Boughn, Stephen P.

Hamilton, W. O.

Pipe, P. B.

Rasband, S. N.

English

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Haverford College Haverford Scholarship Faculty Publications Astronomy 1972 Two Gravity-Wave Detectors: A Comparison S. N. Rasband P. B. Pipe W. O. Hamilton Stephen P. Boughn Haverford College, sboughn@haverford.edu Follow this and additional works at: https://scholarship.haverford.edu/astronomy_facpubs Repository Citation "Two Gravity-Wave Detectors: A Comparison" (with S. N. Rasband, P. B. Pipes and W. O. Hamilton), Phys. Rev. Letters 28, 253 (1972). This Journal Article is brought to you for free and open access by the Astronomy at Haverford Scholarship. It has been accepted for inclusion in Faculty Publications by an authorized administrator of Haverford Scholarship. For more information, please contact nmedeiro@haverford.edu. VOLUME 2S, +UMBER 4 PHYSI CA I. REVIEW LETTERS 24 JaNUARY 1972*Work supported by the U. S. Atomic Energy Commis-sion under contract with Union Carbide Corporation.)Present address: Department of Physics and Astro-nomy, Vanderbilt University, Nashville, Tenn. 37203.J. Rainwater, Phys. Bev. 79, 432 (1950).J. M. Blatt and V. F. Weisskopf, Theoretical Nu-clear Physics (Wiley, New York, 1952).K. Kumar, Phys. Rev. C 1, 369 (1970).A. Bohr, Kgl. Dan. Vidensk. Selsk. , Mat. -Fys. Medd.26, No. 14 (1952); A. Bohr and B.R. Mottelson, Kgl.Dan. Vidensk. Selsk. , Mat. -Fys. Medd. 27, No. 16(1953); B. R. Mottelson, in The Many Body Browed,edited by J. Percus (Wiley, New York, 1958).G. Ripka, in Advances in Nuclear Physics, edited byM. Baranger and E. Vogt (Plenum, New York1,968),Vol. I, p. 183.The two-body operator defined by Eqs. (1) and (2) issimilar to that of the separable, quadrupole-quadrupoleinteraction. However, .the two would be identical onlyif the nuclear mass and charge distributions were iden-tical.We follow the conventions of A. de-Shalit and I. Tal-.mi, Nuclear Shell Theory (Academic, New York, 1963)for the tensor and scalar products of irreducible ten-sors (except for some obvious change of notation).de-Shalit and Talmi, Bef. 7.K. Alder and A. Winther, Coulomb Excitation (Aca-demic, New York, 1966).I. A, Fraser et al. , Phys. Rev. Lett, 23, 1047, 1051(1969); R. G. Stokstad, private communication of com-pilation of the results of several experiments fthesematrix elements can be found in Table I of K. Kumar,Phys. Bev. Lett. 26, 269 (1971)].Kumar, Bef. 10.' P. H. Stelson and L. Grodzins, Nucl. Data 1, 21(1965).Two Gravity-Wave Detectors: A Comparison*S. N. Rasband, P. B. Pipes, and %. 0. HamiltonDepartment of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70808S. P. BoughnDepartment of physics, Stanford University, Stanford, California 04808(Received 3 September 1971)The sensitivity of a dumbbell gravity-wave antenna is compared with that for a cylindri-cal detector. It is concluded that a Weber cylindrical antenna is decidedly the more sen-sitive at mutually accessible frequencies, particularly if the detector is to operate athigher frequencies in additiori to the fundamental. A dumbbell antenna does offer the pos-sibility of sampling the very low-frequency end of the spectrum which is inaccessible tocylinders.In the current period of activity following thepioneering work of Weber" in gravity-wave de-tection, considerable discussion is being gener-ated concerning antenna design. 3 ' As a possiblealternative to a cylinder, a dumbbell (two largemasses connected by a rod) is considered hereas a possible gravity-wave detector. ' We com-pare this alternative antenna to the more familiarcylinder without idealizing either of these detec-tors as two masses connected by a spring. Anal-ysis of longitudinal elastic vibrations resultingfrom gravity pulses and thermal noise showssuperior sensitivity for cylinders, particularlyif a single detector is to operate at several fre-quencies. A dumbbell may, however, providea way to observe low-frequency (- 100-Hz) gravi-tational radiation which is in practice inacces-sible to cylinders.The dimensional parameters for the detectorsare depicted in Fig. 1. All dependence of theelastic oscillations on directions perpendicularto the horizontal symmetry axes of the detectorsis ignored. This is equivalent to sett. ing the Pois-son ratio equal to zero. Any attempt at a realis-tic description of the elastic modes in such de-FIG. 1. .Two gravity. -wave detectors with their re-levant parameters.VOLUME 28, NUMBER 4 PHYSj;CAL RKVIKW LKTTKRS 24 JANUARY 19721(E/3p)1/2277 1(2n + 1)& '=tU '(E/m)"'(M/2m)'"W D =tv '(E/m)'/2~ 'n '.