IL BIUOVO CIMENTO VoL. 9 C, N. 1 Gennaio-Febbraio 1986

Data Analysis for a Gravitational Wave Antenna with

Resonant Capacitive Transducer.

E. A~rAT,DI, O. PIZZELLA, P. R A P A O ~ I a n d F . ~ICCI

D i p a r t ~ n t o di .Fisiea dell'Un~versith ~ L a Sa/pienza ~ - Ron~a, I tal ia Istituto Nazionale di ~ is iea .~uoleare . Se~ione di Ro~a , Italda

P. Bo~-~Az~ Istituto .tYazionale di 2'isiea .Nucleate . Sezione di l~oma, Italia Istituto di 2'isiea detlo Slaazio Interplanetario del C.IY.R. - .Frascati, I talia

G. CAVAT, T , ~

O.E.B.~Y. European Organization /or IYuelear Research - Geneva, Switzerland

E. C o o L ~

D~parti~ento di _Fisica dell' Universith ~ To t Vergata ~ - l~oma, Italia Istituto Nazionale di Fis iea s - Sezione di Roma, I tal ia

G.V. P~LOTTI~O Dipavtimento di ~ is iea dell' Universit~ (i L a Sapienza ~ - l~oma, I tal ia Istituto Nazionale di .Fisica Nucleate - Sezione di t~oma, I tal ia Istituto di ~is ioa dello Spazio Interplanetario del C .N .R . - Frascat/i, I tal ia

(rioevuto il 21 Ottobre 1985)

Summary.- We present here a new approach which simplifies con- siderably the data analysis for gravitational-wave antennas equipped with resonant transducer, based on the representation of the antenna as two independent oscillators. In fact, we can apply to each of the two modes of these antennas the same data analysis procedures already in use for antennas with nonresonant transducer and then compute the coincidences between the outputs of the two modes. The results deduced by such a procedure are in good agreement with the experimental results from the data collected in March 1984 with our 2270 kg 5056 A1 bar cooled at liquid-helium temperature ( T = 4.2 K). The performances of the algorithms are presented in terms of the effective noise temperatures

51

~2 ~. AMALDI, G. PIZZ]~LLA, P. RAPAG~ANI, F. RICCI, P. BONIFAZI ETC.

and sensitivity to short bursts of gravitational waves. With our experi- mental values, obtained by using a FET amplifier, we get an instrumental sensitivity /~(v) of 6 J /m 2 Hz. We have also evaluated the sensitivity of the detector for monochromatic gravitational waves with frequencies in bandwidths of --~ 0.4 Hz around the frequencies v_ and v+, for one month of observation: h 0 _~ 3.10 -2~.

PACS. 04.80. - Experimental tests of general relativity and observations of gravitational radiation. PACS. 95.85. - Astronomical observations (listed by technique of ob- servation).

1 . - I n t r o d u c t i o n .

I n F e b r u a r y 1984 we have cooled to the l iquid-hel ium t e m p e r a t u r e our

g rav i t a t iona l -wave an t enna instal led at the C E R N L a b o r a t o r y in Geneva. A detai led descript ion of the expe r imen ta l appa ra tus has been given in previous publ icat ions (~,2).

W e recal l here t h a t t he an t enna is a cyl indrical ba r of A1 5056 wi th mass M = 2270 kg and length L----2 .97 m, equipped with a resonant capac i t ive

t ransducer . The first symmet r i ca l f lexural mode of t he t r ansduce r is coupled to the first longi tudinal mode of the ba r (8,~). The t r ansducer is b iased wi th a b a t t e r y th rough a high-value resistor (10 G ~ ) provid ing an electric field Eo

in its gap (d = 51 ~m). The t ransducer is v e r y well t uned and therefore t he t r ans fe r of energy be tween the ba r and the t r ansducer is larger t h a n 99 %.

We repor t here on the analysis of the da ta collected f rom March 2 to Apri l 2,

1984, using a }~ET preampl i f ier (s), a s t andard electronic chain and a da ta

acquisi t ion sys t em based on ~ I~ORD 100 compu te r ("). This work has been pe r fo rmed as a t e s t of bo th the opera t ion of the an tenna in this p re l iminary

configuration, and the acquisi t ion and da ta analysis techniques described below. The acquisi t ion and the analysis of the da ta have been per formed, sep-

(1) E. AreAL1)1, P. BO~:FAz:, F. BRO~Z:~I, G. CAV~LLA~I, E. COCCIA, F. FVLIG}{I, G. M~.RTI~LL~, I. ~IOI)~A, G. V. PALLOWTI~O, G. PmZ~LL~, P. RAPAG~A~I, F. RIce1, E. S~R~A~, S. UG~ZIO and G. V A ~ g o ~ I : The gravitational wave ex1~ericaent of the l~ome grou I) at CER2V, CERN/EF 83-17 {28 September 1983). (2) E. _A_MALDI, P. BONIFAZI, 10. BRONZINI, 1 ~ CARELLI, M. G. CAST:ELLANO, G. CAVAL- LARI, E. COCCIA, C. COSMELLI, V. FOGLIETTI, R. HAB:EL, I. MODENA, Y. OGAWA, G.V. P,~LLOTTINO, G. PlZZ~LL~, P. RAPAGN.~NI and 10. RICCI: Nuovo Cimento C, 7, 338 (1984). (3) P. RAPAGNANI: NUOVO Cimento C, $, 385 (1982). (a) Y. OGAWA and P. RAPAG~VANI: 2VUOVO Cimento C, 7, 21 (1984). (s) G.V. PALLOTTINO and G. VANNA-~ON~: I E E E Trans. Instrum. Meas., IM-34, 676 (1985).

