Vr)u~Ma 56 6 JANUARY 1986 Nt. )MOER t

Reanalysis of the Eotvos Experiment

Ephraim Fischbach"institute for IVuclear Theory, Department of Physics, University of Washington, Seattle, Washington 98I95

Daniel Sudarsky, Aaron Szafer, and Carrick TalmadgePhysics DepartmentPurd, ue University, West Lafayette, Indiana 479Q7

S. H. AronsonPhysics Department, Brookhaven Nationa/Laboratory, Upton, New York l) 973

(Received 7 November 1985)

We have carefully reexamined the results of the experiment of Eotvos, Pekar, and Fekete, whichcompared the accelerations of various materials to the Earth. We find that the Eotvos-Pekar-Fekete data are sensitive to the composition of the materials used, and that their results support theexistence of an intermediate-range coupling to baryon number or hypercharge.

PACS numbers: 04.90.+e

Recent geophysical determinations of the Newtonianconstant of gravitation G have reported values whichare consistently higher than the laboratory value Gz. '

Kith the assumption that the discrepancy betweenthese two sets of values is a real effect, one interpreta-tion of these results is that they are the manifestationof a non-Newtonian coupling of the form

V (r) = —G„(I+ e o'I")m)pl2

= VN(r)+A V(r).

o = —(7.2+ 3.6) x10, g= 200+ 50 m. (2)

If 5 V(r) actually describes the effects of a new force,and is not just a parametrization of some other sys-tematic effects, then its presence would be expected tomanifest itself elsewhere as well. Recently, we have

Here VN(r) is the usual Newtonian potential energyfor two masses m

~ 2 separated by a distance r, and 6is the Newtonian constant of gravitation for rThe geophysical data can then be accounted for quanti-tatively if o. and A. have the values

undertaken an exhaustive search for the presence ofsuch a force in other systems. Our analysis, to bepresented elsewhere, 3 leads to the conclusion that ifsuch a force existed it would show up at present sensi-tivity levels in only three additional places: (i) theK -K system at high laboratory energies, where in

fact anomalous effects have previously been reported;(ii) a comparison of satellite and terrestrial determina-tionss of the local gravitational acceleration g; and (iii)the original Eotvos experiment which compared theacceleration of various materials to the Earth. Wenote that the subsequent repetitions of the Eotvos ex-periment by Roll, Krotkov, and Dicke and by Bragin-skii and Panov compared the gravitational accelera-tions of a pair of test materials to the Sun, and hencewould not have been sensitive to the intermediate-range force described by Eqs. (1) and (2). Motivatedby our general analysis, we returned to the Eotvos ex-periment and asked whether there is evidence in theirdata of the presence of 5 V(r) in Eq. (I). Althoughthe Eotvos experiment has been universally interpret-ed as having given null results, we find in fact that thisis not the case. Furthermore, we will demonstrate ex-plicitly that the published data of Eotvos, Pekar and

1985 The American Physical Society

VOLUME 56, NUMBER PHYSICAL REVIEW LETTERS 6 JANUARY 1986

f2/Gom~2 = —a/(I + u), (3)

where m~ is the proton mass. Consider the relative ac-celerations of two objects I and 2 with masses m t 2 andhypercharges (or baryon numbers) 8t 2. Because ofthe presence of 5 V (r) the accelerations at 2 of theseobjects to the Earth will no longer be the universalNewtonian value g, but will differ by an amount4a = a ~

—a2 given by

Fekete6 (EPF) not only suggest the presence of anon-Newtonian coupling 5 V (r), but also strongly sup-port the specific values of the parameters o. and A. in

Eq. (2), which emerge from an analysis of the geo-physical data.

Guided by the observations that (a) n ( 0, whichindicates a repulsive force, and (b) anomalous effectshave been reported in the E -K system as well, weconsider the effects of a massive hypercharge fieldwhose quanta (hyperphotons) have a mass mz

' = I x 10 9 eV. The exchange of a hyperphotonthen gives rise to a potential having the same form as5 V(r) in Eq. (1), with u being related to the unit ofhypercharge f by

Equation (4) can now be compared directly to theresults of EPF, where in their notation Aa/g=Kt—K2= AK. Table I gives AK for each of the nine pairsof materials measured by EPF, exactly as their result is

quoted on the indicated page of Ref. 6. For each ofthe pairs in which the composition of both samples canbe established (see discussion below), we also tabulateb (8/p, ) = 8t/p, , —82/p2u, sing the data of Ref. 10. Inthe computation of 8/p, for each material, care has

been taken to average over all the isotopes of each ele-ment, and to weight the contribution of each elementin a compound according to the appropriate chemicalformula. Among the substances appearing in Table I,Cu, Pt, and water require no further description, crys-talline copper sulfate has the formula CuSO4 5H20,and the CuSO4 solution consisted of 20.61 g of crystal-line copper sulfate in 49.07 g of water. By contrast,magnalium is an aluminum-magnesium alloy of vary-

ing composition, with typical Al:Mg ratios being in therange 95:5-70:30. Although the exact composition ofthe magnalium alloy used by EPF is not given, 8/p,for Al and Mg are very nearly equal so that 8/p, forany magnalium alloy would fall in the narrow range

T 1

4a f'e(R/)t) 8 eGomH2 p y pt P2

(4)1.00845 (pure Mg) & 8/p, (magnalium)

& 1.00851 (pure Al). (6)

Here p, ; denotes the mass m; in units of atomic hydro-gen, with mH= m(tH') =1.00782519(8) u, and wecan take 8@/p, = I for present purposes. e(R/h. )arises from integration of the intermediate-range hy-

percharge distribution over the Earth, assumed to be auniform sphere of radius R, and is given by (x = R/h. )

e(x) = 3(1+x) e "(xcoshx —sinhx).X

(5)

For X oo, e(0) 1, and (4) reduces to the result ofLee and Yang. However, the limit of interest to ushere is x && 1 in which case e(x) = 3/2x.

The results in Table I assume a composition Al:Mg =90:10, which is one of the more common alloys. Theremaining material whose composition can be estab-lished with some certainty is asbestos, since 95'/o of as-bestos production is a fibrous form of the mineral ser-pentine called chrysotile, " whose chemical formula isMg3Si20s(OH)4. In addition to measuring the relativeacceleration of various pairs of materials, EPF alsocompared the accelerations of the reactants before andafter the chemical reaction

Ag2SO4+ 2FeSO4 2Ag+ Fez(SO4) 3.

TABLE I. Summary of EPF results for AK, and page quoted from Ref. 6, along with thecomputed values of d (B/p, ). Ag-Fe-SO4 refers to the reactants before and after the chem-ical reaction described by Eq. (7).

