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Signal to noise ratio in a satellite gravitational wave experiment
Acta Astronautica.
Vol. 2, pp. 535-537.
Pergamon Press 1975.
Printed in the U.S.A.
Signal to noise ratio in a satellite gravitational wave experiment V. B. B R A G I N S K Y Department of Physics, Moscow State University, Moscow, U.S.S.R.
(Received 4 February 1975) Abstract--A system for detection of gravitational radiation is proposed which consists of a widely-separated pair of drag-free satellites. Variations in their relative velocity induced by gravitational waves would be detected by a Doppler-ranging system. It is shown that the sensitivity of such a system can be sufficient to allow detection of gravitational radiation from high-density star clusters near the galactic center.
Introduction
ONE OF THE most interesting problems of modern astrophysics is that of the detection of gravitational waves of nonterrestrial origin. The first experiment to detect pulses of gravitational radiation was performed by J. Weber (1969, 1970a, 1970b). Although his first positive result was not confirmed (Braginsky et al., 1972; Tyson, 1972) experiments of this type are in progress in some laboratories. The main goal of these laboratories is to increase the sensitivity of the antennae remaining in the framework of Weber's scheme and in the band-width 1-10 kHz (Zeldovich and Novikov, 1964; Thorne, 1967). The sources of gravitational radiation within this band-width are rather exotic: the collision of two black holes or two neutron stars. Such collisions may be rare not only in our galaxy but also in other galaxies. For frequencies less than 1 kHz we must expect to find other sources of radiation and we must use other types of detectors, which are nonsimilar to Weber's detector. The goal of this paper is to analyze the possibilities of the realization of an experiment in the bandwidth 0.1-10 Hz using two or more drag-free satellites. Relative satellite velocity induced by gravitational ranging
The gravitational wave produces a field of acceleration (Weber, 1969, 1970a, 1970b): F~t
m
-
2 ct
tx
c l R0~0,
(1)
where 1~ is the distance between the two bodies, R~oo, the a.c. component of the Riemann tensor, and c, the speed of light. As a gravitational antenna it is possible to use two satellites and a system measuring the variation of the relative speed By, produced by the force F " (eqn 1). Braginsky and Hertzenschtein 0967) show 535
536
v . B . BRAGINSKY
that, for the optimal orientation of the c o m p o n e n t s l" toward the wave, •
1/81rG f~\
(2)
where G is the gravitational constant, L the density of energy in the burst of gravitational radiation, and ~, the duration of the burst. Equation (2) is valid if ! <. (,rrc/to~), where to~ is the mean f r e q u e n c y in the burst. Zeldovich and Polnarev (1973) have shown that the high-density star clusters in the center of our galaxy must be sources of gravitational radiation and we m a y expect to o b s e r v e bursts with density I f = 10-3-10erg/cm 2, m e a n f r e q u e n c y to~ = 10 ' - 10rad/sec and duration ~ - ~ 10-0.1 sec. Such bursts m a y appear approximately 25 times per year. Substituting in eqn (2) r = 1 erg/cm 2, "~ = 1 sec, l = 1012 cm (10 million km) we obtain 8v~ --- 2.5 × 1 0 -7 cm/sec. It is evident that to realize such an experiment it is necessary to have a very sensitive Dopplerranging system to measure small variation of speed. Limits to Doppler-ranging The present state of the art corresponds to 8v -~ 10 ~ cm/sec with averaging time ( -~ 10 sec (Anderson, 1972). L e t us consider the prospects of increasing the sensitivity in a Doppler-ranging system. The main p r o b l e m is that of the stability of R.F. or V.H.F. generators. T w o independent conditions for the generator exist: (a) The quasistatic stability of the f r e q u e n c y must be high enough. (b) The spectral density of the f r e q u e n c y deviation must not exceed a definite level. The first condition m a y be described with a simple equation:
