Possible direct method to determine the radius of a star from the spectrum of gravitational

发布于:2021-08-03 21:57:29

Possible direct method to determine the radius of a star from the spectrum of gravitational wave signals II : Spectra for various cases
Motoyuki Saijo
?

Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801-3080

Takashi Nakamura

?

Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan (Received 27 August 2000) We compute the spectrum and the waveform of gravitational waves generated by the inspiral of a disk or a spherical like dust body into a Kerr black hole. We investigate the e?ect of the radius R of the body on gravitational waves and conclude that the radius is inferred from the gravitational wave signal irrespective of (1) the form of the body (a disk or a spherical star) (2) the location where the shape of the body is determined, (3) the orbital angular momentum of the body, and (4) a black hole rotation. We ?nd that when R is much larger than the characteristic length of the quasinormal mode frequency, the spectrum has several peaks and the separation of the troughs ?ω is proportional to R?1 . Thus, we may directly determine the radius of a star in a coalescing binary black hole - star system from the observed spectrum of gravitational waves. For example, both trough frequency of neutron stars and white dwarfs are within the detectable frequency range of some laser interferometers and resonant type detectors so that this e?ect can be observed in the future. We therefore conclude that the spectrum of gravitational waves may provide us important signals in gravitational wave astronomy as in optical astronomy. PACS number(s): 04.30.Db, 04.25.Nx

arXiv:astro-ph/0012061v1 4 Dec 2000

I. INTRODUCTION

Researchers in gravitational physics have been expecting for the direct detection of gravitational waves (GW) using laser interferometers such as LIGO, VIRGO, GEO600, TAMA300, and LISA as well as several projects using resonant type detectors [1]. The direct detection of GW will provide us not only the veri?cation of General Relativity but also plenty of new aspects in many other ?elds, such as nonlinear physics related to the Einstein equations, nuclear and particle physics from the estimation of equation of states of the neutron star (NS), cosmology from the evidence of in?ationary universe and from the measurements of cosmological parameters. Therefore, it will open the frontiers of physics, gravitational wave physics/astronomy in the 21st century. Among many possible sources of GW, coalescing binaries composed of black holes (BHs) or NSs are probably the most promising. From these gravitational wave signals, we can trace the astrophysical event. Comparing the inspiral waveform with the theoretical templates using the matched ?ltering technique, we may determine masses and spins of the BH and star, respectively [1]. On the other hand, di?erent type of information from the inspiral phase may be obtained from the ?nal merging phase such as the radius of a star, the equation of state of a high density matter, and the structure of the stars. Since many physical elements have to be taken into account at once to obtain such information in the merging phase, little numerical and analytical study have been e?ectively investigated. Recently, Vallisneri [2] argued a 1

possibility to extract the radius of a star from GW. If we measure the frequency of GW from the location where the star is tidally disrupted, we may ?nd information about the central density of the star, and from that, we may obtain the radius of a star if we know its equation of state. This method as well as other methods using the results of numerical simulations, however, may need information about the equation of state to determine the radius of the star. For example, Vallisneri [2] used the result of an incompressible homogeneous Newtonian ?uid in a Kerr BH [3]. The ?rst step to tackle this problem of the merging phase might be examined by using a BH perturbation approach. In this paper, we shall consider GW from either an usual star or a NS inspiraling into a BH, emphasizing on the size e?ect of a disk/star on GW using a BH perturbation approach. Many papers along this approach have been published on this topic; GW from a test particle plunging into a BH, since Sasaki and Nakamura [4,5] reformulated Teukolsky equation [6] to make it possible to calculate GW induced by a particle plunging into a Kerr BH from in?nity. Especially, several papers focus on phase cancellation e?ects on GW using this approach. In early studies by Nakamura and Sasaki [7] and Haugan, Shapiro, and Wasserman [8], GW from a deformed shell falling into a Schwarzschild BH were investigated. Saijo, Shinkai, and Maeda [9] extended their model to a Kerr BH and found that when the central BH is rotating, a slightly oblate dust shell minimizes the collapsing energy of GW but with a non-zero ?nite value, which depends on the Kerr parameter. A BH perturbation approach is not only a toy model to

illustrate GW from a binary system. There are two wellknown facts comparing the results from the BH perturbation approach with the full numerical simulation. The research of the head on collision of two BHs which was pioneered by Smarr [10] and updated by Anninos, Price, Pullin, Seidel, and Suen [11] was the ?rst work. They compared their numerical results with that of the BH perturbation approach ; GW from a test particle falling into a Schwarzschild BH by Davis, Ru?ni, Press, and Price [12]. The radiated energy agrees with each other within a factor of two. Especially, the waveform of the ringing tail has a very accurate coincidence between these two cases. The second research was an axisymmetric rotating star collapse by Stark and Piran [13]. They also compared their result with the previous semi-numerical result; ring shaped test particles plunging into a Kerr BH by Nakamura, Oohara, and Kojima [14]. Although these two models are quite di?erent, the waveforms of these two computations give a fairly good agreement. We can interpret the reason why these di?erent two results have a coincidence as follows. The main body of the rotating star ?rst collapses to form a Kerr BH. Then, the remaining part which is essentially regarded as ring-shaped particles fall into the newly formed BH. Therefore, the above agreement is not so strange from this physical picture of a rotating star collapse. These two examples encourage us to apply the BH perturbation approach to the ?nal phase of coalescing binary systems. The model worked in this approach may throw a milestone on what kind of situation provides us a very impressive viewpoint for GW and needs to be followed by the detailed, numerical simulations. The result from this approach also helps those who work in this ?eld to concentrate on such predicted, exciting results. Our purpose in this paper is to focus on the size effect of a star on GW from a coalescing binary BH - star system. We use the BH perturbation approach for the sake of the detailed, perspective investigation to ?nd the property appears in the GW. In Ref. [15], we have already presented the result with a limited case; a disk with one set of parameters. In this paper, we examine the dependence of the results on (1) the form of the body (a disk or a spherical star), (2) the location where the shape of the body is determined, (3) the angular momentum of the body, and (4) a BH rotation in detail. This paper is organized as follows. In Sec. II, we introduce the method how to construct the radial wave function of GW for disk shaped test particles as well as spherical shaped ones using the BH perturbation approach. We present our numerical results of GW from a disk and a spherical dust star in Sec. III. Section IV is devoted to discussion. Throughout this paper, we use the geometrized units of G = c = 1 and the metric sign as (?, +, +, +).

