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The Evolution of the Gravitational Radiation from Stellar Components of Galames V.M. LIPUNOV, E.YU. OSMINKIN, M.E. PROKHOROV Sternberg Astronomical Institute ABSIrRACT This paper discusses the evolution of gravitational waves spectra pro- duced by binary stars, supernova explosions and coalescences of binary compact stars in outer galaxies. These spectra are integrated over a simple model of the universe to give an estimate of the stochastic gravitational waves background due to astrophysical sources. LNWODUCTION Different lands of gravitational waves (GW) exist in nature. The major types of GW are the cosmological GW and GW from astrophysical sources such as binary stars and supernova explosions (SHE) (Thorpe 1987~. The evolution of the ensemble of the binary systems in an arbitrary galaxy has been investigated by the method of statistical simulation. We have taken into account the following evolutionary processes: the mass transfer in close binaries, the evolution inside a common envelope, possible binary disruption due to the SNE, and creation of the compact objects and their evolution (see Lipunov l987~. The method and scenario are considered in detail in Kornilov and Lipunov (1982) for massive stars and in Lipunov and Postnov (1987b, c) for low and moderately massive binaries. In this paper we will consider two different ldods of Got The first type is the continuous gravitational radiation produced by binanes. A binary is assumed to emit the GW as two point-like masses on the circular orbit stricter at twice the orbital frequency. In this case the GW spectrum 261

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262 MEXICAN AND SOVIET PERSPECTIVES from a galaxy is composed of thin "lmes." The projected nonresonant GW detectors with a band Liz' ~ z', will detect such a signal as continuous. Below, we shall calculate the amplitude of the signal on such a detector. The second sort of GW sources can be connected with catastrophic processes occunug during the stellar evolution. First of all, these are the SNE and the coalescence of binary compact objects (white dwarves, neutron stars, or black holes). These GW are believed to be powerful and short In the duration pulse. Due to existing uncertainties in the calculation of this GW, we will use an estimation given by Shapiro and lbukolsky (1983~. Stochastic GW background from the binaries in our galaxy has been calculated by Lipunov and Postnov (1987a). The simple estimations show chat the stochastic GW background from all of the external galaxies should be lower than from our own However, there is a possibility of distinguishing both Apes of signals by using the fact of their different distributions over the sky. This can be done by a GW detector with a narrow diagram (Lipunov et al. 19~7~. In our study we deal with the extragalactic GW sources, taking into account their evolutionary effects. THE METHOD OF CALCUI^TION OF GW E V()LUTION 1b investigate the evolution of GW with time from galaxies having the different star formation laws, the following procedure has been used. At first the evolution of a quantity G(t, r) is calculated (for example, the GW flux or the rate of SNE) for an ensemble of binaries with ~ (t - ~ ~ star formation rate. We assumed the parameters of the scenario do not change with time, and so, the Green function is G(t, r) = G(t-r). For a Gallup with an arbitrary star formation rate ~ (t) the correspond- ing value G(t) can be presented by the convolution t+= G(~) = ~Aught-aide -00 The Green function for a continuous specimen of GW from binaries was constructed by the following. The time internal from 0 to 15 billion years was equally spaced to At = 109 yes bins. The frequency range (10-8 -. 10-3 Hz) was divided by bins with TV = hi+ -hi, so that ig~vi+~/vi) = 0.25. For each dine interval ~tj and frequency bin Levi the average GW flux has bleed calculated FGW (~i, tj') = /\lj J' tti+ ^ti ' FGW (veldt ~i<~<~i+~i

