I L NUOVO C I M E N T O VOL. 4 C, N. 4 Lugl io -Agos to 1981

Background of Gravitational-Wave Antennas of Possible Terrestrial Origin - III (').

E . A,~IALDI~ E . COCCIA, S. FRASCA a n d F. RICCI

Ist i tuto di .Fisica dell' Universitd - Roma, I ta l ia Ist i tuto Nazionale di Fis ica _Vucleare - Sezione di Roma, I ta l ia

P . BONIFAZI, V. IAFOLLA and G. I~ATALI

Ist i tuto Plasma nello Spazio del C .N .R . - Prascati, I ta l ia Ist i tuto Nazionale di .Fisica Nucleate - Sezione di Roma, I ta l ia

G . V . P~LLOTTL~O a n d G. PIZZEI, LA

Ist i tuto di ,Fisica dell' Universit~ - Roma, I ta l ia Ist i tuto Nazionale di .Fisiea Nucleare - Sezione di Roma, I ta l ia Isti tuto Plasma nello Spazio del C..N.R. - .Frascati, I ta l ia

(r ieevuto il 10 Lugl io 1981)

S u m m a r y . - - In two p rev ious papers we h a v e presen ted ev idence for a corre la t ion be tween a pa r t of the background of a g r av i t a t i ona l -wave an t enna ( M = 390kg) and the free osci l la t ions of t he E a r t h . These resul ts are suggest ive of a s imi lar s tudy of t he forced osci l la t ions of the Ea r th . In th is paper we present the resul ts of the Fou r i e r analysis for per iods be tween ~r = 100 and 1667 min of the da t a col lected on the occasion of two runs: one in 1978, the o ther in 1980. T h e y show a few frequencies , some of which are in ve ry good ag reemen t w i th t ida l f requencies . T h e p robab i l i ty for these (~ overlappings)) to be accidenta l is of the order of 10 -2 or less.

(*) The ma in points of the results deduced f rom the (( 1978 d a t a ,) have been presen ted by S. FRASCA at the Workshop on Gravitational Rc~liation Detectors o/ the Tenth Texas Sympos ium on Relativistic Astrophysics, held in Ba l t imore , Md., December 15-19, 1980.

441

442 E. AI~IALDI~ E. COCCIA~ S. FlgASCA~ F. RICCI~ P. BONIFAZI ~ ETC.

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

In two preceding papers (1,~)we have discussed a) a few very large

signals (briefly, events) recorded dur ing the nights May 6 and 7, 1980, by our detectors operated s imul taneous ly- -one in Frascati , the other in Rome- -wi th in delays inferior to a few seconds or tens of seconds (1); b) the Fourier trans-

form of the occurrence times of the lCrascati events which shows peaks - - outside statistical f luc tuat ions--a t values of the period 3- (between 20 and 100 min) which in m a n y cases overlap the periods of the free oscillations of

the E a r t h (3). Similar results were obtained from a 1978 set of data (s). These observations prompted us to search for periods correlated with the

forced oscillations of the Ear th , i.e. with the tides due to the Moon and the Sun. The most impor tan t t idal lines are listed in table I with the corresponding ampli tude coefficients as they are given in the l i terature (8). I n the same table

we give also the values of the corresponding half-periods for reasons tha t will

appear below.

TABLE I. -- Periods and hall-periods o] the six largest tide components.

Symbol 3- (min) Coefficient 3-/2 (min) Denomination

Q1 1612.1 0.07216 806.1 larger lunar elliptic of O~

01 1549.2 0.37689 774.6 principal lunar diurnal

/)1 (*) 1444.0 0.175 84 722.0 principal solar diurnal

K1 (*) 1436.1 0.53050 718.0 luni-solar diurnal

iV~ 759.5 O. 173 87 379.8 larger lunar elliptic of M s

Ms 745.2 0.90812 372.6 principal lunar

S 2 (*) 720.0 0.42286 360.0 principal solar

K s (*) 718.0 0.11506 359.0 luni-solar semi-diurnal

(*) These are pairs of periods unresolvable by means of our analysis.

The data of table I suggest an extension from ~q'= 100 min to above J - = 1612 min of the Fourier analysis of the <~ 1978 data ~> considered in ref. (s).

{1) ]P~. AMALDI I P. BONIFAZI, :E. COCCIA, Ci COSMELLI, ~q. FRASCA, V. GIOVANARDI, V. IAFOLLA, I. MODENA, G. V. PALLOTTINO, G . PIZZELLA, R. RAPAGNANI, F. RICCI, S. UGAZIO and G. VAtCNARONI: Nnovo Cimento C, 4, 295 (1981). (2) E. AMALDI, P. BONIFAZI, S. I~RASCA, G. V. PALLOTTINO and G. PIZZELLA: N~OVO Cimento C, 4, 309 (1981). (3) LANDOLT-BORNSTEIN, I I I Band, Astronomie nnd Geophysik.