Materials compared Page quoted 10 AK 10 d, (B/p, )

Cu-PtMagnalium-PtAg-Fe-S04Asbestos-CuCuSo4. 5H20-CuCuSO4(solution)-CuWater-CuSnakewood-PtTallow-Cu

37343947

45423548

+0.4+0.2+ 0.4+ 0.1

0.0 + 0.2—0.3 + 0.2—0.5 + 0.2—0.7 + 0.2—1.0 + 0.2—0.1 + 0.2—0.6 + 0.2

+0.94+0.50

0.00—0.74—0.86—1.42—1.71

VoLUME 56, NUMBER I PHYSICAL REVIEW LETTERS 6 JAxUARv 1986

= (4.6+ 0.6) x10 e, (9)

Q.4—

0.2—Pt

~Ag- Fe- SOq

- asbestos-Cu

CuSQ~- QHp 0- Cu

—t.Q, H~Q-Cui I I I I

-l.s -I,5 -I.Z -0.9 -0,6 -0.5 0 0.5 0.6 0.9 l.2

~0 6 (B/ )

FIG. l. Plot of A~ vs ~(8/'p. ) using the data in Table I.Ag-Fe-S04 refers to the reactants before and after the chem-ical reaction described by Eq. (7). The solid line representsthe results of a least-squares fit to the data.

Since 8/pi, s the same before and after the reaction,AK should be zero in this case, which is indeed whatEPF found. The remaining materials used by EPF are~chiangenhoiz (snakewood) and talg (tallow, grease,suet, etc.) whose exact compositions cannot be estab-lished. In particular, the amount of ~ater in each ofthese is unknown, and since water has a relatively low

value of 8/p, , the effective value of 8/p, for the sam-

pel could vary over a wide range depending on its wa-

ter content.In Fig. 1 we plot the measured value of AK versus

the computed values of 4(8/p, ) using the data givenin Table I. %e see immediately that the EPF dataclearly exhibit the linear relationship between 4K and

&(8/p) expected from Eq. (4). Furthermore, thesolid line resulting from a least-squares fit to the datapasses through the origin, as it should if Eq. (4) holds.Finally, the slope of the line is in remarkably goodagreement with the value expected from the parame-ters in Eq. (2) which arise from the geophysical data.Specifically, we find from the least-squares fit that theequation of the line is

b, K =a b ( 8/p, ) +b,

a = (5.65 + 0.71) x 10

i =(4.S3+6.44)»o-to,X'= 2. l (5 degrees of freedom).

Combining (4) and (8), we can solve for f e(R/h. ),[f'e(R/)t) ]E,-,„,-, ——GomH2a

~here e is the electric charge in Gaussian units. Thisshould be compared to the value derived from the geo-physical data in Eq. (2),

[f'e(R/z) ]„,„t,„,;,,„=(2.8 + 1.5) x 10 'e'. (10)

The agreement between these two results is surprising-

ly good, particularly in view of the simple model of theEarth that has been used in deriving (4) and (9). If X

is in fact on the order of 200 m, then the details of thelocal matter distribution will clearly modify the func-tional form of e(R/X), and could lead to improvedagreement between (9) and (10). If the potential in

Eqs. (I) and (2) describes a coupling to hypercharge,as we have assumed, then it should also give rise to ananomalous energy dependence of the fundamentalKo-K parameters such as the KL Ks mass -differenceb, m, the Ks lifetime rs, and the CP nonconservingparameter q+ . Here the intermediate-range natureof the coupling is crucial in understanding the effectsthat arise. As we discuss in Ref. 3, the specific valuesof n and h. in (2), which account for both the geophys-ical data and the Eotvos results, may also explain thekaon data as well, both qualitatively and quantitatively.

The possibility that the three effects that we havediscussed do in fact have a common origin can bedirectly tested in several ways. To start with, theEotvos experiment itself should be repeated with

greater sensitivity, and with a variety of materialswhose precise composition is known. As has beennoted elsewhere, '2 the composition dependence of theEotvos anomaly' b, a/g can be used to rule out vari-ous possible explanations of this effect. In particular,we show in Ref. 3 that neither a coupling to leptonnumber nor a recently proposed model of Lorentznoninvariance can account for the data of Ref. 6.While a repeat of the Eotvds experiment with bettersensitivity may be possible with modern techniques, itmay be more practical simply to compare the times offiight of different test masses dropped from the sameheight, in an updated version of the Galileo experi-ment. ' To achieve a sensitivity sufficient for our pur-poses, say Aa/g =10 ', would require measurementof the time of flight to within 0.1 ns over a distance of10 m which is within the realm of feasibility. In addi-tion, one can attempt to improve the measurement of4g, the difference between the locally measured valueof g and that implied by satellite data. Evidently satel-lite measurements would not be sensitive to b, V(r) in

(I) and (2), whereas local measurements would, and itfollows from (1) and (2) that b, g/g should be approxi-mately 2 & l 0 . An analysis of the available data byRapp' gives a value b,g/g = (6 + 10) x 10, but theprospects for improving on this result are somewhatuncertain. Finally, if we take seriously the existenceof a hypercharge field, then one can search directly forhyperphotons yz via their cosmological effects, and in

Vw UME56, NUMB, R 1 PHYSICAL REVIEW LETTERS 6 J~NUAa. v 1986

decays such as Eo 2m+yz. Following %einberg, '

we note that the branching ratio for this mode is

~here E,„&&mz is the maximum hyperphoton en-

ergy detected. For f and my as given in (2) and (3),and E,„ 100 MeV, the branching ratio is 6X10This is safely below the level where hyperphotonscould have been detected in the course of other exper-iments, but at the same time is large enough so that adirect search for this mode may prove possible. Froma cosmological point of view, hyperphotons would actas a massive but very weakly interacting constituent ofinterstellar space, and could thus help account for themissing mass of the Universe.

We are indebted to Frank Stacey for communicatingthe results in Eq. (2) prior to publication, and to PeterBuck for translating parts of Ref. 6. We also wish tothank Mark Haugan, %ick Haxton, Ernest Henley,Fred Raab, and Richard Rapp for helpful conversa-tions. One of us (E.F.) wishes to thank the Institutefor Nuclear Theory at the University of Washingtonfor its hospitality during the course of the research.This work was supported in part by the U. S. Depart-ment of Energy.

Note added. —R. H. Dicke (private communication)has raised with us the question of whether some sys-

tematic effect in the EPF experiment could simulatethe observed correlation between Etc and I (B/p, ). He

proposed an interesting model in which thermal gra-dients could lead to a correlation between AK and thequantity (a + b/p& —c/p2), where pt 2 are the densi-ties of the samples and a, b, and c free parameters.%e have investigated this model, and others involving

pt 2, and have found that none of these show a corre-lation with d, K. These results will be presented in de-tail in Ref. 3, where we will also show that they are aconsequence of two special properties of B/p, . (I) it

has an anomalously low value for hydrogen, and (2) it

has a maximum near Fe and is lower toward either endof the Periodic Table. %e wish to thank ProfessorDicke for stimulating us to investigate this question.

t'&On sabbatical leave from (1985-1986) from PurdueUniversity, %est Lafayette, Ind. 47907.