II. RADIAL WAVE FUNCTION

In this section, we explain the method to construct the radial wave function of a dust disk and a spherical star in Kerr spacetime using the BH perturbation approach. Before explaining the construction method of the radial wave function, let us brie?y introduce the BH perturbation approach [4,5,14]. This approach is applicable only if M ? ?, where M is the BH mass and ? is the particle mass. However, the computational result happens to be well-behaved in some cases even if we extrapolate the result to ? ? M as we have already mentioned before. In order to extract GW at in?nity with su?cient computational accuracy, we have to solve the generalized Regge-Wheeler equation (Sasaki-Nakamura equation) [Eq. (2.28) of Ref. [5]] such as d2 d ? F (r? ) ? ? U (r? ) Xlmω (r? ) = Slmω (r? ), dr?2 dr (2.1) where r? , Slmω (r? ), F (r? ) and U (r? ) are the tortoise coordinate of a Kerr BH, the source term from a test particle of mass ? [Eq. (2.29) of Ref. [5]] and two potential functions [Eqs. (2.12a) and (2.12b) of Ref. [5]], respectively. To solve Xlmω (r? ) using a Green function method, we need two independent homogeneous solutions whose boundary conditions are given by
in(0) Xlmω (r? ) out(0)

=

in out Blmω e?ikr + Blmω eikr r? → ?∞ ? , eiωr r? → ∞ √ where k = ω ? ma/[2(M + M 2 ? a2 )]. The inhomogeneous solution to Eq. (2.1) becomes

e?ikr r? → ?∞ , in ?iωr ? out iωr ? Almω e + Almω e r? → ∞
? ?

?

Xlmω (r? ) =

Xlmω (r? ) = Xlmω

in(0)

∞ r?

Slmω (r? ) out(0) ? Xlmω dr W
r? ?∞

+Xlmω

out(0)

Slmω (r? ) in(0) ? Xlmω dr , W

where W is the Wronskian, W ≡ Xlmω
out(0) in(0) dXlmω dr?

? Xlmω

in(0) out(0) dXlmω . dr?

(2.2)

The asymptotic behavior of the radial wave function Xlmω (r? ) is given by Xlmω (r? ) = Almω eiωr , ∞ Slmω (r? ) in(0) ? Xlmω dr . Almω = W ?∞
?

(2.3)

Using the amplitude of the radial wave function, the energy spectrum and the waveform of GW [Eqs. (3.6) and (3.10) of Ref. [9]] are given by 2

dE dω

= 8ω 2
lmω

Almω c0


2

,
?

(?∞ < ω < ∞), (2.4)
?t)

h+ ? ih× =

8 r ×

dωeiω(r
?∞

l,m

eim? Almω aω , ?2 Slm (θ) √ c0 2π

(2.5)

where r is the coordinate radius from the center of the aω BH, ?2 Slm is spin ?2 weighted spherical harmonics, c0 is a constant given in Ref. [9]. First, we consider a dust disk plunging into a Kerr BH from in?nity. We set three assumptions to construct a disk in Kerr spacetime. (1) The disk is made up of test particles and each of them moves in the equatorial plane plunging into the Kerr BH from in?nity. (2) The shape of the disk is set at the location r = r0 . (3) All component particles of the disk have same energy, angular momentum, and Carter constant. In this paper, we set the disk at the location r0 as 10M or 20M , because our purpose in this paper is to focus on the merging phase of the coalescing binary BH - star [16]. The motion of a test particle in Kerr spacetime is written in general as [18] dt dτ dr Σ dτ dθ Σ dτ dφ Σ dτ Σ ? ? = ?a(aE sin2 θ ? Lz ) + √ = ± R, √ = ± Θ, ? Lz ? = ? aE ? sin2 θ + a P, ? r 2 + a2 P, ? (2.6) (2.7) (2.8) (2.9)

same energy and angular momentum. Then, we can set ? ? another geodesic for same E and Lz with shifting time and azimuthal angle as t = T (r) + ct and φ = Φ(r) + cφ where ct and cφ are constants. After we set the point where the representative particle pass through (r = ri and φ = φi at t = T (r0 )), the orbit of each component particle can be expressed as t = T (r) + T (r0 ) ? T (ri ) and φ = Φ(r) + φi ? Φ(ri ). Therefore, it becomes possible to ? ? set a number of particles with same E and Lz to form the disk of radius R whose center is r = r0 and φ = Φ(r0 ) at t = T (r0 ). Using this character, we can construct the radial wave function only with a function of r [7–9]. We ? ? can also choose arbitrary E and Lz for the disk except for the condition that each test particle plunges into a Kerr ? ? BH. The parameter range of Lz in the case of E = 1 which we use in this paper is ? 2M ? ? 4M 2 + 4M a < Lz < 2M + 4M 2 ? 4M a. (2.13) We construct the radial wave function of GW for a dust disk. Since Kerr spacetime has a property of symmetry for t and φ, we can use this symmetry to construct the radial wave function in order to save computational time as we explained above. Therefore, the radial wave (disk) function of the disk Almω can be constructed from that (particle) of a test particle Almω as Almω = fmω Almω fmω = 2 ? S
(disk) (particle) r0 +R

, sin(mφ0 (r)) m

(2.14)

drr

×e

r0 ?R i[ω[t(r)?t(r0 )]?m[φ(r)?φ(r0)]] 2 r2 + r0 ? R2 , 2rr0

,

(2.15) (2.16)

? ? where E = ?E is the energy, Lz = ?Lz is the orbital angular momentum, C is Cater constant of the particle, Σ = r2 + a2 cos2 θ, ? = r2 ? 2M r + a2 . The symbols P , R and Θ are de?ned by ? ? P = E(r2 + a2 ) ? aLz , ? ? R = P 2 ? ?[r2 + (Lz ? aE)2 + C], (2.10) (2.11) (2.12)

φ0 (r) = cos?1

where ? is the mass of the disk, r0 is the location of the central point of the disk where we set the circular disk, R is the coordinate radius of the disk at r = r0 , S is the normalization factor de?ned by
r0 +R

?z L2 ? Θ = C ? cos2 θ a2 (1 ? E 2 ) + . sin2 θ

S=
S

ds = 2 r2 + 2rr0
(disk)

drrφ0 (r),
r0 ?R 2 r0 ? R2

(2.17) (2.18)

When a particle moves in the equatorial plane or in the constant zenith angle, Carter constant is required to be zero from the stability of its trajectory. Here we explain our e?ective method to construct the radial wave function, instead of computing all the trajectories of each component particle. We note that the geodesic in the equatorial plane of a Kerr BH is charac? terized by two parameters; the speci?c energy (E) and ? z ). Let t = T (r) and the speci?c angular momentum (L ? φ = Φ(r) express the orbit of the geodesic for given E and ? Lz . Since there is time symmetry and azimuthal symmetry in Kerr spacetime, we can ?nd another geodesic with 3

φ0 (r) = cos?1

.

In order to construct Almω [Eq. (2.14)], we only have (particle) to prepare Almω , t(r), and φ(r). Next, we present how to set a spherical dust star at r = r0 (r0 = 10M , 20M ) in order to set a realistic model of BH - star merging. We also set three assumptions to construct a spherical dust star in Kerr spacetime. (1) The star is made up of test particles that move along the constant zenith angle of a Kerr BH. (2) The shape of the star is set at the location of r = r0 , and the center of the star is located in the equatorial plane of a Kerr BH. (3)

All test particles have the same speci?c energy, speci?c orbital angular momentum, and Carter constant. From the assumptions (1) and (3), we cannot take the orbital angular momentum of the particles into consideration. Since the orbital angular momentum has almost lost by the gravitational wave emission in the previous inspiral phase, the zero orbital angular momentum may not be an absurd approximation. Our main purpose in this modeling to construct a spherical star is to con?rm that the di?erence of the body (a disk or a spherical star) little a?ect our main conclusion. We expect this proposition is true because the size e?ect in our model does not depend on the orbit of the star. We also note that the stability of the constant zenith angle trajectory also requires C = 0 ? when we choose E = 1. We use the same technique as the dust disk case to construct the radial wave function of the spherical dust star. We can only adopt this technique to the particles in each constant zenith angle, the radial wave function (star) for the spherical dust star Almω becomes a function of θ as Almω = 2
(star)

