gravitational wave detection by a spherical antenna: the angular sensitivity of resonators in the...

PHYSICAL REVIEW D 72, 104012 (2005)

Gravitational wave detection by a spherical antenna: The angular sensitivity of resonatorsin the truncated icosahedral gravitational wave antenna configuration and its variation with

sidereal time and galactic longitude

Maria Alice GaspariniDepartement de Physique Theorique, Universite de Geneve, 24 quai Ernest-Ansermet, CH-1211 Geneve 4

(Received 25 June 2005; published 18 November 2005)

*Electronic

1550-7998=20

Experimental projects using spherical antennas to detect gravitational waves are nowadays a concretereality. The main purpose of this paper is to give a possible way of interpreting output data from such asystem. Responses of the five fundamental quadrupole modes and of the six resonators in truncatedicosahedral gravitational wave antenna (TIGA) collocations are shown as a function of the incomingdirection of the incident wave. Then, for a source lying in the galactic plane, sidereal time and galacticlongitude dependence is given. Thus, once a candidate source of gravitational waves is considered, we canexactly predict the resonators’ response as a function of time.

DOI: 10.1103/PhysRevD.72.104012 PACS numbers: 04.30.�w

I. INTRODUCTION

Theoretical interest in spherical gravitational wave(GW) antennas dates back to the 1970s. Spherical antennashave a greater cross section than a bar detectors of similardimensions. More importantly, they have both omnidirec-tional and omnipolarization sensitivity, and also the poten-tial to detect the direction of wave provenance.

Interest in experimental research into resonant sphereshas increased over the past 15 years, and today sphericalantennas are recognized to be the new generation of gravi-tational resonant detectors, to complement existing cylin-drical antennas. Two experiments are under way:MiniGRAIL in Leiden (Holland) [1] and The GravitonProject in Sau Paulo (Brasil) [2].

One of the main problems today in detecting gravita-tional waves with resonant spheres is to find the ‘‘best’’location for amplifiers on the detector surface in order tooptimize sensitivity. To date, no theoretical basis has beendeveloped to provide a definitive solution to this problem.Nevertheless many suggestions have recently been pro-posed, including the uncoupled transducer configurationby Zhou and Michelson in 1994 [3], and the PHC configu-ration by Lobo and Serrano in 1996 [4]; both are five-resonator configurations, the first including radial and tan-gential motion of the resonators, the second only radialmotion. In addition, the TIGA configuration, proposed byMerkowitz and Johnson [5] (1995), consists of six trans-ducers moving radially, and it enjoys a peculiar symmetrywhich greatly simplifies equations.

The general theoretical aspects of an ideal simple sphereinteracting only with gravitational radiation have alreadybeen treated in the literature (see for example [6] or [7]). Inthe first part of this paper we will work out briefly some ofthem and obtain the energies stored in the five fundamentalsphere quadrupole modes, as well as their dependence on

address: [email protected]

05=72(10)=104012(13)$23.00 104012

the incident wave direction and polarization. Taking intoaccount of earth’s rotation and the detector’s location, wewill translate the direction dependence into galactic longi-tude and sidereal time dependence for a source lying in thegalactic plane.

We will review the general problem of the sphericaldetector with radial moving resonators on its surface, againin the case of high signal-to-noise ratio (SNR). We thenfocus our attention on the TIGA configuration, finding theoscillation amplitudes for the six transducers as a functionof the incident GW direction. With the same method usedin the case of the simple sphere without amplifiers, we willfind the resonators’ amplitude dependence on sidereal timeand on galactic longitude for a source lying in the galacticplane which emits randomly polarized GW radiation.

The paper is organized as follows. In Sec. II we brieflysketch some general aspects of the physics of GW resonantdetectors without any assumption on the shape of theantenna. In Sec. III we focus our attention on the sphericaldetector, and we present the solutions to the equations ofmotion followed by the five modes angular and siderealtime sensitivity. In Sec. IV results of the previous sectionsare generalized to the case of the spherical detectorcoupled to a set of radial transducers. Section V containsthe summary and conclusions.

II. GENERIC RESONANT DETECTORS

A. The equations of motion

We will work with a solid whose density is ��~r�, where~r � �r1; r2; r3� � �x; y; z� is the position of an infinitesimalmass element relative to the center of the solid and we callthe Lame coefficients of the material � and �.

We suppose that a gravitational wave hits our detector atthe time t � 0. If we call ~u� ~r; t� the small displacementfrom the equilibrium position at the point ~r and time t, theequation of motion turns out to be [8]

-1 © 2005 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.72.104012


�@2 ~u� ~r;t�

@t2��r2 ~u�~r;t�� ~r� ~r� ~u�~r;t�� ~F� ~r;t�;

(1)

where Fj� ~r; t� � �Rj0k0xk� ~r; t� is the jth component of the

tidal force density attributable to the GW [9].As is well known for a resonant detector the spatial

dependence of the Riemann tensor can be neglected.Moreover, as all our summation indices are spatial andwe reasonably assume the background metric to be flat onthe earth’s surface (g�� h��), we can write themall as lowered indices. Thus, in the first order of h, theRiemann tensor can be written Ri0j0 �

12

�hij�t�; which is arank 3 traceless symmetric tensor. It represents then aparticle of spin 2, the graviton. Such a tensor can bedecomposed into a base of 5 matrices. In order to dothat, we follow the procedure used by Lobo [10], workingwith a basis of real matrices instead of imaginary matrices,as is usually done:

M1c �

��15

16�

s 0 0 10 0 01 0 0

0@ 1A;

M2c �

��15

16�

s 1 0 00 �1 00 0 0

0@ 1A;

M1s �

��15

16�

s 0 0 00 0 10 1 0

0@ 1A;

M2s �

��15

16�

s 0 1 01 0 00 0 0

0@ 1A;

M0 �

��5

16�

s�1 0 00 �1 00 0 2

0@ 1A:

(2)

Noting I the identity matrix, the above matrices have thefollowing properties:

M�ijM

�ij �

15

8��; IijM

�ij � 0;

5

2IijIkl �

X�

M�ijM

�kl �

15

16��ikjl � iljk�;

(3)

with �;� � 0; 1c; 1s; 2c; 2s. Then we can rewrite the jcomponent of the force density:

Fj�~r; t� � �Rj0k0�t�rk

��2rk

16�15

Rl0m015

16��mjlk � ljmk�

�8�15

Xs

�M�jkrkM

�lmRl0m0 �

X�

f�j �h��t�: (4)

MARIA ALICE GASPARINI

104012

Repeated indices are summed. Equation (4) introducesthe functions �h��t� � 8�

15 M�lmRl0m0 and f�j � ~r� � �M�

jkrk.