(la)(1b)(1c)The strain amplitude in each of the detectors atthe center istectors must take into account these other direc-tions as well as the detector suspension. For thepresent purposes of comparison this further de-tail is unnecessary .The end masses (M/2) ofthe dumbbell in Fig, 1 are also assumed to bepoint masses without internal degrees of freedom.The boundary conditions which supplement thestandard wave equation of elasticity neglectingdamping are as follows: The center of mass re-mains stationary. For the cylinder the stressvanishes at the ends. For the dumbbell the stressat an end must equal the inertial force on the endmass. These conditions lead to the normal fre-quencies of the detectors in the usual way. Byusing the eigenfunction solutions thereby obtained,the total energy in these detectors may then beexpressed as a sum over the energies in the in-dividual modes. For purposes of comparison wemay choose the initial conditions as is convenientand thus take the time derivative of the displace-ment to be initially zero. With v denoting thelongitudinal sound velocity, E denoting the energyin the nth mode, the superscript C (D) denotingcylinder (dumbbell), n=0, 1, 2, for thecylinder and n=1, 2, ~ - ~ for the dumbbell, weobtain for the displacement amplitudescm ' sec ' Hz ' and o„ is the detector cross sec-tion for the nth mode averaged over direction,We assume that I(v) is approximately constantover any bandwidth where the cross section isappreciable and compute the frequency-averagedcross section following the outline of Refs. 7 and8. Proceeding in this fashion and using Eqs. (1)we obtain a ratio of the signal displacement am-plitude in a dumbbell to that in a cylinder whichdepends only on detector parameters and variesessentially as (l/I-)(M/m)'/2 for comparison oflowest modes. Although the signal displacementamplitude of the dumbbell for 1 & I. and (M/m)»1 ean be larger for the dumbbell than for acylinder, the displacement amplitude due tothermal noise makes the total displacement am-plitude signal-to-noise ratio less for the dumb-bell than for the cylinders as is shown below,The signal strain amplitudes for the lowest modeof the dumbbell and the cylinder are independentof the lengths and masses and are comparablein magnitude. In the higher modes both the strainand the displacement amplitudes are consider-ably reduced in the dumbbell as compared to thecylinder by the ratio m/M as well as a decreasedue to mode number dependence.Using Eqs. (1) and (2) with the equipartitiontheorem, one obtains expressions for the dis-placement and strain fluctuation due to thermalnoise. Defining the signaI-to-noise ratio asS/N, we obtain for detectors at the same temper-ature the following comparisons valid for bothdisplacement and strain amplitudes:S = 2v '(E/%') "2D —2+ - (E/. )1/22 -1/2S D=2v '(E/m)1/2(2a)(2b)(2c)(S/N), ' ~ 2m '"(s/N) ' 4 x(S/N)„1 m rn '/22n'+1(S/Ã) „21/M K(3a)(3b)One notes that for equal amounts of energy de-posited, the strain in a dumbbell exceeds that ina cylinder by essentially the factor (K/m)'/2,where 9R is the same order as M. This fact isof course a two-edged sword. For a given ener-gy deposited from a gravity wave the strains de-veloped in the dumbbell are larger, but thenagain so are those strains resulting from ther-mal-noise energy. We return to the thermal-noise question in a later paragraph and considerfirst the energy deposited from a gravity-wavepulse.The average energy deposited in the nth modein a time b, t is given by E =t1t JI(v)o„dv, whereI(v) is the intensity of the gravity wave in ergsFor the situation considered here where m «M=SR, one readily sees that in all cases the signal-to-noise