BACKGROUND OF GRAVITATIONAL-WAVE ANTENNAS ETC. - I I I 443

We have actual ly extended the analysis up to

~ = 1667 rain = (6 .]0 -4 nlin-~) -~ = (10 ~Hz) -~ .

I n addition, we present here similar results obtained from a new set of data collected during December 1980.

The Frascat i an tenna (M = 389 kg) was operated at its fundamenta l mode

during November and December 1980. We discuss here only the December data, because, during November, a series of ear thquakes occurred in Irpinia some of which were sensed by the Frascat i detector.

I n our two previous papers we have shown tha t the events of interest for this type of investigation have energy such tha t ~2 is in the range of at least 103 K. Therefore, we decided to operate the ~rascat i antenna, during November

and December 1980, at l iquid-nitrogen temperature. The mechanical filters inside the cryostat bad also been changed.

In table I I we give the main parameters of our an tenna for the two sets of data considered in this paper.

TABLE II. -- Main /eatures o/ the Frascati antenna /or the two sets o/ data.

v R ~ Q ~, I'~ s O T To T,,~ At (u,) (V/m) .1o-, (K) (K) (K)

�9 1 0 - 7 \ H z ]

1 9 7 8 1 7 9 5 . 9 3 4 9 5 . 7 8 6 7 . 7 1 2 0 0 . 2 0 1 . 2 2 4 . 2 9 .1 1 .2 1 . 0

1980 1793.475 9.2 5.4 9.5 1.6 (*) 81 77 97 55 1.0

(*) I n t h i s r u n the F E T ampl i f i e r has a g a i n e q u a l to t h r e e . The re fo r e , t he v a l u e of T n is d e t e r m i n e d b y t h e s e c o n d s t a g e .

2 . - D a t a a n a l y s i s .

The analysis has been made with Frasca 's method (2) by adopt ing as in- dependent variable the frequency v (instead of the period 3 " - - v -~) because the resolving power Av is independent of the frequency. _As shown in ref. (2),

1 ( 1 ) A v - - - -

iN-- tl '

where t 1 and t~ are the t imes of the first and l'~st event submit ted to the analysis.

Expression (1) represents the width of the frequency peak at its base. I n later discussions on the overlapping of peaks observed from different sets of data

we will use as peak width

(2) /1~ = A~/2,

4 4 4 ~ . Ai~ALDI~ 1~. COCCIA, S. FI~ASCA, F. I~ICCI, P. BOI~IFAZI, ~TC.

which is s o m e w h a t less t h a n t he F W H M . The full w id th A~- of a peak at t he

per iod 3" is c o m p u t e d b y t he fo l lowing t r iv ia l re la t ion :

(3) ~ r = A~3-~.

W e recall also t he fol lowing definit ions a l r e ady given in ref. (3). The probability o/ observing accidentally at a cer ta in va lue of v = j - - 1 a

va lue of M2(v)> M 2 is g iven b y

(4a) p ( > M s) = exp [ - - MS],

where , accord ing to t he defini t ions (1) t o (3) of ref. (s),

(4b) M s [F(v)ls

IF(v)[ is t he m odu l us of t he Four i e r t r a n s f o r m of t he even t occurrence t imes a n d N is t he n u m b e r of events .

The number o/ resolvable peaks

Vmax- ~mln (5a) nr.p. - - A~

represents t he n u m b e r of in terva ls A�89 c o n t a i n e d in the a d o p t e d f r equency range

(5b) Vma x = (100 rain) -~ , vmi ~ = (1667 min) -1 ;

re la t ionship (5a) has been tes ted wi th the Monte Carlo m e t h o d (4).

s is the number o/ observed peaks with P<Po, where p is defined b y (4a), a n d for Po we a d o p t e i ther 10 -2 or 10 -3.

The p robab i l i t y of obse rv ing N~ or more peaks wi th P<~Po out of n .... re- solvable peaks is g iven b y

__'*~" [n,.p.~,,,,, 1 __ ~ ~,,,.p.-, (6) p l y , - /.:, / N /l,o~ to , �9

The expected number o] overlapping peaks be tween s 1 peaks der ived f r o m

one of our sets of da t a a n d N s t ida l per iods or t ida l hal f -per iods is g iven b y

Av (7 ) n = .~TI.~ s

~ m a x - - Vmln "

(4) Notice that (Sa) differs by a factor 2 from expression (6) of ref. (2). As a con- sequence, the values (16) and (19) of ref. (2) which appear also in table V become io5= 1.4.10 -9 and P19= 0.19. Notice that, in spite of the large value of P19, the cor- responding p (> no.p. ) remains of the order of 10 -3.

B A C K G R O U N D OF G R A V I T A T I O N A L - W A V E ANT~,BTNAS ETC. - I I I 4 ~ 5

For the number o] observed overlapping peaks we use no.p..

The probability o/ observing n > no.p. overlapping peaks is computed assuming a Poisson distribution:

no..--I -

(8) p(n>n .... ) = 1 - - '~ e - . ~ k : o k ! "

Also this expression has been tested by the Monte Carlo method.

