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Estimate of the cosmic ray background in an interferometric antenna for gravitational wave detection
Volume 128, number 5
PHYSICS LETTERS A
ESTIMATE OF THE COSMIC RAY BACKGROUND FOR GRAVITATIONAL WAVE DETECTION Adalberto
4 April 1988
IN AN INTERFEROMETRIC
ANTENNA
GIAZOTTO
INFN, Sez. di Piss, Piss, Italy
Received 5 October 1987; revised manuscript received 4 December 1987; accepted for publication 8 February 1988 Communicated by J.P. Vigier
The effect of the cosmic ray interaction in the test masses of an interferometric antenna for gravitational wave (GW) detection is evaluated. In a 3 km antenna this background, ‘mainly due to muons gives a limit, for 1 ms GW pulses, of h - 8.5 x 10 -Z with a frequency of 2~ 10-l events/year and 8.5X 1O-26 with 4.1 X lo6 events/year. For periodic GW having frequency> 10 Hz the sensitivity limit is h - 1.7~ lo-“. This background seems to allow unshielded operation of the interferometer test masses.
It has been experimentally shown [ I] that charged particles chanical vibrations whose fundamental mode has amplitude
traversing
an aluminum
rod can create me-
a! EL2
C(x)=---c0s(7tx/L), C, M n
(1)
where E is the energy lost by the particle in the bar, a! the thermal linear expansion coefficient, C,, the specific heat at constant volume, L and M are the bar length and mass respectively and x ( Ixl
EixE,
(2)
where E, P are the cosmic ray energy momentum
4Y mirror
lost in the mirrors,
P,,, is the mirror
CMS momentum
in
cosmic
surface
\ry
GLd
/
Fig. 1. The mirror section with a traversing cosmic ray losing the total momentum P, + P2. The momenta difference P, - P2 excites the mirror’s resonance modes.
241
Volume
128, number
5
PHYSICS
LETTERS
A
4 April 1988
the laboratory system, M the mirror mass, Ei the energy inelastically released to the mirror by the cosmic ray. In the second equation the rod CMS and vibrational kinetic energies have been neglected. If P, and Pz are the momenta lost in the mirror for x> 0 and x< 0 respectively, it follows that the mirror first longitudinal mode is excited by the momentum AP,=
(p,2 >x. __
-p2
For the sake of simplicity we can consider the mirror to be a simple harmonic oscillator, composed by a spring of stiffness K/2 and two masses M/2, having circular frequency o. = m and suspended in its CMS by means of a wire, whose pendular circular frequency is w,. The mirror surface displacement Ax due to the cosmic ray interaction is A_x=~(x) exp[ - (t-&)/r01
sin[o,(t-t,)]B(t-t,)+~ 0
1 c exp[--(~-rl)/~,l +‘Mw, Io
sin[o,(t-rl)lfi(q)
I
sin[w(t--rl)lF,
ew[-(t--v)/~~l
(rl) du
0
dq,
(4)
where F, and F2 are the impulsive forces F, =mxS(t-tn
),
FZ = (&)x&f-&)
and t, is the cosmic ray arrival time. We can maximize Ax putting U,from eqs. (4) and (5 ) we obtain
A.% +
,
( PM)X- E/c (c is the speed of light) and x=0; with these conditions
Kr(o)+&> 0
&exp[-_(l--l,)lr,l P
(5)
ew[--(t-t,)/~~l
sin[~o(t-t,)l (6)
sintw,(t-t,)l[8(t-t,)l.