~F. D. Stacey and G. J. Tuck, Nature 292, 230 (1981);S. C. Holding and G. J. Tuck, Nature 307, 714 (1984);G. W. Gibbons and B. F. Whiting, Nature 291, 636 (1981);F. D. Stacey, in Science UndergIound, edited by M. M. Nietoet al. , AIP Conference Proceedings No. 96 (American Insti-tute of Physics, New York, 1983), p. 285.

2F. D. Stacey, private communication.E. Fischbach, D. Sudarsky, A. Szafer, C. Talmadge, and

S. H. Aronson, to be published.4S. H. Aronson, G. J. Bock, H. Y. Cheng, and E. Fisch-

bach, Phys. Rev. Lett. 4$, 1306 (1982), and Phys. Rev. D2$, 476,495 (1983); E. Fischbach, H. Y. Cheng, S. H. Aron-son, and G. J. Bock, Phys. Lett. 116B, 73 (1982).

5R. H. Rapp, Geophys. Res. Lett. 7, 35 (1974), and Bull.Geod. 51, 301 (1977); Report of Special Study Group No.5.39 of the International Association of Geodesy, "Funda-mental Geodetic Constants (August 1983)," in Travaux de/' Association Internationale de Geodesic (International Associ-ation of Geodesy, Paris, 1984), Vol. 27.

R. v. Eotvos, D. Pekar, and E. Fekete, Ann. Phys.(Leipzig) 6$, 11 (1922).

7R. H. Sicke, Sci. Am. 205, 84 (1961); P. G. RopplR. Krotkov, and R. H. Dicke, Ann. Phys. (N.Y.) 26, 442(1964). These authors have pointed out various incon-sistencies in a repetition of the original Eotvos experimentby Renner, and hence we have ignored Renner's results.

8V. B. Braginskii and V. I. Panov, Zh. Eksp. Teor. Fiz. 61,873 (1971) [Sov. Phys. JETP 34, 463 (1972)].

9T. D. Lee and C. N. Yang, Phys. Rev. 9$, 1501 (1955).toHandbook of Physics, edited by E. U. Condon and H. Od-

ishaw (McGraw-Hill, New York, 1967)."McGraw Hill Conc-ise Encyclopedia of Science and Technol

ogy, edited by S. P. Parker (McGraw-Hill, New York, 1984),p. 147.

E. Fischbach, M. P. Haugan, D. Tadic, and H. Y, Cheng,Phys. Rev. D 32, 154 (1985).

It is interesting to note that on p. 65 of Ref. 6 EPF sum-marize their data as if all the indicated samples were actuallymeasured against a Pt standard, notwithstanding the factthat in most of the measurements the actual standard wasCu. The effect of combining, say, An(H20-Cu) andAn(Cu-Pt) to infer hn(HqO-Pt) is to reduce the magnitudeof the observed nonzero effect from 5u to 2o-. Any sugges-tion of a nonzero effect was further reduced by choosing Ptrather than Cu as the standard since, had the opposite choicebeen made, the signs of all the nonzero AK would have beenthe same, and might thus have pointed to a possible sys-tematic effect.

Improvements in the Eotvos and Galileo experimentswill be the subject of a separate paper.

~5R. H. Rapp, private communication.'6S. Weinberg, Phys. Rev. Lett. 13, 495 (1964).

PHYSICAL REVIEW LETTERS 31 M&RcH 1986

ERRATA

Reanalysis of the Eotviis Experiment. EPHRAIM

FtscHEAcH, DANtEt. SUDARsKv, A~RON SzAFER,CARRtCK. TALMADGE, and S. H. ARONSON [Phys. Rev.Lett. 56, 3 (1986)].

Some of the names in Ref. 7 were misspelled. Thecorrect reference is R. H. Dicke, Sci. Am. 205, 84(1961);P. G. Roll, R. Krotkov, and R. H. Dicke, Ann.Phys. (N.Y.) 26, 442 (1964).

Periodic Qnasicrystal. TtN-LUN HO [Phys. Rev. Lett.56, 468 (1986)].

In Eq. (17), the k„~ should be q„~. The followingclause should read, "where the relation exp(i2mhq/ko) =exp(i~$5 ~X„) has been used. "

Classical Limit of Bethe-Ansatr Thermodynamics forthe Sine-Gordon System. NIU-NtU CHEN, MtCHAELD. JoHNsoN, and MtcHAEL FowLER [Phys. Rev. Lett.56, 904 (1986)].

The left-hand side of Eq. (1) should read(1+5,„,)Inq, (P).

VoLUME 56, NUMBER 22 PHYSICAL REVIEW LETTERS 2 JUNE 1986

The Letter to which the following Comments refer attracted unusual interest. Thequestion of the sign, discussed by Thodberg, was also treated in a paper by Kenji Hayashi,Kitasato University, and Takeshi Shirafuji, Saitama University. The effect of the brasscontainers, discussed by Keyser, Niebauer, and Fuller, was also treated in papers by Y. E.Kim, Purdue University, and by J. P. Wurm and K. T. Kndipfle, Max-Planck-Institut fCir

Kernphysik, Heidelberg.

Comment on the Sign in the Reanalysis of theEOtvds Experiment

In a recent Letter' to this journal Fischbach et al.reanalyzed the measurements of Eotvos et a!.2 andfound an intriguing correlation between the effectivegravitational mass and the baryon number of matter.

Fischbach et a/. propose the existence of a fifthforce mediated by hyperphotons coupled to the baryonnumber or hypercharge. This vector force is repulsivebetween matter objects. The authors claim that theEotvos data support this conjecture.

Here it is pointed out that there is a sign error inRef. 1 in Eqs. (4) and (9) for the predicted correlationbetween the gravitational acceleration and baryonnumber according to the hyperphoton force. (In a pre-vious article3 the sign was given correctly. )

To see this, compare the gravitational acceleration ofwater and copper. Water has a smaller nuclear bindingenergy than copper and therefore a smaller ratio ofbaryon number and mass. This leads to a smaller cou-pling to the hyperphoton and thus a weaker repulsionfrom the Earth. Since this repulsion compensates thegravitation, water will have a larger effective gravita-tional mass than copper, according to the hyperforcetheory.

Consider next the Eotvos data which are quotedcorrectly in Fig. 1 and Table I of Ref. 1. It is impor-tant to specify the definition of the sign of K, and thisis given on p. 15 of Ref. 2:

G'= G(1+K).

Here G is gravity for standard matter, e.g. , water, andG' is gravity of another substance, e.g. , Cu, character-ized by ~. The sign convention of G is unimportant; itis the sign of ~ relative to 1 which matters. On p. 42 inRef. 2 it is reported that

K i= Kc„=( —0.010 + 0.002) && 10-';

i.e., water has less gravitational mass than Cu, asindeed quoted in Ref. 1.

In this context it must be realized that Eotvos was ageophysicist, and that his apparatus was mainly used tomeasure the gravitational gradients in mountains.Eotvos was trained to indicate the sign of the measure-ments, since this is essential when making measure-ments in mines in order to know in which direction tosearch for ore.

To summarize, Eotvos has measured that water fallsmore slowly than copper, whereas the hyperforcetheory predicts the opposite.