2"1. Analysis o] the (< 1978 data )>. - The list of these events is given in table I of ref. (~).

Figure 1 shows the statistical distr ibution of the corresponding values of M ~ and its comparison with the theoretical formula (4a). The appreciable de- viation of the observed distribution with respect to the s traight line at M ~ > 6

indicates the presence of events with statistics different f rom eq. (4a).

10 3

10 2

101

10 0 2 4 6 8

i

10 M2 12

Fig. 1. - Statistical distribution of the density tunction of the parameter M ~ for the 1978 data: the experiraental histogram is compared to the theoretical line.

The results of the analysis of these data are shown in table I I I . I n this

as well as in successive tables we give the period 3" (and not the frequency v) in order to conform, in the presentat ion of our results~ to the well-established habits of t ide 's investigators. All computat ions have been made~ however~ in frequency.

The peaks we give in table I I I are those with a probabi l i ty of being ac-

cidental p <Po ---- 10-~.

~ ]~. AMALDI, ~. COCCIA, S. FI~ASCA~ F . RICCI~ P. BONIFAZI, :ETC.

TABLE I I I . -- Analysis o] the 1978 data.

3-(rain) p A3- (rain)

1667 3.5.10 -a 37

1504 5.6" lO -a 30

1154 1.3" 10 -4 18

1001 2.3" 10 -a 14

940 2.4- 10 -4 12

819 7.3" 10 ~a 9

722 1.3" lO -a 7

460.2 2.3" lO -a 2.8

371.8 2.1" 10 -5 1.9

360.6 5.1" 10 -4 1.7

315.2 4.8" 10 -3 1.3

266.0 2.5" 10 -4 1.0

261.2 1.4" 10 -a 0.9

174.5 6.4" 10 -3 0 .4

F o r each pe r iod 5 r we g ive also t h e v a l u e of t h e co r r e spond ing p r o b a b i l i t y p.

F o r t h i s set we h a v e

A~ = 1 .33 .10 -5 ra in -1 .

I n t a b l e IV, first l ine, we g ive t h e n u m b e r N of e v e n t s a n d t h e n u m b e r

n . , . of r e so lvab l e peaks for t he (( 1978 d a t a ~).

T h e re su l t s shown in t a b l e I I I sugges t t h e fo l lowing r e m a r k s :

1) a t l eas t one pe r iod ($2) or a t l ea s t t h r e e ha l f -pe r iods (1K1, �89 M2, �89

o v e r l a p w i t h peaks l i s t ed in t a b l e I ;

2) t h e t h r e e o v e r l a p p i n g ha l f -pe r iods co r re spond t o t h e s t r o n g e s t com-

p o n e n t s of t h e E a r t h t ides (4).

TABL]~ IV. - Numbers. of events and of resolvable peaks.

N nr.9.

!978 85 708

1980; Q ~ 2 5 0 0 K 174 376

5 000 K 96 376

10000 K 59 376

20 000 K 32 376

40 000 K 23 376

BACKC-ROU~ND OF G R A V I T A T I O N A L - W A V E ANT]~NNA8 E T C . - I I I 4 4 7

2"2. Analysis o/ the <~ December 1980 data ~>. - In these data we found 174 events fulfilling the condition

(9) e2>~2500 K .

They have been recorded from December 1 (10h 42 rain 05 s) to Decem-

ber 15 ( 1 1 h 3 0 m i n 29s) for a tota l measuring t ime t = 1213104s . The full list, not given here, can be sent on reqnest.

Figure 2 shows a comparison of the statistical distribution of the corre-

sponding values of M 2 with the theoretical formula (4). For this set of data

At --~ 2.48 .] 0 -~ min -1 .

The results of the analysis are given in table V. They have been obtained with the Frasca 's method for a few values of the Q2-threshold. The corresponding numbers 5T and n.~. are given in table IV.

103

102

10 ~

10' 4 6 8 10 12 M21~4

Fig. 2. - The same as fig. 1 for the 1980 data.

The results corresponding to different thresholds are correlated but in a ra ther involved manner , because clearly the threshold is introduced before the application of l~rasca's method.

We notice, however~ tha t the overall picture shown in table VI does not

change much by vary ing the ~2-threshold in spite of the variat ion undergone by _~ (table IV).

~ ~ . A ~ A L D I ~ E . COCCIA, S. I~RASCA, F , R I C C I , P . BONiFAZI~ ~TC.

c~

I

A ~ c~

A ~

!-o A

A

7

o~

"~ r~

~q

?

o~

7

~ ~l ~

~ ~ i ~

i

&

c~

7

o0

B A C K G - R O U N D OF G R A V I T A T I O N A L - W A V E A N T E N N A S E T C . - I I I 449

T A B L E VI. - - Statistical signi]icance o] the occurrence o/ the peaks and their overlapping with tidal periods and hall-periods.