? V

r0 +R r0 ?R θ(r) ?θ(r)

drr2

sin(mφ(r, θ)) m
(particle)

×

dθ sin θAlmω

(θ) (2.19)

×ei[ω[t(r,θ)?t(r0,θ)]?m[φ(r,θ)?φ(r0,θ)]] ,

where ? is a mass of the star, r0 is the location of the central point of the spherical dust star where we set r0 = 10M , 20M in this paper, R is a coordinate radius of the star, V is the normalization constant of the star de?ned by
r0 +R

10M , 20M ) than before [15]. We place the spectra into three categories, classi?ed by the size of the disk. We brie?y explain the feature of each type of the spectrum. Type 1 spectrum has the same character as that of a test particle; the energy spectrum has one dominant characteristic frequency for each m mode where most of GW are radiated. This frequency corresponds to the quasinormal mode (QNM) of the background BH. The real part corresponds to the resonant oscillation of the BH, and the imaginary part corresponds to the “damping” time scale of GW at the late time. The waveform, which has a damping oscillation at the late time, can also be described by the QNM frequency at that part. Since GW from a disk has a very similar character to that from a test particle, we may ignore the size e?ect on GW as far as the diameter of the disk is smaller than the characteristic length of the QNM frequency. We classify the spectrum to type 2 when the diameter of the disk is comparable to the characteristic length of the QNM frequency. In this case, the sharp peak near the QNM frequency has been weakened. GW still have a damping oscillation in the waveform, though the amplitude of the ringing tail is much smaller than that for a test particle due to the phase cancellation e?ect on GW. When the diameter of the disk is larger than the characteristic length of the QNM frequency, the spectrum has several peaks (type 3). One remarkable feature appears in the spectrum that the separation of the troughs does not largely depend on the orbit. Using this character, we ?nd a direct method to determine the radius of a star [15]. In the waveform, we can hardly recognize the damping oscillation at the late time. Therefore, we cannot ?nd any characteristic frequency of GW, corresponds to the QNM frequency for a test particle in a plunging orbit.
A. Dependence on a BH rotation

V =
V

dv = 2
r0 ?R θ(r)

drr2 φ(r, θ) (2.20) (2.21) (2.22)

×

dθ sin θ,
?θ(r) 2 ?1 r 2 + r0 ? R2 , 2rr0 sin θ(r) 2 r2 + r0 ? R2 . 2rr0 (particle)

φ(r, θ) = cos

θ(r) = cos?1

Therefore, we have to prepare Almω (θ) and t(r, θ), φ(r, θ) for a set of constant zenith angle trajectories in (star) order to construct Almω [Eq. (2.19)].
III. GRAVITATIONAL WAVES FROM THE DISK AND THE SPHERICAL STAR

We show our numerical results for a dust disk plunging into a Kerr BH. We calculate GW for a wide range of ? parameters (Lz /M = 2, 0, ?3, a/M = 0, 0.5, 0.9, r0 = 4

First, we discuss the dependence of the energy spectra and the waveforms on the radius of the disk and the BH rotation, keeping the other set of parameters as r0 = ? ? 10M , Lz /M = 2. Since we choose Lz /M = 2, we only focus on l = m = 2 nonaxisymmetric mode which is the dominant mode in all modes in both spectra and waveforms. For a test particle case, the peak frequency which corresponds to the QNM frequency of the BH is different among these three spectra [Figs. 1 (a), 2 (a), and 3 (a)] . For example, the spectrum has a peak at frequency M ω = 0.33 (l = m = 2) in Fig. 1 (a) (a = 0), frequency M ω = 0.41 (l = m = 2) in Fig. 2 (a) (a/M = 0.5), frequency M ω = 0.63 (l = m = 2) in Fig. 3 (a) (a/M = 0.9), while QNM frequency for a = 0, is M ω = 0.37 ? 0.089i (l = 2), for a/M = 0.5 is M ω = 0.46 ? 0.085i (l = m = 2), and for a/M = 0.9 is M ω = 0.66 ? 0.065i (l = m = 2) [Fig. 1a of Ref. [19] and Fig. 3 (c) of Ref. [20]]. These results indicate the

fact that the vibration of the BH has a dominant e?ect on GW for a test particle in a plunging orbit. When we turn to look at waveforms, Figs. 4 (a), 5 (a), and 6 (a) have damping oscillations at the late time, which can be described by the QNM frequency (Fig. 4 of Ref. [21] and Fig. 4 of Ref. [22]). For type 1 case, both energy spectra and waveforms for a disk have a very similar behavior to the case for a test particle. The small size of the disk (for example R = 0.785M ) does not largely a?ect energy spectra and waveforms. The characteristic behavior in spectra and waveforms is almost the same as the case of a test particle. We con?rm this feature by comparing Fig. 1 (Figs. 2, 3) (a) and (b) for the spectra, and Fig. 4 (Figs. 5, 6) (a) and (b) for the waveforms. The di?erence between the spectrum from a test particle and that from a small size disk appears only in the form factor as we follow (disk) the construction of Almω in Eq. (2.14). When we look at the form factor [Figs. 7 (a), 8 (a), and 9 (a)], |f2ω |2 almost takes the value unity for almost all range of the frequency, which is equivalent to the case of a test particle. The behavior of the form factor lead the conclusion that there is little di?erence between R = 0.785M and R = 0 from the viewpoint of gravitational wave signals in the plunging orbit. For type 2 case, both energy spectra and waveforms have a di?erent feature from those of a test particle. In the energy spectrum, the sharp peak near the QNM frequency has been weakened [Figs. 1 (c), 2 (c), and 3 (c)]. Since we have de?ned the range of the radius for type 2 that the diameter of the disk is comparable to the characteristic length of the QNM frequency, the range depends on the BH rotation. We choose each range for each BH rotation for type 2 case as R/M ? 2 for a/M < 0.5, while R/M ? 1.5 for a/M = 0.9. The di?erence of the radius for each BH rotation is also con?rmed from the form factor [Figs. 7 (b), 8 (b), and 9 (b)] since there is a sharp trough in the form factor appears near the QNM frequency. When we turn to look at the waveform [Figs. 4 (c), 5 (c), and 6 (c)], the amplitude near the merging phase has been weakened due to the phase cancellation e?ect on GW. In spite of this e?ect, we can still ?nd a damping oscillation quite clearly. For type 3 case, both energy spectra and waveforms have a completely di?erent feature from the case of type 1 and type 2; several peaks appear in the spectrum [Figs. 1 (d), 2 (d), and 3 (d)]. These peaks also appear in the form factor [Figs. 7 (c), 8 (c), and 9 (c)] and take the same behavior to the spectrum. We show the relation between the separation frequency of the troughs and the radius of the disk in Table I. Remarkably, the relation R?ω ? 1 holds for a wide range of parameters. This relation make it possible to measure R directly from the observed ?ω. We will discuss this point in more detail in Sec. IV. When we turn to look at the waveform [Figs. 4 (d), 5 (d), and 6 (d)], we can hardly ?nd a damping oscillation at the late time due to the phase cancellation e?ect

on GW. We ?nd that the size e?ect only appears at the merging phase comparing the waveforms with each different radius. For example, we ?nd the di?erence within the range of ?100 < t ? r? /M < ?30 in Fig. 4 (a = 0), within the range of ?100 < t ? r? /M < ?30 in Fig. 5 (a/M = 0.5), within the range of ?130 < t?r? /M < ?30 in Fig. 6 (a/M = 0.9). Although we assume that all the component test particles come from in?nity, the waveforms in the merging phase might not depend on this assumption so much. Since the dominant wave is described by the QNM frequency for a plunging orbit case, irrespective of the particle energy and angular momentum, the phase cancellation e?ect from several QNM frequency waves is responsible to the behavior of the waveforms in this merging phase.
B. Dependence on an orbital angular momentum