If the Eotvos data are correct we can conclude thatthere is a new attractive force between matter objectscoupled to baryon number. This new interaction ismediated by spin-2 particles (or spin-0 particles).

We notice that this disagrees with the interpretationsof the geophysical measurements reported by Fisch-bach et al. ,

' and by Holding, Stacey, and Tuck, 4 whichhint at an extra repulsive force. Of course, the pres-ence of the geophysical data does not allow one tochange the sign of the Eotvos anomaly according tothe interpretation in Ref. l.

I acknowledge conversations with S. H. Aronson andE. Fischbach.

Hans Henrik ThodbergNiels Bohr InstituteDK-2100 Copenhagen, Denmark

Received 24 January 1986PACS numbers: 04.90.+e

iE. Fischbach et ai. , Phys. Rev. Lett. 56, 3 (1986).R. v. Eotvos, D. Pekar, and E. Fekete, Ann. Phys.

(Leipzig) 6$, 11 (1922).3E. Fischbach et ai , Phys. Re.v. D 32, 154 (1985), Eqs. (3)

and (8).4S. C. Holding, F. D. Stacey, and G. J. Tuck, Phys. Rev. D

33, 3487 (1986).

1986 The American Physical Society 2423

VoLUMF 57, NUMBFR9 PHYSICAL REVIEW LETTERS 1 SFPTFMasR 1986

ERRATA

Beta-Decay Asymmetry of the Neutron and gz/gi, .P. BOPP, D. DUBBERS, L. HORNIG, E. KLEMT, J. LAST,H. SCHUTZE, S. J. FREEDMAN, and O. SHA'RPF [Phys.Rev. Lett. 56, 919 (1986)].

The sentence beginning on line 11 of the right-handcolumn on page 921 which reads, "The deviationcounting rate is very small and the combination ofcounting rates plotted in Fig. 2 is very sensitive tobackground" should be replaced by "The deviation forthe highest-energy points occurs near the end pointwhere the P-decay counting rate is very small and thecombination of counting rates plotted in Fig. 2 is verysensitive to background. "

derived parameters, such as the time constant for an-

nealing obtained from the data in Fig. 4, remain unal-

tered.

Frisch, Rivier, and tyler Respond. H. L. FRIScH,N. RIvIER, and D. WYLER [Phys. Rev. Lett. 56, 2331(1986)].

The symbol 5 in Eq. (1) should be p.

Metastable Defects in Amorphous-Silicon Thin-FilmTransistors. A. R. HEPBURN, J. M. MARSHALL,

C. MAIN, M. J. POWELL, and C. VAN BERKEL [Phys.Rev. Lett. 56, 2215 (1986)].

Comment on the Sign in the Reanalysis of theEotvos Experiment. HANS HENRIK THODBERG [Phys.Rev. Lett. 56, 2423 (1986)].

The formula on the bottom of column I should read

~„„„-Kc„=( —0.010 + 0.002) x 10

A limitation in the extrapolation procedure used tocalculate the trapped charge Qo and Qo at zero delaytime leads to unrealistically high values of this parame-ter for the high-temperature ( ~ 350 K) data in Figs. 3and 4.

Experimental measurements of trapped charge Qwere restricted to delay times greater than a few tenthsof a second. At high temperatures, this resulted inmost of the charge being released prior to the mea-surement point [see Fig. 2(a)]. To calculate Qo or Qii,the model of Fig. 2(b) was used to correct the data tozero delay time. The multiplication factor thus ob-tained is strongly dependent upon the value of trapdepth, Eo, to the extent that an inaccuracy of a fewhundredths of an electronvolt produces an error in Qoor Qo of several orders of magnitude at 400 K.

The above effect resulted in calculated values of Qoand Qo in excess of those realistic for the experimentalconditions. Note, however, that (a) data taken belowabout 320 K are not subject to the above problem,since only a limited amount of charge is released in theinitial delay period [see Fig. 2(a)l, and (b) the shapesof the high-temperature curves in Figs. 3 and 4 areunaffected by any computational error, since a con-stant multiplying factor was employed for all data tak-en at a particular temperature. Therefore, other

Density-Functional Theory and Freezing of SimpleLiquids. W. A. CURTIN and N. W. ASHCROFT [Phys.Rev. Lett. 56, 2775 (1986)].

In the sentence following Eq. (5), ". . . for r & r„„is equivalent. . ." should read ". . . for r & r„„/2 isequivalent. . . ."

In Table I, the value of p, o.3 at kT/a=2. 74 reads"'(1.150)"and should read "(1.179)."

Derivation of the Equilibrium Degree of Polarizationin High-Energy Electron Storage Rings. S. R. MANE[Phys. Rev. Lett. 57, 78 (1986)].

On page 78, opening paragraph, the sentence "Thisanalysis does not simplify the mathematics but yieldsnew insights. . ." should read "This analysis not onlysimplifies the mathematics but also yields new in-sights. . . .

On page 81, in Ref. 7, "II+2jn" should read"8+ 2m'. "

1192

Vor.UME 56, NUMBER 22 PHYSICAL REVIEW LETTERS

Fischbach et aL Respond: Thodberg' and alsoHayashi and Shirafujiz raise a valid question about thesigns in Eq. (4) and the subsequent equations in ourpaper. 3 We have defined g= lgl = GoM@/~ is

that Eq. (4) is correct as it stands for the assumedrepulsive force. On the other hand, these authorscorrectly point out that the Eotvos-Pekar-Fekete(EPF) results, taken at face value, imply that the forceactually is attractive, contrary to what we have as-sumed.

We show elsewhere4 that one cannot in fact deducefrom the EPF data whether the force is attractive orrepulsive. The reason for this is that in the presenceof an intermediate-range force, local horizontal massinhomogeneities (e.g. , buildings or mountains) can bethe dominant source in the Eotvos experiment. 4 s

Their effects can be more important than those arisingfrom the Earth as a whole, even though the Earth is

the main source of the gravitational force. For thisreason one cannot infer either the magnitude or thesign of the EPF anomaly in the absence of a more de-tailed knowledge of the local matter distribution at thetime of the experiment than we presently have avail-able.

