MAR.25,1963 0000 i200 MAR.26,1963 15r-- (b) TIME (EST) FIG. 16. Results after subtracting least squares polynomial fits from the torque (L*) and temperature (T2*) data obtained in a typical run. A second order (quadratic) polynomial was subtracted in this case, since the 63-hr run provided three 24-hr-averaged points to be fit. a: w CD '+' 5 0 -I T2 ~ ::l Z . B which could be used to subtract torque: L+(t) = out the temperature-dependent L*(t) B.T*(t). part of the (15) a I -I 0 I TEMPERATURE CORRELATION R COEFFICIENTS It was felt that such a regression coefficient, if it was to have any meaning, should be the same for all data runs. The statistical method used to determine B(T2) and B(T5) involved calculating a regression coefficient for each run. In order to eliminate the possibility of subtracting out a real 24-hr period resulting from a nonzero value of TI, the 24-hr components of both L * and T* were removed before computing the linear regression coefficient. This was accomplished by subtracting least squares fits to func- FIG. 17. Distributions of (a) temperature regression and (b) temperature cnrrelation coefficients for the 39 data runs. In (a), the arrows indicate mean values of the regression coefficients of the torsion balance output with thermistor no. 2 and thermistor no. 5 outputs, respectively. The horizontal bars across the arrows represent the widths (standard deviations) of the two distributions. Note that 1 m\'= 10-10 stu. Although the magnitudes of the regression coefficients shown in (a) are quite small, the distributions of correlation coefficients in (b) indicate that there is, in general, a statistically significant positive correlation between torque on the torsion balance and temperatures T2 and T5. 500 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 501 calculations are given in Fig. 17(a), and are approximately Gaussian in shape. The fact that the widths of these distributions are comparable to the mean values, with even a significant number of regression coefficients having.signs opposite to that of the mean, emphasizes the small size of the temperature effects. In addition to regression coefficients, correlation coefficients R(T2) and R(T5) were computed to provide an estimate of the statistical significance of the temperature regression coefficients. (-1 ~ R ~ 1, with values ofl R I > 0.28 representing correlations which are statistically significant at the 95 % level for a sample size of 50 (the number of data points in a typical run).) The histograms of R(T2) and R(T5), shown in Fig. 17(b), indicate that in general, the regression 'coefficients, though small, are statistically significant. TABLE IV COMPARISON OF TEMPERATURE RUNS REGRESSION COEFFICIENTS FROM TEMPERATURE CYCLING WITH MEAN COEFFICIENTS FROM THE 39 NORMAL DATA RUNS B(T2) (stu/10-3 ec)" B(TS) (stu/10-3 eC)" + Temperature.cycling run no. 1 Temperature cycling run no. 2 Average for temperature cycling runs Mean from 39 normal data runs a 7.1 5.6 6.3 4.5 X X X X 10-11 10-11 10-11 10-11 6.5 4.3 5.4 5.1 X X X X 10-11 10-11 10-11 10-11 1 stu (solar torque unit) = 58.6 dyne cm for the gold.aluminum torsion balance. :> E 6 I- 15 . " ::;) Ilfl el- ::;) 4 2 \0 <:> 5 I- ::;) 0 a -2 -4 w ::;) a -5 - w ::;) a: 0 l- 0 a: 0 0 I- ",'0 p N a -10 ci z r- a:: Ilfl o -20 ::!: a:: w ::c I- 0000 1200 0000 1200 NOV. 23, 1963 0000 NOV.22,1963 TIME (EST) FIG. 18. Torsion balance torque (L*) and thermistor no. 2 (T2*) outputs from one of the temperature cycling runs. A first order polynomial fit has been subtracted from each. From these two curves, a significant correlation between the two quantities is apparent. The size of the temperature fluctuations required to make this correlation stand out should be compared with the size of temperature fluctuations encountered in a normal data run, such as shown in Fig. 16. As an experimental check on the temperature dependence of torque, two runs were conducted in which the temperature was artificially cycled, with a period of about 16 hr and much larger amplitudes than would occur naturally during a normal run. This cycling was carried out by increasing or decreasing every 8hr the voltage applied to the electric blankets just above the bottom of the insulation plug in the instrument pit. The effects of these temperature variations were such as to require reductions in torque sensitivity by factors of 5 and 2, respectively, for the two temperature cycling runs. In Fig. 18 are displayed the outputs of the torque and thermistor bridge NO.2 recorders, after processing through steps 1 and 2 in the usual way. It is evident from these graphs that there is indeed a real dependence of torque on temperature. Regression coefficients were calculated from Eq. (16) for each of the temperature cycling runs, and are compared in Table IV with the mean regression coefficients from the normal data runs. The average temperature regression coefficient from the. temperature cycling runs is close to the mean value from the normal data runs. Hence, the average values from the temperature cycling runs were adopted as the standard temperature regression coefficients to be used in subsequent steps of the data