TIDAL LOVE NUMBERS OF NEUTRON STARS

Transcription

1 The Astrophysical Journal, 677:116Y10, 008 April 0 # 008. The American Astronomical Society. All rights reserved. Printed in U.S.A. TIDAL LOVE NUMBERS OF NEUTRON STARS Tanja Hinderer Center for Radiophysics and Space Research, Cornell University, Ithaca, NY 14853; tph5@cornell.edu Received 007 November 15; accepted 008 January ABSTRACT For a variety of fully relativistic polytropic neutron star models we calculate the star s tidal Love number k.most realistic equations of state for neutron stars can be approximated as a polytrope with an effective index n 0:5Y1:0. The equilibrium stellar model is obtained by numerical integration of the Tolman-Oppenheimer-Volkhov equations. We calculate the linear l ¼ static perturbations to the Schwarzschild spacetime following the method of Thorne and Campolattaro. Combining the perturbed Einstein equations into a single second-order differential equation for the perturbation to the metric coefficient g tt and matching the exterior solution to the asymptotic expansion of the metric in the star s local asymptotic rest frame gives the Love number. Our results agree well with the Newtonian results in the weak field limit. The fully relativistic values differ from the Newtonian values by up to 4%. The Love number is potentially measurable in gravitational wave signals from inspiralling binary neutron stars. Subject headinggs: equation of state gravitation relativity stars: neutron 1. INTRODUCTION AND MOTIVATION 116 A key challenge of current astrophysical research is to obtain information about the equation of state (EOS) of the ultradense nuclear matter making up neutron stars (NSs). The observational constraints on the internal structure of NSs are weak; the observed range of NS masses is M 1:1Y: M (Lattimer & Prakash 007), and there is no current method to directly measure the radius. Some estimates using data from X-ray spectroscopy exist, but those are highly model-dependent (e.g., Webb & Barret 007). Different theoretical models for the NS internal structure predict, for a NS of mass M 1:4 M, a central density in the range of c Y8 ; gcm 3,andaradiusinthe range of R 7Y16 km (Lattimer & Prakash 007). Potential observations of pulsars rotating at frequencies above 1400 Hz could be used to constrain the EOS if the pulsar s mass could also be measured (e.g., Zdunik et al. 008). Direct and model-independent constraints on the EOS of NSs could be obtained from gravitational wave observations. Coalescing binary NSs are one of the most important sources for ground-based gravitational wave detectors (Cutler et al. 1993; Cutler & Thorne 00). LIGO observations have established upper limits on the coalescence rate per comoving volume (Abbott et al. 008), and at the designed sensitivity, LIGO II is expected to detect inspirals at a rate of day 1 (Kalogera et al. 004). In the early, low-frequency part of the inspiral ( f 100 Hz, where f is the gravitational wave frequency), the waveform s phase evolution is dominated by the point-mass dynamics, and finite-size effects are only a small correction. Toward the end of the inspiral, the internal degrees of freedom of the bodies start to appreciably influence the signal, and there have been many investigations of how well the EOS can be constrained using the last several orbits and merger, including constraints from the gravitational wave energy spectrum ( Faber et al. 00) and from the NS tidal disruption signal for NSYblack hole binaries ( Vallisneri 00). Several numerical simulations of the hydrodynamics of NS-NS mergers have studied the dependence of the gravitational wave spectrum on the radius and EOS (see, e.g., Baumgarte & Shapiro 003 and references therein). However, trying to extract EOS information from this late-time regime presents several difficulties: (1) the highly complex behavior requires solving the full nonlinear equations of general relativity together with relativistic hydrodynamics; () the signal depends on unknown quantities such as the spins and angular momentum distribution inside the stars; and (3) the signals from the hydrodynamic merger are outside of LIGO s most sensitive band. During the early regime of the inspiral, the signal is very clean and the influence of tidal effects is only a small correction to the waveform s phase. However, signal detection is based on matched filtering, i.e., integrating the measured waveform against theoretical templates, where the requirement on the templates is that the phasing remain accurate to 1 cycle over the inspiral. If the accumulated phase shift due to the tidal corrections becomes of order unity or larger, it could corrupt the detection of NS-NS signals, or alternatively, detecting a phase perturbation could give information about the NS structure. This has motivated several analytical and numerical investigations of tidal effects in NS binaries (Bildsten & Cutler 199; Kokkotas & Schafer 1995; Kochanek 199; Taniguchi & Shibata 1998; Mora & Will 004; Shibata 1994; Gualteri et al. 001; Pons et al. 00; Berti et al. 00). The influence of the internal structure on the gravitational wave phase in this early regime of the inspiral is characterized by a single parameter, namely, the ratio k of the induced quadrupole to the perturbing tidal field. This ratio k is related to the star s tidal Love number k by k ¼ 3GkR 5 /, where R is the star s radius. Flanagan & Hinderer (008) have shown that for an inspiral of two nonspinning 1:4 M NSs at a distance of 50 Mpc, LIGO II detectors will be able to constrain k to k :01 ; gcm s with 90% confidence. This number is an upper limit on k in the case that no tidal phase shift is observed. The corresponding constraint on radius would be R 13:6 km(15:3 km)foran ¼ 0:5 (n ¼ 1:0) fully relativistic polytrope, for 1:4 M NSs (Flanagan & Hinderer 008). Because NSs are compact objects with strong internal gravity, their Love numbers could be very different from those for Newtonian stars that have been computed previously, e.g., by Brooker & Olle (1955). Knowledge of Love number values could also be useful for comparing different numerical simulations of NS binary inspiral by focusing on models with the same masses and values of k.