Direct computation gives for the ~f�:

~f1c�~r� � �

��15

16�

s r3

0

r1

0BB@

1CCA; ~f1s

�~r� � �

��15

16�

s 0

r3

r2

0BB@

1CCA;

~f2c�~r� � �

��15

16�

s r1

�r2

0

0BB@

1CCA; ~f2s

� ~r� � �

��15

16�

s r2

r1

0

0BB@

1CCA;

~f0�~r� � �

��5

16�

s �r1

�r2

2r3

0BB@1CCA: (5)

We now want to write down the explicit expressions forthe functions �h��t� given a GW propagating along a ge-neric direction with respect to the detector coordinatessystem. In the frame where the GW propagates along thez axis we can pass into the transverse and traceless (TT)gauge in order to rewrite the Riemann tensor as

Ri0j0 �1

2

�h� �h� 0�h� � �h� 00 0 0

0B@1CAij

: (6)

We call the angle from the z axis and � the anglebetween x and the projection into the x; y plane. �x; y; z�is the detector frame basis. For a generic arrival directiongiven by the angle �;��, we have to perform a rotation inorder to obtain the Riemann in the antenna frame:

Ri0j0 �1

2

0B@M �h� �h� 0�h� � �h� 00 0 0

0B@1CAMT

1CAij

(7)

with

M �

cos�� cos�� sin�� sin�� cos��cos�� sin�� cos�� sin�� sin�� sin�� 0 cos��

0@ 1A:(8)

We now are able to compute the components �h��t�which are

-2

GRAVITATIONAL WAVE DETECTION BY A SPHERICAL . . . PHYSICAL REVIEW D 72, 104012 (2005)

�h 0�t� ��5

rsin2�� h��t�; �h1c�t� �

��4�15

s�� cos�� sin�� cos�� h��t� � sin�� sin�� h��t��;

�h1s�t� �

��4�15

s�cos�� sin�� sin�� h��t� � sin�� cos�� h��t��;

�h2c�t� �

��4�15

s ��1� cos2��

�cos2��

1

2

��h��t� � 2 cos�� sin�� cos�� h��t�

�;

�h2s�t� �

��4�15

s��1� cos2�� cos�� sin�� h��t� � cos��cos2�� sin2�� h��t��:

(9)

Note that we have defined the � and � polarization withrespect to the detector frame. In the case where the sourcehas an intrinsic polarization e�; e�, we have to make thesubstitution

h� � cos�2 �e� � sin�2 �e�;

h� � � sin�2 �e� � cos�2 �e�;(10)

where describes the polarization angle. This becomesuseful if we want to take the average of the polarization.

B. The general solution

The solution of Eq. (1) can be expressed formally bymeans of the Green function integral [10], getting:

~u� ~r; t� �X�

XN

!�1N f�Ng�N�t� ~�N� ~r�; (11)

where

f �N � M�1Z

Solid

~�N� ~r� � ~f�� ~r�d3r;

g�N�t� �Z t

0

�h��t0� sin!N�t� t0�dt0;(12)

and ~�N�~r� are the normalized (i.e. they verifiesRSolid

~�N� ~r� � ~�N0 �~r�� ~r�d3r � MN;N0) eigenfunctions

solutions of the corresponding homogeneous equation

�!2N~�N ��r2 ~�N � �� ~r� ~r � ~�N� � 0 (13)

with the appropriate boundary conditions. N indicates acollective quantum number denoting the quantum state.

Putting the above result in a different form, i.e. writingthe solution as

~u�~r; t� �XN

BN�t� ~�N�~r�; (14)

it is obvious, as expected from [9] that the quantity BN �

104012

P�!�1N f�Ng�N�t� satisfies equation

�BN�t��!2NBN�t� � FN�t� �

X�

f�N �h��t�

�X�

M�1Z

Solid

~�N� ~r� � ~f�� ~r� �h��t�d3r

�Z

SolidM�1�� ~r�Rj0k0�t�rk�N;j� ~r�: (15)

Each mode is then formally equivalent to a one-dimensional harmonic oscillator with frequency !N ,driven by a force per unit mass FN �

P�f�N �h��t�: For

such a system, the expression for the energy per massunit adsorbed from the driving force is given by [11]:

Es �1

2

��Z �1�1 FN�t�e�i!Ntdt��2: (16)

Before moving on to the case of the spherical detector, itwould be interesting to use this general method on acylindrical detector. While we omit this exercise here, itcan easily be verified that this general method reproducesthe results for the eigenfunctions and the energy stored inoscillations in the well-known case of the one-dimensionalhomogeneous bar [12].