~Vc= ~>~ Ks
(3)
(T)AT,
where f is the f r e q u e n c y of the generator; Af, the variation of the f r e q u e n c y due to the change of the t e m p e r a t u r e A T of the resonator; a (T), the coefficient of linear thermal expansion; and K, a dimensionless factor, which defines the connection between the relative change of the eigen,frequency of the resonator and the relative change of the Size of the resonator. If the t e m p e r a t u r e is m u c h less than the D e b y e t e m p e r a t u r e a - T -3, and due to this behaviour of a (T) if T = I°K it is possible to reach a - ~ 3 × 1 0 -j3 deg. '. Substituting in eqn (3) K = 0 . 0 1 ; AT = 10-3°K, a = 3 x 10-~3 deg. -1 we obtain 8v ~> 10-Tcm/sec. The second condition is the following
av
a[>~~/[w(.f)/÷]
-;=7
7
(4) '
where W(f) is the spectral density of the f r e q u e n c y deviation:
Signal to noise ratio in a satellite gravitational wave experiment
537
kTff W ( f ) = 2PQ,,'
(5)
k is the B o l t z m a n c o n s t a n t ; P, t h e p o w e r o f t h e g e n e r a t o r ; a n d Qe t h e q u a l i t y f a c t o r of t h e r e s o n a t o r ( L a m b , 1965). S u b s t i t u t i n g in e q n s (4) a n d (5) T = I ° K , P = 103 e r g / s e c , Qe = 5 x 101° (this l e v e l o f Q, w a s o b t a i n e d w i t h a s u p e r c o n d u c t i v e n i o b i u m r e s o n a t o r , as in A l l e n , 1971) w e o b t a i n 8v ~> 3 × 10 -1~ c m / s e c . T h e s e t w o e s t i m a t e s s h o w t h a t a D o p p l e r - r a n g i n g s y s t e m to m e a s u r e t h e v a r i a t i o n o f s p e e d at t h e l e v e l 10 -7 c m / s e c is p o s s i b l e to r e a l i z e . T h e s a t e l l i t e s i n c l u d e d in t h e g r a v i t a t i o n a l a n t e n n a m u s t b e d r a g - f r e e . It is e a s y to s h o w t h a t t h e l e v e l o f c o m p e n s a t i o n o f t h e n o n g r a v i t a t i o n a l a c c e l e r a t i o n w h i c h e x i s t s n o w (De B r a , 1971; J u i l l e r a t , 1971) (~: ~< 1 0 - S c m / s e c 2) is high e n o u g h to d e t e c t b u r s t s w i t h f = 1 e r g / c m 2, ?~ = 1 s e c (this c o r r e s p o n d s to 6 ~ ~-10 7 cm/sec2). It is e v i d e n t t h a t to o b t a i n h i g h e r s e n s i t i v i t y it is n e c e s s a r y to u s e a drag-free system with a better level of compensation. T h e d e s c r i b e d g r a v i t a t i o n a l a n t e n n a s e e m s to b e e a s i e r to r e a l i z e t h a n a h e t e r o d y n e a n t e n n a ( B r a g i n s k y a n d N a z a r e n k o , 1971) a n d t h e a n t e n n a in w h i c h g r a v i t a t i o n a l e l e c t r o m a g n e t i c r e s o n a n c e is u s e d ( B r a g i n s k y , 1972).
References Allen, M. A. (1971) IEEE, Transaction, N 5-18, 168. Anderson, J. (1972) Lecture at "E. Fermi" Summer School, Varenna. Braginsky, V. B. (1972) Lecture at "E. Fermi" Summer School, Varenna. Braginsky, V. B. and Hertzenschtein, M. E. (1967) Pisma JETP 5, 348. Braginsky, V. B. and Nazarenko, V. S. (1971) Proceedings of the Caltech Conference on the tests of Gravitational theories. Braginsky, V. B., Manukin, A. B., Popov, E. I. and Horev, A. A. (1972) Pisma JETP 16, 157; Physics Letters (to be published). De Bra, (1971) Proceedings of the Caltech Conference on the tests of Gravitational theories. Juillerat, R. (1971) Proceedings of the Caltech Conference on the tests of Gravitational theories. Lamb, (1965) Optique et electronique quantitique, Paris. Thorne, K. S. (1967) Non-relativistic pulsation of general relativistic stellar models. Preprint, OAP-167. Tyson, J. (1972) Null search for bursts of gravitational radiation, Bell Lab. Preprint. Weber, J. (1969, 1970a, 1970b) Phys. Rev, Lett. 22, 1320; 24,276; 25, 180. Zeldovich, Ya. and Novikov, I. (1964) Doklady Academii Nauk 155, 1033. Zeldovich, Ya. and Polnarev, Yu. (1973) The radiation of gravitational waves from clusters of high density stars. Preprint, I.A.M.