2 TIDAL LOVE NUMBERS OF NEUTRON STARS 117 In Flanagan & Hinderer (008), the l ¼ tidal Love numbers for fully relativistic NS models with polytropic pressuredensity relation P ¼ K 1þ1/n,whereKand n are constants, were computed for the first time. The present paper will give details of this computation. Using polytropes allows us to explore a wide range of stellar models, since most realistic models can be reasonably approximated as a polytrope with an effective indexintherangen 0:5Y1:0 (Lattimer & Prakash 007). Our prescription for computing k is valid for an arbitrary pressuredensity relation and not restricted to polytropes. In x, we start by defining k in the fully relativistic context in terms of coefficients in an asymptotic expansion of the metric in the star s local asymptotic rest frame and discuss the extent to which it is uniquely defined. In x 3, we discuss our method of calculating k, which is based on static linearized perturbations of the equilibrium configuration in the Regge-Wheeler gauge as in Thorne & Campolattaro (1967). Section 4 contains the results of the numerical computations together with a discussion. Unless otherwise specified, we use units in which c ¼ G ¼ 1.. DEFINITION OF THE LOVE NUMBER Consider a static, spherically symmetric star of mass M placed in a static external quadrupolar tidal field E ij. The star will develop in response a quadrupole moment Q ij. 1 In the star s local asymptotic rest frame (asymptotically mass-centered Cartesian coordinates) at large r the metric coefficient g tt is given by (Thorne 1998) ð1 g tt Þ ¼ M r 3Q ij r 3 n i n j 1 3 ij þ O 1 r 3 þ 1 E ijx i x j þ Or 3 ; ð1þ where n i ¼ x i /r; this expansion defines E ij and Q ij.inthe Newtonian limit, Q ij is related to the density perturbation by Z Q ij ¼ d 3 x (x) x i x j 1 3 r ij ; ðþ and E ij is given in terms of the external gravitational potential ext as E ij j : We are interested in applications to fully relativistic stars, which requires going beyond Newtonian physics. In the strong field case, equations () and (3) are no longer valid, but the expansion of the metric from equation (1) still holds in the asymptotically flat region and serves to define the moments Q ij and E ij. We briefly review here the extent to which these moments are uniquely defined, since there are considerable coordinate ambiguities in performing asymptotic expansions of the metric. For an isolated body in a static situation these moments are uniquely defined: E ij and Q ij are the coordinate-independent moments defined by Geroch (1970) and Hansen (1974) for stationary, asymptotically flat spacetimes in terms of certain combinations of the derivatives of the norm and twist of the timelike Killing 1 The induced quadrupolar deformation of the star can be described in terms of the star s l ¼ mode eigenfunctions of oscillation. The l ¼ tidal moment can be related to a component of the Riemann tensor R of the external pieces of the metric in Fermi normal coordinates at r ¼ 0asE ij ¼ R 0i0j (see Misner et al. 1973). ð3þ vector at spatial infinity. In the case of an isolated object in a dynamical situation, there are ambiguities related to gravitational radiation, for example, angular momentum is not uniquely defined (Wald 1984). For the application to the adiabatic part of a NS binary inspiral, we are interested in the case of a nonisolated object in a quasi-static situation. In this case there are still ambiguities (related to the choice of coordinates), but their magnitudes can be estimated (Thorne & Hartle 1985) and are at a high post-newtonian order and therefore can be neglected. We are also interested in working (1) to linear order in E ij and () in the limit where the source of E ij is very far away. In this limit the ambiguities disappear. To linear order in E ij, the induced quadrupole will be of the form Q ij ¼ ke ij : ð4þ Here k is a constant which is related to the l ¼ tidal Love number (apsidal constant) k by (Flanagan & Hinderer 008) k ¼ 3 GkR 5 : Note the difference in terminology; in Flanagan & Hinderer (008) k was called the Love number, whereas in this paper, we reserve that name for the dimensionless quantity k. The tensor multipole moments Q ij and E ij can be decomposed as E ij ¼ X m¼ Q ij ¼ X m¼ ð5þ E m Y m ij ; ð6þ Q m Y m ij ; ð7þ where the symmetric traceless tensors Yij m are defined by (Thorne 1980) Y m (; ) ¼ Y m ij n i n j ð8þ with n ¼ (sin cos ; sin sin ; cos ). Thus, equation (4) can be written as Q m ¼ ke m : Without loss of generality, we can assume that only one E m is nonvanishing, this is sufficient to compute k. 3. CALCULATION OF THE LOVE NUMBER 3.1. Equilibrium Configuration The geometry of spacetime of a spherical, static star can be described by the line element (Misner et al. 1973) ds 0 ð9þ ¼ g(0) dx dx ¼ e (r) dt þ e k(r) dr þ r d þ sin d : ð10þ The star s stress-energy tensor is given by T ¼ ð þ pþu u þ pg (0) ; ð11þ where u ¼ e t is the fluid s four-velocity and and p are the density and pressure, respectively. Numerical integration of