analysis. In all cases, the regression coefficients displayed in Fig. 17(a) and Table IV are quite small. Since typical daily peroids in temperature were less than 1 millideg C, the maximum effect of temperature on the value of TJ obtained would be less than five parts in 1011. The relation between torque and derivatives of the two temperatures was investigated for the first 8 data runs. It was found that this relation was not statistically significant, and so the effort was dropped. Reference to the temperature cycling runs (Fig. 18) indicates similarly that there is no appreciable phase shift between torque and temperature, as would be required if the time derivative of temperature were to be significant. Stimulus for this particular 502 ROLL, KROTKOV, AND DICKE investigation was provided by the fact that earlier data from the copper-PbCh torsion balance had shown a strong correlation between torque and time derivative of temperature. The correlation between torque and rate of change of temperature in this older apparatus may have been due to two significant factors. First, the gas pressure was greater by two orders of magnitude, and consequently the apparatus was much more sensitive to temperature gradients across the vacuum chamber. Second, the location of the PbCb weight in this older apparatus was such as to destroy the symmetry plane cutting the mirror. In fact, this weight, on the right looking into the chamber, had a substantially larger cross-sectional area. To permit the entrance of the light beam, it was necessary to provide an opening in the multiple thermal insulating jackets (four radiation shields separated by insulation in this older apparatus). This opening may have been the chief port for the transport of heat, resulting in thermal gradients across the vacuum chamber along the telescope axis. While this model has elements of conjecture, the resulting calculated torques have the right sign and roughly the correct magnitude to be explained in this way. In the more sensitive apparatus, the weak dependence upon temperature (not upon rate of change of temperature) could be due to any of a variety of causes; temperature-dependent twist of the fiber, temperature-induced defiections of the detection apparatus, and temperature dependence of the contact potential differences due to temperature dependence of the amount of gas absorbed on the walls. 4. E:d-raction of a Daily Period j1'om the Torque The values of L*(t) (with no temperature regression subtracted) and L+(t) = L *(t) - B. T*(t) were fit to a function of the form (S sin t C cos t K), and the constants S, C, and K evaluated by the method of least squares. From r-Eq.(12), tl1errlagiiitlideof the coefficients S, if they result from a real difference in passive gravitational to inertial mass ratio, will be essentially the magnitude of 'YJ(cos D ~ 1). More precisely, for sin <p = 0, I J ': " I~ ' , ,- I~ERTIAL A~D GRAYITATIONAL MASS .503 ~. ;, 1 TABLE V VALUES 24-HRSINE ANDCOSINEAMPLITl'DESAND C, AS WELLAS GRAVITATION OF S AIr INERTIAL MASSRATIODIFFERENCE '1(Au, AI), COMPUTED FOR I~ACH THE OF 39 DATARUNS Initial d.te Duration (hr) B(T) cos ~~Run {.i ~. = 0 I ~~- B(T2) = 6.3 X 10-' sturC I B(T5) = .1.4 X 10-' sturc No. cp S (mV) --'-- I C (mV) +0.17 +1.11 -1.04 +0.97 +1.17 :-0.19 +0.93 -0.58 +1.60 -0.53 +0.22 -0.49 -0.35 -0.73 -1.01 +0.92 +0.70 -0.25 +0.71 +0.59 -1.70 -2.03 -1.08 +0.9fi +0.88 +0.73 -2.2fi -0.83 -0.54 +1.05 -0.23 -0.22 -3.81 -1.59 -0.47 -1.07 -1.23 -0.10 +0.88 '1(XIO-Il)S (mV) C (mY) '1(XIO-Il)S (mV) C (mV) '1 (XIO-ll) -~ --~ ~- ~,~- -- --0.11 +0.85 -'0.97 +0.80 +0.77 -0.52 +1.'1.1 -- ~ 1 ~{; 2 { 3 ): 4 ~. 5 ~;. 6 i: ~ 7 ( 8 9 ~;.'10 '>:.. o' ,J " .J + + 11 12 13 ... 14 '; 15 ,f 16 '~ 17 ,~. 18 l 19 20 21 ;ii 22 23 ~f 24 ,',;. 5 2 ~l * > .' 26 ."\ 27 28 29 30 .31 ...32 33 34 35 36 'l/i(Au, AI) = (Slcos <p cos D),: (17) for the ith run. The cosine coefficient C, on the other hand, should represent noise. Values of Si , Ci, and 'l/i, obtained by fitting L* without and with subtraction of temperature regressions, are given for each data run in Table V. Histograms of the distributions of 'YJiand Ci are shown in Fig. 19; for the cases of no temperature regression subtracted (Fig. 19(a)), regression with T2 subtracted (Fig. 19(b), and regression with T5 subtracted (Fig. 19(c. 7/7/62 7/9/62 7/13/62 7/20/62 7/28/G2 8/3/62 8/5/62 8/7/62 8/10/62 8/25/62 9/2/62 9/8/62 9/11/62 9/14/62 9/18/62 9/25/62 9/28/62 10/1/62 10/6/62 10/10/62 11/11/62 12/11/62 12/19/62 12/27/62 1/5/63 1/7/63 1/10/63 1/13/63 1/29/63 2/28/63 3/11/63 3/17/63 3/19/63 3/22/63 3/24/63 3/29/63 4/1/63 4/5/63 4/8/63 42 38 40 66 54 40 47 65 55 65 55 72 64 69 67 51 64 47 75 52 55 68 44 59 48 54 54 45 47 86 72 51 48 48 63 84 70 63 77 -1 -1 -1 +1 +1 +1 +1 +1 +1 -1 -1 -I ~1 -1 -I +1 +1 +1 +1 +1 +1 +1 +1 +1 -1 -I. -1 -1 -1 -1 -1 -1 -I -1 -I +1 +1 +1 +1 +0.15 +0.71 +0.52 +0.03 +0.98 -0.82 -0.20 +0.84 +1.30 +1.94 -0.48 +0.53 +0.46 +0.44 +0.51 +0.fJ8 -0.15 +0.59 -1.lG -0.4G +3.09 +0.69 +0.39 +1.11 +0.29 -0.20 +0.21 +0.24 -1.47 +0.8fi +0.78 -0.52 -1.34 -0.25 +0.48 -1.33 -0.57 -0.46 -0.50 +1.70 +8.20 +5.98 -0.31 -11.20 +8.57 +2.32 -9.50 -14.72 +23.75 -5.32 +5.84 +5.12 +L95 +5.G7 -7.65 +1.70 -6.59 +13.08 +5.27 -37.40 -8.74 -5.02 -1.'1.38 +3.70 -2.53 +2.70 +3.11 -18.69 +10.79 +9.79 -6.55 -16.78 -3.10 +6.04 +16.88 +7.27 +5.93 +6.43 -0.02 +0.55 +0.18 +0.22 +1.06 -0.47 +0.45 +1.23 -0.14 +1.97 -0.54 +0.20 +O.Il -0.42. -0.21 +0.11 -1.35 -0.56 -2.63 -1.44 +0.77 +4.06 +0.88 +0.98 +0.16 -0.13 +0.20 +0.01 +0.33 +0.G1 +0.72 -0.09 -1.75 +0.43 -0.10 -0.67 -1.79 -0.25 -0.39 -0.10 +0.51 -1.07 +0.84 +0.84 +0.08 +2.20 -0.73 +0.98 -0.74 +0.47 -0.64 +0.00 -0.46 -1.82 +1.53 +1.19 +0.29 +1.62 +0.40 -0.40 -0.02 -0.99 +0.G5 +0.11 +0.58 -1.43 -0.66 +0.34 +0.47 -0.fi9 -0.G4 +0.57 -0.11 -0.G8 +0.01 -0.45 -0.10 +0.34 -0.20 +6.36 +2.07 -2.5G -12.12 +5.38 -5.l(j -13.93 +152 +24.10 -6.12 +2.22 +1.25 -4.74 -2.34 -1.19 +15.11 +6.34 +29.70 + 1(j.4fj -9.35 -51.70 -10.30 -12.fifi +2.07 -1.65 +2.fi4 +0.14 +4.1fi +7.64 +9.02 -1.W -22.00 +5.40 -1.21 +8.fifi +22.75 +3.lfi +5.05 -0.05 +0.71 +0.29 -0.08 +0.62 -1.04 -0.13 +0.81 +0.38 +1.90 -0.30 +0.20 +0.27 +0.1;~ +O.Hi +0.92 -0.77 +0.47 -1.51 -0.(;8 +0.83 -0.38 +1.23 +1.19 +0.14 -0.32 +0.18 -0.02 -0.45 +0.59 +0.fi7 -0.32 -1. 71 +0.38 -0.18 -1.02 -1.83 -0.50 -0.59 -0.74 +1.48 -0.87 +0.27 -0.71 -0.15 -0.(;8 -1.45 +1.41 +1.0(i -0.02 +1.24 +0.40 -o.no -0.97 -0.fi8 +0.(i9 +0.24 +0.5fi -1.72 -0.53 +0.01 +0.41 -0.71 -0.G9 -0.3fi -0.40 +0.77 -0.23 -0.64 -O.OG +0.44 -0.62 +8.21 +3.42 +0.95 -7.10 +11.78 +1.47 -9.14 -4.17 +23.25 -;3.29 +2.20 +2.9f; +1.4(; +1.7fJ -10.:3:3 +858 -5.30 +17.22 +7.7fj -1O.0G +4.90 -15.74 -1.5.38 +1.77 -4.09 +2.2fi -0.32 -5.fi!) +7.44 +8.34 -4.07 -21.50 +4.81 -2.27 +12.88 +23.15 +6.32 +7.55 504 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 505 TABLE 12 10 TELESCOPE POINTING; (</> =0) (</> ~ VI ERRORS OF THE MEAN" o SOUTH NORTH (0) Pojnts MEAN VALVES o~' '7(Au, AI), WITH PROBABLE 8 = 180) more than 3 std. dey. from mean excluded -II Telescope orientation No temperature regression subtracted (B(T2) = OJ (B(T2) T2 regression subtracted = 0.3 X 10-' sture) T5 regression subtracted (B(TS) ~ 5.4 X 10-' sturC) . ~':!i (D 6 4 -2 0 2 4 -4 -2 0 C(rnV) 2 -40 -20 0 7] (Au, At) 20 40.10 5(rnV) B ( T 2 ) = 6.3 x 10- 8 stu / c n 0 C(rnV) +. 0 No No No Yes Yes Yes (b) :::E ::> Z 2 o ...----.--- I I I -4 . . -2 . 2 -40 -20 . North South North + South North South North + South (+2.22:f: (-2.53:f: (-0.09:f: (+2.22:f: (-0.59:f: (+0.89:f: 1.42) X 10-11 (1.38 :f: 1.27) X 10-11 (1.30 :f: 1.25) X 10-1' 1.94) X 10-11 (-0.25 :f: 2.70) X 10-]] (1.33 :f: 1.73) X 10-11 1.20) X 10-11 (0.59 :f: 1.45) X 10-]] (1.32:f: 1.04) X 10-11 1.42) X 10-11 1.51) X ID-II 1.03) X ID-II (1.38 :f: 1.27) X ID-II No values more t.han (2.44 :f: 2.00) X 10-]] 3 std. dev. from (1.94 :f: 1.15) X ID-]] mean 20 40.10 -II 7] (Au. AI) I:t [j 4 2 -8 B (T 5) = 5.4 x 10 stu/oc II . I . (e) a The means and their probable errors for the 20 runs 'with the optical lever telescope oriented north and the 19 runs with it oriented south are presented separately in rows 1, 2, 4, and 5, while rows 3 and 6 contain the corresponding results for all 39 runs averaged together. Results are given with no temperature effects removed (column 3), after sub. tracting the regression with thermistor no. 2 output (column 4), or after subtracting the regression with thermistor no. 5 output (column 5). The two runs with values of '7 more than three standard deviations from the mean have been excluded in rows 4, 5, and li, but are included in the first three rows. O. -2 0 S (rnV) 2 4 -4 -2 0 C{rnV) 2 -40 -20 0 7](Au,AI) 20 40.10-11 FIG. 19. Distributions of amplitudes of sine and cosine coefficients Sand C in the least squares fit of (S sin t + C cos t + K) to the torsion balance torque output. t is the local solar time in radians (local solar noon = 0, 2"., "'). The third column shows histograms of values of '7(Au, AI) = (Slcos D cos ",) for the 39 data runs, with the arrows and horizontal cross bars indicating mean values and widths (standard deviations) of the distributions, ;- - respectively. Orientation of the optical lever telescope is indicated by the shading of the bars in the histograms. (a) Distributions obtained without subtraction of a temperature regression from the torque. (b) Distributions obtained after subtracting a regression with thermistor no. 2 (T2) from the torque. (c) Distributions obtained after subtracting are. gression with thermistor no. 5 (T5) from the torque. Numerical values displayed on the above histograms are given in Table V.. 5. Combining the Results of Individual Runs In order to obtain a single estimate of '1 (Au, AI) from all of the data runs, it is necessary to combine the 39 values of '1 in some way. The simplest method is to form an arithmetic mean and assign to it a precision equal to the probable error of the mean of the 39 observations. This has been done separately for the 20 north -oriented (cos tp = -1) and 19 south -oriented (cos tp = 1) runs, as well as for the 39 runs together. Furthermore, such calculations have been made for all three of the cases considered in Table V and Fig. 19: B(T) = 0, 8 B(T2) = 6.3 X 10- stu;oC, and B( T5) = .5.4 X 10-8 stu/oC. These results are displayed in the first three rows of Table VI. In two of these three rows, it was found that a single observation differed