The AX Fourier transform, in the approximation ~,t, B 1, and wore B 1, is Ax(s2, t,)&Yexp(iQt,)
1 ZaLw, 1 ~ [( C,Mx +z > -sZ2+2iS/ro+o~
For a quartz mirror having [3] M=400 vP=wP/27c=0.5 Hz, eq. (6) gives hx(Q, t,)=exp(iQt,)E
3x10-* -.Q2+2is2/ro+w~ (
kg, LcO.6
+L
m, (r/C,=7x
8x lo-l2 -Q2+2iQ/rP+wi
1
CM -Q2+2iQ/rP
> ’
+og
1 ’
lo-lo kg “C/J, ~~=w~/2n=5x
(7) lo3 Hz,
(8)
Since eq. (8 ) has poles for 52~ w. and 52~ w, we limit the frequency interval to the region wP< Qn
242
Volume
128, number
5
PHYSICS
LETTERS
A
4 April 1988
where A is the interferometer arm length and h the GW amplitude. The measurability condition for h is 3x10-~+8x10-1* .@
4
We can choose a frequency, with this value h becomes
>
,
W,
(10)
such that the GW impulse area is only slightly reduced, for example Q= 1/ 1OAt;
h>1.6~10-~+ Assuming h>8.5~
(11)
A=3x
lo3 m, At= lop3 s, we obtain
10-26Eo,,
,
(12)
where EGeV is E measured in GeV. Eq. ( 12) shows that a cosmic ray event giving in the antenna pulses comparable with those expected by the Virgo Cluster (h N 10 -*I ), should release 1O4 GeV. These events are mainly produced by muons [ 21 releasing into the mirrors almost entirely their incident energy by means of four processes: knock-on electrons [ 41, bremsstrahlung [ 5 1, direct pair production [ 41 and photo nuclear interaction
[61. The number N of events per second which deposit into the mirror an energy > E is evaluated the double integral:
by means of
(13) where p= 2.2 x lo3 kg/m3 is the mirror density, N, the Avogadro number, S- 3.6 X 10-I m2 the mirror projected area, Ai and Zi are the atomic weight and number of the ith atomic component of the mirror, 0, is the cross section of the jth process, Wand E, are the deposited and the incident muon energies respectively, E,( W) is the minimum muon energy necessary to release the energy W by means of the jth process and dZ( E,) /dE, is the muon differential intensity spectrum at sea level [ 71 integrated over the solid angle. In table 1 the sensitivity limits on h for 1 ms pulses for a few relevant values of EGeV together with N(ev/s) and NY (ev/year) are given. For periodical GW of circular frequency L&n,=27ryg, ]S(Q) ] becomes IS(s;$)l=+,
(14)
where T is the measurement 3x lo-‘+8x w:
time. Comparing
lo-‘* a,2
>I
T ew(iQt,)W,
eq. ( 13) with eq. (8) we obtain ,
q-Qg
the measurability
condition (15)
Table 1 E Ge”
h ( 1ms pulses)
N (w/s)
1 10 lo* 10 3 lo4
8.5x 8.5X 8.5x 8.5x 8.5x
1.4x 10 -’ 1.9x 10 --3 7.2x 10 -6 7 x10-9 7 x10-14
10 10 10 10 10
-26 -25 -B -*l --22
4.1x106 5.7x 10 4 2.3~10’ 2 x10-l 2 x10-6
243
Volume 128, number 5
PHYSICS LETTERS A
4 April 1988
W,, is the energy of the nth cosmic ray and C, is extended to cosmic ray events having 0 < t, < T. Since t, is a random variable we obtain
where
U*= T exp(iQt,) W, 2=x
WfAni )
(16)
I
where Ani is the number of cosmic ray events having energy between Wi and W,, , . Substituting the sum with the integral we obtain (17) where W, is a low energy cut off that we have assumed to be equal to the total ionization energy loss in the mirror due to minimum ionizing particles i.e. E, -0.26 GeV. This choice is justified by the fact that energy losses lower than W, are taken into account by the ionization losses. A numerical computation of eq. ( 17 ) gives U= 1.3 x 1O-6 J for T= 3 x 10’ s; with this value we obtain from eq. (15) h>L
3x lo-‘+8x
AT
0s
lo-‘* Q,z
Uz0.6~
l&-31 ,
>
(18)
where we have assumed vg= 10 Hz. The ionization losses can be evaluated putting in eq. ( 16) Wi=EI; since the charged cosmic ray flux at sea level is [ 8 ] I, = 2.4 x 1O2( m2 s) - ’ we obtain, assuming T= 3 X 10’ s, U= 2.2 x 10e6 J. With this value, eq. ( 15 ) evaluated at v,= 10 Hz, gives h> 1.1 x 10-J’ .