Next, we discuss the dependence of energy spectra and waveforms on the radius of the disk and the orbital angular momentum, keeping the other set of parameters as r0 = 10M , a/M = 0.9. For a test particle case, the spectrum de?nitely depends on the orbital angular momentum [Figs. 3(a), 10 (a), 11 (a), and 12 (a)]. In fact, GW are e?ectively radiated near the QNM frequency and the spectrum takes an additional bump in the case of ? Lz /M = 2.63 [Fig. 10 (a)]. Since we do not take into account of the radiation reaction force e?ect in our model, the amount of radiated energy diverges when the particle takes a circular orbit. Therefore, GW are radiated e?ectively when the amount of the orbital angular momentum approaches to the value of a circular orbit. In the case ? of Lz /M = 2, m = 2 mode dominates in l = 2 spec? trum while for Lz /M = ?3, m = ?2 mode dominates in the spectrum within the positive frequency range. This tendency turns completely contrary within the negative frequency range. For example, m = ?2 mode dominates ? in the case of Lz /M = 2, while m = 2 mode dominates ? z /M = ?3. There is another feature in the specfor L trum; each ±m mode has a re?ection symmetry at the zero frequency. This comes from the fact that the system has an equatorial plane symmetry. When we turn to look at waveforms (Figs. 6, 14, and 15), some unclear damping oscillation appears in Figs. 14 (a) and 15 (a). These behavior might come from the superposition of several waves, each of them is described by a single QNM frequency. The classi?cation of the type, depending on the radius of the disk, and the behavior of each type are the same as the case of Subsec. III A, we will brie?y describe the results in each category. Spectra, waveforms, and form factor for type 1 case are shown in Figs. 3 (b), 10 (b), 11 (b), 12 (b), in Figs. 6 (b), 13 (b), 14 (b), 15 (b), and in Figs. 9 (b), 16 (b), 17 (b), 18 (b). Since this type little a?ects the size on GW, the disk can almost be regarded as a test particle from the viewpoint of a

5

gravitational wave signal. Spectra, waveforms, and form factor for type 2 case are in Figs. 3 (c), 10 (c), 11 (c), 12 (c), in Figs. 6 (c), 13 (c), 14 (c), 15 (c), and in Figs. 9 (c), 16 (c), 17 (c), 18 (c). The characteristic QNM frequency, as appeared in the spectrum for type 1 case, has been weakened due to the phase cancellation e?ect on GW, but we can still look at the dumping oscillation in the waveform. Spectra, waveforms, and form factor for type 3 case are in Figs. 3 (d), 10 (d), 11 (d), 12 (d), in Figs. 6 (d), 13 (d), 14 (d), 15 (d), and in Figs. 9 (d), 16 (d), 17 (d), 18 (d). We can hardly ?nd the characteristic frequency in both spectra and waveforms. We also show the snapshot of the disk form in Figs. 19 and 20, focusing on the modi?cation of the disk. When we turn to look at R?ω ≡ C in Table I, the dependence of the orbital angular momentum on C is larger than that of the BH rotation although it takes 0.87 < C < 1.16 in any case. The value C indeed depends on the orbit, as we will explain the structure of C in Sec. IV in detail.
C. Dependence on the location where we set the circular disk

D. Spherical star

Let us consider the e?ect of the radius and the location, where we de?ne the circular disk, on energy spectra and waveforms, keeping the other set of parameters as ? Lz /M = 2, a/M = 0.9. The circular disk at r0 = 20M change the shape and when it comes to r = 10M the aspect ratio between the radial direction and azimuthal angle direction becomes 3 : 1 in the case of Fig. 21. It is, therefore, natural to investigate the dependence on r0 . We show the spectra, the waveform, and the form factor in the case of r0 = 10M in Figs. 3, 6, and 9 and in the case of r0 = 20M in Figs. 22, 23, and 24. Since the spectra with the size range of 1.57 < R < 6.18 have ? ? several peaks in r0 = 20M case, it may be possible to determine the radius of the disk. From Table II, we have R?ω ≡ C ? 0.9 for r0 = 20M case. Since C ? 1 for r0 = 10M , the value of C slightly depends on r0 . We can estimate the radius of the disk only using the separation of the peaks appeared in the spectrum. We will explain this point in Sec. IV. For r0 = 20M case (Fig. 24), there is one remarkable di?erence in the form factor from r0 = 10M case (Fig. 9); the form factor almost has ω = 0 re?ection symmetry for l = m = 2 mode with the existence of the orbital angular momentum. This means that the system is almost axisymmetry. For r0 = 20M case, the late stage of the shape turns out to be quite axisymmetry. In fact, at the end of the evolution, say r ? 2M , the location of the particles approaches to axisymmetric state (Fig. 21). This causes less gravitational wave emission and weakens the amplitude at the late time.

Finally, we show energy spectra and waveforms of GW from spherical dust stars in the case of type 2 (a/M = 0.9, R = 1.56M ) and type 3 (a/M = 0.9, R = 5.88M ). We cannot express the radial wave functions as the product of a test particle part and of a form factor part in the (particle) spherical dust star, because Almω (θ), t(r, θ), φ(r, θ) depends on both r and θ. However, the main property of the spectrum from the spherical dust star is the same as that from the disk. When the radius of a spherical dust star is comparable to the characteristic length L of the QNM frequency, a phase cancellation e?ect on GW appears in the spectrum [Fig. 25 (a)]. Therefore, the spectrum has a di?erent behavior from a test particle, which is the same to the disk case as we mentioned in Subsec. III A. When the radius of the spherical dust star is much larger than L, several peaks appear in the spectrum (Fig. 25 (b)). The behavior of the spectrum for a spherical star is similar to that for a disk case. We also show the dependence on r0 in Fig. 25 (b), (c). These results have the same behavior to the disk case; the spectrum has smaller amplitude for r0 = 20M than for r0 = 10M . Since all of the characters appeared in the spectrum are very similar to the disk case, it is also possible for a spherical star to determine the radius directly. We summarize the estimation of the radius in Table III and IV using the same method as the disk case. The method still works quite well, but the relative error rate becomes large when we compare the result with disk case. The reason for the large error may account for θ dependence of the orbit. Since all of the component particles of the disk have the same E and Lz for the disk case, the di?erence of the orbit comes only from t and φ. While, we have additional θ dependence on the orbit for the dust star case, we cannot estimate a simple relationship between R and ?ω as we will discuss in Sec. IV. Therefore, the relative error rate for the dust star case becomes larger than that for the dust disk case. In spite of the above reason, R?ω is still constant within 10% error. We also show the waveform of GW from the spherical dust star in Fig. 26. The waveform is also similar and has the same property to the disk case.