To see why this is the case„ let us consider the ef-fects of a building and its basement on the Eotvos bal-ance. We approximate the former by a sphere of massMand radius R, and the latter by a sphere of missingmass M' and radius R'. The centers of each sphereare located at angles $ and @' relative to the horizon-tal, and the experiment is performed at a latitude 8Finally, we locate these spheres to the north of the ap-paratus (itself oriented east to west), since we knowfrom the EPF description of the location of their ex-periment that they were in a room facing south. Thenthe component of the net torque v„„about the fiberaxis x3 is given by4

a sin()y+yb i4; s cos(P+P) —yi', ie cos(@'+P),

P2 g

where a is the Earth's centrifugal acceleration,P ~Pi —P2 is the angle that a plumb line makes (formass 1,2) with the vertical, and 2li is the length of thetorsion bar. The hypercharge fields due to the Earthand the building are denoted by y and y', respectively,with y=g(p ~i. /ipiti)(Bis/Mis)«(&/~)g, y' —4g'{where (—=f/G m02, and g and g' are respectively themagnitudes of the gravitational accelerations towardsthe Earth and the building), and all other notation is asin Refs. 3 and 4. For typical institutional buildingsM ~ (2-5) X 106 kg, M'~ (8-20) X10s kg, and fromthe known dimensions of the basement above whichthe EPF experiment took place, M' —14x 106 kg, onthe assumption of an average local mass densitypi, i

= 2750 kg/m3. The significance of the basementis that it acts as a "hole" in the otherwise uniformmatter distribution of the Earth, whose effects aredescribed by the first term in Eq. (1). Thus its effectscould contribute with a sign opposite to that of theEarth as a whole, and can be substantially larger. Ifthe picture of the matter distribution given in Ref. 4were correct, then the actual EPF data would indeedcorrespond to a repulsive force.

The simple picture described above should not betaken too literally, since there were in fact other(perhaps more important) sources of local matter in-homogenei ties. Ho~ever, the preceding discussionmakes it clear that local mass anomalies can be impor-tant because the magnitude of the horizontal hyper-charge force component may be enhaced by as muchas a factor of I/sinP with respect to that originallymodeled in Ref. 1. Moreover, the sign of the horizon-tal component can be either positive or negativedepending on the location of the anomalies relative to

the apparatus. These arguments make it clear that theonly unambiguous consequence of the intermediate-range coupling to baryon number or hypercharge is thepattern of points along a line plotting h~ vs 4(B/p, ),and not the sign or magnitude of the correspondingslope.

Ephraim FischbachPhysics DepartmentUniversity of WashingtonSeattle, Washington 98195

Daniel Sudarsky, Aaron Szafer„and Carrick TalmadgePhysics DepartmentPurdue UniversityWest Lafayette, Indiana 47906

S. H. AronsonEP. Division, CERNGeneva, Switzerland

Received 28 February 1986PACS numbers: 04.90.+e

iH. Thodberg, preceding Comment [Phys. Rev. Lett. 56,2423 (1986)].

K. Hayashi and T. Shirafuji, Kitasato University ReportNo. 86-2, 1986 (to be published).

3E. Fischbach, D. Sudarsky, A. Szafer, C. Talmadge, andS. H. Aronson, Phys. Rev. Lett. 56, 3 (1986).

4C. Talrnadge, S. H. Aronson, and E. Fischbach, Universi-ty of Washington Report No. 40048-12-N6, 1986 (to be pub-lished).

5P. Thieberger, Phys. Rev. Lett. 56, 2347 (1986) (this is-

sue); A. Bizzeti, University of Florence Report No. &6, 1986(to be published); M. Milgrom, Institute for AdvancedStudy Report, 1986 (io be published); D. A. Neufeld, Phys.Rev. Lett. 56, 2344 (1986) (this issue).

VOLUME 56, NUMsEx 22 PHYSICAL REVIEW LETTERS 2 JUNE 1986

Comment on "Reanalysis of the Eotvos Experi-ment"

Fischbach et al. present an analysis of the Eotvos,Pekar, and Fekete (EPF) data2 from which they sug-gest the presence of a non-Newtonian coupling tobaryon number (i.e., hypercharge). We find twofiaws: (a) They misinterpret or omit some of the EPFdata and (b) they reject the work of Jinos Renner. 'To make Fischbach et al. 's Eq. (10) consistent with

Eq. (9) for a uniform-density Earth would require thatn be increased from —0.0072+0.0036 to —0.11, orby 28 times the quoted uncertainty. Allowing for theEarth's lower surface density would require twice thisincrease.

EPF used a number of composite test bodies inwhich a brass container held the material of interest,and correct their data to account for use of these brasscontainers. When we take the raw b, tc data of EPF andcalculate the values of B/p, including these brass con-tainers, we find that four of the seven points used byFischbach et al. are moved toward the origin by vary-

ing amounts (along the fitted line of Fischbach et al.).The tallow-copper and snakewood-platinum points

were omitted by Fischbach et al. on the grounds thatthe composition of the first material was unknown. 4

Schlangenholz (lignum vitae) is a wood, and all woodsare composed primarily of cellulose (a known polysac-charide ) and lignin (a known two-amino-acid copoly-mer6). The Talg was purified beef or mutton tallow,which are known fats (triglycerides). 8 Most serious isthe unnoted omission of a datum. EPF measured b, K

for a O. l-g glass vial containing 0.1 g of RaBr2 in a25.2-g brass container versus a 25.4-g platinumcylinder. 9 This test body was 99% brass by mass, sothat this test amounts to a brass-Pt test. In fact,b, (8/p, ) here differs from the 5(8/p, ) of Cu-Pt byonly 0.006x 10 3, yet its value of b, ~ differs by some3.5a from the b, tc of the Cu-Pt datum.

Fischbach et al. 'o rejected the work of Renner be-cause of "various inconsistencies" noted by Roll,Krotkov, and Dicke" (RKD). On the basis of exam-ining Renner's original data, RKD suggest that hisstandard deviations should be multiplied by a factor of3. RKD also point out the anomalously large ratio ofhis (corrected) standard deviations to the deviations ofhis means. Though these issues cannot be resolvedunambiguously "after the fact, " we nevertheless feelthese data should not be wholly ignored in regards tothe d ~ values which can be derived from them.

Figure 1 shows all Etc data from Refs. 2 and 3. Theline is from Fischbach et al. for comparison. (The o.

values for Renner's data have been increased by thefactor of 3 suggested by RKD.) The experimental datado not appear to support Fischbach et ai. %'e are look-ing at various possible contemporary Eotvos andGalileo tests.

0.8

0.6—

0+ -0.2o

-04

p

-06—-08—-10—

I I I I j i l I i

-18 -1 5 -1.2 -0.9 -0,8 -0, 5 0 05 06 09 1.2

lo 6(8/p)

FIG. 1. Graph of bK vs 5(8/p, ) data. From Ref. 2 (tri-angles): (A) 4'/0 Mg-Al alloy vs Pt; (B) Schiangenholz vs Pi;(C) Cu vs Pt; (D) Ag2SO&+2FeSO4 vs 2Ag+Fe2(SO4)3',(E) H20 in brass vs Cu; (F) Cu2SO4 ~ SH20 in brass vs Cu in

brass; (G) solution in brass vs Cu in brass; (H) Asbest in

brass vs Cu in brass; (K) Talg in brass vs Cu in brass; (M)RaBr2 ampule in brass vs Pt with EPF misprint corrected.From Ref. 3 (circles): (1) Pt vs 30% Zn brass; (2) glass in

brass vs brass; (3) paraffin in brass vs brass; (4) NH4F in

brass vs Cu; (5) Cu vs 30'/o Mn-Fe alloy; (6) brass vs Bi.

We thank Dr. D. Bartlett, Dr. P. Bender, and Dr.C. Wieman for helpful discussions and the NationalBureau of Standards, Sensor-Technology Division ofBelvoir RD&E Center, and The Air Force GeophysicsLaboratory for support.