DATA ANALYSIS FOI~ A GttAVITATIONAL WAVE ANTENNA :ETC. ~

arately, at the two modes of the antenna-transducer system, with frequencies v_ and v+. In other words we have used two lock-in amplifiers driven at v_ and v+, respectively, and we have analysed the two resulting streams of data, as if they were the outputs of two different antennas with nonresonant trans-

ducers.

2. - D a t a a n a l y s i s at t h e t w o m o d e s .

When dealing with antennas with nonresonant transducers the acquisition of the data is performed with a lock-in amplifier driven at the resonance fre- quency. The analysis can be done with the difference (ZOP, zero-order predic- tion) algorithm or with the Wiener-Kolmogoroff algorithm. The former provides a sensitivity close to tha t of the latter if the integration t ime is near $o the optimum value (e).

This, however, is not t rue in the case of antennas equipped with a resonant transducer, as required for obtaining higher values of the energy coupling

factor ft. I t has been shown that , when a single lock-in driven at the frequency v = (v_ + v+)/2 is used, the difference algorithm (v) provides a much smaller sensitivity than the more complex Wiener-Kolmogoroff algorithm, t~urtherly, a rather large observation bandwidth, such as to include both modes, is re- quired in this ease. This implies a fairly high sampling rate of the data.

A different approach, however, allows us to simplify considerably the data analysis (8,~). This is based on the representation of an antenna with resonant transducer as two separate oscillators. The deviation of this model, described by two independent second-order differential equations~ from the model de-

scribed by two coupled second-order equations, is negligible over the entire frequency axis, with the exception of a very small region half-way between

v_ and v+. Here the transfer functions of the coupled oscillators exhibit a pair of complex zeros (untiresonance), while those of the independent oscillators, of course, do not.

According to the simplified model~ the bilateral noise spectrum at each of the two modes can be expressed as

(2 .1 ) S ( a ) = S o ( a ) + - - V~b 4~o~

602 (2)2) 2 T~ ( ~ - - + 4~2/~ ' } '

(e) Pc BONIFAZI, V. FERRARI, S. FRASCA, G. V. PALLOTTINO and G. PIZZELLA: NUOVO Gime~to C, 1, 465 (1978). (~) G.V. PALLOTTINO and G. PlZZeLLA: .Y~tOVO Cimento C, 4, 237 (1981). (s) G, V. PALLOTTINO: Analysis o] some ]iltering and data analysis methods ]or gravita- tional wave antennas with resonant transducers, Internal Report IFSI-82-11, Frascati (September 1982). (~) G.V. PALLOTTINO and G. PlZZELLA: Data analysis ]or gravitational wave antenna with resonant transducer, Internal Report IFSI-82-12, Fraseati (October 1982).

54 E. A~IALDI, G. PIZZELLA, P. I~APAGlgA:NI, i e. RICCI, P. BONIFAZI ETC.

where ~o R is the resonance pulsat ion (o~ = w_ or co R = ~o+) and ~v the ampl i tude decay t ime of the mode,

(2.2) V~ -- flkT~ C~

is the variance of the nar row-band noise (Brownian plus back-action), k is the Bol tzmann constant , and

(2.3) ~o(O~) = v~(a,) + I*.(a,)/,~r

is t he wide-band noise spectrum due to the preamplifier, wi th variance

(2.4) v ~ = go(~)/to

in the observat ion band-width of a lock-in amplifier with integrat ion t ime to.

As customary, fl represents the energy coupling factor, t h a t is t he rat io between the electrical energy in the t ransducer and the to ta l energy in the antenna. The quan t i ty fl is given for each of the two modes (~0) b y

(2.5)

where

(2.6)

~nd

(2.7)

~ j = { ? t E ~ 1 6 2 1 \ ~o• / m+C~'

M 2 M/2 ~ 7t ,2• m • = ( 2 . - - 1)~

;t~ = ~ - ~ - ~) t/~ t

We recall t ha t M/2 is the reduced mass of the an tenna and 7tMt is the equiv- alent mass of the t ransducer (we obta in 7t = 0.865 for our t ransducer) . G t is the capacitance of the t ransducer . V and / are the voltage and current noise spectra of the preamplifier, and T e is the equivalent t empera tu re of the mode:

flI~vv (2.8) 2", = T ~ A T = T q- 4kC,~

where T is the t empera tu re of the antenna . The response to a short burs t of gravi ta t ional radia t ion with constant

spectral energy densi ty -~(v) [J /m 2 Hz] in a region including v_ and v+ can be espressed as

(2.9) v(t) ~ V, [exp [-- t/v+v] sin eo+t -- exp [-- t/vT] sin eo_t],

(lo) ~ r 0GAWA a n d P . •APAGNANI: private c o m m u n i c a t i o n .

DATA ANALYSIS FOR A GRAVITATIONAL WAR ANTENNA ETC. ~ 5

where

(2.10) V~ = %/flXFO,)/C,2

is the amplitude of the signal at each of the two modes, and X is the cross- section of the antenna.