III. THE SPHERICAL DETECTOR

A. The normal modes

The normal eigenfunctions of Eq. (13) for the sphere canbe found in the literature (see, for example [7]). Note thatthey are usually given in terms of imaginary sphericalharmonics. Here we will put all results in terms of realspherical harmonics, but the shape of the solutions is thesame. Let us briefly recall the main points. There are twokinds of solutions with boundary conditions �ijnj � 0 atr � R, with �ij the stress tensor, conventionally defined as�ij � �kkij� 2�uij, with uij �

12 �ui;j � uj;i�. These so-

lutions are called the toroidal ones and the spheroidal ones:

~� Tnl��~r� � Tnl�r�i ~LYl��;��;

~�Snl�� ~r� � Anl�r�Yl��;��n� Bnl�r�in� ~LYl��;��;

(17)

-3

MARIA ALICE GASPARINI PHYSICAL REVIEW D 72, 104012 (2005)

where each mode n; l; � corresponds to the generic N inEq. (13), n is a positive integer which represents the energylevel for a fixed angular momentum l and Yl� are the realspherical harmonics with kinetic momentum l obtained bythe imaginary ones Yl;�m in the following way:

Yl;0 � Yl;0; Yl;mc �1��2p �Yl;�m � ��1�mYl;�m�;

Yl;ms �i��2p �Yl;�m � ��1�m�1Yl;�m�

(18)

with m � 0; . . . ; l. So defined, all the spherical harmonics(18) are real quantities. For l � 2 this gives:

Y2;0 �

��5

4�

s �3

2cos2�

1

2

�;

Y2;1c �

��15

4�

s1

2sin2 cos�;

Y2;1s �

��15

4�

s1

2sin2 sin�;

Y2;2c �

��15

4�

s1

2sin2 cos2�;

Y2;2s �

��15

4�

s1

2sin2 sin2�:

(19)

Also, given a generic unity vector in spherical coordinates,n � �sin cos�; sin sin�; cos� we can see that

Y2;��;�� M�ijninj; (20)

where M�ij are the matrices defined in Eq. (2).

Anl�r�; Bnl�r�, and Tnl�r� are scalar functions of r:

Tnl�r� � C0�n; l�jl�knlr�;

Anl�r� � C�n; l��3�knlR�j

0l�qnlr�

� l�l� 1�qnlknl

�1�qnlR�jl�knlr�knlr

�;

Bnl�r� � C�n; l��3�knlR�

jl�qnlr�qnlr

�qnlknl

�1�qnlR�knlrjl�knlr�0

knlr

�; (21)

where the jl are spherical Bessel functions and

�0�x� �jl�x�

x2 ; �1�x� �ddx

jl�x�x

;

�2�x� �d2

dx2 jl�x�;

�3�x� �1

2�2�x� �

�l�l� 1�

2� 1

��0�x�:

(22)

104012

The k and the q are related to the quantized ! of Eq. (13)by

k2nl �

�!2nl

�; q2

nl ��!2

nl

�� 2�: (23)

B. Interaction with a gravitational wave

We start by computing f�nl�0 of Eq. (12) for our sphericaldetector for both toroidal and spheroidal eigenfunctions:

f�Snl�0 � M�1Z

Sphere

~�Snl�0 �~r� � ~f

��~r�d3r � anl;2�;�0 ;

f�Tnl�0 � M�1Z

Sphere

~�Tnl�0 �~r� � ~f

��~r�d3r � 0;

(24)

with

an � �1

M

Z R

0�r3�An2�r� � 3Bn2�r��dr: (25)

It is clear from Eq. (24) that toroidal modes do not enterinto play, so we will work from now on only with spheroi-dal modes and we will drop the S for simplicity.

Once the Riemann tensor and the �h��t� are known for agravitational wave coming from a generic direction �;��,it is easy to compute g�nl�0 �t� using Eq. (12). The solution ofEq. (1) can be written:

~u� ~r; t� �X1n�1

~un� ~r; t� �X1n�1

an!n2

X�

~�n2��~r�g�n2�t�: (26)

For spheroidal modes, BN�t� � Bn;l;��t�; as defined in theprevious paragraph. In this case,

BN�t� � Bn;l;��t� �X�

!�1nl f��n;l;�g��n;l;� � !�1

n2 ang�n2�;

and its equation of motion (15) is

�B n;l;� �!2n2Bn;l;� � l;2an �h��t�: (27)

The energy stored in the mode � at frequency !nl is(from Eq. (16)):

Es�n; l; �� 1

2

��Z �1�1 Fnl��t�e�i!nltdt

��2: (28)

Using Eqs. (15) and (24), Eq. (28) becomes

Es�n; l; �� 1

2a2nl;2

��Z �1�1 �h��t�e�i!n2tdt��2: (29)

The only contributions have, as expected, l � 2. For these,we fix the value of n and replace �h� using Eq. (9).

-4


For the � � 0 mode, we haveZ �1�1

�h�0��t�e�i!n2tdt �Z �1�1

��5

rsin2 �h��t�e

�i!n2tdt

�

��5

rsin2!2

n2

Z �1�1

h��t�e�i!n2tdt

�

��5

rsin2!2

n2~h��!n2�; (30)

104012

thus

Es�n; 2; 0� ��10a2nsin4!4

n2j~h��!n2�j

2: (31)

Similarly, for the other modes, we get:

Es�n; 2; 1c� �2�15a2n!4

n2cos2sin2cos2�j~h�j2 � sin2sin2�j~h�j2 � cossin2 sin2�<�~h� ~h��;

Es�n; 2; 1s� �2�15a2n!

4n2cos2sin2sin2�j~h�j

2 � sin2cos2�j~h�j2 � cossin2 sin2�<�~h� ~h��;

Es�n; 2; 2c� �2�15a2n!

4n2

��1� cos2�2

�cos2��

1

2

�2j~h�j

2 � 4cos2sin2�cos2�j~h�j2

� 4 cos sin� cos��1� cos2��cos2��

1

2

�<�~h� ~h��;

Es�n; 2; 2s� �2�15a2n!4

n2�1� cos2�2cos2�sin2�j~h�j2 � cos2�cos2�� sin2��2j~h�j2

� 2 cos sin� cos��1� cos2��cos2�� sin2��<�~h� ~h��:

(32)

We have assumed that h�;��t� and _h�;��t� decrease fast enough at �1 to allow the integration by parts, ~h��f� and~h��f� stand for the Fourier transform of h��t� and h��t�: In this paper we use the notation ~a�!� �

R�1�1 dta�t�e

�i!t for theFourier transform and a�t� �

R�1�1 d!~a�!�e�i!t for the inverse. In Eq. (32) all Fourier transforms are evaluated at the

frequency fn2 � !n2=2�.If the source emits randomly polarized radiation, we can rewrite h� and h� in terms of one possible intrinsic

polarization of the source (Eq. (10)) and average over . We finally obtain

�E s�n; 2; 0� ��15a2n!4

n2�j~e�j2 � j~e�j2�

3

4sin4;

�Es�n; 2; 1c� ��15a2n!