1978 1980

~2 > 2500 K q3 > 10000 K

p < 10 -2 p < 10 -a p < 10:3 p < 10 -a p < 10 =3 p < 1 0 -a

-Yl 14 5 10 5 14 8

P~'I 7.5" 10 -a 7.3" 10 -4 3.5.10 -a 4.3" 10 -s 2.7" 10 -5 6.4.10 -9

a ) no. , ) . 1 - - 4 2 4 4

0.237 0.0847 0.319 0.160 0.447 0.255

p ( > no.,).) 0.21 - - 3.4.10 -4 1.1.10 -~ 1.2.10 -a 1.4.10 -4

b) no.~. 3 2 3 1 3 3

0.237 0.0847 0.319 0.160 0.4~:7 0.255

p(>~ no.,.) 1.9" 10 -3 3.4" 10 -a 4.3" 10 -8 0.15 1.1" 10 -* 2.3" 10 -8

c) no.p. 4 2 5 2 5 5

0.425 0.155 0.585 0.292 0.819 0.468

p ( > no.').) 1.1.10 -a 1.1.10 -3 3.5.10 -4 3.5.10 -3 1.6.10 -3 1.3.10 -4

Al so in t h i s case we no t i ce t h a t

1) four t i d a l pe r i ods (Q1, K3, 2q3 a n d $3) or t h r e e t i d a l h a l f - p e r i o d s (Q1/2, 01/2, K1/2) o v e r l a p w i t h pe r i ods l i s t e d in t a b l e V, for e x a m p l e for ~3> 10 000 K ;

2) t h e s t r o n g e s t c o m p o n e n t s of t h e E a r t h t i d e s a r e p r e s e n t a m o n g t h e

o v e r l a p p i n g pe r iods .

2"3. Statistical analysis. - T a b l e V I s u m m a r i z e s t h e r e su l t s of t h e a n a l y s i s

g i v e n in subsee t . 2"1 a n d 2"2. I t is d i v i d e d in fou r p a r t s . T h e u p p e r p a r t shows

t h a t t h e p r o b a b i l i t y P~I (eq. (6)) is v e r y s m a l l for b o t h se ts of d a t a a n d for

b o t h va lue s of Po ( 10-3 a n d 10-3). The t h r e e o t h e r p a r t s r e fe r to t h e o v e r l a p -

p i n g of t h e p e a k s d e d u c e d b y m e a n s of F r a s c a ' s m e t h o d w i t h a) t h e IY3 = 6

t i d a l p e r i o d s of t a b l e I , b) t h e -Y3 ---- 6 t i d a l ha l f -per iods~ c) t h e iV~ = 11 d i s -

t i n g u i s h a b l e (5) t i d a l p e r i o d s a n d ha l f -pe r iods .

W e g ive a lso in a l l t h r e e cases t h e e x p e c t e d ~ of o v e r l a p p i n g s c o m p u t e d

f r o m (7).

N o t i c e t h a t t h e p r o b a b i ! i t y p(n>no.').) (eq. (8)) is a l w a y s b e t w e e n 10 -~

a n d 10 -a e x c e p t in t h r e e cases.

2"4. Correlation with tidal phases. - T h e r e s u l t s p r e s e n t e d in t h e p r e v i o u s

sec t ions sugges t a c o r r e l a t i o n b e t w e e n t h e s t r o n g e s t t i d e p e r i o d s a n d t h e oc -

(~) K 1 / 2 i s undistinguishable from K s .

4 5 0 :E. AMALDI, 1~. COCCIA, S. FRASCA, F. RICC], P. BO/NIFAZI, ETC.

TABLe. VI I . - Results o] phase analysis.

