IL NUOVO CIMENTO VOL. 9 C, h ~. 2 Marzo-Aprilc 1986

Estimate of the Background of a Gravitational-Wave

Detector Due to Cosmic Rays.

E. AMALDI a n d G. PIZZELLA

Dipar t imen to di .~'isiea dell' Universit& <~ L a Sap ienza ~> - ]~oma I s t i tu to IVazionale di l~isica Nucleare - Sezione d~ l~oma

(ricovuto 1'1 Luglio 1985)

Summary . - - The authors examine once more the effect of cosmic rays on a resonating gravi tat ional-wave antenna in view of the very high sensitivities tha t are required for detecting the supernovae of the Virgo Cluster. They show that , at sea-level, the secondaries generated in the bar by the electromagnetic interact ion of high-energy muons produce signals with rates much larger than tha t expected from supernovae. This inconvenience is el iminated in an underground laboratory.

PACS. 0 4 . 8 0 . - Exper imental tests of general re la t iv i ty and observa- tions of gravi ta t ional radiat ion. PACS. 94.40.Tc. - ~iuons and neutrhms.

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

BERON a n d HO~'STADTE~ (~), a l r e a d y in 1969, ca r r i ed o u t e x p e r i m e n t s

a i m i n g to d e t e c t r ~ d i a l ~nd c o m p r e s s i o n a l m o d e s of osc i l l a t ion (40 a n d 158 k H z )

of p i ezoe lec t r i c discs e x c i t e d b y ~ 1 (or 0 . 2 ) G o V e lec t ron b e a m c o n t a i n i n g

10~+10 e pa r t i c l e s p e r pu l se of 1 ~.s d u r a t i o n . The i r r e su l t s b r o u g h t t h e a u t h o r s

to sugges t t h a t <, ~ v e r y la rge c o s m i c - r a y e v e n t cou ld exc i t e m e c h a n i c u l v i b r a -

t ions in ~ m e t a l l i c c y l i n d e r a t i t s r e s o n a n c e f r e q u e n c y a n d t h e y cou ld p r o v i d e

a n acc iden taJ b a c k g r o u n d for e x p e r i m e n t s on g r a v i t a t i o n a l w~ves ,>.

(*) B . L . B):~oN and R. ~Io)'STADT)i]r P h y s . t~ev. Le t t . , 23, 184 (1969); B. L. BE~O~, S . P . B()uGll~-, W. 0. IIA~ILTO~', R. HOFST~DT~.:I~ and T . W . T~RT]I,,': I E E E Trans . N u c l . Sc i . , NS-17, 65 (1970).

612

]~STIMA'I']'~ OF TII]s BACKGROUND OF -4. GRAVI'I'ATIONAL-WxkV~E DETI,~CTOR ETC. 613

A few months la ter WEBER et al. (o.) searched for coincidences between a large cosmic-ray telescope placed immedia te ly over the 96 cm d iameter gravi ta t ional an t enna and the ou tpu t of the an tenna and concluded t h a t (( a gravi ta t ional rad ia t ion detector does not produce a signal when bi t by showers of 9art icle densi ty of the order of ]00 p~rt icles/m 2 ~).

The mechanical response of a small (20 cm long and 3 cm diameter) alu- min ium alloy ba r to pulses of 30 MeV protons has been invest igated b y a Milan

g r o u p (~).

F r o m these results, all the people in teres ted in the search for gravi ta t ional waves years ago arr ived a t the conclusion t ha t any effect due to cosmic rays was below the best sensibility of the avai lable detectors. This was our a t t i tude, also when, some t ime ~go, Pau l MUSSET posed the question to us.

The problem of the influence of cosmic rays on grav i ta t iona l -wave detec- tors should, however, be re-examined tod~y in consideration of the much higher sensi t ivi ty already reached, and even more in view of the ampl i tude expected f rom supernovae taking place in the Virgo Cluster. The ampl i tude of the grav i ta t iona l waves due to this type of events is es t imated to be of the order of h-- -10-2~ 10 - ~ depending on the mass conver ted into gravi ta t ional radia t ion a n d the band width covered b y the signal.

2. - Exc i ta t ion o f the a n t e n n a f u n d a m e n t a l mode due to a h i g h - e n e r g y particle .