We notice that , at each mode, the characteristics of both the signal and the noise are similar to those of an antenna with nonresonant transducer. There- fore, the standard data analysis techniques developed for the lat ter case can be applied, separately, to each of the two modes of an antenna with resonant transducer.

We remark, however, tha t when considering only one mode the value of the effective temperature is twice that obtained by using the information of both the modes, which is 2Tn in the case of perfect matching, where T---- = In VJk represents the noise temperature of the amplifier.

This can be shown by applying the WK algorithm at one mode. In this case the signal-to-noise ratio (6) is

(2.11) SNR = y ~oV~b "

If the matching conditions are fulfilled (7), that is the wide-band spectrum is 8o----V=' and

$ 2 (2.12) v . 'b- fl A~v

40,'

i.e. the back-action noise is much larger than the Brownian noise, we obtain

(2n3) SNR --/:F(~) 4kTn

by substituting eqs. (2.10) and (2.12) in eq. (2.11). The effective temperature is, therefore, putt ing SNR ---- 1

(2.14) T..-- l:/~(~)/k ---- 4T~.

The full sensitivity To, ~ 2 T can be recovered by performing coincidences between the data of the two modes. The joint probability density function of the filtered data of the two modes is given by the product of the two density functions, if the data are independent. I f the noise variances at the two modes are equal, then the noise variance of the combined data is one half of tha t of a single mode. Since the signal is present on the data of both modes, the signal to noise ratio for the combined data will be twice that of a single mode.

~6 E. AMALDI, O. PIZZELLA, P. RAPAGIVANI, F. RICCI, P. BONIFAZI :ETC.

3 . - B r o w n i a n - n o i s e m e a s u r e m e n t s .

T h e a n t e n n a was in o p e r a t i o n f r o m M a r c h 2, 1984 t h r o u g h A p r i l 2, w i t h

two i n t e r r u p t i o n s ( r e s p e c t i v e l y f rom M a r c h 6 to M a r c h 11 a n d f rom M a r c h 24

to M a r c h 29) w h e n t h e a n t e n n a was u s e d for o t h e r t e s t s (sect . 5). T h e m o s t

i m p o r t a n t c h a r a c t e r i s t i c s of t h e a p p a r a t u s a r e g i v e n in t a b l e I .

TABLE I. -- Characteristics o] the experimental apparatus.

Frequency of the bar (first longitudinal mode) Mass of the bar Frequency of the t ransducer (first fundamental flexural mode) Mass of the t ransducer (disk) Capacity of the t ransducer Electr ical field in the t ransducer Frequency of the modes Merit factors of the modes Decay t imes of the modes Energy coupling factors Total gains of the electronic chain (preamplifier, tuned amplifier, lock-in) Capacity of the t ransducer plus cables and amplifier Voltage noise of the amplifier Total current noise (amplifier + cables) Noise tempera ture of the amplifier Sampling t ime interval In tegra t ion t ime of the lock-in amplifiers

v_ =908.160 Hz, Q_ -- 5.20. l0 s T~ = 30.41min, fl_ = 5.8" 10 -5 , A _ = 1.23.108

vb = 915.765 Hz M = 2270 kg vt = 916.6 Hz

M t = 0.402 kg Gt = 3891 p F E o = 8.7"105V/m v+ = 924.234 Hz Q+ = 7.25. l0 s "r + = 41.64 rain fl+ = 6.5' 10 -5 A+ = 1.17" l0 s

C2 = 5495 p F v~ = 0.31 n V / V - f f ~ x~ = 4 .88 f A / v " g / T n = 0.110 K At = 1.0 s to = 1.0 s

W e h a v e r e c o r d e d on m a g n e t i c t a p e t h e t w o o r t h o g o n a l c o m p o n e n t s p(t) a n d q(t) a t t h e o u t p u t of t h e l ock - in ampl i f i e r s , t h a t r e p r e s e n t t h e a n t e n n a

m o t i o n , for t h e m o d e v (908 Hz) a n d for t h e m o d e v+ (924 Hz) , s e p a r a t e l y .

W e h a v e a lso r e c o r d e d t h e o u t p u t s p(t) a n d q(t) of a lock- in a m p l i f i e r d r i v e n

a t 916 H z , f a r f r o m t h e two r e s o n a n c e s , in o r d e r t o m o n i t o r t h e w i d e - b a n d

no ise a n d d e t e c t p o s s i b l e i m p u l s i v e d i s t u r b a n c e s . F r o m t h e r e c o r d e d d a t a we

d e r i v e t h e fo l lowing q u a n t i t i e s for t h e m o d e v_:

(3.1) r~_(t) = pS( t ) + qS(t) ,

t h a t r e p r e s e n t s t h e e n e r g y s t a t u s of t h e a n t e n n a a n d

(3.2) Q~(t) = [p_(t) - - p_( t -- At)] 2 + [q_(t) - - q_(t - - At)] 2 ,

t h a t r e p r e s e n t s t h e v a r i a t i o n of t h e e n e r g y s t a t u s for t h e m o d e v_. S i m i l a r l y

we d e r i v e t h e q u a n t i t i e s r~ a n d ~ for t h e m o d e v+.

DATA ANALYSIS FOR A GRAVITATIONAL WAVE A N T E N N A :ETC. ~ 7

The hour ly averages of the var iab les r ~ and ~ of the two modes, for the

per iod of m e a s u r e m e n t here considered, show the occurrence of a few large ampl i tude (~ events ~> well above the background. These events are l isted in t ab le I I : bo th modes of the an tenna are excited, wi th values of energy va-

TABL~ II. - Events eliminated ]vom the data.