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HIGH-ENERGY ASTROPHYSICS 2f53 where FGW(~i) is the total gravitational flux from all binaries. All results are presented in terms of dimensionless metric strain am- plitude (Lipunov, Postnov 1987a) /4G 1 he = ~ = N/ 7rC3 (FEW) An The number of events from the sources of GW pulses has been calcu- lated inside each time bin At. Time interval was taken to be At = 106, 107,108, 109 years. RESULTS The spectral evolution of gravitational radiation from a galaxy with [- like star formation rate is plotted in Figure 1. The results are for a galaxy, containing 3 10~i stars, with a half of their total number entering the binaries. A distance to the galaxy is assumed to be 1 Mpc. The simulations have been earned out with the following parameters of the scenario. the masses of initially more massive stars in binary are supposed to be distributed according to Salpeeter's law dN or M-2.35dM O.1Mc' < M: < 120M<~. the mass ratios q _ M2/M: and semimajor axes a are distributed as and dN or dq, O < q < 1 dN or da/a, 1R. A full description of the evolutionary scenario parameters can be found in Kornilov and Lipunov (1982) and in Lipunov and Postnov (1987b,c, 1988~. Note that the GW spectrum changes significantly during the first two billion years. The magnitude h in the frequency range 10-6 10-3 Hz decreases approximately by a factor of 30 (gh ~ -19.7) at t = 109 yr and igh ~ -21.5 at 10~ yr). This drop can be explained both by the binanes' disruption due to the SEE and by a diminishing mass in binary stars in mass through the stellar wind and common envelope processes. ~ obtain the GW spectrum dependence upon time for an arbitrary star formation function, the [unction in Figure 1. must be integrated with star formation function (t). Me star formation rate is supposed to be

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264 AMERICAN AND SOREST PERSPECTIVES ~ ~ 19\ 1~ t J gyro 1, ~ , / -8 1 6, . ) . . . I, -22.0 -23.5 . . J. . . 9 7 ,,/ I, 109 yrs -6 -5 -4 8 1 _ -22 0 _ -22.5 -23.0 _ 19`, Hz 1 _ ' 1 3 J 4 _ 5 FIGURE 1 The evolution of GW spectrum from a galaxy with [-like star formation rate. A metric strain amplitude hi, has been calculated at distance 1 ME The galas contains 3 1011 stars (t) = for elliptical, and for spirals. coast, 0, t ~ lO9yrs (t) = const O_t<109yrs

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HIGH-ENER~ ~TROP~ICS -20 265 _ -21 ~v2t: ~ _ -22 _ . 213 Led 1 l 1 1 1 ~1 1 1 1 1 1 1 -8 -7 -5 -4 -3 19v Hz FIGURE 2 1-he GW background Tom binaries in outer spirals (I) and ellipticals (II). The GW spectrum emitted by ellipticals (II) and spirals (I) from the modeling universe are plotted in Figure 2. The galaxies are assumed to be distributed homogeneously within ~ < z < 3. The contributions of ellipticals and spirals are 30% and 70% respectively. The density of the visible matter is assumed to be ,B = 1/30 of the average density of the universe. The Hubble constant is taken to be H = 75 km/sec/Mpc, Q -P/Pcr = 1' and zero pressure equation of state. In these calculations we have taken into account the redshift of gravitational radiation and a curvature of the universe (Zel'dovich and Novikov 1970~. The influence of massive black holes (BH) on the spectrum shape has been considered separately. We assumed that such objects result from massive SN progenitors with masses M > 35 Me,. The collapse into such black holes occurs without significant mass loss. The spectra of GW from a galaxy having the same parameters as above in age of 7 109 years with (I) and without (IT) massive BH are presented in Figure 3. From this figure it follows that in the second case there is a considerable increase of gravitational radiation at frequencies below 10-5 5 HZ. This can be due to the massive binary black holes. Its evolution is fully determined by orbital decay caused by GW radiation on the time scale