(19)
The sum of eqs. ( 18 ) and ( 19 ) gives, for periodic GW having vg> 10 Hz, the limit h > 1.7 x 10 -‘I, well below the sensitivity of the future interferometric antennas. I am very grateful to G. Pizzella for having raised the problem, and to L. Bracci for illuminating discussions
References [ I] A.M. Grassi Strini, G. Strini and G. Tagliaferri, J. Appl. Phys. 51 ( 1980) 948. [ 2 ] E. Amaldi and G. Pizzella, Nuovo Cimento 9 ( 1986) 6 12; F. Ricci, Nucl. Instrum. Methods A 260 ( 1987) 49 1, [ 31 Antenna Interferometrica a grande base per la rivelazione di onde gravitazionali, Italian-French Collaboration, INFN PI/AE 87/ 1, Pisa, 12 May 1987. [4] B. Rossi, High energy particles (Prentice-Hall, Englewood Cliffs, 1952). [ 51A.A. Petrukhin and V.V. Shestakov, Can. J. Phys. 46 (1968) S377. [ 6 ] L.B. Bezrukov and E. Bugaev, Sov. J. Nucl. Phys. 33 ( 198 1) 635. [ 71 O.C. Allkofer and P.K.F. Grieder, Physics data, Fachinformationszentrum Energie-Physik-Matematik 0344-8407, No. 25-1 (1984). [ 81 Particle Data Group, M. Roos et al., Review of particle properties, Phys. Lett. B 111 ( 1982) 1.
244
GMBH Karlsruhe, ISSN
PHYSICS LETTERS A
ESTIMATE OF THE COSMIC RAY BACKGROUND FOR GRAVITATIONAL WAVE DETECTION Adalberto
4 April 1988
IN AN INTERFEROMETRIC
ANTENNA
GIAZOTTO
INFN, Sez. di Piss, Piss, Italy
Received 5 October 1987; revised manuscript received 4 December 1987; accepted for publication 8 February 1988 Communicated by J.P. Vigier
The effect of the cosmic ray interaction in the test masses of an interferometric antenna for gravitational wave (GW) detection is evaluated. In a 3 km antenna this background, ‘mainly due to muons gives a limit, for 1 ms GW pulses, of h - 8.5 x 10 -Z with a frequency of 2~ 10-l events/year and 8.5X 1O-26 with 4.1 X lo6 events/year. For periodic GW having frequency> 10 Hz the sensitivity limit is h - 1.7~ lo-“. This background seems to allow unshielded operation of the interferometer test masses.
It has been experimentally shown [ I] that charged particles chanical vibrations whose fundamental mode has amplitude
traversing
an aluminum
rod can create me-
a! EL2
C(x)=---c0s(7tx/L), C, M n
(1)
where E is the energy lost by the particle in the bar, a! the thermal linear expansion coefficient, C,, the specific heat at constant volume, L and M are the bar length and mass respectively and x ( Ixl
EixE,
(2)
where E, P are the cosmic ray energy momentum
4Y mirror
lost in the mirrors,
P,,, is the mirror
CMS momentum
in
cosmic
surface
\ry
GLd
/
Fig. 1. The mirror section with a traversing cosmic ray losing the total momentum P, + P2. The momenta difference P, - P2 excites the mirror’s resonance modes.