4n2�j~e�j

2 � j~e�j2�sin2�cos2cos2�� sin2��;

�Es�n; 2; 1s� ��15a2n!

4n2�j~e�j

2 � j~e�j2�sin2�cos2sin2�� cos2��;

�Es�n; 2; 2c� ��15a2n!

4n2�j~e�j

2 � j~e�j2��1� cos2�2

�cos2��

1

2

�2� 4cos2sin2�cos2�;

�Es�n; 2; 2s� ��15a2n!

4n2�j~e�j

2 � j~e�j2��1� cos2�2cos2�sin2�� cos2�cos2�� sin2��2:

(33)

As before, all Fourier transforms are evaluated at thefrequency !n2.

It is important to stress at this point that if we sum theenergies of Eq. (33) we obtain a constant value:

�E s�n; 2; tot� �2�15a2n!4

n2�~e��!2n�2 � ~e��!2n�

2�: (34)

This means that the total sensitivity of our detector isalways the same, independently of the GW direction.This is to be expected because of the symmetry of thedetector, and it is one of the most important propertiesdifferentiating spherical detectors from cylindrical ones.Together with its greater sensitivity (keeping the same

dimensions), this feature makes the new generation ofresonant detectors much more advantageous than bars.

Figure 1 shows, for each mode n, how the energy storedin each � varies as a function of and �; the angles thatgive the direction of the incoming unpolarized GW. Therate

� �15 �Es�n; 2; ��

�a2n!

4n2�~e

2� � ~e2

��

is zero where the graph is blue, and maximal ( 34 ) where the

graph is red.The total sensitivity is the sum of contributions coming

from the five modes � and the sensitivity of each mode

-5

–3

–2

–1

0

1

2

3

b

0 5 10 15 20t

ε1s

–3

–2

–1

0

1

2

3

b

0 5 10 15 20t

ε1c

–3

–2

–1

0

1

2

3

b

0 5 10 15 20t

ε2s

–3

–2

–1

0

1

2

3

b

0 5 10 15 20t

ε2c

–3

–2

–1

0

1

2

3

b

0 5 10 15 20t

ε0

FIG. 2 (color online). Sensitivity of the five fundamentalmodes as a function of the GW source position in the galacticplane, �b�; and of sidereal time in sidereal hours, (t), for a spherenear Leiden, NL (latitude l � 52:16�N, longitude L � 4:45�E).As in Fig. 1, the lightest zones indicate maximal values.

0

0.5

1

1.5

2

2.5

3

theta

0 1 2 3 4 5 6phi

ε1s

0

0.5

1

1.5

2

2.5

3

theta

0 1 2 3 4 5 6phi

ε1c

0

0.5

1

1.5

2

2.5

3

theta

0 1 2 3 4 5 6phi

ε2s

0

0.5

1

1.5

2

2.5

3

theta

0 1 2 3 4 5 6phi

ε2c

0

0.5

1

1.5

2

2.5

3

theta

0 1 2 3 4 5 6phi

ε0

FIG. 1 (color online). Sensitivity of the five fundamentalmodes as a function of the GW arrival direction �;��: Theratio � is zero where the graph is darkest, and maximal ( 3

4 )where the graph is lightest.


depends on the orientation of the source with respect to thechosen frame. This means that for different angular posi-tions of the source with respect to the detector referenceframe, there will be different distributions among the fivemodes of the total energy transferred from the GW to thesphere. In contrast, if the source is fixed in space, and thesphere moves because of the earth’s rotation, the energyratio stored in each of the five modes will be a periodicfunction of the arrival time, with period one sidereal day,while the sum of the five contributions will be time-independent.

The expected sources for GW resonant detectors aretraditionally distinguished into: (1) burst of relatively greatintensity, given by cataclysmic events as binary coalescing,or supernovae explosions; (2) sources which emit GW

104012

bursts occasionally but repeatedly in time and with pecu-liar statistic features, the so-called ‘‘GW-bursters’’ [13];(3) periodic sources, of longer duration and smaller spreadin frequency, as pulsars or inspiraling binaries. Such as-trophysical objets are present in galaxies, and as the energyhitting a detector decreases as the distance of the sourcesquared [14], it is then reasonable to expect a GW signalcoming from our galaxy to be more easily detected.

Once the position of the source is known in galacticcoordinates �b; lg�; a set of space rotations gives the posi-tion of the source in the detector frame �xd; yd; zd�: Theserotations are derived and explained in detail in the appen-dix of [14]. For a given �b; lg� one can then easily find theprecise position ��t�; ��t�� of the source with respect tothe detector frame, with sidereal time t varying from 0 to24.

As example, Fig. 2 displays the energy rates � stored inthe five modes � of a sphere near Leiden, Holland (latitudel � 52:16�N, longitude L � 4:45�E, as functions of

-6

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20t

FIG. 3 (color online). 0 of Fig. 2 as a function of t in siderealhours for b � 0 (source near the galactic center).


Greenwich sidereal time (in sidereal hours) and of galacticlongitude b for a randomly polarized source lying in thegalactic plane (lg � 0), where stars population is moreimportant (and then their remnants population also). Asbefore, red zones indicate maximal values. Finally, theexample of a section (at b � 0, i.e. for a source near theGC) for the � � 0 mode is represented in Fig. 3. Thesegraphics can be specially interesting for sources of type 2and 3 mentioned above, for which a signal from the samelocation persists in time.

IV. THE SPHERICAL GW DETECTOR WITH A SETOF RESONATORS ON THE SURFACE

Until now we have looked at the spherical detector as aunique object interacting with GW. Nevertheless, becauseof the smallness of h, the magnitude displacement of thedetector as a result of this interaction is tiny, of the order ofhL, L being the detector size.

For this reason small resonators of equal mass mr Mand elastic constant kr are put on the surface of the GWresonant detector. The small resonators are tuned to thefundamental frequency of the detector, in order to amplifydisplacements of its surface at that chosen frequency,which for the sphere turns out to be !1;2. In this way,frequencies other than the fundamental mode and nearbyfrequencies will not be amplified by the system.