by more than three standard deviations of a single observation from the mean (Run No. 21 with B(T) = 0 and Run No. 22 with B(T2) = 6.3 X 1O~8 stu;oC). Hence, the second three rows of Table VI present the same results with these suspicious measurements excluded. Since certain runs contain considerably more noise in the torque output than others, it was felt that perhaps the 39 individual values of '1 should be weighted in some way before calculating a mean. Two possible weighting schemes sug. gested themselves. (a) The noisier the torque data, the larger should be its fluctuations about a least squares fit to the function (8 sin t + C cos t + K). If rr/ is the mean square difference between L * (t) (or L + (t and this function for the ith run, then an appropriate weighting function Wi for the ith value of 'YJ might be Wi = l/rr/ noise in a run is the total duration + (b) Another measure of the (nonGaussian) 506 ROLL, KROTKOV, ND DICKE A 1.00, 0 .... . , INERTIALAND GRAVITATIONAL MASS 507 II 00 .0 o o ~o 00 0.90 (1_ ~t) 0.80 01 o 0 !' ~ [I' (0) The validity of both of these weighting schemes can be checked crudely by considering scatter diagrams of (T, and (1 - !!.t,/T,) as functions of the rms amplitude S/ C/ of the 24-hr component of torque. Such diagrams are shown in Fig. 20, and suggest that neither weighting scheme is of much value. Indeed, when weighted means and probable errors are calculated in this manner, they seem to be larger in magnitude than the unw~ighted results. + 0.75 E. ANALYSIS DATA FROMTHE COPPER-PbCb TORSIONBALANCE OF The significant difference in the data obtained from the gold-alurninumand copper-PbCb torsion balances was the lengths of data runs. With the former, the 39 runs ranged in length from 38 to 86 hr, while the older copper-PbCl2 balance yielded 5 runs between 180 and 425 hr long. The small number of runs from the older balance meant that distributions of results such as 1/, , B, , etc. would not be significant. Because the longer runs were not seriously disturbed by nearby construction activity during the day, it was not necessary to omit any data points from the analysis, or worry about the effects of daytime nonGaussian noise on a daily period in the torque. Hence, analysis to extract the 24-hr period could be done by means of numerically-evaluated Fourier integrals, yielding a frequency spectrum of torque amplitudes. The Fourier components corresponding to frequencies adj.acent to (:/z4 hr) provided a measure of the noise level of the data, and any positive effect (1/ ~ 0) would cause a peak to stand above this noise at a frequency of (:/z4 hr). In general, data analysis followed the same procedure as that used for the gold-aluminum torsion balance. Step 1, the extraction of data from the charts, was identical in both cases. Since there were no serious daytime disturbances in the torque, the low order polynomial fit of Step 2 was made to all of the data rather than to 24-hr averaged points. The length of the runs made it feasible to remove polynomial drifts of up to fourth order in this step. In computing temperature regression coefficients for the individual runs with the copper-PbCl2 torsion balance, the 24-hr periods in torque and temperature were first removed by calculating and subtracting the 24-hr Fourier components of these signals, rather than by the least squares fitting technique employed for the newer data. Regression coefficients were calculated for thermistor bridge outputs T2 and T5, and their time derivatives T2 and T5. Since the runs were quite long and instrument sensitivities changed between runs (see Table VII), the regression coefficients computed for a given run were used to subtract the temperature effects from that run, and no effort was made to establish a single set of regression coefficients for all 5 runs. The significance of the relation between torque and temperature, as well as the results of removing temperature regressions, are illustrated by the Fourier spectra of Fig. 21. Prominent components of torque, temperature, and/or temperature derivative occur with periods of 12 and' a 4 3 cr (mV) 2l 0 , .. , ...., .. 0 0 0 0 0 0 0 , .1.: ~-!I' "f .~ o 0 .~\ (b) p.i- J a FIG. ! I 2 ../S2+CZ OF 24-HR. I ! j.; 4 3 RMS AMPLITUDE COMPONENT 20. Scatter diagrams of the rms amplitude of the 24-hr period in torsion balance _output, versus two quantities which may be related to the statistical weights of the results from individual data runs. These two quantities are: (b) <T, the rms difference between L * (t) and the function (8 sin t C cos t K), and (a) (1 - t:.t/T), where T is the duration of the run and t:.tis the total duration of all non-Gaussian noise pulses longer than 15 min. The more or less random scatter of points in both diagrams suggests that the nns amplitude of the 24-hr period in torque is not related to either of the quantities u or (1 - t:.t/T) , and therefore that neither of them are useful weighting functions for combining results of the 39 data rims. + + !!.t, of all nonGaussian noise pulses longer than 15 min or so. If T, is the length of the ith run, than a weighting function of the form w, = (1 - !!.t;jT,r might be suitable, with the exponent n a small positive number (say 1 ~ n ~ 10). 