Paul T. Keyser, Timothy Niebauer, and James E.Faller

Joint Institute for Laboratory AstrophysicsUniversity of Colorado and National Bureau of StandardsBoulder, Colorado 80309

Received 6 February 1986PACS numbers: 04.90.+e

iE. Fischbach, D. Sudarsky, A. Szafer, C. Talmadge, andS. H, Aronson, Phys. Rev. Lett. 56, 3 (1986).

R. Eotvos, D. Pekir, and E. Fekete, Ann. Phys. (Leipzig)68, 11 (1922).

3J. Rennet, Mat. Termeszettud. Ert. 53, 542 (1935).E. Luschin, Berg-Huttenminn. Jahrb. 38, 87 (1890), says

that Asbest is fibrous amphibole (tremolite to actinoliie), sothat 8/p, —1= (8.470+0.096) &10 s, 2.6o. 1arger than8/p, —1 for chrysotile. Uncertainties on 8/p, (Asbest,Schlangenholz, Talg) cover the full range of values.

5CRC Handbook of Chemistry and Physics, edited byR. Weast (CRC Press, Boca Raton, Fla. , 1982), 63rd ed. , p.C223.

68ioIogy Data Book, edited by P. L. Altman and D. S. Ditt-mer (Federation of American Societies for ExperimentalBiology, Bethesda, Maryland, 1974), 2nd ed. , Vol. 3, pp.1562-1563.

Meyers Lexikon (Bibliographisclms Instiiut, Leipzig,1929); see "Talg."

SRef. 6, Vol. 1, pp. 348-349.9Ref. 2, pp. 59-61.

ioRef. 1, p. 6, footnote '7.

P, G. Roll, R. Krotkov, and R. H. Dicke, Ann. Phys,(N.Y.) 26, 448 (1964).

1986 The American Physical Society 2425

VoLUME 56, NUM@BR 22 PHYSICAL REVIEW LETTERS 2 JUNE 1986

Fischbach et al. Respond: In the accompanying Com-ment' Keyser, Niebauer, and Faller (KNF), claim thatwe misinterpreted the E6tv6s-Pekar-Fekete (EPF)data by using the results quoted by EPF rather thanthe raw b, K values. They fail, however, to note thatone can equivalently use either the raw data or the fi-nal corrected data for the investigation of a correlationwith 8/p„since (8/p, )b„„—(8/p, )c„. EPF correctedtheir data (in this case, legitimately) by assuming thatthe brass container did not contribute to the accelera-tion anomalies. Furthermore, even if one were study-

ing a property in which brass was distinctly differentfrom copper, only the H20-Cu datum would requirespecial treatment. This is because for all of the othercomparisons in which the sample was contained in abrass container, the Cu reference was also contained in

a brass cylinder similar in size to that used to hold thesample. Moreover, a fit to the data with the brass vials

yields very nearly the same parameters as were ob-tained in Ref. 2.

KNF quote 8/p, for various materials which differsomewhat from ours. For asbestos, KNF correctlynote that it can contain a large amount of actinolite.However, the closest source of asbestos to Hungarywould have been the Asbest region in the Ural moun-tains, which "as far as is known, produces only chryso-tile, "3 which is what we used in Ref. 1.

KNF note that the RaBr2 datum was not included.The reason for this is that in the same section of thepaper in which EPF discuss their measurements of h~for RaBr2, they also discuss a series of experimentswhich demonstrate the various anomalous effectswhich can arise from thermal heating of the apparatusby the RaBr2 sample. It was for this reason that thisdatum was excluded from the overall analysis in Ref.1. Furthermore, a subsequent recheck of the EPF pa-

per has revealed a typographical error in the sign EPFquote for this value. With the corrected sign AK isnow (+1+2)x10 9, and this result agrees with thevalue obtained for Cu-Pt within the quoted errors.One can infer from this comparison that their ap-paratus was not excessively sensitive to thermal ef-fects.

KNF also comment that the Renner data should beincluded along with the EPF data in a search for a pos-sible correlation with b, (8/p, ), although they do notgive any compelling reasons for our disregarding thecriticism of these data by Roll, Krotkov, and Dicke"(RKD). RKD's primary criticism was that Renner'squoted errors "are not consistent with his claimed —„scale-division reading error. " Second, RKD also notethat Renner incorrectly evaluated his errors, whichRKD give as a possible explanation of the inconsisten-cy between Renner's quoted errors and his scale-reading accuracy. Thirdly, RKD note that the meanvalues are too small compared even to his quoted er-rors. Finally, we note that even if Renner's data were

2426

completely valid, it would still be incorrect to plotthem on the same graph with those of EPF. Thereason for this is that, as we and others have noted,the magnitude and sign of the slope bK/6(8/p, ) thatone obtains from a particular Eotv6s experiment ishighly sensitive to the local matter distribution. TheEPF and Renner experiments were not in fact per-formed in the same location. Hence if the latter werecarried out in a relatively symmetric environment, itcould very well be the case that Renner would havebeen expected to see an (almost) null result.

The apparent discrepancy between the geophysicaland the EPF data can be resolved by noting that the ef-fect of the nearby matter distribution is to change boththe sign and magnitude of the E6tvcis anomaly com-pared to what one obtains from the naive sphericalmodel of the Earth used in Ref. 2. Moreover, since(9) and (10) of Ref. 2 effectively involve the producta)t, these equations can be made consistent by an in-crease of either X or a. Recent work of Holding, Sta-cey, and Tuck8 indicates that the value of X impliedfrom their data is so uncertain that there is at presentno conflict in the magnitudes of the effects implied bythe EPF data.

E. Fischbach, "D. Sudarsky, ' A. Szafer, 'C. Talmadge, ' and S. H. Aronson'"

'Department of PhysicsUniversity of WashingtonSeattle, Washington 98195'Department of PhysicsPurdue University%est Lafayette, Indiana 47907'Brookhaven National LaboratoryUpton, New York 119734CERN1121-Geneva 23, Switzerland

Received 17 March 1986PACS numbers: 04.90.+e

&Paul T. Keyser, Timothy Niebauer, and James E. Faller,preceding Comment [Phys. Rev. Lett. 56, 2425 (1986)].

2E. Fischbach, D. Sudarsky, A. Szafer, C. Talmadge, andS. H. Aronson, Phys. Rev. Lett. 56, 3 (1956).

3McGraw Hill Concise Ency-clopedia of Science and Technalogy, edited by S. P. Parker (McGraw-Hill, New York, 1984),p. 147.

4P. G. Roll, R. Krotkov, and R. H. Dicke, Ann. Phys.(N.Y.) 26, 448 (1964).

5C. Talmadge, S. H. Aronson, and E. Fischbach, Universi-ty of Washington Report No. 40048-12-N6, 1986 (to be pub-lished) .