M2 P = ~ ~ t -~s N~ ANs A(M 2) 59 = e x p [ - - M 2] (~ (~

Ma/2

J u n e 1978 2.13 1.1.10 -3 44 44 14 30 3.9 0.68 15

J u l y 1978 2.28 5.5.10 -3 17 41 15 26 3.6 0.63 14

December 1980 1.32 0.18 295 59 10 49 4.9 0.83 26

all d~ta 2.41 3.0.10 -a 9 144 29 115 7.6 0.80 15

82/2

J u n e 1978 ].64 6.8- 10 -2 353 44 11 33 4.1 0.75 21

J u l y 1978 2.24 6.6.10 -3 25 41 14 27 3.7 0.66 14

December 1980 0.23 0.95 266 �9 59 2 57 5.4 0.97 72

all dat,~ 1.99 1.9.10 -3 7 144 24 120 7.8 0.83 18

t(1/2

J u n e 1978 1.75 4.6.10 -2 344 44 12 32 4.0 0.73 19

J u l y 1978 1.69 5.7.10 -3 47 41 11 30 3.9 0.73 20

December 1980 3.56 3.1.10 -6 172 59 27 32 4.0 0.54 8

all d a t a 1.01 0.36 134 144 12 132 8.1 0.92 34

01/2

J u n e 1978 0.71 0.60 82 44 5 39 4.4 0.89 43

J u l y 1978 1.49 0.11 354 41 10 31 4.0 0.76 23

December 1980 2.64 9.4.10 -4 128 59 20 39 4.4 0.66 12

all d a t a 1.65 6.6.10 -2 97 144 20 124 7.9 0.86 22

Q1/2

J u n e 1978 2.24 6.6.10 -s 233 44 15 29 3.8 0.66 14

J u l y 1978 1.70 5.6.10 -2 264 41 11 30 3.9 0.73 20

December 1980 2.65 8.9. l0 -3 64 59 20 39 4.4 0.66 12

all d a t a 1.94 2.3.10 -2 256 144 23 121 7.8 0.84 19

2V2/2

J u n e 1978 2.02 1.7.10 -2 249 44 13 31 3.9 0.70 16

J u l y 1978 0.32 0.90 305 41 2 39 4.4 0.95 65

December 1980 1.51 0.10 95 12 12 47 4.9 0.80 23

all d a t a 0.44 0.82 210 144 5 139 8.3 0.97 58

B A C K G R O U N D O F G R A V I T A T I O N A L - W A V E A N T E N N A S :ETC. - I I I 451

�9 cur rence t imes of a large p a r t of t he even ts recorded b y t he F ra sca t i an t enna . U n f o r t u n a t e l y , we h a v e no t y e t succeeded in iden t i fy ing the l ink be tween t he

slow osci l lat ion of t he E a r t h (free as well as forced) and t he exc i t a t ion of our a n t e n n a at a m u c h h igher f r e q u e n c y (o) R-~ 10 4 rad/s). ) ' o r a few models , a m o n g

the m a n y t h a t t e n t a t i v e l y can be env isaged (1), a fixed (or a lmos t fixed) re la t ion could exist be tween t he phase of a single well-defined t ide oscil lat ion a n d t he

occurrence t ime of the events recorded by t h e F r a s c a t i a n t e n n a . Clearly the phase ~ is o b t a i n e d f rom the va lues of (2)

F~(v) /~'Av) s in = vL, v i + cos v +

The express ion of t he var ious t ida l c o m p o n e n t s used for the c o m p u t a t i o n of the phases in F ra sca t i arc g iven in the append ix . I n the case of M2 and $2

t he phase 0 ~ cor responds r o u g h l y to the i n s t a n t of passage t h r o u g h the F ra sca t i mer id ian p lane of t he b o d y responsible for t he cons idered forced oscillation.

The resnlts of t he phase ana lys is are shown in t ab le VII~ in which for each

c o m p o n e n t we give, in add i t ion to the a m p l i t u d e M , t h e p robab i l i t y (4a) and the phase ~, t he fo l lowing qua n t i t i e s :

N t = t o t a l n u m b e r of even t s ;

Ns = expec t ed n u m b e r of even ts c o n t r i b u t i n g to the c o m p o n e n t u n d e r cons idera t ion (6):

(10) N8 = M V / ~ ;

N~, = n u m b e r of even ts t h a t do no t con t r ibu t e to the c o m p o n e n t

u n d e r cons ide ra t ion :

(11) N.~ = N t - - N s ;

(~) From eq. (4) of ref. (2) we have

~-~ sin2~vt k + ~ cos2~vt k ~ k = l k = l J

Nt

which, in the most favourable case, becomes

N 2 ( 1 0 ' ) M 2 = ~ ' s .

Nt

In connection with (10') and (I0), we point out that a single event can contribute to more than one peak of the Fourier analysis. A detailed discussion of this point could throw light on the generation mechanism of the events.

452 E. A1KALDI, E. COCCIA, S. FRASCA~ F. RICCI, P. B0.NIFAZI, ETC.

AN~ = statistical error on Nx:

(22) AN~ = V�89162

A(M~) = statistical error on M2:

.~T N (13) A(M2) = Art ;

A 9 = statistical error on the phase:

(14) A~ = a r c t g - - - - N~

From table V I I we see tha t the data relative to the components M2/2 and

$2/2 are consistent with 9 -~ 0~ This is what is expected for a local mechanism of excitation of the antenna, tr iggered by the tide.

The K1/2 line deserves special consideration since it concides with one- half of the sideral period. Because of the radiation pa t te rn of the antenna, the

observation of this period can be interpreted in two different ways: a) the ex- citation of the an tenna is due to a local mechanism triggered by the tide,

b) because of the East -West orientation of the antenna, its excitation is due to a gravi ta t ional source located at the centre of the Galaxy. In case a) the phase ~0 is expected to be 180 ~ In case b) the phase should be

= 2 ~ o c = 172 ~ ,

where ~oc is the r ight ascension of the galactic centre. From table V I I we see tha t the K1/2 line is statistically significant only for the December 1980 run for which we obtain

(i5) ~e.p(K1/2) ---- (172 - - 8 ) ~ .

The statistical error does not allow the distinction between the two cases a) and b).

The data of table V I I concerning the three other tide lines (01/2 , Q1/2, N2/2) do not offer any simple interpretation, since they appear to be distributed a t random.

