General Relativity and Gravitation, Vol. PS, No. 5, 1996

Earth-based Gravitational Wave Detection from Pulsars Walter Velloso, 1,2,3 Fabrizio Barone, 4,5 Enrico Calloni,5 Luciano Di Fiore, 5 AnieUo G r a d o , 4'5 Leopoldo Milano, 4'5 a n d Guido Russo s,6 Received August I, 1995. Rev. version November I, 1995

The main features of continuous gravitational radiation bathing the Earth has been evaluated for a set of 558 pulsars. In particular, the maximum gravitational wave background and the maximum gravitational wave emission have been evaluated for each source and compared with the projected sensitivities of the planned Earth based very long baseline interferometric antennas for gravitational wave detection, like VIRGO and LIGO. This study shows that such detectors have a good chance of detecting gravitational waves emitted from this class of astrophysical sources.

1. I N T R O D U C T I O N G r a v i t a t i o n a l wave (ow) detection is certainly one of the most challenging goals for t o d a y physics, providing a very strong proof in favour of the E i n s t e i n ' s G e n e r a l R e l a t i v i t y for describing p h e n o m e n a related to the

1 2 3 4

InstitutoAstronSmico e Geoffsico,Universidad de SRo Paulo, Brazil Faculdad Hebraico-BrasileiraRenascen~a, Universidad de SRo Paulo, Brazil Faculdad Sant' Anna, Universidad de S~o Paulo, Brazil Universit~ di Napoli "Federico II", Dipartimento di Scienze Fisiche,Mostra d'Oltreg

mare Pad.19, L80125 Napoli, Italy 5 Istituto Nazionale di Fisica Nucleare, sez. Napoli, Mostra d'Oltremare Pad.19, 180125 Napoli, rtaly. E-maih fbaroneC}na.infn.it 6 Universit~ della Calabria, Dipartimento di Fisica, Arcavacata di Rende, L87036 Cosenza, Italy {}13

0~01-7701/96/0500-0613509.50/0(~

1996 Plenum PublishingCorporation

614

Velloso, Barone, Calloni, Di Fiore, Grado, Milano, and Russo

dynamics of gravitation and opening a completely new channel of information on astrophysical objects [1,2]. In the last decades great efforts have been made to implement detectors sensitive enough for cw detection [3,4]. In particular, interferometric detectors are very promising because their projected high sensitivities (h ~ 10 -24 § 10 -26 for integration times of the order of 107s) and an intrinsic large measurement band (from few Hz to many kHz) make them suitable for detection of Cw emitted from different classes of astrophysical objects such as coalescing binaries, pulsars (PSRS) and supernovae explosions. The construction of many of these large detectors (VIRGO, LIGO, GEO600, etc.) has already started: they should become operational by the end of this century [5-7]. At the ssme time, many researchers are trying to obtain useful information to identify possible candidates for cw interferometric detection and to implement suitable detection algorithms. Within this framework, we began a systematic study of GW emission for classes of astrophysical objects, trying to draw a general scenario based on observational data, and pointing out real objects worth observing. In this study we followed a quantitative approach, aiming to obtain all the quantities useful for cw detection, to look for possible good candidates and to give a first rough estimate of the cw background at Earth [8-10]. In particular, we evaluated elsewhere [9] the cw emission from a sample of about 330 galactic PSRS, these being at that time all but six of the known sources located beyond the 10 Hz region. These statistics together with the total number of estimated PsRs in the Galaxy (up to 105 or more; Refs. 11,12) allowed us to predict more than 103 PSRSemitting cw in the detection band of the interferometric detectors. This estimate is being confirmed by the large number of PSRS discovered in this band in the last five years, few of them also in the millisecond region (millisecond pulsars) [13]. These Psas are also becoming relevant Gw sources for modern resonant detectors, whose projected sensitivity seems to be sufficient to detect a subset of this population, that lies within 0.1 kpc of the Sun [14]. We have now extended our previous analysis of cw emission from PSRS, using a larger sample (558 psP.s from the Catalog of Taylor, Manchester and Lyne; Refs. 13,15-23) with the main goal of obtaining reliable data to compare with the theoretical predictions and with the projected sensitivities of cw interferometric detectors like VIRGO and LIGO. In the following sections we will briefly describe the theoretical model used for the evaluation of continuous gravitational radiation from PSmS, which is the most relevant way of cw emission for interferometric detection. Then we will describe the results obtained using all the available observational data. In particular, we have estimated the maximum expected