508 ROLL, KROTKOV, AND DICKE PERIOD 19296 48 24 16 INERTIAL AND GRAVITATIONAL MASS 509 (HR) 12 9.6 8 6.87 'Ol\ A.'~RS;~~i~~~~~E. 1 (oJ (f) w a ::l f-.J \. ., / '-.,.'-'. THERMISTOR (T2-) ,..' ' ... .....-.' NO.2 -I a.. :2 <[ (b) a:: w a:: ::l d ( .-.-. -,- .~... ...., 27.5 hr. After subtracting out the regression on temperature derivative, both of these prominent components of torque were reduced essentially to the noise level. However, this was not true when the regression on temperature itself was subtracted. The end result of the numerical analysis of the torque data from a run with the copper-PbCh torsion balance was a Fourier spectrum with the temperature derivative regression removed, such as Fig. 2l( e). From these spectra, a mean 2 square noise amplitude A at a frequency of 0-'2'4 hr) was extracted, which repre. sents a lower limit to the size of the positive effect which could be detected. From Eq. (11) the root mean square amplitude of the 24.hr period produced by a non. zero value of 1J would be A = 11J (Cu, PbCl2 + pyrex I cos DYI - cos2 A sin2 cpo (18) o I.J... w a:: <[ ::l a (f) 0 __ I i\ . .' \ I'. dt T2 ) =12* (e) 8 Z <[ w :2 f- '06 - o !X o I .. \../\ ! ! .~\! . .... . \/ . /\ eO' , , ... I . \_............ . ' :--_ In none of the five data runs did the rms amplitude of the 24.hi: Fourier compo. nent appear to be larger than the general rms noise at frequencies near (~i 4 hr) . Figure 21 (e) is typical of the spectra obtained. Hence, the value of 1Ji deduced for the ith run can be interpreted as 1Ji I! .J T2 L*.-B{T2)' J = O::f::Hi, ,/ .',,'" /.... 1\'. , ~ ..- ... 0.14 (d) where Hi = Ai/[cos DiYl - cos2 Asin2 cpl. (19) '~"""-""'" 0' ;\........\ ... . L."-B (1'2).+2'" /. 1 '...... . _., /\l..... ! (e) Ai is again the measured rms amplitude of the 24.hr Fourier component of torque, measured in stu (solar torque units). The mean value of 1J for the five runs is then the weighted mean of the five individual values, with 1/ H/ as the weighting factor: fi I-I [" o 0.02 0.04 0.06 0.08 0.10 0.12 1 = O::f:: [ ; 5 (1/H/) J-1/2 (20) FREQUENCY (HR ) FIG. 21. Rms Fourier amplitude spectra from the 300-hr data run beginning March 27, 1961, using the copper-PbC!, torsion balance. (a) Spectrum of torsion balance torque output L*. (The asterisk signifies that a fourth,order least squares polynomial fit has been subtracted from the data.) (b) Spectrum of the output T2* of thermistor bridge no. 2. (c) Spectrum of the time derivative 1'2* of the output of thermistor bridge no. 2. The time derivative was calculated by simple differences. (d) Spectrum of torsion balance torque after subtracting the regression with T2*. (e) Spectrum of the torsion balance torque after subtracting the regression with 1'2*. The regression coefficients used in (d) and (e) were calculated from Eq. (16), using data from this 300.hr run only. Note the effectiveness of the regression with 1'2* in removing the prominent spectral components with periods of 12hr and 27.5hr. The regression with T2*, on the other hand, leaves these components unaffected, since the coefficient B(T2) is almost negligible. Table VII displays the individual values of Hi obtained from each run, as well as some of the other significant parameters associated with the individual runs. The resulting mean value for fi is, according to Eq. (20), fi (Cu, PbCb + pyrex) = (0 ::f:: 1.6) X 10-10 Using the mass ratios of PbCb and pyrex given in the caption to Fig. 14 and the known composition of pyrex (80 % 8i02 , 14 % B20a , 4 % Na20, plus traces of other oxides), the above result can be further decomposed into limits on the passive gravitational to inertial mass ratio differences between copper and the elements lead, oxygen, silicon, and chlorine. (The validity of this decomposition requires the additional assumption that the anomaly results solely from one of 510 ROLL, KROTKOV, AND DICKE TABLE RESULTS OF ANALYSIS OF DATA VII TORSION A INERTIAL AND GRAVITATIONAL MASS 511 by a temperature FROM THE COPPER-PbCI, BALANCE H (lO-lO stu) from the mean), an error which was apparently fluctuation. We therefore consider the result 'I induced Date (beginning of run) Length of run (hr) Torque sensitivity (mV/dyn em) (mV/stu)a B(T2) <sture b,-I) (10-10 stu) (Au, AI) = (1.3 ::I:: 1.0) X 10-11, Aug. Sept. Dec. Jan. Mar. .a 18, 1960 1, 1960 28, 1960 16, 1961 27, 1961 116 274 145 420 299 2.0 8.6 4.3 4.3 4.3 X X X X X 10' 10' 108 108 108 2.6 1.1 5.6 5.6 5.6 X X X X X 108 10' 10' 10' 10' 1.1 0.43 0.98 0.34 0.76 = X X X X X 10-' 10-' 10-. 10-6 10-6 26.7 19.4 4.00 1.52 2.82 33.6 23.7 6.7 2.5 2.8 For this torsion balance, 1 solar torque unit (stu) TABLE UPPER LIMITS ON '7(Cu, VIII 13.1 dyn em. X), FROM THECOPPER-PbC12 TORSION BALANCE X 1'7(Cu, X)I Lead Oxygen Silicon Chlorine ;:;; (0::1::4.0) X 10-10 ;:;; (0::1:: 4.8) X 10-10 ;:;; (0 ::I:: 7.1) X 10-10 ;:;; (0::1::11.6) X 10-10 the four elements. Such an assumption would seem to be most legitimate for lead, since oxygen, silicon, and chlorine are all of relatively low atomic number, for which factors such as the nuclear electrostatic energy and the neutron excess are smalL) Table VIII lists the upper limits obtained in this manner for 'I (Cu, X), where X represents one of the four above-mentioned elements. VI. CONCLUSIONS .