3 . - D i s c u s s i o n .

The observation of some of the t idal components in the background of the

Frascat i antenna supports the idea put forward in two previous publications (1.,)

BACKCxROUND OF C~RAVITATIONAL-WAVF~ ANTENNAS ETC. - III 4~3

t h a t the Frasca t i an t enna is sensit ive to small m o v e m e n t s of the Ear th . The fact t h a t the phases of the t idal components 01/2, Q1/2, N2/2 have different values in different runs could be due to the complex i ty of the phenomenon. A s i tuat ion in some way similar has been found by KLv.T_~ (7) in the s tudy of the ea r thquakes which appear to be t r iggered by the t ide with a phase depend- ing on the t ype of ea r thquake and the location of ils focus.

I n the f r ame of these considerations we also ment ion the results obta ined by Moon seismologists t h a t ((the moonquakes occurred mos t f requent ly near the t imes of m i n i m u m (perigee) and m a x i m u m (apogee) distance be tween the E a r t h and Moon dur ing each mon th ly revolut ion of the Moon abou t the Ea r th , suggest ing t ha t the moonquakes are t r iggered b y t idal stresses ,~ (*).

I f we compare table I I I and table V (the la t te r for ~ > 1 0 0 0 0 ) referring, respect ively, to the ]978 and the ]980 data , we notice t h a t 7 periods (each wi th p < 1 0 -2) overlap. Only two of these appear to be correlated with t ide components (M~/2, K~/2). The probab i l i ty of finding 7 over lapping periods amoun t s to 2.10 -5, and, therefore, one can suspect the existence of E a r t h ' s periods due nei ther to the tides nor to free oscillations of the E a r t h as a whole. I t appears interest ing to point out t h a t the values of these periods (for example , ~-----1640 min and 3 - = 997 min, etc.) are in the range of the computed values for the free oscillation of the E a r t h core (~).

The most impor t an t and not ye t clarified aspect of the phenomena consid- ered in ref. (~.~) as well as in this th i rd pape r is the l ink be tween the (slow) oscillations of the E a r t h (free as well as forced) and the exci ta t ion of the detector.

At the end of ref. (~) we l isted th ree classes of possible mechanisms, the descript ion of which can be summar ized as follows:

a) a pure ly ins t rumen ta l effect due to the ve ry slow mot ion of the cryo- s ta t carried by the Ea r th , the surface of which is displaced up to a few centi- me te r in one hour;

b) a mechanical burs t produced locally in the crust (propagat ing th rough the E a r t h with seismic velocity) which is t r iggered b y the E a r t h oscillations, and passes th rough the mechanica l filters p ro tec t ing the an tenna ;

e) a gravi tat ional-f ield var ia t ion due to a ter res t r ia l source.

(~) F . W . KI, EI.~: Geophys. J . R. Astron. Soc., 45, 245 (1976). (s) G. V. LATHAM, M. EWING, F . PRESS, G. SUTTOK, J . DOMAI~', Y. NAKAMURA, N. TOKSOZ, D. LAMMLEIN and F. DUENNEBIER: Apollo 15, Preliminary Science Report, NASA SP-289 (1972), p. 8.1. (9) See, for example, C. L. PEKERI8 and Y. ACCAD: Trans. 1~. Soe. f~ondon, 273, 237 (1972); D. E. SMYLIE: I I International Symposi~m on Geodesy and Physics o] the Earth, G.D.R., Po.~tdam, May 7-10, 1973, VerSff. Zentralinst. f. Phys. der Erde, Postdam (1977) 52.T.2.

30 - II Nuovo Gime~to G.

~ . ~ :E. AMALDI, ~. COCCIA, S. FRASCA, F. RICCI, 1". BONIFAZI, ETC.

We still consider mechan i sm a) very unl ikely (1) and we do not discuss here mechanisms of t ype v) since, according to previous es t imates (lo), t hey are expec ted to be too weak.

Class b) can be split into var ious subclasses according to the f requency of the seismic signal and the response of the electronic appara tus , which can r ema in in the l inear region or introduce nonl inear i ty because of saturat ion.

Concerning the f requency of the seismic signal we consider here low fre- quencies ( v ~ ] Hz) and resonance frequencies (~-~ 1800 ttz).

Low-frequency signals should be considered as a possible mechan i sm because of the inadequacy of the mechanical filters for frequencies below a few tens of Hz. The sa tura t ion of any of the amplifiers should be kept in mind because it generates h igh-frequency components which fall in the f requency band near the an t enna resonance f requency tha t is accepted by the PSDs. The l inear regime also should be examined because all the amplifiers (including the PSDs), in spite of appropr ia te filtering stages, t r a n s m i t low-frequency signals with a finite a t t enua t ion (table V I I I ) . Such a signal can be mis in terpre ted as due to a signal a t the an tenna resonance frequency.