Earth-based Gravitational Wave Detection from Pulsars

615

cw emission for each PSR of the sample, in connection with the interferometric detector sensitivities. At the same time we have tried to evaluate possible lower limits in using current physical models of PsRs. We have also found some useful general empirical laws (e.g. an expression describing the gravitational luminosity, LGw, as function of the PSR spinning period, P), but we have also found that for further steps (e.g. to obtain a general expression of the gravitational flux at Earth, F e w , or of the dimensionless amplitude, h) the set of PSRS is not good, mainly because of observational selection effects. Finally, we have compared the best expected cw emission from PsRs with the planned sensitivity of the two largest interferometric detectors, VIRGO and LIGO, showing that PSRS exist which may be good candidates for cw interferometric detection. 2. T H E O R E T I C A L C O N T I N U O U S G W E M I S S I O N F R O M P U L S A R S

As we underlined above, in this paper we will show the results of the analysis of the available observational data in connection with the theoretical approach. In fact, theoretical predictions must be always supported by experimental evidence if they are used to direct the experimental part, although often an accurate theoretical description of the phenomena involved cannot be justified by the lack of experimental data or by their poor accuracy. This is the case of Cw emission from PSaS. Therefore, we think that at this stage of the work (where we need only the ranges of possible values of h due to cw emission for each pulsar) the use of accurate physical models for each PSR is far beyond the purpose of this paper. There are three relevant physical quantities for GW detection: the gravitational luminosity, LGw, the gravitational flux at Earth, FGW, and the dimensionless amplitude, h, which are necessary for the evaluation of the Cw emission from each Psa and of the cw background. But, as said above, a global scenario of cw emission from PsRs built using poor observational data has many limitations due to the many strong approximations needed. We have made a first relevant approximation limiting our analysis to continuous gravitational radiation (cw), which, anyway, is the most important way of cw emission for the first detection. In fact, as is well known, spinning neutron stars should emit cw via two different mechanisms. Newly born PSaS should radiate away most of their deviation from axisymmetry in strong bursts of gravitational radiation [24,25], while non-axisymmetric PSRS should emit cw at a frequency equal to twice the spinning frequency [26]. According to cw approximation, the gravitational luminosity (both x and + polarization contributions included) is

616

Velloso, B a r o n e , C a l i o n i , D i F i o r e , G r a d o , M i l a n o , a n d R u s s o

given by [27] 32 G e202 52(21rf)6,

(1)

where G is the gravitational constant, c is the velocity of light in vacuum, f is the spinning frequency, J = J l l d- J22,

(2)

Jn,J22 being the xx and yy quadrupole moments (rotation around 0E assumed) and = 0~-

~o,

(3)

where Ow is the so-called wobble angle, which describes the misalignment between the star's rotation and symmetry axes and eo is the oblateness defined by eo =

Jn

J22

-

j

(4)

Actually, the evaluation of cw emission for each polarization would be much more useful, because, as is well known, gravitational radiation from PSRS is not isotropic. This anisotropy of cw emission is well described by the radiation diagrams defined in [8] as

( (

z

F i g u r e 1. Radiation diagram for a PSR (~, is the rotation axis).

