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OCR for page 261
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
OCR for page 262
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
OCR for page 263
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
OCR for page 264
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
OCR for page 265
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
OCR for page 266
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.
OCR for page 267
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
OCR for page 268
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~.
OCR for page 269
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.
Representative terms from entire chapter:
formation rate