~ , '~i obtained after removal of the T5 temperature regression, to be the most significant of those shown in Table VI. Secondly, in view of the probable error associated with this mean value of "f/, it is not significantly different from zero. We tHerefore do not consider the results of the experiment to provide any evidence whatsoever for a lack of equivalence between passive gravitational and inertial mass. Finally, it may be noted that, . regardless of whether temperature regressions have been removed, or whether the two points more than three standard deviations from the mean have been excluded, none of our estimates of the probable error in the mean of the 39 values of 'I exceeds ::f: 1.45' X ~0-11. . It seems reasonable to assume that, if a large number of similar ensembles of 39 measurements of "f/ were carried out, the resulting mean values would be distributed normally about the true value of 'I, with half of these mean values lying within about ::1::1X 1011of the true value. Under such an assumption, we can state, with 95 % confidence on the basis of our measurements, that the true magnitude of'J (Au, AI) is less than about three times the probable error of our mean. Specifically, 1'1 (Au, AI) 1< 3 X 10-11 (95% confidence limit). There are several limitations to the theoretical significance of this result which should be mentioned. First of all, the active gravitational mass or source of gravitational field in the experiment was the sun, consisting largely of the light elements hydrogen and helium. A similar experiment using materials with high atomic number to produce the gravitational field could conceivably lead to different results. However, the original Eotvos experiment used the acceleration toward the earth, which as the active gravitational mass contains a much larger fraction of heavy elements. Hence, it sets stringent limits to a possible anomaly of this type. Second, the laboratory in which the experiment was performed is known to be moving rather slowly relative to the coordinate system in which the universe appears isotropic. Had it been moving with a velocity close to that of light relative to this coordinate system, the results conceivably could also have been different. While certain types of relativistic theories predict effects of this type, such theories are unpalatable and fortunately implausible on other experimental grounds (16,17). Third, because the contribution of the weak interaction to nuclear binding is so small, the present experiment does not serve as a test of the equivalence of inertial to passive gravitational mass for weak interaction energy. Finally, the gravitational self-energies of the torsion balance Is i~ ~ The most precise results, from the gold-aluminum torsion balance, have been summarized in Table VI above. In evaluating the significance of these results, three observations may be made concerning the numbers presented in that table. First, the removal of temperature regressions seems to eliminate the small difference in the values of 'I obtained with the torsion balance oriented in the north and south directions (compare, for instance, the first two entries in columns 3,4, and 5 of Table VI). This suggests a small effect, perhaps due to a temperature gradient across the vacuum chamber, which rotates with the torsion balance. The effect; however, is too small to affect the results substantially whim the north and south values are averaged together (rows 3 and 6 of Table VI). The removal of the T5 temperature correlation reduces the probable error slightly and eliminates one point with a large error (more than three standard deviations :~ ~, ;.,~~; r(~ 512 ROLL, KROTKOV, AND DICKE r~ INERTIAL AND GRAVITATIONAL MASS 513 as (A6) weights were negligible, so that the experiment this form of energy of inertial to gravitational APPENDIX A. EQUATIONS does not test the equivalence mass. for ~ ~'t '~ ~, the torsion balance from its equilibrium Le position, this may be rewritten 20), iJ OF MOTION OF THE TORSION BALANCE 11 = - K' 8 + eVB * VB(l - 1. ~,. where K' is the feedback 2. torsion constant, which is in general complex. ELECTROSTATIC TORQUE ON THE TORSION BALANCE Referring to Fig. 6 of the text, the electrostatic torque Le on the torsion balance can be seen to result from the stresses on the gold weight induced by the electric field between the feedback electrodes. If V1 and V2 are the potentials of the"two electrodes, and C1 and C2 the capacitances between them and the grounded gold weight, respectively, then the energy stored in the resulting electric field will be U 'If. ~~~. iE! DYNAMIC BEHAVIOR OF TORSION BALANCE ~(. The complete equation of motion of the torsion balance may be written J(j as (A7) + K8 = Le +L ext , ~, Lext any externally ): = ,72'[C1V12 + C V/]. 2 (AI) Jli where I is the moment or inertia of the balance, K the fiber torsion constant, and applied torque. Ignoring constant terms in this equation, we can set Lext = 0 and, using Eq. (A6), write jj Displacing the gold weight by a small amount dx in the direction perpendicular to the electrodes will require an amount of work F dx = -dUo Hence, the force on the weight is F + wo (1 + A)8 2 = 8. (A8) ~ = -7~[V/(aCt! ax) + V2 ( aCd ax)]. 