ratio of a dumbbell is less than that ofa cylinder. In particular, when the higher modesof a dumbbell are compared with the lower modesof a cylinder the signal-to-noise ratio of thedumbbell compared to the cylinder is even worsebecause of the mode number factors in Eq. (3b).That the higher mode response of a dumbbellshould be severely reduced is readily understoodwhen one views the dumbbell as approximatedby two masses connected by a massless spring.Such a system has only a fundamental mode withfrequency ~ determined by the spring and no high-er modes. The right-hand side of Eq. (3a) may in254VOLUME 28, NUMBER 4 PHYSICA L REVIE%' LETTERS 24 J&N Upas 1972fact be written in terms of the parameters ofsuch an equivalent mass-spring system asm'4 '(M/29P)'"(t/1. )(e/&uc'),where ~,' is the fundamental frequency of thecylinder. As m/ivl becomes smaller one wouldexpect such an approximation to be even moremeaningful. "In making these comparisons of signal to noisein various modes it has been assumed, for rea-sons of simplicity and lack of observational in-formation, that the gravity-wave intensities atthe corresponding frequencies are equal. Forcomparisons corresponding to quite differentfrequencies the signal-to-noise ratios in Eqs. (3)are multiplied by the ratio of intensities for anyreal spectrum. If this ratio strongly favors lowfrequencies, we see from Eq. (3a) that it is pos-sible for a dumbbell operating in its lowest modeto have sensitivity comparable to a cylinder oper-ating at a much higher frequency.Since both displacement'" and strain' ' detec-tion have been proposed, both have been con-sidered here. The question of Q dependence,phase relationship between signal and noise, timeresolution, and transducer effects have not beenincluded in the present work since the point ofinterest here is in a sensitivity comparison be-tween two types of gravitational-wave detectorsand not in the calculation of absolute sensitivities.A clever detection system could conceivably im-prove the relative sensitivity of a dumbbell to acylinder. "%e conclude that cylinders are likely to besuperior to dumbbells as gravity-wave antennasin any frequency range which is mutually acces-sible.It is a pleasure to acknowledge helpful com-ments from Kip S. Thorne.*Work supported in part by the National Science Foun-dation.J. Weber, Lett. Nuovo Cimento 4, 653 (1970).J. Weber, General Relativity and Gravitational 8'aves(Interscience, New York, 1961).3D. H. Douglass and J. A. Tyson, Nature 229, 34(1971).D. H. Douglass, in Proceedings of a Conference onExperimental Tests of Gravitational Theories, Califor-nia Institute of Technology, 11 November 1970 (unpub-lished) .G. W. Gibbons and S. W. Hawking, "The Detection ofShort Bursts of Gravitational Radiation, " 1971 (to bepublished) .The possibility that dumbbells may have applicationas gravity-wave antennas was originally suggested byI. Rudnick (private communication) as a means of look-ing at 60-Hz radiation from the Crab pulsar. More re-cently dumbbells have been discussed by D. H. Douglassand J. A. Tyson.'R. Ruffini and J. A. Wheeler, "Relativistic Cosmologyand Space Platforms" (to be published)."C. W. Misner, K. S. Thorne, and J. A. Wheeler,Gravitation (Freeman, San Francisco, 1971).V. B. Braginskii and V. N. Rudenko, Usp. Fiz. Nauk13, 395 (1970) tSov. Phys. Usp. 13, 165 (1970)] .W. O. Hamilton, Bull. Amer. Phys. Soc. 15, 1361{1970).One is tempted to expect these results as ~—~ toapproach those calculated using eigensolutions appro-priate to a "clamped" bar. This is not the case be-cause a "clamped" bar is unphysical in the context ofgeneral relativity where the motion of the "clamps"must be considered in calculating the frequency-aver-aged cross section.In a private communication J. A. Tyson reports ex-perimentally measured relative sensitivities for thedumbbell modes to be worse using strain gauges thansuggested by the considerations of this Letter.255

Two Gravity-Wave Detectors: A Comparison

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Two Gravity-Wave Detectors: A Comparison

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