D+'• (0, r -- (dld~)I~,, -#-,x

Lcw/41r

(5)

Earth-based Gravitational Wave Detection from Pulsars

61"/

where D +'x is the angular power density in the (8, r directions divided by the mean angular power density, and f~ is the solid angle. Using the corresponding angular luminosity-distribution (both polarization included) defined in [9] as,

dLGWd6- ~rc sG j2eo282 (2~rfg)6

[4(1 - sin s 8) q- ~1 sin4 8]

(6)

and specializing eq. (5) by including both the polarizations, we have obtained the radiation diagram shown in Figure 1, which clearly shows the low directivity of Gw radiation from PSRS (the ratio between Dmax and Drain is equal to 8). Therefore, although the inclination of the PSR rotation axes cannot be obtained from observations, the low directivity of their cw emission has allowed us to assume isotropic GW radiation for the evaluation of the GW flUX at Earth, F e w , that is LGw

F e w - 4~R2,

(7)

where R is the Earth distance from the Gw source. Then, because for a monochromatic gravitational wave with frequency fg, the Cw flUXis related to the dimensionless amplitude h via [9]

(" G ,~1/2 rl/2 h ..~ \-~-~] fgR ' 9 ~ G W

(8)

substituting eq. (1) into eq. (8) we have obtained a general expression for h, that is,

h=

\ V.]

-~ -~

(9)

3. EXPECTED GW EMISSION FROM PULSARS

The application of the theoretical model described in the previous section to the evaluation of the dimensionless amplitude h still requires knowledge of parameters not directly measurable, ~like the wobble angle, 8w, the oblateness, co, and the quadrupole moment, J. The solution of this problem needs to define reliable values of these quantities for each PSR, on the basis of physical models of PSRS, which apply to a large number of cases, although not to the generality, since these models depend on the age and structure of the PSR, etc. This approach introduces an uncertainty in

618

Velloso, Barone, Calloni, Di Fiore, Grado, Milano, a n d R u s s o

the determination of the dimensionless amplitude which may be large and not easy to quantify. Therefore, we have preferred to try to define a possible range of dimensionless amplitude, h, which characterizes each pulsar, defining an upper limit and a presumable lower limit, which has allowed us to locate each pulsar on the sensitivity diagrams of interferometric detectors. At the same time we have looked for empirical laws to try to obtain a global description of Gw emission from PSRS, which would make it easier to solve the problem of cw detection and would contribute to understanding the physics of these objects better, by analyzing the correlations existing among physical quantities. 3.1. GW Upper Limit A reliable real upper limit for h is easily evaluated by assuming an observed spindown rate completely due to cw emission, as is done by several authors [8,28-30]. In fact, although many concurrent mechanisms contribute to the PSRS slowdown at different levels, this assumption surely represents a physical upper limit to cw emission. In particular, this hypothesis makes it possible to indirectly estimate the maximum values of L o w ~ J , F c w R 2 / j , h R / v ~ and eoS~V~ using the spinning period variation P, a quantity which can be measured with high precision for each PSR.

For this purpose, we have modelled a PSR as a rigid body rotating around a fixed axis. Hence, the energy competing to its rotational motion

can be classically written as E = ~ Zw = ~ Y

(10)

while the energy loss per unit time can be written as dt - 2 ~ \ - - P ]

- (27r)2J

"

(11)

This energy loss may be due to many physical causes, which depend on the evolutionary state of the pulsar under study. Only a fraction of this energy loss may be attributed to cw emission, so that dE > Low dt -

(12)

should always result. In order to obtain an estimate of the maximum cw radiation emission for each PSR, LoW, and an estimate of the maximum

E a r t h - b a s e d G r a v i t a t i o n a l W a v e D e t e c t i o n from P u l s a r s

619

values of FGw, h and of the product eo0w we have equated the rotational energy loss to the gravitational energy emission. A further necesary hypothesis is relative to the variation of the quadrupole moment with time, dJ/dt, which is ~either measurable nor easily estimated. Actually, the term containing this quantity can be considered negligible, at least for those pulsars which are not in the formation phase (which is the majority of the known ones), so that it is possible to approximate, (13)

dt

This expression has allowed us to evaluate the rotational energy variation as a function of two directly measurable quantities, the spinning period P and its temporal variation ]~, if a suitable estimate of J is provided. Actually, the value of J must be in the range 3.1044 < J[gcm2] _< 3" 1045, from a hadronic equation of state corresponding to masses from 0.2 M o to 3.0 M o [31]9 This range of uncertainty has been substantially confirmed by recent studies on neutron-star properties using meson-exchange potential models [32,33]. On the basis of what has been said above and using eqs. (1), (7), (9) we have obtained the expressions for LGW. . . . FGW. . . . (eoSw)max and hma• that are Lcwm~ = ( 2 r ) 2 J P ,