2 (A2) If the gold weight is symmetrically located, then (act! ax) '"" - (aCd ax) ~ canst. for small displacements. In this case, the torque on the balance is approximately (A3) Le ~ canst. (V22 - V12). The potentials Here,'wo = VK/l is the natural frequency of the balance and A = K'/K is the open loop feedback gain. From Eq. (4) and Fig. 6, it is apparent that A will be proportional to VB *, the feedback gain control setting g, and the feedback filter transfer function (3(w). This latter is a complex function of frequenc}. iw Letting 8 = 8oe t, Eq. (A8) reduces to _w2 + wo [1 + A(w)] 2 = O. (A9) V1 and V2 of the electrodes are given by V2 V1 = V + (1 - o)VB 011B + Va2 When the known frequency-dependence (3(w) of A is substituted, a cubic equation in w results, which may be solved for complex angular frequency as a function of the dc open loop feedback gain A (0). The results of such a calculation for the filter circuit of Fig. 6 are shown in Fig. 8 of the text. = 11 - + Val (A4) 3. STATIC BEHAVIOR OF THE TORSION BALANCE , where 0 is the setting of the electrode bias potentIometer (0 ~ 0 ~ 1, and 0 = 0 ~_corresponds to the potentiometer wiper connected to electrode no. 1). Also, , VB * = VB Va2 "- 1181 = VB Va , where Val and V82 are the contact potential differences between the surfaces of the two electrodes and that of the gold weight. Combining Eqs. (A3) and (A4), Neglecting now the time-dependent the torsion balance is K(l terms in Eq. (A7), the static equation 20) of + +~ + A)8 = CVBVB*(l that +L ext (A1O) From Eqs. (4) and (A6), it is apparent K'8 Furthermore, the recorder output feedback voltage V: Le = const. X 11B *[(211 + Val + Va2) + (l + (l - 2o)VBJ, - 20) VB]' (A5) = 2eVB*V. (All) to (1/ g) times the (AI2) Ignoring the contributions of the (constant) brackets, this reduces to Eq. (4) of the text, Le = surface potentials inside the square (4) 8 of voltage V R is proportional fg VR = eVB*[2V V. where e is a constant. Since the feedback voltage V is proportional to the angular displacement As defined in Eq. (5) of the text; the torque sensitivity ST of the torsion balance is the change in recorder voltage per unit change in externally-applied torque. 514 Equations (AlO) through ST ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 515 (A12) can be used to show that (xl == .1YR/.1Lext = O'2cYB*jg)[A/(1 + A). (A13) N (0) This result is equivalent to Eq. (6) of the text, with Cl = (}/zcj). Finally, it was convenient in Section lV,E of the text to define the ratio R of change in recorder voltage .1Y R to change in electrode bias potentiometer setting .10, In this case, Eqs. (AlO) through (A12) can be used to express R in the form R == .1YR/.1o = 2 E (YB/jg)[A/(l + A). (7) APPENDIX B. TORQUE EXERTED ON THE TORSION BALANCE BY AN ANOMALOUS GRAVITATIONAL-TO-INERTIAL MASS RATIO Consider the torsion balance to be located in a polar coordinate system fixed to the earth, with z-axis directed toward the zenith and the azimuthal plane containing the centers of mass of the three balance weights (Fig. 22(a. If masses 1 and 2 are identical, the azimuthal orientation angle of the balance is the angle tp formed by the radius vector to the third mass and the north direction. In this coordinate system the angular coordinates of the sun are the zenith angle and the azimuthal angle A. (A is the supplement of the conventionally defined azimuth.) Now, if all three masses are identical, the torque on the balance due to the gravitational attraction of the sun will just balance the torque resulting from the centrifugal forces produced by the motion of the earth in its orbit about the sun: 't ( b) s L = L: m,l, i=l 3 X [(M/m)iaG + ad = O. (Bl) \EO Here, "AI and m represent passive gravitational and inertial masses, respectively, aG and aI are the above-mentioned gravitational and inertial (centrifugal) accelerations, respectively, and 1;is the radius vector of the center of mass of the ith weight. lLthe_gravitational-to-inertial mass ratio of the third weight, of material A, differs from that of the first two, constructed of material B, then (M/m)A (M/mh = 'I7(A,B), (B2) FIG. 22. Diagrams illustrating the coordinates used in calculating the torque on the torsion balance resulting from an anomalous gravitational-to-inertial mass ratio. (a) Top view of the coordinate systern fixed at the center of the torsion balance, The azimuthal coordinates A and", of the sun and the third weight, respectively, are measured from the north direction. Weights 1 and 2 are made of material B, while weight 3 (black) is composed of a different material A. Since the Z-axis points toward the zenith, the colatitude of the sun in this coordinate system is the zenith angle r. (b) Top view of the celestial sphere, -showing the pole P, the zenith Z, and the sun S. The great circles which form the astronomical triangle through these three points are dotted in. The sides of this triangle are the zenith angle r, and the complements a = (1r/2) - A and b = (1r/2) - D