A. The general case of K resonators

Imagine having K such resonators on the surface of thesphere. We are interested in what happens when a GWinteracts with such a system, in the ideal case of high SNR(i.e. where noises are negligible). In line with previouswork, in deriving the equations of motion for such a systemwe will use the formalism and notations of [5], where theproblem has already elegantly been posed, and a proposalfor a six-detector configuration, the TIGA configuration, ispresented.

104012

The equation of motion (1) for small displacements nowbecomes

�@2 ~u�~r; t�

@t2��r2 ~u� ~r; t� � �� ~r� ~r � ~u�~r; t��

�X�

~f��~r�g��t� (35)

where � is a parameter that takes the values0; 1c; 1s; 2c; 2s; 1; 2; . . . ; K � f�; ig: It is clear that the firstcomponents of this force density,

~F GW �X�

~f�� ~r�g��t� �X�

~f��~r�h��t�; (36)

are the usual ones (see Eq. (4)) attributable to GW, while

~F res �XKi�1

~f�i�� ~r�g�i��t� (37)

are new contributions given by the elastic terms ofresonators.

Let us focus now on one resonator, which we assume tohave only radial motion, linked with a tuned spring to thesurface of the sphere, say at the position ~Rk. We define thescalar-time dependent quantity

zi�t� � ~u� ~Ri; t� � ri (38)

as the radial projection of the surface displacement at thispoint. We also define qi�t� to be the relative distancebetween the resonator and the sphere surface. Note thatqi�t� is not an inertial coordinate, but zi�t� � qi�t� is. Thenthe equation of motion for this last quantity is

mr� �qi�t� � �zi�t�� krqi�t� � FGWi �t�; (39)

where FGWi is the radial component of the GW force

density acting on the resonator, which will be neglectedas we know that mr M.

Since

gi�t� � qi�t�; ~fi� ~r� � kr3� ~r� ~Ri�Ri; (40)

the force density component (37) can be written:

~F res �Xi

krqi�t�3� ~r� ~Ri�Ri: (41)

Generalizing the same method used above (in particular,see Eqs. (11), (12), and (14)) the solution of Eq. (35) can beexpressed as

~u� ~r; t� �X�

Xn;l;�

!�1n;l;�f�n;l;�g�n;l;��t�

~�n;l;��~r�

�Xn;l;�

Bn;l;��t� ~�n;l;�� ~r� (42)

with

-7


f�n;l;� �1

M

ZSphere

~�n;l;��~r� � ~f��~r�d3r;

g�n;l;��t� �Z t

0g��t0� sin�!n;l;��t� t0��dt0:

(43)

The last equality of Eq. (42) defines

Bn;l;��t� �X�

!�1n;l;�f�n;l;�g�n;l;��t�: (44)

In our case we get

f �n;l;� ��an;l�;�0 if � � �0krM An;l�R�Yn;l;��i; �i� if � � i:

(45)

It is then straightforward to derive the equations ofmotion for the sphere modes Bn;l;��t�; by insertingEq. (42) into Eq. (35) :

�Bn;l;��t� �!2n;l;�Bn;l;��t� �

X�

f�n;l;�g��t�

� an;l �h��t� �XKi�1

krMAn;l�R�

� Yn;l;��i; �i�qi�t�: (46)

When noises are negligible, the motion is dominated byquadrupole modes (l � 2; � � 0; 1c; 1s; 2c; 2s), the onlyones interacting with GW. We limit ourselves to the n � 1modes, for which the cross section is the most significant[15]. Note that for l � 2, an;2 � an of Eq. (25). From nowon, we simplify the notation, writing

B1;2;��t� � B��t�; a1;2 � a; A1;2�R� � A;

Y1;2;��i; �i� � Y��i; �i� � Y�i:(47)

This last equality defines the 5� K matrix Y.Multiplying for the sphere mass M, and calling the five

equal quantities M!21;2;� � M!2

s � ks; Eq. (46) can berewritten

M �B��t� � ksB��t� � Ma �h��t� �Xi

krAY�iqi�t�: (48)

Defining now the vectors

fqi�t�g � q�t�; fMa �h��t�g � f �H��t�g � �H�t�;

fB��t�g � b�t�;(49)

we can again simplify the expression for these equations:

MARIA ALICE GASPARINI

104012

MI5�b�t� � ksI5b�t� � krAYq�t� � �H�t�: (50)

It is evident that b and h are 5-component vectors, q is aK-component vector and I5 the identity matrix 5� 5. ThusEq. (50) is a system of 5 equations. In the same way we canrewrite Eq. (39) in matrix form. To do that, we need toexpress zi as a function of B�. Using the approximationthat only the n � 1 modes of the sphere are excited at highSNR, from Eq. (38) we have

zi�t� � ~u� ~Ri; t� � ri �X

n�1;l�2;�

B��t�AY�i: (51)

Defining the vector fzig�t� � z�t�,

z �t� � AYTb�t�: (52)

Then Eq. (39) becomes

mr� �q�t� � AYT �b�t�� krq�t� � 0: (53)

The K equations for q (53) and the 5 equations for b (50)can be rewritten as a system of K � 5 equations

MI5 0

mrAYT mrIK

" #�

�b

�q

" #�t� �

ksI5 �krAY

0 krIK

" #

�b

q

" #�t� �

�H�t�

0

" #: (54)

These are the equations of motion for a system com-posed of a homogeneous sphere with K resonators movingradially on its surface, all interacting with a GW.

In the remainder of this paper, we will solve theseequations once a configuration for the positions of theresonators is given. We first look at what happens in thecase with only one resonator (k � 1), in order to test theequations by looking at a simple example. Then we moveon to the TIGA configuration (K � 6) proposed byMerkowitz and Johnson in [5].

B. The case of one resonator at � � 0

In this particular case we choose to put only one reso-nator on the surface of the sphere. Obviously each positionis equivalent, but for simplicity let us put the resonator atthe north pole of our coordinates system (or, equivalently,at the south pole because of the symmetry of the problem).