IV. DISCUSSION

Using a BH perturbation approach, we discuss the size e?ect on GW in the spectra from a dust disk and a spherical dust star spiraling into a Kerr BH for a wide ? range of parameters (Lz : spiraling case, 0 ≤ a/M ≤ 0.9, r0 = 10M, 20M ). First, we ?nd that when the radius of a disk or a spherical star is larger than the characteristic length of the QNM frequency, the phase cancellation e?ect from the waves generated from each di?erent location of the particle appears in the spectrum. Therefore, it is meaningful 6

to compare the diameter of the disk or the spherical star with the characteristic length of GW, which is described by the QNM frequency for the plunging orbit case. We de?ne the characteristic length L that the time delay between the earliest particle at r0 ? L and the latest one at r0 +L is equivalent to the time period of the characteristic frequency, t(r0 + L) ? t(r0 ? L) = 2π . ωQNM (4.1)

ωn ?

(n + 1)π , ′ Tr=r0 R

n = 0, 1, 2, . . . ,

(4.4)

which agrees quite well with the numerical results. Equation (4.4) suggests that the separation of peaks in the spectrum ?ω may be in proportion to R?1 . In Table I and II, we show ?ω for various R/M and we ?nd the following relation R=C 1 for R ? L. ?ω (4.5)

Since the shape of the disk changes from the circle to a long and narrow shape (Figs. 19, 20, and 21) and then swallows into a BH, the de?nition of L is indeed represents the typical length scale of GW. For the disk case, the time lag between the earliest and the latest component particle plunging into a Kerr BH is ?t = t(r0 + R) ? t(r0 ? R). We ?nd that when R is larger than L, the energy spectrum of GW has a di?erent feature from that for a test particle. This character comes from the phase cancellation e?ect on GW from the earliest and the latest particle of the disk plunging into a Kerr BH. We summarize our result in Table V. For the spherical star case, it is rather di?cult to argue with such a simple interpretation because the orbit has another dependence θ. In spite of the complicated structure of the geodesic for a spherical dust star, C is almost constant in Tables III and IV, and we can interpret them in a similar manner to the disk case. We should also argue the bound orbit case whether this phase cancellation e?ect works in the inspiral phase of a coalescing binary system. Although the characteristic frequency is di?erent from the QNM frequency, the basic idea (phase cancellation e?ects on GW) would also be the same in that case. Therefore, the rule found in this paper can also apply to the binary stars in the inspiral phase and it will be con?rmed in the future. Next, we propose a possibility to determine the radius (disk) of a star. The energy spectrum (dE/dω)lmω of GW from a disk is expressed as dE dω
(disk) lmω

The value C does not depend on the radius and take a similar value of ? 1, but depends on the orbits and the initial data, where we set the location of the circular disk. In fact, C depends on them within ? 20%. In the physical unit, ?ν ≡ ?ω/(2π) is given by ?ν = 5kHz R 10km
?1

= 8Hz

R 7000km

?1

,

(4.6)

∝| fmω |2

dE dω

(particle)

,
lmω

(4.2)

where (dE/dω)lmω is the spectrum from a single test particle. The spectrum from a test particle has only one peak at the frequency ωQNM , so that the square of the form factor | fmω |2 is responsible for this behavior. The existence of several peaks in Figs. 7 (c), 8 (c), 9 (c), 16 (c), 17 (c), 18 (c), 24 (b), (c) can be understood by the approximate estimation of fmω as fmω ∝
′ sin(ωTr=r0 R) , ω

(particle)

(4.3)

where T ′ = dT (r)/dr, assuming that only eiωt term depends on t in Eq. (2.15). The frequency where fmω takes zero is 7

assuming that C = 1. Therefore, for NSs and white dwarfs, the frequency band is within the detectable frequency range of some laser interferometers and resonant type detectors Finally, we argue from the observational point of view whether we have a real situation to determine the radius of a tidally disrupted star from GW in our model itself. In the ?nal phase of a coalescing binary system, the e?ect of the tidal force and the deformation of the star should be taken into account. Carter and Luminet [23] demonstrated the deformation of the star using an a?ne star model with Newtonian particle dynamics (bound orbit). They separate the deformation stage into ?ve phase, according to the position of the star, and point out that when the star moves inside the Roche radius with the condition of β ≡ (RRoche /Rp ) > 3, where RRoche is ? Roche radius and Rp is a periastron radius of the orbit, the tidal force term rapidly grows and the pressure and self gravity terms can be neglected. This leads us to the conclusion that we can neglect the e?ect of the pressure and the self-gravity when β > 3. Laguna et al. [24] ? extended the analysis of Carter and Luminet [23] using smooth particle hydrodynamics in Schwarzschild spacetime. They concluded that when β > 10, we can neglect ? the e?ect of the self-gravity and the pressure. From these previous calculations, we may naturally set the model that a star is tidally disrupted by a BH in a certain distance and then plunge into a BH. In this case, we may neglect the pressure force and the self-gravity e?ect in the ?rst approximation so that each of the component ?uid element follows the geodesic. To apply our results to the realistic case, three conditions should be satis?ed. (1) The star is tidally disrupted at rdisrupt > 6M , which is expressed as ? M > 3? ρstar = , 3 rdisrupt ? 4πR3 (4.7)