241
Volume
128, number
5
PHYSICS
LETTERS
A
4 April 1988
the laboratory system, M the mirror mass, Ei the energy inelastically released to the mirror by the cosmic ray. In the second equation the rod CMS and vibrational kinetic energies have been neglected. If P, and Pz are the momenta lost in the mirror for x> 0 and x< 0 respectively, it follows that the mirror first longitudinal mode is excited by the momentum AP,=
(p,2 >x. __
-p2
For the sake of simplicity we can consider the mirror to be a simple harmonic oscillator, composed by a spring of stiffness K/2 and two masses M/2, having circular frequency o. = m and suspended in its CMS by means of a wire, whose pendular circular frequency is w,. The mirror surface displacement Ax due to the cosmic ray interaction is A_x=~(x) exp[ - (t-&)/r01
sin[o,(t-t,)]B(t-t,)+~ 0
1 c exp[--(~-rl)/~,l +‘Mw, Io
sin[o,(t-rl)lfi(q)
I
sin[w(t--rl)lF,
ew[-(t--v)/~~l
(rl) du
0
dq,
(4)
where F, and F2 are the impulsive forces F, =mxS(t-tn
),
FZ = (&)x&f-&)
and t, is the cosmic ray arrival time. We can maximize Ax putting U,from eqs. (4) and (5 ) we obtain
A.% +
,
( PM)X- E/c (c is the speed of light) and x=0; with these conditions
Kr(o)+&> 0
&exp[-_(l--l,)lr,l P
(5)
ew[--(t-t,)/~~l
sin[~o(t-t,)l (6)
sintw,(t-t,)l[8(t-t,)l.
The AX Fourier transform, in the approximation ~,t, B 1, and wore B 1, is Ax(s2, t,)&Yexp(iQt,)
1 ZaLw, 1 ~ [( C,Mx +z > -sZ2+2iS/ro+o~
For a quartz mirror having [3] M=400 vP=wP/27c=0.5 Hz, eq. (6) gives hx(Q, t,)=exp(iQt,)E
3x10-* -.Q2+2is2/ro+w~ (
kg, LcO.6
+L
m, (r/C,=7x
8x lo-l2 -Q2+2iQ/rP+wi
1
CM -Q2+2iQ/rP
> ’
+og
1 ’
lo-lo kg “C/J, ~~=w~/2n=5x
(7) lo3 Hz,
(8)
Since eq. (8 ) has poles for 52~ w. and 52~ w, we limit the frequency interval to the region wP< Qn
242
Volume
128, number
5
PHYSICS
LETTERS
A
4 April 1988
where A is the interferometer arm length and h the GW amplitude. The measurability condition for h is 3x10-~+8x10-1* .@
4
We can choose a frequency, with this value h becomes
>
,
W,
(10)
such that the GW impulse area is only slightly reduced, for example Q= 1/ 1OAt;
h>1.6~10-~+ Assuming h>8.5~
(11)
A=3x
lo3 m, At= lop3 s, we obtain
10-26Eo,,
,
(12)
where EGeV is E measured in GeV. Eq. ( 12) shows that a cosmic ray event giving in the antenna pulses comparable with those expected by the Virgo Cluster (h N 10 -*I ), should release 1O4 GeV. These events are mainly produced by muons [ 21 releasing into the mirrors almost entirely their incident energy by means of four processes: knock-on electrons [ 41, bremsstrahlung [ 5 1, direct pair production [ 41 and photo nuclear interaction
[61. The number N of events per second which deposit into the mirror an energy > E is evaluated the double integral:
by means of