(14)

~rJ P FGwm~x - R2 pa ,

(15)

1

(s

= ~

/5e 5 _1 p

V3')G

(16)

(17)

It is easy to see that the uncertainty on J is not relevant because it would produce differences of only a factor 3 in the computation of hmax and a factor 10 in the computation of Fcwm~x and LGW. . . . which are surely well within the approximations we made for the model up to now. In our calculations we have assumed J = 1045 g cm 2. Using the simplified model described above we have looked for possible correlations among the quantities of the available sample. In Figure 2, the quantity emax = (eoSw)max is plotted against the period P. This figure shows a linear correlation between the logarithms of these two quantities,

620

Velloso, Barone, Calloni, Di Fiore, Grado, Milano, and Russo

which can be expressed by the empirical law, obtained by fitting the data linearly in the least square sense, log(emax) = log((coO~,)max) = (a 4- o'a). l o g P + (b 4- O'b) = (2.10 4- 0 . 0 5 ) - l o g P -t- (-1.92 4- 0.03), (18) and characterized by a correlation coefficient of c = 0.90 (in principle a better correlation may be supposed if we take into account the fact the we have used a value of J equal for all the PSRS). This relation simply means that it seems possible to predict emax by the simple knowledge of the PSR spinning period, P, according to

emax = (eoOw)max = 10(-1"924-~176 9p(2.104-0.05).

(19)

This equation indirectly suggests that there may also be a correlation between the spinning period P and its temporal variation 15. We have substituted this relation into eq. (16), obtaining an expression which relates P and P (see Figure 3), that is, /5 =- (2r") 4 LI"~32GJ )"~" lO(2(b4-ab)) " p(2a-3:l:2a,,) 2.75" 10 -11 9 10 (-3.844-0.o6) 9p(1.2o+o.xo).

(20)

At the same time, we have looked for a direct linear correlation between P and/5. We have found a similar relation, /5 =

10(-14.41-1"-0.06). p(1.21=1:0.09),

(21)

characterized by a correlation coefficient of c = 0.51 which does not make the existence of such a correlation certain. On the other hand, eq. (20) has allowed us to obtain a general expression for Lcw~..x in terms of P, or better, in terms of fg = 2/P, LGwm~x = 1.09 9 1 0 ( - 2 5 " 1 6 4 - 0 " 0 6 ) 9 p ( 0 . 2 0 4 - 0 . 1 0 ) = 2.18.10 (25'164-0'06) 9jf(-0.204-0.10) g

(22)

The computation of FGwm,x and hmax requires the knowledge of the distribution of the PSR distances from Earth. Actually, the distances of all the psas (necessary for the evaluation of FGWm~x and hmax) have been estimated for all the PSRS in the sample by dispersion measure, although with a very large uncertainty [13]. In Figures 4-7 we plot the relative

Earth-based Gravitational Wave Detection from Pulsars

I

621

I

== . .

o" ==-s

-10

,

,

,

I

-4

,

,

-2

,

I

*

,

,

0 Log Period (sec)

Figure 2. ~.,,~ vs. P distribution of the PSR sample. I

I

-10

~s .~.,~. ;. -15

a

,j, -20

-25. 4 - - '

,.