Following the approach of previous authors (a) we assume tha t the v ibra t ion of the fundamen ta l mode of the bar (o) - - z v / L , Z - - - b a r length, v = sound velocity in the bar material) originate f rom the local t he rma l expansion caused by the warming up duo to the energy lost by a high-energy part icle crossing the ba r perpendicular ly to its axis a t a distance x f rom the centre. According to the Milan group the ampl i tude of the fundamen ta l mode is

a W L 2 ~x (1) ~ ~ c , M z~ c~162

where W is the energy lost b y the part icle in the bar , a the the rma l l inear ex- pansion coefficient, c~ the specific bea t a t constant vo lume and M the mass of the bar. The corresponding to ta l energy of the ba r is

1 (2) ~ - - ~ ~ o ~ ~ - - ~ ~ eos~ ~x w e

(2) D . H . ]~zRow, • . S. WALL, J . ~V~EBEI~ a n d G . B . YODtt : Phys. LRev. Lett., 24, 945

(1970). (3) A. )/I. G~Assl S'rlu~,*I, G. Srrm,~ and G. TAGnL~FWlUU: J. A1~pl. l)hys., $1, 849 (1980).

614 ~. AI~ALI)I and G. I'IZZ~LLA

In t roduc ing into (2) the mass of our bar ( M - - 2 3 0 0 kg of A1) the sound

velocity (v ---- 5400 m/s at liquid-helium temperature) and assuming, according to the Griineisen law, ~/c~ independent of the temperature, we obtain

~ X (3) # : 1 .54 .10-sW 2 c o s 2 ~ - - - L / 2 < x < L / 2 ,

where # is expressed in kelvin and W in GeV. The same problem has been treated in recent times by ALLEGA and CA-

BIBBO (4) and by D~, R~JULA et el. (~) with more refined procedures. The numerical values given by (3) agree within factors of a few units with those deduced in ref. (4,5).

I n order to detect the supernovae in the Virgo Cluster the gravitat ional wave detector should reach a sensibility of the order of # ~ 10 -7 K.

l~rom expression (3) we see tha t any cosmic-ray event releasing in the bar an energy W ~ 3 GeV contributes to the background of the antenna.

3. - The effect of cosmic rays.

A single high-energy particle like a muon at minimum ionization looses in aluminium 2 MeV/(g/cm~). The diameter of our bar is 60 cm corresponding"

to 162 g/cm ~ and an energy loss of 324 MeV. According to cq. (3) it gives a

signal of 10 -9 K, and therefore indetectable for many years to come. Extensive air showers, at sea-level have an integral density spectrum (6)

(4) H(>A) = 0.2"d -L5 s -1 ,

where d is the density defined as the number of particles per m s. 95 % of the

shower particles are electrons and photons almost all with an energy very close to the critical energy in air, let us say 100 MeV. Therefore, the total energy dissipated in the bar by an air shower of density d is

(5) W : d �9 S" 100 McV,

where S = 1.8 m ~ is the geometric cross-section of the bar. This energy 9 how- ever, is dissipated over all the bar and, therefore, its mechanical effect is cer- tainly smaller tha t tha t computed from eq. (3) taken with x = 0. For this

overestimated upper limit we obtain, f rom (3)-(5), 17 events per day with energy d~>~5.10 -6 K and 0.5 per day with d~>10 -4 K.

(4) A.M. ALL~GA and N. CABIBBO: Lett. s Cimento, 3B, 263 (1983). (5) C. B~RNAnI), A. D~ R~JULA and B. L A ~ P : s Phys. B, 242, 93 (1984). (6) G. Cocco~r and V. Cocco~I TONGIORGI" Phys. Roy., 75, 1058 (1949).

;ESTISIAT~E OF TI IE BACI(GI~OU:ND OF A GI~AVITATIONAL-WAV;E D:ET:ECTOIr ;ETC. 615

These events cun be easily identified b y an appropr ia te set of counters. A grav i ta t ion wave detector is such a mass ive body tha t o ther effects

p roduced by cosmic-ray p~rticles should be considered. I n par t icular the product ion of electronic and photonic secondaries b y the muons in the b~r s ta r t e lect romagnet ic showers t ha t are in great purr absorbed in the ba r itself. These processes are of four different types : p roduct ion of knock-on electrons, bremsst rahlung, direct pai r p roduc t ion and photonuclear interactions.