TABLE VIII . - Ere~ueney dependence o] the attenuation o/ the electronic chain.

v (Hz) 1978 data, A(v)/A(18OOHz) 1980 data, A(v)/A(1800Hz)

10 7.4.10 -11 5.1 �9 10 -8

100 7.4.10 -~ 1.5 �9 10 -6

1800 1.0 1.0

A(1800 Hz) 2.3.108 1.66.107

I n order to avoid the sa tura t ion we had observed in the da ta of May, 1980, we have s t rongly reduced the amplif icat ion of the low-noise F E T preampli- fier (see footnote of table I I ) .

Therefore, we are inclined to exclude such a class of events also because of a direct observat ion concerning low frequencies ment ioned below.

For the low-frequency case we have cal ibra ted the Frasca t i detector at the first ha rmonic in the configuration used for the December 1980 run, by means of a series of 7 ea r thquakes t ha t occurred in I rp in ia (at about 300 k m f rom Frascat i ) in the period ~ 'ovember 23-25, ]980.

We found t h a t the m a x i m u m value of Q2 was connected to the m a x i m u m displacement ~ of the E a r t h surface at frequencies near 1 Hz by the rough empir ical relat ionship

(]6) Q~K.,.,~ = 20$~m(around 1 t t z ) .

.(so) See, for example, M. RE~]S, R. R~'FFL~'I and J. A. WH~.LER: Black Holes, Gravita- tional Waves and Cosmology (New York, N.Y. , 1976), p. 135.

BACKOI~OUND OF GRAVITATIONAL-WAVE ANTENNA8 ]ETC. - III 4~5

From this re lat ion we see tha t , in order to genera te events of ~2>~2500 K b y this mechanism, a d isplacement of the E a r t h surface of a t least 11 Ezm is required.

Dur ing the run of December 1980 there was no ea r thquake in the per iod of da ta tak ing with a d isplacement grea ter t h a n 11 ~m and, therefore, we con- clude t ha t the December ]980 events cannot be due to this k ind of process.

I f we consider now the frequencies near the resonance of the first mode of the antenna , we recall t h a t the mechanical filters placed below and inside the c ryos ta t have a global t ransfer funct ion of 240 dB. I n considerat ion of the Q-value of the ba r ( table I I ) the overall a t t enua t ion is reduced to 145 dB. There- fore, in order to get ~ >~ 2500 K, a d isplacement of the E a r t h surface is required of the order of 2.5.10 -2 ~m which is about five or six orders of magn i tude above the seismic noise measured in the ki lohertz region.

I n conclusion, we consider ve ry unl ikely t h a t the events discussed here orig- inate f rom low-frequency seismic waves, while an in te rpre ta t ion in t e rms of a mechanical pe r tu rba t ion a t the resonance f requency should be fur ther inves- t igated. Fu r the r exper imenta l invest igat ions are in prepara t ion .

We dedicate this paper to A. SAg_m~ov as a cont r ibut ion to the celebra- bra t ion of his 60-th b i r thday .

This work has been financed by the CNIr and I N F N . We express our t hanks to our colleagues F. BOI~DONI and F. F~LIG~'I for crit ical discussions, to Mr. A. So~cE for his contr ibut ion to the da ta analysis and to Drs. C. GASPAICLNI and A. I~OVELLI of I s t i t u to Nazionale di Geofisica for providing the seismological data .

APPENDIX

Tide-generatlng potential (11).

We use the following nota t ions in t roduced b y DO0DSO~:

L ----terrestrial longitude of the observer, computed eas tward ;

v ----local lunar m e a n t i m e ;

S ----mean lunar longi tude;

h ~ mean solar longi tude;

(11) X. T. DOODSOK: Proc. R. Soc. London Ser. A , 10O, 305 (1922).

~ ~,. A B~ALDI~ ~,. COCCIA~ S. FI~ASCA~ F. RICCI~ P. BOI~IFAZI~ :ETC.

p ~ t h e l o n g i t u d e of the l u n a r p e r i g e e ;

N ' - - ~ - - N ~ t h e n e g a t i v e of t h e l o n g i t u d e of t h e a s c e n d i n g node /~;

p~ ~ l o n g i t u d e of t h e so la r pe r igee .

T a k i n g t h e G r e e n w i c h t i m e (U.T.) in m i n u t e s s t a r t i n g f rom 0 h J a n u a r y 1 1900 (t) we h a v e t h e fo l lowing p h a s e s :

(A.1)

~-- 0 .25 . t -~ h - - S ~ Z~

S ~ 277.02 ~ ~ 0 . 0 0 9 1 5 0 2 7 5 5 0 6 . t

h ~ 280.19~ - 0 . 0 0 0 6 8 4 4 7 7 3 1 6 1 . t ~

p --~ 334.38 ~ -~- 0 . 0 0 0 0 7 7 3 6 3 9 4 4 6 . t

N ' - - 100.84 ~ ~- 0.000 036 773 555 711. t

p l ---- 281.22~ - 0 . 0 0 0 0 0 0 0 3 2 6 8 . t .