-2

0 Log Period (sec)

Figure 3. jb vs. P distribution of the PSR sample,

2

622

Velloso, Barone, Calloni, Di Fiore, Grado, Milano, and Russo

PSR abundance against distance from Earth, the relative PSR abundance against the distance from the galactic centre, the relative PSR abundance against the distance from the galactic plane and the relativePSR abundance against the spinning frequency, respectively. It is possible to see that there is a general agreement between these results and the results obtained with smaller samples (149 PSRS by Manchester and Taylor, Ref. 34; 330 PSRS by Barone et al., Ref. 9) which m a y mean that the sample is statistically homogeneous or, more likely,characterized by the same selection effects! Therefore, no general distribution law for the distances can be obtained because of the evidence of these selection effects,which are particularly obvious in Fig. 6. In fact according to this picture, the PSRS do not appear to be distributed in equal number on both sides (Zmean ~ -320 pc) and there are some privileged distances in which a particularly large number is detected. O n the other hand, although a general distribution law for PSaS can not be obtained, the single objects can be studied and a state-of-the-art c w background can be built. 3.2. G W Lower Limit

We have followed a different approach for the evaluation of a lower limit for h. In fact, in this case it is necessary to give a reliable estimate of the oblateness and of the wobble angle, trying to use the existing models. Actually, the oblateness eo is a quantity quite difficult to estimate, because it depends on the PSR velocity of rotation. Existing estimations are based on the starquake theory and predict values in the range 10 -3 -" 10 -2 [35]. These values were already used in similar calculations made by Zimmerman [30] and by Barone et al. [9]. However, Araujo et al. [36], using a new set of state equations, derived by Rufa et al. [37], showed that the expected values for eo could be one or two orders of magnitude greater, depending on the angular velocities (Aranjo model). On the other hand, values of the wobble angle ~w are affected by a much larger uncertainty, due to the lack of a reliable microquake model [38]. According to Zimmermann [30] the assumption 10 -3 ~ ~Wtr.dI ~ 10 -2

(23)

would yield a conservative estimate of PSR cw luminosities. It has been suggested, however, that significantly larger wobble angles could result from the existence of internal toroidal magnetic fields, tending to align perpendicular to the spin axis. Mountains and/or locM crust inhomogeneities could have an effect as well [30]. On the other side, at large rotational velocities, we can expect that the PSRStend to align the rotational and the

E a r t h - b a s e d G r a v i t a t i o n a l W a v e D e t e c t i o n from P u l s a r s

Lok[ IO'

lo+l

iO-a 0.~

6.00

I0.00

nlHmMH H

I$.00

20.OO

25.00

Distance from Earth (k'pc)

F i g u r e 4. Relative PSR abundance vs. distance from the Earth.

Log~'l ~ lO-I

[

o.oo

5.00

I0.00

15+00 20.00

~&oo

30.00

,35.00

40.00

Distance from Galactlc Center (Kpc)

F i g u r e 5. Relative PSR abundance vs. distance from the Galactic Center.

623

624

Velloso, Barone, Calloni, Di Fiore, Grado, Milano, and Russo

io4'

Log~. lO-t

10-3

lO-+ -3.90

-2.00

-I.00

0.00

1.00

2.00

3.00

Distance from GalacticPlane (kpc) Figure 6. Relative PSR abundance vs. distance from the Galactic Plane. to9

Log~. lO-I

IO-S

lO'a IO'z

tO-t

tOI

lOt

10~

103

104

Log SpinninS Frequency (sec-') Figure 7. Relative PSR abundance vs. spinning frequency.