In this case, the only nonvanishing component of the 5�

1 matrix Y is Y01 ��5

4�

q� y. Then Eq. (54) can be written:

M 0 0 0 0 00 M 0 0 0 00 0 M 0 0 00 0 0 M 0 00 0 0 0 M 0

myA 0 0 0 0 m

2666666664

3777777775�

�B0�B1c�B1s�B2c�B2s

�q

2666666664

3777777775�

ks 0 0 0 0 �krAy0 ks 0 0 0 00 0 ks 0 0 00 0 0 ks 0 00 0 0 0 ks 00 0 0 0 0 kr

2666666664

3777777775�

B0

B1c

B1s

B2c

B2s

q

2666666664

3777777775�

�H0

�H1c

�H1s

�H2c

�H2s

0

2666666664

3777777775: (55)

It is clear from Eq. (55) that the only mode of the sphere which couples with our resonator is B0, while all the othersbehave as if the resonator was not present. Their equations of motion are the same as given above in Eq. (27). So we can

-8

0

2e–18

4e–18

6e–18

8e–18

1e–17

.2e–17

.4e–17

998.5 999 999.5 1000 1000.5 1001 1001.5

omega

FIG. 4. j~q�!�j2 with a sphere damping of 10�4, a resonatordamping of 10�2, !0 � 103 Hz, � 10�6, and ~h0�!� �10�19Hz�1.


reduce our system of equations to a system of 2 coupledequations with 2 unknowns, taking only the first and thelast lines of (55):

�B 0�t� �!20B0 � !2

0Ay q� a �h0�t�;

�q�t� �!20q�t� � �Ay �B0�t�:

(56)

In Eq. (56) we consider that the resonator frequency isperfectly tuned with the fundamental frequency of thesphere, i.e. that

ksM�krm� !2

0;

and we call the (small) ratio between the resonator massand the sphere mass.

In order to solve Eq. (56) we move to Fourier coordi-nates, giving:

~q�!� ��!4aAy~h0�!�

�!20 �!

2�2 �!2!20A

2y2 ;

~B0�!� � a!2 ~h0�!�!2 �!2

0

�!20 �!

2�2 �!20!

2A2y2 :

(57)

As expected, the resonant frequencies !� for both thecoupled quantities are the same, and they are the splittingof the original frequency!0. They can be found setting thedenominators of Eq. (57) equal to zero:

!2� � !2

0

�1� Ay

�� 1�

A2y2 4

�s�A2y2

2

�

� !20

�1�

�� A2y2

q�A2y2

2��A2y2 �3=2

8�O� 2�

�:

(58)

At the frequency !0 we have ~B0�!0� � 0, and ~q�!0� �aAy

~h0�!0� is proportional to the ratio between the spheremass and the resonator mass, so we will have the greatestoscillations if we make the resonator mass as small aspossible.

Note that here we neglected the damping of both thecoupled oscillators. This is because we are interested inburst signals, where the time-scale for damping an oscil-lation of the system is much longer than the time durationof a signal. Taking account of damping means changingEq. (57) to cancel divergences which arise at ! � !� andmake them smoother. In particular, ~B0�!0� will no longervanish. Computations adding damping terms are analogand easily done. Figure 4 shows a plot of j~q�!�j2 with asphere damping of 10�4, a resonator damping of 10�2,!0 � 103Hz, � 10�6, and ~h0�!� � 10�19Hz�1.

The quantity we are really interested in is the one we canmeasure, ~q�!�; or its squared absolute value. Rewriting ~h0

to show its dependence on ~h�, ~h� and on the wave direc-

104012

tion (see Eq. (9)) we get:

j~q�!�j2 �!8a2A2

4�!20 �!

2�2 �!20!

2A2y2 2sin4j~h��!�j2

� ��!�sin4j~h��!�j2; (59)

or, assuming that the GW is unpolarized, and averagingover polarizations,

j~q�!�j2

��!��j~e�j2 � j~e�j2�=2� sin4��; (60)

with

��!� �!8a2A2

4�!2 �!2��!

2 �!2��: (61)

This turns out to be the same angular sensitivity obtainedfor the cylindrical bar [12], for which one finds that sensi-tivity is optimal for a plane orthogonal to the bar and dropswith a factor sin4 too. Having only one transducer on thesphere surface is a simple example showing that the wayresonators are positioned can result in the loss of the main

-9


initial properties given by the spherical symmetry of thedetector. In order to keep these properties as far as possible,we have to put more resonators on the sphere in appropriatepositions.

C. The TIGA configuration

With the hope that, in future, we will be able to placeresonators moving radially on the sphere surface, and thatwe will be able to measure experimentally the displace-ments q of these masses, and assuming that all otherinteractions with the sphere-resonators system are negli-gible1 with the exception of gravitational waves, in thissection we find the response of ~q�!� in the TIGA configu-ration [5] to a gravitational wave coming from a genericdirection �;��: In order to do so, we have to solve theequations of motion (54) in Fourier space.

In the TIGA configuration there are a total of six reso-nators: three of them have azimuth angle 1 � 2 � 3 �A and�1 � 0; �2 �

2�3 ; �3 �

4�3 and the other three have

4 � 5 � 6 � B and �4 ��3 ; �5 � �;�6 �

5�3 . The

angles A and B are between 0 and �=2 and they aresolutions of the equation

45cos4� 30cos2� 1 � 0:

With this information, the Y matrix for this configuration iseasily found:

1��4�p 1��

4�p 1��

4�p � 1��

4�p � 1��

4�p � 1��

4�p��

3��5p

6�

q�

��3�

��5p

24�

q�

��3�

��5p

24�

q ��3�

��5p

24�

q�

��3�

��5p

6�

q ��3�

��5p

24�

q0

��3�

��5p

8�

q�

��3�

��5p

8�

q ��3�

��5p

8�

q0 �

��3�

��5p

8�

q��3�

��5p

6�

q�

��3�

��5p

24�

q�

��3�

��5p

24�

q�

��3�

��5p

24�

q ��3�

��5p

6�

q�

��3�

��5p

24�

q0 �

��3�

��5p

8�

q ��3�

��5p

8�

q ��3�

��5p

8�

q0 �

��3�

��5p

8�

q

2666666666664

3777777777775:

(62)

It is important to recall a very useful property of thismatrix: YYT � 3

2� I5 is proportional to the identity 5� 5matrix. Nevertheless, the inverse is not true: YTY has alldiagonal values equal to 5

4� and all non diagonal valuesequal to � 1

4� : It is easily seen that sums over lines orcolumns are always zero. Actually, YTY is a 6� 6 matrixobtained from two 5� 6 matrices, so it is clear that itcannot be invertible.