6P. Thieberger, Phys. Rev. Lett. 56, 2347 (1986) (this is-sue); A. Bizetti, University of Florence Report No. 86, 1986(to be published); M. Milgrom, Institute for AdvancedStudy Report, 1986 (to be published); D. A. Neufeld, Phys.Rev. Lett. 56, 2344 (1986) (this issue).

7J. Barnothy, private communication.8S. C. Holding, F. D. Stacey, and 6, J. Tuck, Phys. Rev. D

33, 3487 (1986).

VOLUME 57, NUMBER 15 PHYSICAL REVIEW LETTERS 13 OcToBER. 1986

Alternative Explanations of the Edtvbs ResultsOur recent reanalysis' of the experiment by

Eotvos, Pekar, and Fekete (EPF) has uncovered inthe EPF data a correlation between the fractional ac-celeration difference b, K and the quantity LL(8/p, ),where 8 is baryon number and p. is the mass in unitsof m(iiH). Although this correlation agrees with whatone would expect from the presence of an inter-mediate-range force whose source is baryon number ofhypercharge, the possibility remains that the EPFresults could be explained in terms of conventionalphysics. The only alternative model we know of atpresent which has a serious chance of explaining theseresults is the "thermal-gradient" model of Chu andDicke4 (CD), and for this reason the CD modeldeserves to be taken seriously. In what follows we ex-amine the strengths and limitations of the CD modelin light of the EPF data.

The premise of this model is that if horizontal ther-mal gradients were present in the EPF apparatus, theycould produce a gentle "breeze" which could exert aforce on the samples being compared. Since the sam-ples, or the containers they were in, had differentphysical dimensions, the forces exerted on oppositesides of the apparatus would not be equal, and a nettorque could result. In practice, this would lead to acorrelation between b, tt and 5(1/p) or hS, where p isthe density of the sample and S is the cross-sectionalsurface area of the sample or its container. This nicelyillustrates how a systematic effect can appear todepend on a property of the samples, such as I/p.

It is useful to picture any acceptable alternative tothe baryon-number or hypercharge hypothesis as satis-fying two separate criteria: (i) To start with, themechanism in question should have the property thatthe torque which it produces must change when theEPF apparatus is rotated through 180', (ii) the torqueproduced by the alternative mechanism must not onlybe composition dependent, but also must vary fromone material to another in a manner that (at least ap-proximately) simulates a variation with 8/p, . The sig-nificance of the first criterion, when applied to the CDmodel, is to suggest that if a thermal gradient (orbreeze) is to simulate the effects of a matter distribu-tion (e.g. , the presence of a mountain or building), itmust produce a force which is similarly constant onaverage temporally relative to the apparatus. Howev-er, it is unclear how any likely heat source (e.g. , a win-dow or radiator) would always produce a gradient withboth a fixed direction and fixed magnitude, independentof time of day or year over the period of months oryears that the experiment took place. The challenge tothe CD model arising from the second criterion is toexplain adequately the Pt data, which tend not to fitthat well, irrespective of how the model is formulated.As CD correctly point out, there is a clear suggestion

of a correlation between b~ and AS or 5(1/p) evi-denced by the double-torsion-balance data alone, par-ticularly for those comparisons carried out using EPF'smethod III, which happened not to involve Pt. Thereason for this is illuminating, and provides an insightinto other possible correlations as well. If we examinea plot of 8/p, as a function of atomic number Z, wesee that for the EPF samples 8/p, is an approximatelymonotonically increasing function of Z, provided weexclude Pt. It follows that since 8/p. does in factcorrelate with tt, so will any other nearly monotonicallychanging variable, provided that the Pt data are ex-cluded. Hence it is precisely these data which test thecharacteristic shape of 8/p, as a function of Z, andwhich thus discriminate between a correlation with

8/p, and one involving some other variable. It followsthat the suggestion of an approximate correlationbetween hatt and 5(1/p) is not surprising, since —I/pis also an approximately increasing function of Z forthe substances studied by EPF.

In summary, the CD model is very clever and suffi-ciently promising to warrant more detailed study,should ongoing experiments fail to confirm the origi-nal EPF results. The issues that it must address morefully are mechanisms for producing a temporally con-stant thermal gradient over a long period of time, andthe behavior of the Pt data.

E. Fischbach, "'""'D. Sudarsky, "' A. Szafer, '"C. Talmadge "' and S. H. Aronson"'"'"'

"'Department of Physics, Purdue University%est Lafayette, Indiana 47907'2 Department of Physics, University of WashingtonSeattle, ashington 98195

Brookhaven National LaboratoryUpton, New York 11973~4~EP Division, CERNCH-1211 Geneva 23, Switzerland

Received 7 July 1986PACS numbers: 04.90.+e

|&~Present address'. Department of Physics, PurdueUniversity, %est Lafayette, IN 47907.

&b~Present address: Brookhaven National Laboratory, Up-ton, NY 11973.

&E. Fischbach, D. Sudarsky, A. Szafer, C. Talmadge, andS. H. Aronson, Phys. Rev. Lett. 56, 3 (1986).

2E. Fischbach, D. Sudarsky, A. Szafer, C. Talmadge, andS. H. Aronson, in Proceedings of the Second Conference on En

teractions Bebveen ParticIe and Nuclear Physics —1986, editedby D. Geesaman, AIP Conference Proceedings No. 1SO(American Institute of Physics, New York, 1986), p. 1102.

R. v. Eotvos, D. Pekar, and E. Fekete, Ann. Phys.(Leipzig) 6$, 11 (1922), and in Roland Eotvos GesammelteArbeiten, edited by P. Selenyi (Akademiai Kiado, Budapest,1953), pp. 307-372.

4S. Y. Chu and R. H. Dicke, this issue [Phys. Rev. Lett.57, 1823 (1986)].

VOLUME 57, NUMBER 22 PHYSICAL REVIEW LETTERS 1 DECEMBER 1986

Comment on "Reanalysis of the Eotvos Experiment"

To the zeroth order, the Earth is a sphere held togeth-er by gravitation, and the plumb line is directed towardthe center of the sphere. To the first order, the Earth isan ellipsoid of revolution held together by gravitation butdeformed by the centrifugal forces of its rotation, andthe plumb line is not, in general, directed toward thecenter of the ellipsoid. The ellipsoid is an equipotentialsurface: Horizontal gravitational forces (northward in

Hungary) on passive gravitational masses are exactlybalanced by opposing centrifugal forces (southward in

Hungary) on inertial masses. The plumb line is normalto this surface. If the ratio of passive gravitational toinertial masses differed between two materials, saycopper and platinum, then proof masses of copper andplatinum would have different plumb lines and thereforedifferent ellipsoids. The basic concept of the Eotvos ex-periment is that if the copper and platinum proof massesof a torsion balance are aligned in an east-west direction(along the intersection of the copper and platinum ellip-soids) and the balance is allowed to rotate, then eachproof mass will be able to drop with respect to its own el-lipsoid until the gravity and elastic torques of the mecha-nism are balanced. Eotvos, Pekar, and Fekete (EPF)and Rennerz performed the experiment for various pairsof proof masses and concluded that the ratio of inertialto passive gravitational mass is independent of the com-position of the mass.