Event Day h.m.s, l~ode ~_ Mode v+

number r s_ (K) @~_ (V 2) @2 (K) r~. (K) @~_ (V s) @~_ (K)

1 63 0 3 : 0 7 : 0 6 10561 19.3 1 0 9 9 7 1 1 2 1 4 18.0 1 0 0 1 4

2 63 0 5 : 5 3 : 1 0 > 4 5 3 0 8 > 2 3 2 . 9 > 1 3 2 2 5 3 > 4 4 4 3 0 > 9 8 . 3 > 5 3 7 6 0

3 73 0 1 : 1 3 : 3 0 > 4 5 3 3 0 > 83.3 > 4 7 3 1 3 > 4 4 5 3 0 > 8 0 . 3 > 4 4 8 0 2

4 75 0 3 : 5 0 : 0 8 > 3 0 1 8 0 > 45.1 > 2 5 6 2 4 > 2 9 5 5 2 > 2 6 . 3 > 1 4 6 4 9

5 79 2 3 : 1 4 : 3 0 > 3 4 3 3 6 > 51.2 > 2 9 0 9 1 2 1 0 8 9 27.8 15531

6 80 0 4 : 3 1 : 4 9 7 5 1 2 8 .4 4 7 7 0 5 5 4 6 8 .34 4 6 5 7

7 81 1 1 : 5 4 : 4 9 1 7 3 5 8 14.9 8 8 4 8 1 6 9 7 9 13.9 7 7 7 1

8 82 1 7 : 0 7 : 0 3 8 6 9 0 9.4 5 3 4 2 5 4 9 8 5.9 3 3 0 9

9 83 0 8 : 1 4 : 0 0 2 0 2 1 2.7 1 5 5 6 1 5 3 6 1.5 834

10 83 0 8 : 2 6 : 0 7 5 2 1 9 3.5 1 9 9 7 3 9 5 5 4 .6 2 5 7 3

11 83 1 1 : 2 9 : 0 7 2 2 5 3 7 23.6 1 3 4 1 8 15432 21.0 1 1 6 9 3

During March 1984 there have been at CERN several explosions of mines (14 at distance of 1 k m and 27 at about 5 k m from the antenna), because of the excavation works for LEP. None of the events l isted above occurre4 in coincidence with these explosions.

f ict ions of the same order of magni tude . Bo th modes decay with the p roper decay t imes given in tab le I . Al though it appears interest ing to s tudy these

events (most of t h e m occur by night), we decided not to consider t h e m in the present s tudy. Therefore, we have e l iminated f rom the da ta the hours

(integer hours for simplicity) containing these <, events ~ for a t o t a l of 15 hours of data . The to t a l r emain ing period is 399 hours.

The expe r imen ta l dis t r ibut ions of the var iables r ~ and @3 for the edited

da ta (399)<3600----1436 400 samples) , which are the es t imat ions of the prob-

ab i l i ty dens i ty funct ion (PDF), at the mode v_ and ~+, respect ively, are given in fig. 1-4. We recall t ha t , in the absence of ex te rna l noise the P D F s of the var iables r ~ and @~ are given b y the Bo l t zmann funct ions

(3.3) ](r , ) ~_ exp [-- r2 /~] exp [-- @21~]

F r o m the slope of the h is tograms, and using the values of A_ and A+ given in table I , we der ive

(3.4) rL-~ 6.46 (nV) ~ , @L---- 0.57 (nV) 2 ,

~+ = 6.88 (nV) ~ , @-~+--~ 0.59 (n~) 2 .

58

10 6

E. AMALDI, G. PIZZELLA, P. RAPAGlVAlVI, F. RICCI, P. BONIFAZI :ETC.

'1 106r

10 5 ~ ~ I 10 5

0.2 0.4. 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 r~(v 2) r2+(v 2)

Fig. 1. Fig. 2.

Fig. 1 . - Experimental distribution (estimate of the probability density function), of the variable r 2_ for the edited data (1436400 samples).

Fig. 2. - The same as fig. 1 but for the mode v+.

We recall (e) t ha t , when the decay t i m e ~v is much grea ter t h a n the in tegra t ion t ime to of the lock-in amplifiers, ~ and Q--~ are given b y

(3.5)

and

(3.6) o~_ ~ 4 r ~ 1 + + ~ v ~ 1 - . T r

F r o m the values of ~v g iven in t ab le I and the expe r imen ta l values of

and ~ given b y (3.4), we compute , for each mode, the es t imates of the quant i t ies

V ~ and V~b: nb

(3.7) ~ ~ b = 3.00 (nV) ~ , ]~b = 0.22 (nV) 2 a t v_,

[ ?~b = 3.21 (nV) 2 , ~ b = 0.23 (nV) ~ a t v+.

The exper imen ta l values of the equiva len t t e m p e r a t u r e s of the modes are ob ta ined b y in t roducing (3.7) into (2.2). We obta in

(3.8) (Te)~ ~ = 20.6 K , (T~)~xP = 19.7 K

DATA ANALYSIS FOR A GRAVITATIONAL WAV~ ANTENNA ETC.