Using this property it is straightforward to solve theequations for the so-called mode-channels [16]

1In particular, we consider the case where another amplifica-tion device, such as SQID, is not tuned to the frequency ofresonance of the sphere.

104012

~p�!� � Y~q�!�; (63)

~p ��!� ��Aa!4 ~h��!�

2�3 �!

20 �!

2�2 � A2!2!20

; (64)

and for ~b�!�

~B��!� ��!2 �!2

0�!2a~h��!�

�!20 �!

2�2 � 32� A

2!2!20

: (65)

The denominators vanish for the two split frequencies

!2� � !2

0

�1�

3

4�

��8�3 A2 �

2A4

!20

s�

3

4� A2

�; (66)

so one can rewrite Eq. (64) as

~p ��!� � �3

2�Aa!4 ~h��!�

�!2 �!2��!

2 �!2��

� �

��

!2 �!2�

��

!2 �!2�

�3

2�!2A~h�; (67)

with

�� !2�

!2� �!

2�

; �� !2�

!2� �!

2�

: (68)

We then have an expression giving a linear combinationof resonator displacements as a function of the sphericalcomponents of a gravitational wave, or we have eachspherical component of ~h as a function of different combi-nations of displacements.

Nevertheless we are also interested in the inverse rela-tion: we want to write each ~qi as a combination of the ~h�:In order to do that we have to express 6 quantities as afunction of 5, which means inverting the matrix Y; a 5� 6matrix. We then need a supplementary condition for q. Thecondition to be added is true for the TIGA configurationonly, in the case of a high SNR. In that case, the sum of allthe six displacements is zero, and this can be seen bylooking at the only nontrivial solution of Yq � 0. Thismeans that we can add a sixth line to the Y matrix, a linemade up of six unities, and a sixth null component to the pvector in Eq. (63). We will call the new 6� 6 matrix Y andthe new 6-vector p.

Y is invertible, and we can easily find q using q �Y�1p; getting:

-10


~q 1�!� �

��p

3~p0�!� �

��3�p

9��5p� 1�~p1c�!� �

��3�p

9��5p� 1�~p2c�!�;

~q2�!� �

��p

3~p0�!� �

��3�p

18��5p� 1�~p1c�!� �

��p

6��5p� 1�~p1s�!� �

��3�p

18��5p� 1�~p2c�!� �

��p

6��5p� 1�~p2s�!�;

~q3�!� �

��p

3~p0�!� �

��3�p

18��5p� 1�~p1c�!� �

��p

6��5p� 1�~p1s�!� �

��3�p

18��5p� 1�~p2c�!� �

��p

6��5p� 1�~p2s�!�;

~q4�!� � �

��p

3~p0�!� �

��3�p

18��5p� 1�~p1c�!� �

��p

6��5p� 1�~p1s�!� �

��3�p

18��5p� 1�~p2c�!� �

��p

6��5p� 1�~p2s�!�;

~q5�!� � �

��p

3~p0�!� �

��3�p

9��5p� 1�~p1c�!� �

��3�p

9��5p� 1�~p2c�!�;

~q6�!� � �

��p

3~p0�!� �

��3�p

18��5p� 1�~p1c�!� �

��p

6��5p� 1�~p1s�!� �

��3�p

18��5p� 1�~p2c�!� �

��p

6��5p� 1�~p2s�!�:

(69)

0

0.5

1

1.5

2

2.5

3

theta

0 1 2 3 4 5 6phi

A1

0

0.5

1

1.5

2

2.5

3

theta

0 1 2 3 4 5 6phi

A2

0

0.5

1

1.5

2

2.5

3

theta

0 1 2 3 4 5 6phi

A3

0

0.5

1

1.5

2

2.5

3

theta

0 1 2 3 4 5 6phi

A4

0

0.5

1

1.5

2

2.5

3

theta

0 1 2 3 4 5 6phi

A5

0

0.5

1

1.5

2

2.5

3

theta

0 1 2 3 4 5 6phi

A6

FIG. 5 (color online). Sensitivity of the six resonators in TIGAconfiguration as a function of the GW arrival direction �;��. Asin Fig. 1, the ratio Ai is zero where the graph is darkest, andmaximal where the graph is lightest.

Taking the squared absolute value, and rewriting ~h��!� asa function of ~h�; ~h� and of the direction �;�� (seeEq. (9)) we can find the explicit behavior of the amplitudeof the oscillations for each resonator as a function of theGW direction and polarization.

Supposing the source to be randomly polarized, we canreplace ~h�;� with the expressions in Eq. (10), and averageover polarizations (as we have already done for the spherewithout resonators). This is equivalent to replacing

j~h�j2 !j~e�j2 � j~e�j2

2j~h�j

2 !j~e�j2 � j~e�j2

2:

(70)

The ratios

Ai�;�� j~qi�!�j

2

��!��j~e�j2 � j~e�j2�=2

with ��!� � A2a2!8

3=5�!2�!2��

2�!2�!2��

2 , give the sensitivities of

the six resonators as a function of the direction of GWprovenance. The explicit expression for one of the Aipreviously defined is

A2�

� ��3p

6sin2�

��3p

18��5p�1��cossincos��

1

6��5p�1�

��cossinsin��

��3p

18��5p�1��1�cos2�

��cos2��1=2��1

6��5p�1��1�cos2�cos�sin�

�2

�

��

��3p

18��5p�1��sinsin��

1

6��5p�1��sincos��

�

��3p

9��5p�1�coscos�sin�

�1

6��5p�1�cos�cos2��sin2��

�2: (71)

The other five Ai have analogous expressions. Their angu-

104012

lar dependence is shown in Fig. 5, while Fig. 6 shows theangular dependence of their sum.