where ρstar is the density of the star. (2) In order that the phase cancellation is e?ective in the spectrum, R should be larger than L. This condition is described as t(r0 + R) ? t(r0 ? R) > T = 2π ωQNM . (4.8)
[1] K. S. Thorne, in Black Holes and Relativistic Stars, edited by R. M. Wald (University of Chicago Press, Chicago, 1998), p. 41. [2] M. Vallisneri, Phys. Rev. Lett. 84, 3519 (2000). [3] M. Shibata, Prog. Theor. Phys. 96, 917 (1996). [4] M. Sasaki and T. Nakamura, Phys. Lett. 89A, 68 (1982). [5] M. Sasaki and T. Nakamura, Prog. Theor. Phys. 67, 1788 (1982). [6] S. A. Teukolsky, Astrophys. J. 185, 635 (1973). [7] T. Nakamura and M. Sasaki, Phys. Lett. 106B 69 (1981). [8] M. P. Haugan, S. L. Shapiro and I. Wasserman, Astrophys. J. 257, 283 (1982); S. L. Shapiro and I. Wasserman, Astrophys. J. 260, 838 (1982); L. I. Petrich, S. L. Shapiro, and I. Wasserman, Astrophys. J. Suppl. 58, 297 (1985). [9] M. Saijo, H. Shinkai, and K. Maeda, Phys. Rev. D 56, 785 (1997). [10] L. Smarr, in Sources of Gravitational Radiation, edited by L. Smarr (Cambridge Univ. Press, Cambridge, 1979), p. 245. [11] P. Anninos, R. H. Price, J. Pullin, E. Seidel, and W.-M. Suen, Phys. Rev. D 52, 4462 (1995). [12] M. Davis, R. Ru?ni, W. H. Press, and R. H. Price, Phys. Rev. Lett. 27, 1466 (1971). [13] R. F. Stark and T. Piran, Phys. Rev. Lett. 55, 891 (1985); R. F. Stark, in Frontiers in Numerical Relativity, edited by C. R. Evans, L. S. Finn, and D. W. Hobill (Cambridge Univ. Press, Cambridge, 1989), p. 281. [14] T. Nakamura, K. Oohara, and Y. Kojima, Prog. Theor. Phys. Suppl. 90, 171 (1987). [15] M. Saijo and T. Nakamura, Phys. Rev. Lett. 85, 2665 (2000) (Paper I). [16] If we only want to consider the merging phase of a coalescing binary BH - star system, it is very useful to evolve the Teukolsky equation with setting a suitable initial data (Close limit approximation [17]). In this case, we have to compute a set of partially di?erential equations. This may guide the result with less accuracy. We also know the result in the case of a test particle that most of gravitational waves are radiated in the merging phase. In our approach, we only need the analysis of Fourier transformation method with calculating the set of ordinary di?erential equations. [17] R. H. Price and J. Pullin, Phys. Rev. Lett. 72, 3297 (1994); J. Pullin, Prog. Theor. Phys. Suppl. 136 (1999) 107. [18] B. Carter, Phys. Rev. 174, 1559 (1968). [19] S. Detweiler, Astrophys. J. 239, 292, (1980). [20] E. W. Leaver, Proc. Roy. Soc. London Ser. A 402, 285 (1985). [21] Y. Kojima and T. Nakamura, Phys. Lett. 96A, 335 (1983). [22] Y. Kojima and T. Nakamura, Prog. Theor. Phys. 71, 79 (1984). [23] B. Carter and L. -P. Luminet, Astron. Astrophys. 121,

(3) The star does not contact to the BH. This condition can be written as R < rdisrupt ? rhorizon . (4.9)

We apply the NS of mass ? = 1.4M⊙ with R = 10km and the white dwarf of mass ? = 0.5M⊙ with R = 7000km to our model. Then, two conditions [Eqs. (4.7) and (4.8)] are rewritten as M< ? 4πR3 3? rdisrupt M
?3/2

≡ M1 ,

(4.10) (4.11)

1 dt M< ? π dr

r=r0

R(M ωQNM ) ≡ M2 ,

where we use the ?rst order approximation of Taylor expansion to expand the left hand side of Eq. (4.8). And the other condition [Eq. (4.9)] gives the lower limit of the BH mass as M> ?
rdisrupt M

R ?

rhorizon M

≡ M3 .

(4.12)

Taking rdisrupt = 6M we tabulate these three conditions [Eqs. (4.10), (4.11) and (4.12)] in Tables VI and VII. From Tables VI and VII, we indeed have a possibility to determine the radius of the NS for M ? 2M⊙ and the radius of the white dwarf for M ? 1000M⊙. In the real situation, we should also take into account of the pressure and self-gravity e?ect on GW so that it is urgent to con?rm our proposal by full 3D numerical simulations that we can indeed extract the form factor from GW and determine the value of C. In any case, it is quite possible that the spectrum of GW may give us important information in gravitational wave astronomy as in optical astronomy.
ACKNOWLEDGMENTS

M. S. thanks Masaru Shibata for discussion. He also thanks the visitor system of Yukawa Institute for Theoretical Physics and acknowledges Stuart Shapiro at University of Illinois at Urbana-Champaign for his hospitality. The Numerical Computations are mainly performed by NEC-SX vector computer at Yukawa Institute for Theoretical Physics, Kyoto University and FUJITSUVX vector computer at Media Network Center, Waseda University. This work was supported in part by a JSPS Grant-in-Aid (No. 095689) and by Grant-in-Aid of Scienti?c Research of the Ministry of Education, Culture, and Sports, No.11640274 and 09NP0801. 8

97 (1983). [24] P. Laguna, W. A. Miller, W. H. Zurek, and M. B. Davis, Astrophys. J. Lett. 410, L83 (1993).

FIG. 1. Energy spectrum of gravitational waves from a disk moving on an equatorial plane in Kerr spacetime whose ? radius is set up at r = 10M in the case of a = 0, Lz /M = 2 [(a) R = 0 (test particle), (b) R/M = 0.785, (c) R/M = 2.33, (d) R/M = 5.88]. We only show l = 2 mode. Solid, dashed, dash-dotted, dotted, dash-three dotted line denotes the case of m = ?2, ?1, 0, 1, 2, respectively.

FIG. 2. Energy spectrum of gravitational waves from a disk moving on an equatorial plane in Kerr spacetime whose radius is set up at r = 10M in the case of a/M = 0.5, ? Lz /M = 2 [(a) R = 0 (test particle), (b) R/M = 0.785, (c) R/M = 2.33, (d) R/M = 5.88]. We only show l = 2 mode. Solid, dashed, dash-dotted, dotted, dash-three dotted line denotes the case of m = ?2, ?1, 0, 1, 2, respectively.

FIG. 3. Energy spectrum of gravitational waves from a disk moving on an equatorial plane in Kerr spacetime whose radius is set up at r = 10M in the case of a/M = 0.9, ? Lz /M = 2 [(a) R = 0 (test particle), (b) R/M = 0.785, (c) R/M = 1.56, (d) R/M = 5.88]. We only show l = 2 mode. Solid, dashed, dash-dotted, dotted, dash-three dotted line denotes the case of m = ?2, ?1, 0, 1, 2, respectively.

FIG. 4. Waveform of gravitational waves from a disk moving on an equatorial plane in Kerr spacetime whose radius is ? set up at r = 10M in the case of a = 0, Lz /M = 2 [(a) R = 0 (test particle), (b) R/M = 0.785, (c) R/M = 2.33, (d) R/M = 5.88]. We only show l = m = 2 mode, setting the observer at the in?nity of θ = π/2, φ = 0.

FIG. 5. Waveform of gravitational waves from a disk moving on an equatorial plane in Kerr spacetime whose radius is ? set up at r = 10M in the case of a/M = 0.5, Lz /M = 2 [(a) R = 0 (test particle), (b) R/M = 0.785, (c) R/M = 2.33, (d) R/M = 5.88]. We only show l = m = 2 mode, setting the observer at the in?nity of θ = π/2, φ = 0.

FIG. 6. Waveform of gravitational waves from a disk moving on an equatorial plane in Kerr spacetime whose radius is ? set up at r = 10M in the case of a/M = 0.9, Lz /M = 2 [(a) R = 0 (test particle), (b) R/M = 0.785, (c) R/M = 1.56, (d) R/M = 5.88]. We only show l = m = 2 mode, setting the observer at the in?nity of θ = π/2, φ = 0.

FIG. 7. Form factor of the dust disk moving on the equatorial plane of a Kerr spacetime whose radius is set up at ? r = 10M in the case of a = 0, Lz /M = 2 [(a) R/M = 0.785, (b) R/M = 2.33, (c) R/M = 5.88]. We only show the case for l = m = 2.