I06r. 10 6

59

105

10 ~

105

104.

w i t h a 4 % s t a t i s t i c a l e r ro r . The c o r r e s p o n d i n g t h e o r e t i c a l v a l u e s c a n b e

c o m p u t e d u s i n g t h e p a r a m e t e r s of t h e a n t e n n a l i s t e d in t a b l e I a n d f o r m u l a (2.8) w i t h T ---- 6 K . W e o b t a i n

(3.9) = 1 4 . 3 K , (To) = 1 8 . 7 K .

W e n o t i c e t h a t (3.8) a n d (3.9) a r e close one to each o t h e r for t h e m o d e ~+ b u t

t h e y di f fer s i gn i f i c an t l y for t h e m o d e L . I n ou r p r e v i o u s p a p e r (2) we f o u n d

Fig. 3. - The same as fig. I but for the variable eL.

Fig. 4. - The same as fig. 3 but for the mode ~+.

1030[ I I ~1 I 103 0.02 0.04 0.06 0.08 o.lo o 0.02 o.o~ 0.06 o.oe o.lo

e~(v 2) 8~.(v 2) Fig. 3. Fig. 4.

6 0 :E. AMALDI, G. PIZZELLA, P. I~APAGNANI, F. I~ICCI, P. BONIFAZI ETC.

(T~)~P-- - 1 9 . 1 K and (To)~'-- - -14.8 K over a much smaller period of t ime

(38 hours). Therefore, we conclude tha t for the present data some disturbances

are present t ha t act only on the mode ~_.

4. - D a t a a n a l y s i s o f t w o s e l e c t e d periods .

4"1. Brownian noise. - We have tested the data analysis methods of the

previous section, on two continuous periods (each one 24 hours long), one

at the beginning (period ONE: March 12, 1984, from UT 00:00, to UT 23:00)

and the other at the end of the global period of measurement (period TWO:

from April 1, UT 15:00, to April 2, UT 14:00).

TABL]~ III . - Results o/ the analysis.

Period ~ode ~ 02 ~2 ^2 - - V~b V~b (T~) th (T~) "~ [(nV)2J [(nV) 2] [(nV) 2] [(nV)2j [K] [K]

ONE v_ 4.27 0.56 1.91 0.22 14.3 13.1 4- 2.1 ONE ~+ 6.15 0.64 2.83 0.25 18.7 17.3 ~ 2.8 ONE 916 Hz 0.41 0.54 - - 0.21 - - - -

TWO v_ 6.02 0.41 2.85 0.16 14.3 19.6 4- 3.1 TWO v+ 4.63 0.41 2.14 0.16 18.7 13.1 4- 2.1 TWO 916 Hz 0.32 0.42 - - 0.16 - - - -

I n table I I I we report the est imates of the narrow-band noise V ~ (Brownian nb

plus back-act ion noise) and of the wide-band noise V" (electronic noise) for wb

these two periods of measurement , computed from eqs. (3.5) and (3.6). The statistical error of ^ 2 Vnb in this case is of ~ 16 % (because we have only one day of data for each period and the values of ~v are relatively high).

Table I I I also shows the values of the quantit ies r ~ and ~2 derived from

the data at 916 Hz. (Here the totul gain of the electronics is A ---- 8.77-10").

We recall tha t the variables r 2 and ~2 at 916 Hz are given by (3.5) and (3.6)

with V~b --~ 0. We see tha t the experimentul values of the variables 92 and r 2 at 916 Hz, computed for the two periods, are consistent according to eqs. (3.5)

and (3.6), and the est imates of V~b at 916 Hz arc in agreement with those at

~_ and ~+. We notice, however, t ha t the est imate of V * derived from the wb

data of the two periods, as well as those based on 399 hours (see (3.7)), are

appreciably larger t han the value computed by subst i tu t ing in eqs. (2.3) and (2.4) the experimental values of V and I obtained by direct measurements on the I~ET amplifier, t ha t is

(4.1) v~b = 0.12 (nV)~.

We believe tha t this discrepancy is due to external disturbances act ing on

the cables (6 metres long) connecting the t ransducer to the F E T amplifier.

DATA ANALYSIS FOR A GRAVITATIONAL WAVE ANTENNA ETC.

TABLE IV. - Results o] analysis.

61

Period l~ode (A T)'~P l~ l? [K] [fA]vrH-~] [nV/~/-H~]

ONE v_ 7.1 4- 1.1 4.52 =t= 1.53 0.45 ONE v+ 11.3 4- 1.8 4.60 4- 1.46 0.48 TWO v_ 13.6 • 2.2 5.60 4- 1.81 0.34 TWO v+ 7.1 4- 1.1 3.00 4- 0.93 0.38

Table I V gives the es t imates of t he cur rent and of the vol tage noise obtain-

ed f rom the expe r imen ta lva lne s of the quant i t ies AT, b y means of eq. (2.8), and

V~,b, b y means of cqs. (2.4) and (2.3), when assuming T = 6 K and for the

p a r a m e t e r s fl and G~ the values g iven in t ab le I . We not ice t h a t the values of

~, are v e r y close to the corresponding values g iven in t ab le I . On the o the r hand the values of 1~ n are grea ter t h a n the corresponding values g iven in the same table , par t i cu la r ly as regards the d~ta of the per iod ONE.