Vol. 2, pp. 535-537.
Pergamon Press 1975.
Printed in the U.S.A.
Signal to noise ratio in a satellite gravitational wave experiment V. B. B R A G I N S K Y Department of Physics, Moscow State University, Moscow, U.S.S.R.
(Received 4 February 1975) Abstract--A system for detection of gravitational radiation is proposed which consists of a widely-separated pair of drag-free satellites. Variations in their relative velocity induced by gravitational waves would be detected by a Doppler-ranging system. It is shown that the sensitivity of such a system can be sufficient to allow detection of gravitational radiation from high-density star clusters near the galactic center.
Introduction
ONE OF THE most interesting problems of modern astrophysics is that of the detection of gravitational waves of nonterrestrial origin. The first experiment to detect pulses of gravitational radiation was performed by J. Weber (1969, 1970a, 1970b). Although his first positive result was not confirmed (Braginsky et al., 1972; Tyson, 1972) experiments of this type are in progress in some laboratories. The main goal of these laboratories is to increase the sensitivity of the antennae remaining in the framework of Weber's scheme and in the band-width 1-10 kHz (Zeldovich and Novikov, 1964; Thorne, 1967). The sources of gravitational radiation within this band-width are rather exotic: the collision of two black holes or two neutron stars. Such collisions may be rare not only in our galaxy but also in other galaxies. For frequencies less than 1 kHz we must expect to find other sources of radiation and we must use other types of detectors, which are nonsimilar to Weber's detector. The goal of this paper is to analyze the possibilities of the realization of an experiment in the bandwidth 0.1-10 Hz using two or more drag-free satellites. Relative satellite velocity induced by gravitational ranging
The gravitational wave produces a field of acceleration (Weber, 1969, 1970a, 1970b): F~t
m
-
2 ct
tx
c l R0~0,
(1)
where 1~ is the distance between the two bodies, R~oo, the a.c. component of the Riemann tensor, and c, the speed of light. As a gravitational antenna it is possible to use two satellites and a system measuring the variation of the relative speed By, produced by the force F " (eqn 1). Braginsky and Hertzenschtein 0967) show 535
536
v . B . BRAGINSKY
that, for the optimal orientation of the c o m p o n e n t s l" toward the wave, •
1/81rG f~\
(2)
where G is the gravitational constant, L the density of energy in the burst of gravitational radiation, and ~, the duration of the burst. Equation (2) is valid if ! <. (,rrc/to~), where to~ is the mean f r e q u e n c y in the burst. Zeldovich and Polnarev (1973) have shown that the high-density star clusters in the center of our galaxy must be sources of gravitational radiation and we m a y expect to o b s e r v e bursts with density I f = 10-3-10erg/cm 2, m e a n f r e q u e n c y to~ = 10 ' - 10rad/sec and duration ~ - ~ 10-0.1 sec. Such bursts m a y appear approximately 25 times per year. Substituting in eqn (2) r = 1 erg/cm 2, "~ = 1 sec, l = 1012 cm (10 million km) we obtain 8v~ --- 2.5 × 1 0 -7 cm/sec. It is evident that to realize such an experiment it is necessary to have a very sensitive Dopplerranging system to measure small variation of speed. Limits to Doppler-ranging The present state of the art corresponds to 8v -~ 10 ~ cm/sec with averaging time ( -~ 10 sec (Anderson, 1972). L e t us consider the prospects of increasing the sensitivity in a Doppler-ranging system. The main p r o b l e m is that of the stability of R.F. or V.H.F. generators. T w o independent conditions for the generator exist: (a) The quasistatic stability of the f r e q u e n c y must be high enough. (b) The spectral density of the f r e q u e n c y deviation must not exceed a definite level. The first condition m a y be described with a simple equation:
~Vc= ~>~ Ks
(3)
(T)AT,
where f is the f r e q u e n c y of the generator; Af, the variation of the f r e q u e n c y due to the change of the t e m p e r a t u r e A T of the resonator; a (T), the coefficient of linear thermal expansion; and K, a dimensionless factor, which defines the connection between the relative change of the eigen,frequency of the resonator and the relative change of the Size of the resonator. If the t e m p e r a t u r e is m u c h less than the D e b y e t e m p e r a t u r e a - T -3, and due to this behaviour of a (T) if T = I°K it is possible to reach a - ~ 3 × 1 0 -j3 deg. '. Substituting in eqn (3) K = 0 . 0 1 ; AT = 10-3°K, a = 3 x 10-~3 deg. -1 we obtain 8v ~> 10-Tcm/sec. The second condition is the following
av
a[>~~/[w(.f)/÷]
-;=7
7
(4) '
where W(f) is the spectral density of the f r e q u e n c y deviation:
Signal to noise ratio in a satellite gravitational wave experiment
537
kTff W ( f ) = 2PQ,,'
(5)
k is the B o l t z m a n c o n s t a n t ; P, t h e p o w e r o f t h e g e n e r a t o r ; a n d Qe t h e q u a l i t y f a c t o r of t h e r e s o n a t o r ( L a m b , 1965). S u b s t i t u t i n g in e q n s (4) a n d (5) T = I ° K , P = 103 e r g / s e c , Qe = 5 x 101° (this l e v e l o f Q, w a s o b t a i n e d w i t h a s u p e r c o n d u c t i v e n i o b i u m r e s o n a t o r , as in A l l e n , 1971) w e o b t a i n 8v ~> 3 × 10 -1~ c m / s e c . T h e s e t w o e s t i m a t e s s h o w t h a t a D o p p l e r - r a n g i n g s y s t e m to m e a s u r e t h e v a r i a t i o n o f s p e e d at t h e l e v e l 10 -7 c m / s e c is p o s s i b l e to r e a l i z e . T h e s a t e l l i t e s i n c l u d e d in t h e g r a v i t a t i o n a l a n t e n n a m u s t b e d r a g - f r e e . It is e a s y to s h o w t h a t t h e l e v e l o f c o m p e n s a t i o n o f t h e n o n g r a v i t a t i o n a l a c c e l e r a t i o n w h i c h e x i s t s n o w (De B r a , 1971; J u i l l e r a t , 1971) (~: ~< 1 0 - S c m / s e c 2) is high e n o u g h to d e t e c t b u r s t s w i t h f = 1 e r g / c m 2, ?~ = 1 s e c (this c o r r e s p o n d s to 6 ~ ~-10 7 cm/sec2). It is e v i d e n t t h a t to o b t a i n h i g h e r s e n s i t i v i t y it is n e c e s s a r y to u s e a drag-free system with a better level of compensation. T h e d e s c r i b e d g r a v i t a t i o n a l a n t e n n a s e e m s to b e e a s i e r to r e a l i z e t h a n a h e t e r o d y n e a n t e n n a ( B r a g i n s k y a n d N a z a r e n k o , 1971) a n d t h e a n t e n n a in w h i c h g r a v i t a t i o n a l e l e c t r o m a g n e t i c r e s o n a n c e is u s e d ( B r a g i n s k y , 1972).
References Allen, M. A. (1971) IEEE, Transaction, N 5-18, 168. Anderson, J. (1972) Lecture at "E. Fermi" Summer School, Varenna. Braginsky, V. B. (1972) Lecture at "E. Fermi" Summer School, Varenna. Braginsky, V. B. and Hertzenschtein, M. E. (1967) Pisma JETP 5, 348. Braginsky, V. B. and Nazarenko, V. S. (1971) Proceedings of the Caltech Conference on the tests of Gravitational theories. Braginsky, V. B., Manukin, A. B., Popov, E. I. and Horev, A. A. (1972) Pisma JETP 16, 157; Physics Letters (to be published). De Bra, (1971) Proceedings of the Caltech Conference on the tests of Gravitational theories. Juillerat, R. (1971) Proceedings of the Caltech Conference on the tests of Gravitational theories. Lamb, (1965) Optique et electronique quantitique, Paris. Thorne, K. S. (1967) Non-relativistic pulsation of general relativistic stellar models. Preprint, OAP-167. Tyson, J. (1972) Null search for bursts of gravitational radiation, Bell Lab. Preprint. Weber, J. (1969, 1970a, 1970b) Phys. Rev, Lett. 22, 1320; 24,276; 25, 180. Zeldovich, Ya. and Novikov, I. (1964) Doklady Academii Nauk 155, 1033. Zeldovich, Ya. and Polnarev, Yu. (1973) The radiation of gravitational waves from clusters of high density stars. Preprint, I.A.M.