Note that the sum of the six sensitivities is no longer aconstant. This is because, as we noted before, putting

-11

0

0.5

1

1.5

2

2.5

3

theta

0 1 2 3 4 5 6phi

FIG. 6 (color online). The same as Fig. 5 for the sum of the Ai.

1.315

1.32

1.325

1.33

1.335

0 5 10 15 20t

FIG. 8 (color online). Sum of the Ai of Fig. 7 as a function ofsidereal time for b � 0, i.e. a source near the galactic center.


resonators on the surface of the sphere means destroyingthe original spherical symmetry. Nevertheless, the totalsensitivity is never zero, so for the direction of a GWfrom any provenance, there is at least one resonator whichwill be excited significantly. Omnidirectional sensitivity isthen preserved in the TIGA configuration, even if omnidir-ectional symmetry is broken.

0

5

10

15

20

t

–3 –2 –1 0 1 2 3b

A1

0

5

10

15

20

t

–3 –2 –1 0 1 2 3b

A2

0

5

10

15

20

t

–3 –2 –1 0 1 2 3b

A3

0

5

10

15

20

t

–3 –2 –1 0 1 2 3b

A4

0

5

10

15

20

t

–3 –2 –1 0 1 2 3b

A5

0

5

10

15

20

t

–3 –2 –1 0 1 2 3b

A6

FIG. 7 (color online). The same as Fig. 5, but as a function ofsidereal hours (t) and galactic longitude (b) of the GW source,for a spherical detector near Leiden, NL, l � 52:16�N, L �4:45�E.

104012

In Fig. 7 we take into account the earth’s rotation, asexplained in the appendix of [14], for a fixed source nearthe galactic center. Angular sensitivities for the six reso-nators and for their sum are shown as a function of siderealhour and galactic longitude, while Fig. 8 shows the sectionb � 0 of the sum of the Ai with respect to sidereal time. Asbefore, we take as an example the sphere near Leiden,Holland (latitude l � 52:16�N, longitude L � 4:45�E).

V. SUMMARY AND CONCLUSIONS

In this work, the general problem of the interaction of aspherical detector with GWs has been analyzed, in both thecases of a simple elastic sphere and of a system sphere�transducers. The five responses of the quadrupole funda-mental modes and of resonators motions for the TIGAconfiguration have been found, explicitly identifying an-gular dependence and, for GW radiation coming from asource in the galactic plane, sidereal time dependence aswell.

We find the expected omnidirectional constant sensitiv-ity for the ideal case of a simple sphere. This omnidirec-tional sensitivity is shown to be preserved in the case of thesphere equipped with TIGA transducers, nevertheless it isinteresting to note that it is no longer a constant. Thisconfirms the fact that putting resonators on the surface ofthe sphere means choosing some favorite directions, break-ing the omnidirectional initial symmetry.

For current experiments, the results of this paper allowone to predict exactly how the resonators should behaveonce the position of a candidate source is known. Viceversa, relations between the resonators’ amplitude and��t�; ��t�� can be inversed and whenever there is a statis-tically significative reason for thinking that a gravitationalwave signal has been received, one can estimate the sourcedirection by looking at the ratio between resonators’ ex-citations and at the GW arrival time.

Nevertheless, all this work is based on an ideal sphereand ideal radial moving resonators. Deviations and errorson the above predictions exist in a real spherical detector

-12


experiment. In particular, the asymmetry caused by thehole drilled through the center for the suspension systeminduces a degeneracy of the quadrupole five fundamentalmodes, which in [17] is estimated to be less than 1%. Wehave not derived the consequences of this degeneracy inthis paper, but it would be interesting to do that in order tobetter match theoretical predictions with experiments.

104012

ACKNOWLEDGMENTS

I would like to thank Florian Dubath and Stefano Foffafor many helpful discussions. A special thank to AngelaHaden, Danielle Chevalier, Andreas Malaspinas, MartiRuiz-Altaba, and Fabio Dubath. This work is partiallysupported by the Swiss National Fund (FNS).

[1] G. Frossati et al., www.minigrail.nl.[2] O. D. Aguiar et al., www.das.inpe.br/~graviton.[3] C. Z. Zhou and P. F. Michelson, Phys. Rev. D 51, 2517

(1995).[4] J. A. Lobo and M. A. Serrano, Europhys. Lett. 35, 253

(1996).[5] S. M. Merkowitz and W. W. Johnson, Phys. Rev. D 51,

2546 (1995).[6] R. V. Wagoner and H. J. Paik, Gravitazione Sperimentale

(Accademia Nazionale dei Lincei, Rome, 1977).[7] N. Ashby and J. Dreitlein, Phys. Rev. D 12, 336 (1975).[8] L. Landau and E. Lifshitz, Theorie de l’elasticite,

Physique Theorique (Mir, Moscou, 1967).[9] C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation

(Freeman, San Francisco, 1970).[10] J. A. Lobo, Phys. Rev. D 52, 591 (1995).

[11] L. Landau and E. Lifchitz, Mecanique, PhysiqueTheorique (Mir, Moscou, 1960), Vol. 1.

[12] K. S. Thorne, in 300 Years of Gravitation, edited by S.Hawking and W. Israel (Cambridge University Press,Cambridge, England, 1987).

[13] F. Dubath, S. Foffa, M. A. Gasparini, M. Maggiore, and R.Sturani, Phys. Rev. D 71, 124003 (2005).

[14] E. Coccia, F. Dubath, and M. Maggiore, Phys. Rev. D 70,084010 (2004).

[15] E. Coccia and J. A. Lobo and J. A. Ortega, Phys. Rev. D52, 3735 (1995).

[16] S. M. Merkowitz, Ph.D. thesis, Louisiana State University,1995.

[17] S. M. Merkowitz and W. W. Johnson, Phys. Rev. D 56,7513 (1997).

-13

gravitational wave detection by a spherical antenna: the angular sensitivity of resonators in the...

Documents