Fischbach et al. reanalyzed the EPF data and con-cluded that the results are sensitive to the proof-masscompositions and that the data support the existence ofan intermediate-range (-200 m) coupling to baryonnumber. Keyser, Niebauer, and Faller contend that theresults are not sensitive to the proof-mass compositions,and I contend that in the absence of local mass inhomo-

geneities the Eotvos experiment is quite insensitive toany intermediate-range (small compared with theEarth's radius) coupling of any nature. The effect of alocal coupling would be to change the magnitudes of thedownward forces on the proof masses and on the torsionbalance wire, but it would not change the direction of theplumb lines for different composition proof masses; in ef-fect there would be no horizontal "fifth force" com-ponent, so the torsion balance would sense nothing. Thisargument does not preclude the possible existence of anintermediate-range coupling, such as is hinted at by theconsistent discrepancy between the Newtonian gravita-tional constant determined in laboratories and the con-stant determined from mine and bore-hole data. A dif-ferential Galileo experiment may be sensitive to such aneffect.

Donald H. EckhardtAir Force Geophysics LaboratoryHanscorn AFB, Massachusetts 01731

Received 5 May 19S6PACS numbers: 04.90.+e

R. Eotvos, D. Pekar, and E. Fekete, Ann. Phys. (Leipzig)68, 11 (1922).

J. Renner, Matematikai es Termeszettudomanyi Ertesito53, 542 (1935).

3E. Fischbach, D. Sudarsky, A. Szafer, C. Talmadge, andS. H. Aronson, Phys. Rev. Lett. 56, 3 (1986).

~P. T. Keyser, T. Niebauer, and J. E. Faller, Phys. Rev. Lett.56, 2425 (1986).

5Precision Measurement and Fundamental Constants II,edited by 8. N. Taylor and W. D. Phillips, National Bureau ofStandards Circu1ar No. 459 (U.S. GPO, Washington, D.C.,1984), pp. 565 and 598.

Work of the U. S. GovernmentNot subject to U. S. copyright

VoLUME 57, NUMsER 22 PHYSICAL REVIEW LETTERS 1 DEcEMsER 1986

Fischbach et al. Respond: Since the publication of ourreanalysis of the experiment of Eotvos, Pekar, and Fek-ete (EPF), considerable attention has been devoted tothe question of what the actual source of the Eotvosanomaly is. The answer to this question is important not

only in the design of experiments to check the EPF re-sults, but also in establishing a theoretical connection be-tween the EPF experiment and other systems which aresensitive to the putative "fifth force. " These include thegeophysical data and the Ko-K~ system (see Ref. 1),where anomalies have been reported, as well as the satel-lite data from which interesting limits can be inferred.

The heart of the EPF apparatus consists of two dis-similar test masses attached to a bar, which is itselfsuspended from a torsion fiber. A force (such as thefifth force) which affects the test masses differently cancause a torque about the fibre axis, which is what EPFlook for. However, even if the fifth force existed, its pres-ence could go undetected if it were directed along thefiber axis. Since the fiber axis is determined by the localacceleration field go a, +giv, where a, is the centrifugalacceleration due to the Earth's rotation, and giv is theNewtonian gravitational contribution due to the Earth, itfollows that the fifth force can be detected in the EPFexperiment only if the acceleration field y that it produces is not parallel to gp. * The following examples il-lustrate the implications of this observation.

(1) Suppose, following Ref. 1, that the Earth is pic-tured as a uniform rotating sphere, and that y arisesfrom a force whose nominal range is 200 m. In this case

y is parallel to gn, but y and go are not parallel sincea, &0. However, the angle p between y and go is verysmall, P~tanP asi n8/go~1/581, where 8~45' is thelatitude where the EPF experiment took place.

(2) Assuming the same picture for the Earth, let usadd in the effects of local matter anomalies such asbuildings (and their basements) and mountains. Forthese contributions y is nearly horizontal, so that y pro-duces a relatively large torque about the fiber axis. Thisobservation, which has been made previously by usz 3 andothers, 4 indicated the advantages of carrying out EPF-type experiments near mountains, and also demonstrat-ed23 the importance of the building (and basement)where the EPF experiment took place. If we examineEq. (1) of Ref. 3 we see that if the strength and range ofthe fifth force are as suggested by the geophysical data,then the effects of the building (and particularly itsbasement) are much larger than that of the Earth as awhole, even though the Earth is the main source of thegravitational force.

(3) We next consider the effects of a nonsphericalEarth by assuming that the Earth elastically deforms sothat its surface lies along an equipotential of go. As we

noted in Ref. 2, this is a good assumption since the devi-

ation of go from vertical is only 70" even in the vicinity ofMt. Everest. (A similar point was made previously byBizzeti and is the content of the accompanying Com-ment by Eckhardt. 5) Since y is determined by thematter within a hemisphere of radius = 200 m, y will belocally perpendicular to the Earth's surface, and if thissurface is also an equipotential of go, y and go will bestrictly paralleL Under these circumstances the Earth asa whole makes no contribution whatever to the EPFanomaly, as Eckhardt has also noted. However, this ob-servation has limited practical significance, since wehave already demonstrated in Refs. 2 and 3 that if theforce is of short range then the dominant contributions in

the EPF experiment will come from local departuresfrom the Earth's geoid.

(4) Finally we can consider the same situation as in

(3) above, but under the assumption that y arises from aforce whose range is comparable to (or larger than) R@.In this case y is parallel to giv, as in (1), and the angletween y and ga will again be of order P. Here localdepartures from the geoid are no longer important, sincethe whole Earth contributes to both y and gN, and theEPF anomaly depends on the circumstance that a, &0.This was the picture that EPF had in mind in the designof their experiment.

This work was supported in part by the U.S. Depart-ment of Energy.

Ephraim Fischbach and Carrick TalmadgePhysics DepartmentPurdue University%est Lafayette, Indiana 47907

S. H. AronsonPhysics DepartmentBrookhaven National LaboratoryUpton, Ne~ York 11973

Received 28 October 1986PACS numbers: 04.90.+e

'E. Fischbach, D. Sudarsky, A. Szafer, C. Talmadge, and

S. H. Aronson, Phys. Rev. Lett. 56, 3 (1986).C. Talmadge, S. H. Aronson, and E. Fischbach, in Proceed-

ings of the 21st Rencontre de Moriond, Les Arcs, France, 9-16March 1986 (to be pubhshed).

E. Fischbach, D. Sudarsky, A. Szafer, C. Talmadge, and

S. H. Aronson, Phys. Rev. Lett. 56, 2424 (1986).4P. Thieberger, Phys. Rev. Lett. 56, 2347 (1986); A. Bizzeti,

University of Florence Report No. 86, 1986 (to be published);M. Milgrom, Nucl. Phys. (to be published); D. A. Neufeld,

Phys. Rev. Lett. 56, 2344 (1986).SD. Eckhardt, preceding Comment [Phys. Rev. Lett. 57,

2868 (1986)].

2869