9

FIG. 8. Form factor of the dust disk moving on the equatorial plane of a Kerr spacetime whose radius is set up ? at r = 10M in the case of a/M = 0.5, Lz /M = 2 [(a) R/M = 0.785, (b) R/M = 2.33, (c) R/M = 5.88]. We only show the case for l = m = 2. FIG. 9. Form factor of the dust disk moving on the equatorial plane of a Kerr spacetime whose radius is set up ? at r = 10M in the case of a/M = 0.9, Lz /M = 2 [(a) R/M = 0.785, (b) R/M = 1.56, (c) R/M = 5.88]. We only show the case for l = m = 2. FIG. 10. Energy spectrum of gravitational waves from a disk moving on an equatorial plane in Kerr spacetime whose radius is set up at r = 10M in the case of a/M = 0.9, ? Lz /M = 2.63 [(a) R = 0 (test particle), (b) R/M = 0.785, (c) R/M = 1.56, (d) R/M = 5.88]. We only show l = 2 mode. Solid, dashed, dash-dotted, dotted, dash-three dotted line denotes the case of m = ?2, ?1, 0, 1, 2, respectively.

FIG. 15. Waveform of gravitational waves from a disk moving on an equatorial plane in Kerr spacetime whose radius ? is set up at r = 10M in the case of a/M = 0.9, Lz /M = ?3 [(a) R = 0 (test particle), (b) R/M = 0.785, (c) R/M = 1.56, (d) R/M = 5.88]. We only show l = m = 2 mode, setting the observer at the in?nity of θ = π/2, φ = 0.

FIG. 16. Form factor of the dust disk moving on the equatorial plane of a Kerr spacetime whose radius is set up ? at r = 10M in the case of a/M = 0.9, Lz /M = 2.63 [(a) R/M = 0.785, (b) R/M = 1.56, (c) R/M = 5.88]. We only show the case for l = m = 2. FIG. 17. Form factor of the dust disk moving on the equatorial plane of a Kerr spacetime whose radius is set up at ? r = 10M in the case of a/M = 0.9, Lz = 0 [(a) R/M = 0.785, (b) R/M = 1.56, (c) R/M = 5.88]. We only show the case for l = m = 2. FIG. 18. Form factor of the dust disk moving on the equatorial plane of a Kerr spacetime whose radius is set up ? at r = 10M in the case of a/M = 0.9, Lz /M = ?3 [(a) R/M = 0.785, (b) R/M = 1.56, (c) R/M = 5.88]. We only show the case for l = m = 2. FIG. 19. Deformation of the shape of the spherical disk whose radius is set up at r = 10M in the case of a/M = 0.9, ? Lz /M = 2 [(a) R/M = 1.56, (b) R/M = 5.88]. Solid line shows the geodesic for the center of gravity of the disk, while circle, square, diamond, and triangle show the edge of the disk where the location of the center is at r = 10M , 6M , 4M , 2M , respectively.

FIG. 11. Energy spectrum of gravitational waves from a disk moving on an equatorial plane in Kerr spacetime whose ? radius is set up at r = 10M in the case of a/M = 0.9, Lz = 0 [(a) R = 0 (test particle), (b) R/M = 0.785, (c) R/M = 1.56, (d) R/M = 5.88]. We only show l = 2 mode. Solid, dashed, dash-dotted, dotted, dash-three dotted line denotes the case of m = ?2, ?1, 0, 1, 2, respectively.

FIG. 12. Energy spectrum of gravitational waves from a disk moving on an equatorial plane in Kerr spacetime whose radius is set up at r = 10M in the case of a/M = 0.9, ? Lz /M = ?3 [(a) R = 0 (test particle), (b) R/M = 0.785, (c) R/M = 1.56, (d) R/M = 5.88]. We only show l = 2 mode. Solid, dashed, dash-dotted, dotted, dash-three dotted line denotes the case of m = ?2, ?1, 0, 1, 2, respectively.

FIG. 13. Waveform of gravitational waves from a disk moving on an equatorial plane in Kerr spacetime whose radius ? is set up at r = 10M in the case of a/M = 0.9, Lz /M = 2.63 [(a) R = 0 (test particle), (b) R/M = 0.785, (c) R/M = 1.56, (d) R/M = 5.88]. We only show l = m = 2 mode, setting the observer at the in?nity of θ = π/2, φ = 0.

FIG. 20. Deformation of the shape of the spherical disk whose radius is set up at r = 10M in the case of a/M = 0.9, ? Lz /M = 2.63 [(a) R/M = 1.56, (b) R/M = 5.88]. Solid line shows the geodesic for the center of gravity of the disk, while circle, square, diamond, and triangle show the edge of the disk where the location of the center is at r = 10M , 6M , 4M , 2M , respectively.

FIG. 14. Waveform of gravitational waves from a disk moving on an equatorial plane in Kerr spacetime whose radius ? is set up at r = 10M in the case of a/M = 0.9, Lz = 0 [(a) R = 0 (test particle), (b) R/M = 0.785, (c) R/M = 1.56, (d) R/M = 5.88]. We only show l = m = 2 mode, setting the observer at the in?nity of θ = π/2, φ = 0.

FIG. 21. Deformation of the shape of the spherical disk whose radius is set up at r = 20M in the case of a/M = 0.9, ? Lz /M = 2 [(a) R/M = 3.13, (b) R/M = 6.17]. Solid line shows the geodesic for the center of gravity of the disk, while circle, square, diamond, and triangle show the edge of the disk where the location of the center is at r = 20M , 10M , 4M , 2M , respectively.

10

FIG. 22. Energy spectrum of gravitational waves from a disk moving on an equatorial plane in Kerr spacetime whose radius is set up at r = 20M in the case of a/M = 0.9, ? Lz /M = 2 [(a) R/M = 1.57, (b) R/M = 6.27]. We only show l = 2 mode. Solid, dashed, dash-dotted, dotted, dash-three dotted line denotes the case of m = ?2, ?1, 0, 1, 2, respectively.

TABLE I. Comparison with the characteristic length which is appeared in energy spectrum of gravitational waves to the radius of the disk in the case of r0 = 10M . We only focus on l = m = 2 mode. a/M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 ? L/M 0 0 0 0 0 2 2 2 2 2 ?3 ?3 ?3 ?3 ?3 0 0 0 0 0 2 2 2 2 2 ?3 ?3 ?3 ?3 ?3 0 0 0 0 0 2 2 2 2 2 ?3 ?3 ?3 ?3 ?3 R/M 3.09 3.82 4.54 5.22 5.88 3.09 3.83 4.54 5.22 5.88 3.09 3.83 4.54 5.22 5.88 3.09 3.83 4.54 5.22 5.88 3.09 3.83 4.54 5.22 5.88 3.09 3.83 4.54 5.22 5.88 3.09 3.83 4.54 5.22 5.88 3.09 3.83 4.54 5.22 5.88 3.09 3.83 4.54 5.22 5.88 M ?ω 0.370 0.300 0.255 0.218 0.193 0.340 0.275 0.235 0.203 0.177 0.295 0.243 0.203 0.173 0.148 0.340 0.275 0.235 0.203 0.177 0.335 0.270 0.233 0.198 0.177 0.300 0.248 0.210 0.177 0.160 0.370 0.300 0.255 0.220 0.197 0.335 0.270 0.233 0.200 0.177 0.310 0.250 0.213 0.183 0.163 1/M ?ω 2.70 3.33 3.92 4.59 5.18 2.94 3.64 4.26 4.93 5.65 3.39 4.12 4.93 5.78 6.76 2.94 3.64 4.26 4.93 5.65 2.99 3.70 4.29 5.05 5.65 3.33 4.03 4.76 5.65 6.25 2.70 3.33 3.92 4.55 5.08 2.99 3.70 4.29 5.00 5.65 3.23 4.00 4.69 5.46 6.13 R?ω 1.14 1.15 1.16 1.14 1.13 1.05 1.05 1.07 1.06 1.04 0.911 0.929 0.921 0.904 0.870 1.05 1.05 1.07 1.06 1.04 1.03 1.03 1.06 1.03 1.04 0.927 0.949 0.953 0.924 0.940 1.14 1.15 1.16 1.15 1.16 1.03 1.03 1.06 1.04 1.04 0.957 0.956 0.967 0.956 0.958