The ~bove resul ts show t h a t the ex te rna l d is turbances cont r ibu te mos t ly

to the vol tage noise, r a the r t h a n to the cur rent noise. This is in agreement wi th the resul ts of expe r imen ta l t es t s on the effect of the mechanica l v ibra t ions

act ing on the cables connect ing the t r ansducer wi th the preamplifier . Figures 5 and 6 show the h is tograms of the var iable r 2, respect ively for the

per iod ONE at the mode ~_ and for the per iod TWO at the mode v+, as ex- amples . We not ice t h a t these expe r imen ta l d is t r ibut ions follow ve ry closely the exponen t ia l law.

4"2. Data analysis algorithms. - We have used two a lgor i thms: the zero- order predic t ion a lgor i thm (ZOP) and the Wiener-Kolmogoroff a lgor i thm (WK). They provide a lmost the same SNR only if the sampl ing t ime At (equal to the in tegra t ion t i m e to of the lock-in) is equal to the o p t i m u m value:

(4.2) At = (At)op~ = 1/[/(e-- 1 )Sozz . V~b V

For the da ta of the two periods considered in this section this condit ion is not fulfilled (see t ab le V). Therefore, we concent ra te our analysis on the W K

T A B L E V . - Results o] analysis.

Period Mode T~ ~p Tt~ h T~ xp 8a (At)opt ~'. [K] [K] [K] [rad/s] [s] [K]

ONE v_ 4.80 0.84 1.46 0.07 19.0 0.15 TWO v_ 3.50 0.86 1.30 0.10 19.0 0.16 ONE ~+ 4.87 0.83 0.59 0.07 15.3 0.14 TWO v+ 3.15 0.58 0.49 0.07 15.3 0.08

62 ]~. AMALDI, G. PIZZ]~LLA, P. RAPAG~ANI, F. RICCI, P. BONIFAZI ETC.

105[

10 ~

103

10;' -

10 ~

10 0 I \ 1 I : - 0 100 200 300 2

r_ (K)

Fig. 5. - Experimental distribution of the variable r ~_ for the period ONE.

algori thm and use the ZOP algor i thm only to improve the check of the math- emat ical approach sketched in sect. 3.

We recall briefly in what follows the W K algor i thm me thod (e), which provides the op t imum est imat ion of the input force acting on the bar, b y performing suitable linear combinat ions of the data.

Le t us consider the two discrete sequences p(K) and q(K), obtained by sampling at t ime intervals At the signals at the outputs of the lock-in, at the t imes t, = K A t . I t can be shown tha t the signal-to-noise rat io (SNR) is max- imum if we form the l inear combinat ions:

(4.3)

+co

xw(J) -- ~, p ( J - - K ) w ( K ) , k = - - co

+ c o

gw(J) = ~ q(J- - g ) w ( K ) k = - - o o

by using the discrete sequence w(K), t ha t we call <~ weights )> of the W K filter.

DATA ANALYSIS FOR A GRAVITATIONAL WAVE ANTENNA "R, TO.

1051

10 4

103

102

101

10 ~ I i 0 100

4 ( K )

200

Fig. 6. - Experimental distribution of the variable r~_ for the period TWO.

These weights are computed by

w(K) = f w(a) d a , (4.4) (K--�89

where W(u) is the opt imum weighting function according to W K theory. By solving the integral we find

(4.5) w(K)--_ {fll(fl~d-fl3)(flld-fl3)[exp[~l--exp[_~]]exp[~ KAtfls]} 2rfl0fl~

where the sign ~- holds for K < 0 and the sign -- for K > 0, and

r st/~]/ r ~

(4.7) fll = l/vr, (4.8) fl~ = llto,

(4.9) F : fllSolV~b.

64

Here

(4.1o) ~=~,V ~+F17

]E. AMALDI, 6}. PIZZ]~LLA, P. RAPAGNANI, F. I~ICCI, P. BONIFAZI :ETC.

gives the t ime resolution of the filtered data. The number of the weights we use is established by the condition w(K)/w(O)> 10 -3.

The opt imum way to detect an innovat ion due to a force acting on the bar is to use the quan t i ty

= 2 j (4.11) e~(J) x~(J) -~ y.( )

tha t , in the absence of signal, follows the Bol tzmann distr ibution

1 (4.12) /~(~.) ----- ~ exp [-- e~/Q~] �9

The effective temperature of the W K filter is

(4.13)

with

(4.14)

= -~.K ~ T, , e,,/ ,, (K)

K~ ---- flfl' kA 2 2 G ~ 3 r V ~ + r) '

where A is the overall amplification of the electronic chain at the frequency of the mode.

The ZOP filter is realized by computing the quant i ty given by (3.2). The effective temperature T~, expressed in Kelvin degrees, is

(4.15) where

(4.16)

m: = ~ I K ~ (K),

K~ -- 2flkA~ [ exp [ - fl~ At] - exp [ - fl~-- fll fl212.

We recall (5) tha t , if the sampling t ime At is equal to the opt imum value (see (4.2)) and the matching conditions are fulfilled, we have

(4.17) T~ = 1.21T~.

4"3. Experimental results. - We have applied the analysis technique discussed in the previous section, to the measurements taken during the two periods defined in subsect. 4"1. The parameters used to compute the W K weights are V~b , V~b , Tv, to and At. We have used, for each of the two modes, the values of ]~b and ~ b given in table I I I and for the parameters vv, to and At the values given in table I , obtaining the results shown in table V.