Earth-based Gravitational Wave Detection from Pulsars

625

symmetry axes, resulting in smaller wobble angles, which would reduce the gravitational radiation emission. Combining these two quantities, we get 10 -6 ~_ e[radI ~_ 10 -4. (24) The lower bound in eq. (24) is consistent with the Melosh model of PSR deformation due to poloidal magnetic pressure [39,40]. It is worth noting that, at least for those PSaS with large magnetic field, the variation in the periods could be caused by the effect of a magnetic slow-down. In this case, the ow emission could be considerably smaller than the calculated upper limit. However, since the magnetic fields themself are estimated from the pulsar slow-down [13,34], it is difficult to understand which of the two processes is dominant. Actually, these estimates for e can be considered acceptable for slow pulsars, but, as we said above, we can expect a decrease of this value for fast pulsars. An indirect proof of the validity of this assumption is given by the analysis of the function /5 = p ( p ) . It is easy to see that if P decreases P decreases so much that the assumption of e = 10 - 6 would yield, for P < 10 - 2 S, hmin ~ hmax. Therefore, it is possible to affirm that the deviation from axisymmetry for fast PSaS (millisecond PSaS) may be so small to compensate the increase of Gw emission due to a lower spinning period. Moreover, it is also clear that it is not possible to obtain a real lower limit for cw emission unless we analyze each single PSR. Another way could be that of assuming that, for example, only 0.01% of the energy loss is due to gravitational wave emission, which would simply decrease the quantity h by a factor of about 100. 4. G W B A C K G R O U N D FOR A POPULATION OF 427 PSRS

The results of our analysis on the GW background from PSaS are collected in Figures 8-10, where we show the PSR relative abundance against maximum flux on Earth, the maximum Gw flux on the Earth against ow frequency and the maximum GW dimensionless amplitude on Earth against frequency, respectively. Note that we have limited our analysis to PSRS with known period variation. Assuming that this sample is statistically significant (the shapes of the distributions are similar to those reported in Ref. 9), then the effect of including the whole PSR population would simply raise the flux levels by a factor of the order ~ 100 and, correspondingly, the dimensionless amplitudes by a factor ~ 10, at most. Our computations do not include the effect of Doppler frequency shift due to PSR recession from the galactic centre. The average estimated ve-

626

Velloso, B a r o n e , Calloni, D i Fiore, G r a d o , M i l a n o , a n d R u s s o

lO-O

Log~ I0"

I0-:'

|0": I0"~~

m lO-~,

IO-~e

lO-io

lO-lS

Total G W Flux on Earth[erg/cm 7. sec] Figure 8. Relative PSR abundance vs. marnirnum G W flux at Earth.

10-'

Maximum

OWFlux at Earth

[~.rg/cm~- sect jO-a

IO-IZ

lO-.U

| O-:'o

llllll ,H,,, llllll nnnntn

lO-+a I@~

I0 0

I01

I0 . l

iO-+'

I0-~

10-4

GWPrequency(It'~) F i g u r e 9. Ma~xirnum GW flux at Earth vs. frequency.

Earth-based Gravitational W a v e

627

Detection from Pulsars

Maximum Dimensionless Amp|itude lO-M

iO~=e

I01

I0*

lO-S

io-i

io-~

GW Frequency (Hz)

II 10.4

Figure 10. M a x i m u m G W dimensionless amplitude at Earth vs. frequency. 10-1'

.......

I

'

'

'''"'1

''"'1

'

.......

I

.......

10-1.

10..-I 10-m ~10-m

Virgo

~lO-n

! ~ lO...u ~ lO..-m 10-,,~

,,~N lo"U 10,-a 100.1

1

10 100 Log Frequency (Hz)

1000

104

Figure II. Expected m a x i m u m dimensionless amplitudes, h, compared with the sensitivity curves of V I R G O and L I G O interferometric antennas.