FIG. 23. Waveform of gravitational waves from a disk moving on an equatorial plane in Kerr spacetime whose radius ? is set up at r = 20M in the case of a/M = 0.9, Lz /M = 2 [(a) R/M = 1.57, (b) R/M = 6.27]. We only show l = m = 2 mode, setting the observer at the in?nity of θ = π/2, φ = 0.

FIG. 24. Form factor equatorial plane of a Kerr at r = 20M in the case R/M = 1.57, (b) R/M = case.

of the dust disk moving on the spacetime whose radius is set up ? of a/M = 0.9, Lz /M = 2 [(a) 6.27]. We only show l = m = 2

FIG. 25. Spectrum of gravitational waves from a star in Kerr spacetime in the case of a/M = 0.9 [(a) r0 = 10M , R/M = 1.56, (b) r0 = 10M , R/M = 5.88, (c) r0 = 20M , R/M = 6.27]. We only show l = 2 mode. Solid, dashed, dash-dotted, dotted, dash-three dotted line denotes the case of m = ?2, ?1, 0, 1, 2, respectively.

FIG. 26. Waveform of gravitational waves from a star in Kerr spacetime in the case of a/M = 0.9 [(a) r0 = 10M , R/M = 1.56, (b) r0 = 10M , R/M = 5.88, (c) r0 = 20M , R/M = 6.27]. We only show l = m = 2 mode, setting the observer at the in?nity of θ = π/2, φ = 0.

11

TABLE II. Comparison with the characteristic length which is appeared in energy spectrum of gravitational waves to the radius of the disk in the case of r0 = 20M . We only focus on l = m = 2 mode. a/M 0 0 0 0 0 0 0 0 0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 ? L/M 0 0 0 2 2 2 ?3 ?3 ?3 0 0 0 2 2 2 ?3 ?3 ?3 0 0 0 2 2 2 ?3 ?3 ?3 R/M 3.13 4.67 6.18 3.13 4.67 6.18 3.13 4.67 6.18 3.13 4.67 6.18 3.14 4.67 6.18 3.14 4.71 6.27 3.13 4.67 6.17 3.13 4.67 6.17 3.13 4.67 6.17 M ?ω 0.300 0.192 0.154 0.290 0.182 0.146 0.270 0.172 0.130 0.300 0.188 0.141 0.290 0.183 0.138 0.275 0.172 0.127 0.300 0.195 0.141 0.285 0.185 0.133 0.275 0.173 0.150 1/M ?ω 3.33 5.21 6.49 3.45 5.49 6.85 3.70 5.81 7.69 3.33 5.32 7.09 3.45 5.46 7.25 3.64 5.81 7.87 3.33 5.13 7.09 3.51 5.41 7.52 3.64 5.78 6.67 R?ω 0.938 0.896 0.951 0.907 0.849 0.902 0.844 0.803 0.803 0.938 0.877 0.871 0.907 0.854 0.852 0.860 0.803 0.785 0.938 0.910 0.871 0.891 0.863 0.822 0.860 0.807 0.927

TABLE IV. Comparison with the characteristic length which is appeared in energy spectrum of gravitational waves to the radius of the star in the case of r0 = 20M . We only focus on l = m = 2 mode. a/M 0 0.5 0.5 0.9 0.9 R/M 6.18 4.67 6.18 4.67 6.18 M ?ω 0.260 0.285 0.250 0.260 0.245 1/M ?ω 3.85 3.51 4.00 3.85 4.08 R?ω 1.61 1.33 1.54 1.21 1.51

TABLE V. Characteristic length of gravitational waves for a plunging orbit. We de?ne the characteristic length L with satisfying Eq. (4.1). We solve geodesic equation numerically in order to ?nd the characteristic length. We only use l = m = 2 mode of the QNM frequency. a/M 0 0 0 0.5 0.5 0.5 0.9 0.9 0.9 ? Lz /M 0 2 ?3 0 2 ?3 0 2 ?3 Lr0 =10M 2.78 3.03 2.42 2.22 2.44 1.99 1.54 1.71 1.42 Lr0 =20M 2.31 2.42 2.16 1.85 1.94 1.74 1.29 1.36 1.22

TABLE III. Comparison with the characteristic length which is appeared in energy spectrum of gravitational waves to the radius of the star in the case of r0 = 10M . We only focus on l = m = 2 mode. a/M 0 0 0.5 0.5 0.9 0.9 R/M 5.22 5.88 5.22 5.88 5.22 5.88 M ?ω 0.175 0.145 0.175 0.145 0.170 0.150 1/M ?ω 5.71 6.90 5.71 6.90 5.88 6.67 R?ω 0.914 0.852 0.914 0.852 0.888 0.881

TABLE VI. Allowed mass of the BH (Eqs. (4.10) and (4.11)) for a NS (? = 1.4M⊙ , R = 10km) to determine the radius of the star. M1 and M2 denote the upper limit of the mass from Eqs. (4.10) and (4.11) and M3 denote the lower limit of the mass from Eq. (4.12), respectively. a/M 0 0 0 0.5 0.5 0.5 0.9 0.9 0.9 ? Lz /M 0 2 ?3 0 2 ?3 0 2 ?3 M1 /M⊙ 2.07 2.07 2.07 2.07 2.07 2.07 2.07 2.07 2.07 M2 /M⊙ 2.07 2.06 2.05 2.55 2.75 2.28 3.56 4.12 3.00 M3 /M⊙ 1.69 1.69 1.69 1.64 1.64 1.64 1.48 1.48 1.48

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TABLE VII. Allowed mass of BH (Eqs. (4.10) and (4.11)) for a white dwarf (? = 0.5M⊙ , R = 7, 000km) to determine the radius of the star. M1 and M2 denote the upper limit of the mass from Eqs. (4.10) and (4.11) and M3 denote the lower limit of the mass from Eq. (4.12), respectively. a/M 0 0 0 0.5 0.5 0.5 0.9 0.9 0.9 ? Lz /M 0 2 ?3 0 2 ?3 0 2 ?3 M1 /M⊙ 64300 64300 64300 64300 64300 64300 64300 64300 64300 M2 /M⊙ 1450 1440 1440 1780 1920 1600 2490 2880 2100 M3 /M⊙ 1180 1180 1180 1150 1150 1150 1040 1040 1040

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