DATA AI~YALYSIS FOR A GRAVITATIO~YAL WAVE ANTEN~YA ETC. 65

r l I % %

@ @

I I

0 .4

I

t ~

0 "~

o ~ .~

0

o~

,

lz tz ~ o ~ .~

5 - II N u o ~ o C'tmento C.

6 6 E , A M A L D I , G. P I Z Z E L L A , P~ R A P A G N A N I , F . R I C C I , P . B O N I F A Z I ~,TC.

�9 O

% O O O O O

%

e Q �9 �9

O % % o o ~

A

I I

DATA ANALYSIS :FOR A G R A V I T A T I O N A L WAV~ ANT]ENNA :ETC. 6 7

We note tha t the values of the theoretical effective temperature T are

greater than 4 ~ (where the values of ~ have been computed by using the estimates of I~ and V given in table IV). This occurs because ~he matching conditions are not fulfilled. As a consequence of this~ as well as because the

values of the optimum sampling time (At)opt (shown in table V) are very far from the actual value of the sampling time At ---- ls, the values of T, are much higher than 4 T . Figures 7-10 show, as examples, the experimental distributions of the variables @2 and @2, in degrees kelvin, respectively at the mode v_ and v+ for the same data of fig. 5 and 6.

Figure 11 shows %he sequence of Che WK weights for the mode ~_. Figure 12a) shows a sequence of r 2 data in the presence of a short disturbance (nonmech-

WK

- 5

--10

TTTTTTTTTTTT1 I

t]-20 ~]-~0 l

Fig. 11. - The WK weights for the mode v_.

anical) above the background, and the corresponding output sequence ob- tained by applying the WK filter and the ZOP filter. We notice tha t the latter provides a large-@ 2 signal~ while the former gives ahnost no signal at all. This is because the WK filters reconstructs the input force signal applied to the bar; for the case shown in fig. 12a) such a signal is, in fact, negligible or, from a more refined analysis, composed at most by two delta functions of opposite sign.

The results of combining the data of the two modes of the detector are shown in fig. 13. This is the experimental joint probability density function of the data of the two modes, filtered wi~h the ZOP and WK algorithms, /or the period TWO. The effective noise temperatures of the combined data arc in

68 E. AMA.LDI, G. PIZZELLA, P. I~APAGNANI, F. RICCI, P. BONIFAZI ETC.

10 2

lo 1

10 0

10 2

~" 10 1

10 0

10 -1

a.) b)

I I

t \ ZOP

I

l,,,,I,,,,l t i_s t i t;+ s

II

\

i

I/

i ,,~,IlZ t. I-5

I

r t/+5

Fig. 12. - A sequence of da t a in the presence of a shor t d i s turbance (nonmechanical ) : a) run T W O ; b) run wi th h igh electr ical field.

�9 ~greement with the theoret ical values:

(1 1)-1 (~.18) (~..)* = ~ + ~ .

Figure 13 shows how the W K filter oliminates all the residual disturbances appearing in the ZOP distr ibution.

DATA AIVALYSIS FOR A GRAVITATIONAL WAV~ ANTENNA ETC.

10 s

69

10

10

10

10

l

10 : : I 0 15 30 /,,5

0 2 (K)

Fig. 13. - The experimental joint probability density functions of the data of the two modes filtered with the �9 ZOP and o WK algorithms for the period TWO.

5. - Measurement wi th an higher value o f the electrical field.

During the two interrupt ions in the data acquisit ion (mentioned in sect. 3), various measurements were carr ied out , among which Brownian noise meas- urements with an higher value of the electrical field in the capacit ive t ransducer . In this case the matching condit ion (2.12) is near ly fullfilled. We have made this measurement for a short period (from March 29, 1984, UT 23:00 through March 30, UT 06:00). Table VI shows the values of the parameters changed ~s consequence of the new value of the electrical field Eo, and the values of the gain of the electronic chain at the frequencies v_ and ~+. We recall t h a t t he value of fl is propor t ional to the square of Eo (see eq. (2.5)).

70 E. AI~IALDI, G, PIZZELLA~ P. I%APAGI~ANI, F. I~ICCI, P. B01qlFAZI ETC.

TABLE VI. - Antenna's parameters /or light electric /ield.

Electrical field in the transducer Frequencies of the modes ~eri t factor of the modes Decay times of the modes The fi parameters Total gains of the electronic chains

E o = 2.4.10 ~ V/m v_ = 908.044 Hz v+ = 924.105 tIz Q_ = 4.9. l0 s Q+ = 7.4-106 z~ = 28.6rain T + = 42.stain fl_ = 4.3.10 -a fl+ = 4.8.10 -a A_ = 3.07.107 A+ = 2.92.10 ~

As a consequence of increasing the fl pa r ame te r , the value of V~b (see eq. (2.2)) is much grea te r t h a n the value of V~b and therefore it is not possible to e s t ima te

these quant i t ies as we have done in sect. 3. We have used the values of I=

and V= given in tab le I to compute the value V2,b uti l ized for the W K weights.