They are considered in detail below. Their discussion requires the energy spec t rum of the muons which can be obta ined f rom the exper imenta l da ta given b y MENO~ and I~A3/fAiNA (7) for E ~ 2 0 GeV and ALLKOFER and To-

YascTx (s) for lower energies. F igure i shows the expe r imen ta l points and the curve represents the following pa ramct r i zu t ion of the ver t ica l integrul spec- t r u m I (E~):

(6) in I+,(Ev) : - - 0.15~9 (ln Ev) ~ - - 0.6718 (ln Ev) - - 4.826.

The in tegra t ion over the solid anglo 2~ gives a factor 2~/3 ~ 2.

Knock-on electrons. For the product ion of knock-on electrons of energy be tween W and W + d W we use the well-known formula (9)

(7) dW--2 Z w:

E~ - - (m~ c~) ~ W m a x : 2 ~ e 0 2

( m , 02) 2 ~ - (m~e 2)~ ~_ 2me c 2 E v "

The corresponding (( in tegra ted ~) mean free p a t h in a lumin ium

(s)

A 1 2(E~, > W) - - qN~ a(E~, >~ W) '

Wmax(E~)

a(E~, > W ) = ~ d W ,

W

is p lo t ted in fig. 2. The rat io of the d iameter of the bar to 2(E~, ~ W) gives a rough idea of how often a known-on electron is produced in the bar.

(7) M . G . K . ~/IENON and P. V. RA~AN~ MVl~THY: Cosmic ray intensities deep q~nder- ground, in Progress in Elementary Particle and Cosmic t~ay Physics, edited by J. G. WIL- sON and S.A. WOVTHVYSEN, Vol. 9 (North-Holland PubL Co., Amsterdam, 1967), p. 220. (8) 0. C. ALLKOFER and H. TOKISCH: 2~uovo Cimento A, 15, 371 (1973). (9) B. RossI: High Energy Particles (Prentice-Hall Inc., New York, N.Y. , 1952).

616 ~. AMALDI a n d o. I'IZZELLA

id 2

10 --~

10 -(

T

~,,,~ 10 --= r

io -~

lo -7

10 -0 l I

10 0 101 10 2 I@ 10 ~ EI~(GeV)

Fig. 1. - I n t o g r a l s p o c t r u m in t h e vo r t i ca l d i rec t ion of m u o n s ~t sea- level : exper i - m o n t a l po in t s ~ o m ref. (7,s), cu rve r ep rosen t s eq. (6).

For bremsstrahlung wo use the formul~ b y PE~RU~m~ and Sm~STAKOV (~o)

d W - - ~ 3 2Z~e , ~ 3 E ~ § r

whore, lor Z > 10,

(189m~/m~) Z - i ~b((~) = In 1 + ((189v'-e)/m, e") (~Z-i'

W - - (m~ e2) ~ 2 E ~ ( 1 - - W/E~) "

(lo) A . A . PE'r~UKHIN a n d V . V . SI{ESTAKOV: Can. J. Phys., 46, $377 (1968).

ESTIMATE OF THE BACKGROUND Ole A GRAVITATIONAL-WAVE DETECTOR ETC.

i06r -

E~=5OOeV

=100

617

10 ~-

10 ~

=20

=200

= 3 0 0

"E 3 ~io

I02~/// /60 bar vl/ameter

101

I I 1 I I I I I 10 10 20 30 a0 50 60 70 80

W(GeV)

Fig. 2. - In tegra ted mean free pa th in A1 of muons of energy E~ for production of a knock-on electrons of energy > W.

F o r pair production we use t h e f o r m u l u e (5) a n d (6) g iven u t p a g e 85 of Rossi~s

b o o k (9) a n d for n u c l e a r i n t e r a c t i o n t h e f o r m u l a d e r i v e d b y B E R z v ~ 0 u a n d

BUGAEV (11) in t h e f r a m e w o r k of t h e v e c t o r d o m i n a n c e m o d e l a n d g iven b y

LOmgAN~r et al. (~).

I n d o i n g t h e c o m p u t a t i o n w e a s s u m e t h a t a l l t h e (< secondar ie s ~) p r o d u c e d

(n) L . B . BEZRVKOV and E .V . BUGAEu See. J. NucL Phys., 33, 635 (1981). (13) W. LO~IMA~r162 R. Koe~ and R. Voss: J~nergy loss o] muons in the energy range 1-10000 GoV, CERN 85-03 Exper imental Physics Division 21 March 1985.