A c c o r d i n g to DooDso.~'~ a l l t e r m s of t h e h a r m o n i c e x p a n s i o n of t h e t i de - g e n e r a t i n g p o t e n t i a l h a v e p h a s e s w h i c h a re l i n e a r c o m b i n a t i o n s of t h e phase s (A.1) w i t h i n t e g e r coeff ic ients .

T h e a m p l i t u d e of e ach of t h e s e t e r m s can be f a c t o r i z e d in t h r e e p a r t s :

(A.2) A ~ Aa 'Al , t 'G*

w h e r e Aa d e p e n d s o n l y on t h e c o n s i d e r e d term~ A~t d e p e n d s a lso on t h e ter- r e s t r i a l l a t i t u d e a n d

(A.3) G* ---- 26.2 cm~/s 2

is t h e so -ca l l ed D o o d s o n c o n s t a n t . T h e va lues of t h e v a r i o u s p a r a m e t e r s for t h e t i d e t e r m s c ons ide r e d b y us

a r e g i v e n be low. .Notice t h a t some of t h e m a re of t h e s ine t y p e , o t h e r s of t h e cos ine t ype . The

a b b r e v i a t i o n s tess . a n d sect . m e a n t e s s e r a l a n d s e c t o r i a l t i de . F o r t h e h a r m o n i c s of t h e a b o v e - c o n s i d e ; e d lines~ t h e p h a s e s have been

o b t a i n e d b y m u l t i p l y i n g t h e p h a s e of t h e c o r r e s p o n d i n g l ine b y t h e h a r m o n i c o rder .

Line Origin Type Phase A a A ~ sin/cos

QI L tess. 3 - - 2s ~- p 0.07216 sin (2 lat) sin

01 L tess. 3 - - s 0.37689 sin (2 lat) sin

K 1 L, S tess. 3 + s - -0 .53050 sin (2 lat) sin

N~ L sect. 2v - - s ~- p 0.17387 cos ~ lat cos

M S L sect. 23 0.90812 cos 2 la t cos

$2 S sect. 23 ~ 2s - - 2h 0.42286 cos 2 la t cos

BACKGROUI~ 'D OF G R A V I T A T I O N A L - W A V E A N T E N N A S ETC. - I I I 4 5 7

�9 R I A S S U N T O

In due p receden t i lavori sono s t a t i p r e sen t a t i var i a rgomen t i in favore di u n a eorrela- zione fra una p a r t e dei segnal i di rondo di u n ' a n t e n n a g rav i t az iona le ( 3 / = 390 kg) e le oscil lazioni l ibere del la Terra. Questi r i su l t a t i h a n n o sugger i to di eseguire un analogo s tudio nei r iguard i delle oscil lazioni fo rza te del la Terra . In questo lavoro sono presen- t a t i i r i su l ta t i de l l ' anal is i di Four ie r pe r per iodi eompres i f ra ~ - = 100 e 1667 rain dei da t i raccol t i in due serie di misure , esegui te l ' una nel 1978, l ' a l t r a nel 1980. Ess i p re sen tano var ie f requenze alcune delle quali sono in o t t imo accordo con le f r equenze delle maree . L a probabi l i t~ che ques~e (~ sovrappos iz ioni ,) s iano p u r a m e n t e easual i de l l 'o rd ine o infer iore a 10 -2.

Cleon aHTeml rpalnrralDlomlblx soml BO3MO~CJIOUO 3eMHOUO npoycxo:~,,~eln~.

Pe3IoMe (*). - - B ~Byx npe/~,t~ymnx CTaTb~rx MbI r~pnaeylrt RoKa3aTeYmcTBa cymec'rBo- BaltH~ roppeaaLr~n M e ~ y ~taCTb~O qbOHa aHTeltUI, I rpaBnTaIlttoHnbIx BOYIH ( M = 3 9 0 r r ) r~ CBO60~ttb~MH octtnnnaLra~Ma 3eMna. 3TU pe3ynbTaTbt yraabmamT Ha ~renaTen1,Hocrl, ananorHqHJ, Ix nccne:ioBaaatt BJ, tuy~eHnbIX octlanJmua~i 3eM~m. B aTO~t craTI, e MbI npH- BO~J~M pe3yJ~bTarbt ~ypbe-aHanHaa ~aHHb~X, co6paHH~,IX B ~/ByX cepr~x naMeperta~: O2IHa B 1978, apy raa B 1980, n~a neprio~oB B HHTepBane Me>r~y ~ r : 1 0 0 a 1667 MHHyT. ~r r t peayn1,raTbt n o r a a u s a m r neciconi, r o qaCTOT, HerOTOpbm n3 rOTOpI, IX xopotuo COFJIacyIoTC~I C HpHJII4BHbIMH ~IaCTOTaMH. BepO~THOCTb TOrO, qTO 3TH ~< nepeKpr~tTHn >7 $1B$IHIOTCH cnyqa~HbIMH, HMeeT r t o p ~ o K l0 -2 HUrl MeHbIHe.

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