(13) where p= 2.2 x lo3 kg/m3 is the mirror density, N, the Avogadro number, S- 3.6 X 10-I m2 the mirror projected area, Ai and Zi are the atomic weight and number of the ith atomic component of the mirror, 0, is the cross section of the jth process, Wand E, are the deposited and the incident muon energies respectively, E,( W) is the minimum muon energy necessary to release the energy W by means of the jth process and dZ( E,) /dE, is the muon differential intensity spectrum at sea level [ 71 integrated over the solid angle. In table 1 the sensitivity limits on h for 1 ms pulses for a few relevant values of EGeV together with N(ev/s) and NY (ev/year) are given. For periodical GW of circular frequency L&n,=27ryg, ]S(Q) ] becomes IS(s;$)l=+,
(14)
where T is the measurement 3x lo-‘+8x w:
time. Comparing
lo-‘* a,2
>I
T ew(iQt,)W,
eq. ( 13) with eq. (8) we obtain ,
q-Qg
the measurability
condition (15)
Table 1 E Ge”
h ( 1ms pulses)
N (w/s)
1 10 lo* 10 3 lo4
8.5x 8.5X 8.5x 8.5x 8.5x
1.4x 10 -’ 1.9x 10 --3 7.2x 10 -6 7 x10-9 7 x10-14
10 10 10 10 10
-26 -25 -B -*l --22
4.1x106 5.7x 10 4 2.3~10’ 2 x10-l 2 x10-6
243
Volume 128, number 5
PHYSICS LETTERS A
4 April 1988
W,, is the energy of the nth cosmic ray and C, is extended to cosmic ray events having 0 < t, < T. Since t, is a random variable we obtain
where
U*= T exp(iQt,) W, 2=x
WfAni )
(16)
I
where Ani is the number of cosmic ray events having energy between Wi and W,, , . Substituting the sum with the integral we obtain (17) where W, is a low energy cut off that we have assumed to be equal to the total ionization energy loss in the mirror due to minimum ionizing particles i.e. E, -0.26 GeV. This choice is justified by the fact that energy losses lower than W, are taken into account by the ionization losses. A numerical computation of eq. ( 17 ) gives U= 1.3 x 1O-6 J for T= 3 x 10’ s; with this value we obtain from eq. (15) h>L
3x lo-‘+8x
AT
0s
lo-‘* Q,z
Uz0.6~
l&-31 ,
>
(18)
where we have assumed vg= 10 Hz. The ionization losses can be evaluated putting in eq. ( 16) Wi=EI; since the charged cosmic ray flux at sea level is [ 8 ] I, = 2.4 x 1O2( m2 s) - ’ we obtain, assuming T= 3 X 10’ s, U= 2.2 x 10e6 J. With this value, eq. ( 15 ) evaluated at v,= 10 Hz, gives h> 1.1 x 10-J’ .
(19)
The sum of eqs. ( 18 ) and ( 19 ) gives, for periodic GW having vg> 10 Hz, the limit h > 1.7 x 10 -‘I, well below the sensitivity of the future interferometric antennas. I am very grateful to G. Pizzella for having raised the problem, and to L. Bracci for illuminating discussions
References [ I] A.M. Grassi Strini, G. Strini and G. Tagliaferri, J. Appl. Phys. 51 ( 1980) 948. [ 2 ] E. Amaldi and G. Pizzella, Nuovo Cimento 9 ( 1986) 6 12; F. Ricci, Nucl. Instrum. Methods A 260 ( 1987) 49 1, [ 31 Antenna Interferometrica a grande base per la rivelazione di onde gravitazionali, Italian-French Collaboration, INFN PI/AE 87/ 1, Pisa, 12 May 1987. [4] B. Rossi, High energy particles (Prentice-Hall, Englewood Cliffs, 1952). [ 51A.A. Petrukhin and V.V. Shestakov, Can. J. Phys. 46 (1968) S377. [ 6 ] L.B. Bezrukov and E. Bugaev, Sov. J. Nucl. Phys. 33 ( 198 1) 635. [ 71 O.C. Allkofer and P.K.F. Grieder, Physics data, Fachinformationszentrum Energie-Physik-Matematik 0344-8407, No. 25-1 (1984). [ 81 Particle Data Group, M. Roos et al., Review of particle properties, Phys. Lett. B 111 ( 1982) 1.
244
GMBH Karlsruhe, ISSN