TABLE VII. - Results o/ analysis.

lYlode (T~) th (T~) ~p (T,) th (T,) ~-D (Tw)~h (T,,),~ f13 (At)opt [K] [K] [K] [K] [K] [K] [rad/s] [s]

v_ 64.0 73 ~= 18 0.84 0.68 0.51 0.46 0.57 2.3 v+ 102.7 148 • 44 0.78 0.55 0.51 0.27 0.63 2.1

Table V I I shows the results of the analysis re la t ive to this cont inuous

per iod of 8 hours (8• ~ 28.800 samples). We repor t in this t ab le the

theore t ica l and expe r imen ta l values of Te, T and Tw, and the values of the

p a r a m e t e r s f13 and (At)opt, for the two modes. The d isagreement be tween the theore t ica l and expe r imen ta l values of T for the mode v+ is p robab ly due to a sys temat ic error. The exper imenta l dis t r ibut ions of the var iables ~: and ~2, for each mode, follow ve ry closely the exponent ia l law as expected. Figure 14

shows the exper imen ta l dens i ty funct ions of the var iab le ~ for the da ta of the

modes v_ and v+ and the corresponding joint p robab i l i ty densi ty function. The value of the effective noise t e m p e r a t u r e deduced f rom the joint densi ty

funct ion is 0.17 K, 23 % less t h a n the theore t ica l one ((To~t)~i n ~ 2 T ~ 0.22 K). This is a consequence of the above-ment ioned d isagreement for the mode v+

for the W K filter (see t ab le VI I ) . F igure 12b) shows, as fig. 12a), the response

of t he two filters to a short nonmechanica l d is turbance. We note t ha t , in this

case, the W K filter is not so efficient, as we had seen before.

This is due to the fac t t h a t in our theore t ica l analysis we have considered only cont inuous processes, while the da ta sampl ing procedure makes the da ta discrete. Therefore, the results of our analysis hold exac t ly only for A t < 1/fl8, since 1/fl3 is the character is t ic resolut ion t i m e of the W K filter, when we consider the average values of the filtered noise. When we have such a signal, composed a t mos t b y two opposi te delta funct ions, a much larger band width (At<< 1/f13) is required.

DATA ANALYSIS FOR A GRAVITATIONAL WAVE ANTENNA ETC.

105[

I

71

104

10 3

10 2

101 O-~V--

x00J o 2 ~O2w(K ) 4 6

Fig. 14. - The WK experimental distributions for the variables Q~_, ~ and Q~ for the run with high electrical field.

I t is interest ing to recall t h a t t he an tenna bandwidth Av is not equal to

(vR/Q) ~ 150 t~Hz, bu t is approximate ly defined as the inverse of (At)o,t and it is equal to 2fl3. This result , obta ined in our previous papers (6) dealing with the W K theo ry for short-burst detect ion, has been again obta ined in a recent work (11) on the detect ion of monochromat ic gravi ta t ional waves, as expected, because the bandwid th is a feature of the exper imenta l appara tus and does not depend on the exci ta t ion (12).

(11) G.V. PALLOTTIN0 and G. PIZZELLA: ./~0~)o Cintento C, 7, 155 (1984). (13) G. PIZZELLA: Left. .~tovo Ci~ento, 40, 240 (1984).

7~ F,. &MALDI, G. PIZZELLA, P. RAPAGNANI, F. RICCI, P. BONIF&ZI :ETC.

The a n t e n n a b a n d w i d t h is

H z [ v . ~ fl__ 10 ' T 1 0 - ' 1 ) ' ] (5.1) A v = 166 [900 H z ~ 1 0 -2 Q 4.2 T . ~-o

where

v. 1 (5.2) ~o -

F o r t h e d a t a of th i s r u n we o b t a i n

(5.3) Av_---- 363 m H z , Av+ ~ 401 m H z .

W e recal l (11) t h a t t he sens i t iv i ty of t he g.w. a n t e n n a , for m o n o c h r o m a t i c

g r a v i t a t i o n a l waves wi th f r e q u e n c y in its b a n d wid th , is

(5.4) ho>~.sT.lo-~, [~ + r~(Z9/2 + ~/~9) 2~oo kg ~o, 9oo ~ ~ day]* [ 4 . 2 K M Q v~ t,~ J '

where t m is t he m e a s u r i n g t ime . F r o m (5.4) we deduce for one m o n t h of observa-

t i on a de tec tab le va lue of h o < 3 . 1 0 -~3.

�9 R I A S S U N T 0

Si presenta un nuovo approccio che semplifiea notevolmente l'analisi dei dati per antenne gravitazionali con trasduttore risonante, basato sulla rappresentazione dell 'antenna come due oseillatori indipendenti. A eiascuno dei modi di tali antenne si applieano le procedure di analisi dei dati gi~ usati per le antenne con trasduttore risonante e si calcolano quindi le eoincidenze tra le useite dei due modi. I risultati dedotti mediante tale procedura sono in buon aceordo con i rlsultati sperimentali reIativi ai dati raeeolti nel marzo 1984 con la nostra sbarra eli A1 5056 da 2270 kg raffreddata alla temperatura 4ell'elio liquido (T = 4.2 K). Si presentano le prestazioni degli algoritmi in termini 4elle temperature efficaci di rumore e della sensibilit~ per brevi fiotti di onde gravita- zionali. Con i nostri valori sperimentali, ottenuti usando un amplifioatore a PET, si ha una sensibilit~ strumentale ~(v) di 6 J /m s Hz. Si i~ anohe ealeolata la sensibflit~ del rivelatore per onde gravitazionali monoeromatiche in bande di frequenza di circa 0.4 Hz attorno alle frequenze v_ e v+, per un mese di ossevazione: h 0 ~ 3.10 -~s.

DATA ANALYSIS FOR A GRAVITATIONAL WAVE ANTENNA ~TC. 73

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