628

Velloso, Barone, Calloni, Di Fiore, Grado, Milano, and Russo

locity must be ~ 100 km s -1, thus producing a fractional Doppler shift of 3 . 1 0 -4 in the worst case, which is negligible in the present context. In Figure 11 the maximum dimensionless amplitude coefficient h for the whole sample is plotted against the gravitational frequency. In the background of this figure, the sensitivity curves of VIRGO and LIGO interferometric antennas are shown for comparison. Specifically, we have shown the sensitivity curves for each detector which correspond to the dimensionless amplitude of a periodic gravitational wave of known frequency detected with a unity signal-to-noise ratio after integration for a day, a month and a year (from top to bottom). In Table ! all those Psas are listed which show a maximum value of the dimensionless amplitude h at Earth greater than 10 -26 and are, in principle, detectable by interferometric antennas like VIRGO or LIGO. 5. CONCLUSIONS We can summarize the results obtained in this paper as follows. (i) We have shown that good known candidates for GW interferometric detection exist, which are within the projected sensitivity of the two largest interferometric detectors, VIRGO and LIGO. At the same time we want to remark that the approximations made in our analysis are very large and not easy to quantify, so that the numbers reported must be handled with care. Nevertheless, we have pointed out some objects which need now a very careful analysis for extracting all the useful information for GW detection. (ii) We have shown that millisecond pulsars are not necessary good candidates for GW detection. In fact, the increase of cw emission due to a lower spinning period is strongly compensated by a smaller e, as we have indirectly obtained by the analysis of the period change /5. As a consequence, the number of known millisecond PSRS candidates for cw interferometric detection is still very small. (iii) We have shown that, although the Psa set used is clearly affected by observational selection effects which have prevented us from obtaining further relevant information, we have obtained some useful correlations which have made it possible to express emax = (Co- 0~)m~ as function of the spinning period, P, the period change, P, as function of the spinning period, P, and the maximum gravitational luminosity, Lcwm~, as function of the spinning period P. Further steps, like the determination of Fg~m~ and hmax have not been possible with this set of Psas because of a lack of reliable statistical distribution of their distance from the Earth, due to observational selection effects.

P (sec)

0.0893 0.0334 0.2371

0.1502 0.1024 0.1237

0.1336 0.1249 0.1015

0.0395 0.1014 0.0504

0.0058 0.0062

NAME

P S R 0833-45 P S R 0531+21 P S R 0633+17

P S R 1509-58 P S R 1706-44 PSlq, 1046-58

P S R 1800-21 P S R 1757-24 P S R 1823-13

P S R 1951+32 PSR 0114+58 P S R 0540-69

PSR 0437+47 P S R 1257+12

0.120 10 -18 0.121 10 -38

0.585 10 -14 0.584 10 -14 0.479 10 -12

0.134 10 -12 0.128 10 -12 0.749 10 -13

0.154 10 -11 0.930 10 -13 0.959 10 -13

0.125 10 -12 0.421 10 -12 0.110 10 -13

l5

347.38 321.62

50.59 19.72 39.70

14.97 16.02 19.71

13.31 19.52 16.17

22.40 59.87 8.44

f~ (Hz)

140 620

2500 2120 49400

3940 4610 4120

4400 1820 2980

500 2000 150

R (pc) FGw (erg cm-2s -I) 0.231 10 -o5 0.933 10 -05 0.121 10 -o7 0.775 10 -~ 0.863 10 -~ 0.189 10 -~ 0.120 10 -~ 0.102 10 -~ 0.140 10 -~ 0.500 10 - ~ 0.412 10 -09 0.507 10 -09 0.106 10- ~ 0.432 10-o9

LGW (erg s-1) 0.691 10 +37 0.446 10 +39 0.325 10 +35 0.179 10 +33 0.342 10 +37 0.200 10 +37 0.222 10 +37 0.259 10 +37 0.283 10 +37 0.374 10 +37 0.221 10 +36 0.148 10+39 0.248 10+35 0.199 10+35

~ = ~o 98w (rad) 0.180 10 - ~ 0.757 10 -03 0.231 10 -02 0.138 10 - ~ 0.191 10 - ~ 0.257 10 -02 0.342 10 -02 0.302 10 -02 0.169 10 -02 0.115 10 -03 0.472 10 -~ 0.150 10 -02 0.289 10 - ~ 0.326 10 -07

h 0.121 10 -23 0.906 10 -24 0.732 10 -24 0.371 10 -24 0.267 10 -24 0.151 10 -24 0.130 10 -24 0.112 10 -24 0.106 10 -24 0.785 10 -25 0.578 10 -25 0.319 10 -25 0.166 10 -25 0.363 10 -26

T a b l e 1. S u b s e t of PSl~ which exihibit a m a x i m u m value of t h e dimensionless a m p l i t u d e , h, within t h e sensitivity curves of V I R G O a n d L I G O interferometric detectors.

o

tD

cr

t~

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