618 ~. ~tMALI)I and G. eIZZnLLA

b y a rouen ins ide the b a r gene ra t e a n e l ec t romagne t i c shower which is com-

p le te ly absorbed in to the ba r itself. This s impl i f i ca t ion (is) gives rise to an

105 r - - - - - x - - -

10 a

10 3

~ 10 2

10 ~

10 ~

b)

lo 10 -e 10 -7 10 .6 10 -5 10 -~ 10 -3

e(K)

Fig. 3. - N u m b e r of events per day of energy > ~ (in ke ]~b l ) p roduced b y muons at sea-level in an Al bar of M = 2300 kg, through: a) knock-on electrons, b) bremsstrah- lung, v) nuclear interaction, d) direct pair production.

(la) We recall that the depth within which 50% (98%) of the incident energy is de- posited, is given (in X 0 units) by

0.4 for electrons, tme d = in(W/s) + a , a = 1.2 for y-rays,

tgs% --~ 3t=ea, e = critical energy.

E S T I M A T E OF Tn :E BACI~G:EOUND O F A GI~AVITATIO~ffAL-WAV:E :D~T:ECTO]~ :ETC. 619

overes t imate b y a factor of the order of two or three, which, however , is par- t ial ly compensa ted b y the fac t t h a t we neglect the secondaries produced in the walls of the c ryos ta t and o ther surrounding materials .

The number of events of a given type i wi th an energy greater t han We due to the muons is given b y

f dz,(E~, W)/(E.) dE (9) N~(>~ We) : d W ntot dW '

WO J~min(W)

where rite t ~- 5.11.1028 is the to t a l num be r of A1 a toms in the bar, ](E~) = z - (27e/3)dI~(E~)/dE~, i distinguishes the four processes listed above and /~ ~(W) is the m i n i m u m energy t h a t should have a m u o n for producing a secondary of t y p e i with to ta l energy W.

For deriving the n u m b e r of events due to processes of t ype i which give signals in the ba r with energy greater than # we should use eq. (3) and consider t h a t the event can take place for any value of x between :~Z/2:

+L]'~

O-oa) . ~ ( > ~) = ~ ~(>W(x))dx, --L/2

(10b) W(x) = .5.10 -s cos 2 2 z x / L "

The integrals (9) and (10a) have been computed numerical ly. The results are shown in fig. 3.

4 . - C o n c l u s i o n s .

F r o m fig. 3 we see t h a t for the resonat ing bars of the present generat ion (which can reach, a t best , an effective noise t empera tu re of I inK) cosmic rays do not create problems at sea-level. One could have 1 cosmic-ray event per day, which can be identified b y means of coincidences wi th another g.w. antenna. Coincidences wi th counters t r iggered b y single muons cannot be used because their flux crossing the an tenna amounts to several hundred per second.

For g.w. an tenna sensitivities approaching the q u a n t u m limit, the (~ cosmic- r ay events ~> go up to the order of 5.104 per day. At this level the coincidences wi th other an tennas of a similar t ype are of the order of 5.104 for a sampling t ime At of 1 second and 5000 per day with At ~ 0.1 s.

620 ~. AMALDI and o. I'IZZ:ELLA

C o m p u t a t i o n s m a d e for t h e u n d e r g r o u n d G r a n Sasso L a b o r a t o r y show

t h a t cosmic rays do n o t c rea te p r o b l e m s e v e n a t t he q u a n t u m l imi t . The n u m b e r

of m u o n s crossing t h e b a r is of t he o rder of 1 pe r h o u r and cou ld eas i ly iden t i f i ed

b y an a p p r o p r i a t e c o u n t e r a r ray .

�9 R I A S S U N T O

Gli autori osamiuauo nuovamento l 'effotto dei raggi cosmici su di una antenna gravi- tazionale risonauto, in considerazione della elevata sensibilits che b necessario rag- giungere per rivelare lo supernovae dol Virgo Cluster. Essi mostrano cho al livello del mare i socondari genorati nolla sbarra dalla iuterazione elet~romagnetica dei muoni di a l ia onergia produeono segnali con frequenza sta~istica maggiore di quella previs~a per lo supernovae. Quosto inconvonionto 6 eliminate in un laboratorio sottorraneo.

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