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266 ghv -21 -22 -23 AMERICAN AND SOVIET PERSPECTIVES J _ ~ , ~ . 1 1 -7 it, rLr ~ 1 -6 -5 -4 -3 1g`, Hz FIGURE 3 The spectra of GW from galaxy in age of 7 109 years with (0 and without (II) massive BH. tow = (1.5 lO8yrs)~/~ which exceeds a Hubble time for pairs with a>70R<~. The shape of this spectral feature strongly depends on the assumptions about the massive BH formation and can sufficiently differ from this rough model However, the existence of such a spectral feature could in principle be used as an indicator for massive BH presence. In Figure 4, the event rate for GW pulses versus time is presented for a galas with the parameters described above and with a [-like star formation function The type II SN results from the collapse of massive stars (M > TOME) at final stages of its evolution. So the SNII explosions in such a galaxy occur only dunog the first ~ 40 million years. In the time range 1 - 10 million years, this rate can be approximated by the formula: fsN~(t) _ 50~01l ~)-0.46 Another ldod of event we considered is the coalescences of WDs, which occur within the integral of 107 - 1.5 ~ 101 years. This event rate weakly depends on time and vanes from 1 to 0.01 yr-l.

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HIGH-ENERGY ASTROPHYSICS IgV y 1 o -1 -2 -3 267 - BAA _ i- ~ ~to.46 - SN II ~ SN I o WD + WD .W~ oO o Oo \ ~ \ o o o o ~ o o O o OoO o\ Oo o ~- . ~t~93O oo \~- . `e ':. i. . i ~ Illll ~11111111 1 107 1o8 109 1010 t,yrs FIGURE 4 The rate of supernova explosions and of white dwarves coalescences in a galaxy with [-like star formation rate. From all coalescing WDs we choose a group of binanes, whose total mass exceeds the Chandrasel~ar limit. We consider such binaries as capable of producing the SNeI explosions. These events may be accompanied by supplementary GW pulses arising during a collapse into NS. The rate of SNeI between 6 106 and 1.5 109 years can be approximated as

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268 gh,v -21 -22 -23 -24 AMERICAN AND SOVIET PERSPECTIVES _ L , I I I , , I 4 -3 -2 -' O 1 1g`, Hz FIGURE 5 The dependence of the rate of GW pulses registration ~ on a GW detector with the given sensitivity hi,. fSHI (I) = 1.8 ~ ol1 { 106yr } Assuming the effectiveness of mass-energy conversion to be ~ = 10% of the total mass of a collapsing star we have estimated the dimensionless metric strain amplitude. Integrating these expressions witch the star formation rate functions for ellipticals and spirals and integrating over the whole space where the galaxies are thought to exist (z < 3) and using cosmological parameters described above, we have obtained the rate of arriving pulses with an amplitude not less than that given by the different ldods of SNe and of galaxies, be., the rate of me event's registration on a GW detector with the given sensitivity. This dependence is presented in Figure 5. The results for spirals were obtained analytically by Lipunov, Postnov e! al. (1987~.

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HIGH-ENERGY ASTROPHYSICS 269 ACKNOWLEDGEMENTS We would like to thank Dr. K~ Postnov for useful discussions and comments. REFERENCES Kornilov, V.G., and V.M. Lipunov 1982. Sov. Astron. 27: 163. Lipunov, V.M. 1987. Astrophys. Sp. Sci. 132: 1 Lipunov, V.M., and KA Postnov. 1987. Sov. Astron. 31: 228. Lipunov, V.M., and KA Postnov. 1987. Sov. Astron. 31: 288. Lipunov, V.M., and KA Postnov. 1987. Sov Astoron. 31: 4W. Lipunov, V.M., and K~ Postnov. 1988. Astrophys. Sp. S~ 145: 1. Lipunov, V.M., KA. Postnov, and M.E. Prokhorov. 1987. Astron. Astrophys. 176: L1. Shapiro, S.L, and SA Teukolsky. 1983. Black Holes, White Dwarves, and Neutron Stars. Cornell University, Ithaca. Ihorne, KS. 1987. In: Hawking, S.W., and W. Israel (eds.~. 300 Years of Gravitation. Cambndge Universibr Press, Cambridge. 330. Zel'dovich, Ya.B., and I.D. Novikov. 1971. Relatnistic Astrophysics Chicago Universi~ Press. V. 1.