of the latitude A of the torsion balance and the solar declination D respectively. The interior angle at the pole is the hour angle t, or the local solar time in radians, witht = 0 at noon. The,interior angle at the zenith is the azimuthal coordinate A of the sun. (However, the azimuth of the sun, as conventionally defined in celestial coordinates, would be the supplement of :1 or the exterior angle at Z.) j where 'I7(A,B) 1 is defined by Eq. (lOa) of the text (Section V,A), with units chosen so that }/z[(M/m)A (M/mh) = 1. If '17A, B) =F 0, then the net torque ( on the torsion balance is not zero, as in Eq. (Bl), but + In terms of the coordinates illustrated in Fig. 24(a), this reduces to the anomalous torque on weight number 3 alone: in magnitude, L L = L: nL;1. i=l 2 = 'I7(A,B)(GMs/Rs)ml sin S sin (A - tp), (B4) X [(M/m),aG + arJ + 'I7(A,B)m313 X aG (B3) = 'I7(A, B)msla X aG' where Ms is the active gravitational mass of the sun and Rs the earth-sun distance. The coordinates (A, of the sun may be reduced to more readily useful quantities by considering the spherical triangle formed on the celestial sphere by n ~ j16 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 517 the zenith Z of the observer, the pole P, and the sun S. As shown in Fig. 22(b), the sides of this triangle are a = (71"/2) - REFERENCES 1. H. BONDI,Rev. Mod. Phys. 29, 423 (1957). 2. F. W. BESSEL, Pogg. Ann. 25, 401 (1832). 3. H. H. POTTER,Proc. Roy. Soc. 104,588 (1923). 4. R. V. EOTVOS, ath. u. Naturw. Ber. aus Ungarn 8, 65 (1890). M 5. R. V. EOTVOS, . PEKAR,ANDE. FETEKE,Ann. Physik 68, 11 (1922). D 6. J. RENNER,Mat. es termeszettudomanyi ertsitO 53, 542 (1935). 7. R. H. DICKE,Proc. Intel'Ti. School Phys. "Enrico Fermi," Course 20, Evidence for Gravitational Theories, G. Polyani, ed., pp. Iff. Academic Press, New York, 1962. 8. J. L. SYNGE,"Relativity: The General Theory," preface. North-Holland, Amsterdam, 1960. 9. T. D. LEE ANDC. N. YANG,Phys. Rev. 98, 1501 (1955). 10. L. SCHIFF,Proc. Nail. Acad. Sci. U.S. 45, 69 (1959). 11. R. V. EOTVOS,"Gesammelte Arbeiten," P. Selenyi, ed., p. 190. Akademiai Kiado, Budapest, 1953. 12. Private communication from the P. R. Mallory Company. 13. S. LIEBES, private communication (to be published). 14. "Standard Handbook for Electrical Engineers," 9th ed., A. E. Knowlton, ed., p.1907. McGraw-Hill, New York, 1957. ( 15. Private communication from Rev. J. Joseph Lynch, S.J., Fordham University, Seismic Observatory, New York City, New York. 16. P. J. E. PEEBLESANDR. H. DICKE,Phys. Rev, 127,629 (1962). 17. P. J. E. PEEBLES,Ann. Phys. (N. Y.) 20,240 (1962). A, b = (71"/2) - D and .\, where A is the latitude of the apparatus and D the solar declination. The angles opposite sides Band .\ are respectively the azimuthal angle A and the hour angle t (local solar time, with t = 0 at noon). Applying the law of sines for spherical triangles, it is easy to show that sin .\ sin A' = cos D sin t. (B5) Using the law of cosines, together with a half-angle relation for spherical triangles, it can be shown with considerably more difficulty that sin .\ cos A = sin A cos D cos t - cos A sin D. (B6) of a solar Combining Eqs. (B4) through (B6), and noting that the magnitude torque unit (stu) is (GM./Rs)ml, we arrive at Eq. (11) of the text: L(stu) = 17(A, B) [cos <p cos D sin t sin <p sin A cos D cos t + sin <p cos A sin DJ. (11) ACKNOWLEDGMENTS During the years this experiment and its various precursors have been in progress, most of the staff members and graduate students associated with the Pri~ceton gravity research group, present and past, have contributed to its evolution and sucbess. However, special notice must be taken of the very substantial contribution of the authors' colleague Barry Block, who devised a special technique for pulling quartz fibers of controlled length and diameter. He also supervised the construction of the final gold-aluminum torsion balance. Its fine performance was inlarge measure due tothecare with which the balance was kept free of magnetic contamination. Notice must be taken also of the contribution of R. D. Moore, who designed and constructed most of the electronic systems used in the earlier forms of the apparatus. David R. Curott tooka ~~j;r share of the responsibilityfOf'"thToperation of the copper:"-PbCb balance and the analysis of data from it. ' We are also indebted to Dr. Janos Renner of Budapest, a student and, colleague of Eotvos, who kindly made available to us some of the detailed results of his torsion balance measurements. Data analysis was carried out with the facilities of the Princeton University Computer Center, which is partially supported by a grant from the National Science Foundation. Finally, the facilities Ofthe Radio Corporation of America's David Sarnoff Research Center were kindly made available to us for some of the more difficult materials preparation and fabrication procedures. To these and the many other individuals who offered their assistance and counsel during the experiment, the authors would like to express their sincere appreciation. RECEIVED: - -_ _. -....;;,., .. .......... -- November 18, 1963 ...
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