3 118 HINDERER Vol. 677 the Tolman-Oppenheimer-Volkhov equations (see, e.g., Misner et al. 1973) for NS models with a polytropic pressure-density relation P ¼ K 1þ1=n ; ð1þ where K is a constant and n is the polytropic index, gives the equilibrium stellar model with radius R and total mass M ¼ m(r). 3.. Static Linearized Perturbations Due to an External Tidal Field We examine the behavior of the equilibrium configuration under linearized perturbations due to an external quadrupolar tidal field following the method of Thorne & Campolattaro (1967). The full metric of the spacetime is given by g ¼ g (0) þ h ; ð13þ where h is a linearized metric perturbation. We analyze the angular dependence of the components of h into spherical harmonics as in Regge & Wheeler (1957). We restrict our analysis to the l ¼, static, even-parity perturbations in the Regge-Wheeler gauge (Regge & Wheeler 1957). With these specializations, h can be written as (Regge & Wheeler 1957; Thorne & Campolattaro 1967) h ¼ diag e (r) H 0 (r); e k(r) H (r); r K(r); r sin K(r) Ym (; ): ð14þ The nonvanishing components of the perturbations of the stressenergy tensor (eq. [11]) are T0 0 ¼ ¼ (dp/d) 1 p and Ti i ¼ p. Weinsertthisandthemetricperturbation(eq.[14]) into the linearized Einstein equation G ¼ 8T and combine various components. From G G ¼ 0itfollowsthat H ¼ H 0 H, theng r ¼ 0 relates K 0 to H, andafterusing G þ G ¼ 16p to eliminate p, we finally subtract the rr-component of the Einstein equation from the tt-component to obtain the following differential equation for H 0 H (for l ¼ ): H 00 þ H 0 r þ m(r) ek r þ 4rðp Þ þ H 6ek r þ 4ek 5 þ 9p þ þ p 0 ¼ 0; ð15þ dp=d where the prime denotes d/dr. The boundary conditions for equation (15) can be obtained as follows. Requiring regularity of H at r ¼ 0 and solving for H near r ¼ 0 yields H(r) ¼ a 0 r 1 7 þ (0) þ p(0) (dp=d)(0) 5(0) þ 9p(0) r þ Or 3 ; ð16þ where a 0 is a constant. To single out a unique solution from this one-parameter family of solutions parameterized by a 0, we use Fig. 1. Relativistic Love numbers k. the continuity of H(r) and its derivative across r ¼ R. Outside the star, equation (15) reduces to H 00 þ k0 H 0 6ek r þ k0 H ¼ 0; ð17þ and changing variables to x ¼ (r/m 1) as in Thorne & Campolattaro (1967) transforms equation (17) to a form of the associated Legendre equation with l ¼ m ¼, x 1 H 00 þ xh 0 6 þ 4 x H ¼ 0: ð18þ 1 The general solution to equation (18) in terms of the associated Legendre functions Q (x) and P (x) is given by H ¼ c 1 Q r M 1 r þ c P r M 1 ; ð19þ where c 1 and c are coefficients to be determined. Substituting the expressions for Q (x) and P (x) from Abramowitz & Stegun (1964) yields for the exterior solution r H ¼ c 1 1 M " M(M r) ð M þ 6Mr 3r Þ M r r (M r) þ 3 # log r r þ c 1 M : ð0þ r M M r The asymptotic behavior of the solution from equation (0) at large r is H ¼ 8 M 3! M 4 r c 1 þ O þ 3 c þ O r ; ð1þ 5 r r M M where the coefficients c 1 and c are determined by matching the asymptotic solution from equation (1) to the expansion from equation (1) and using equation (9), c 1 ¼ M 3 ke; c ¼ 1 3 M E: ðþ

4 No., 008 TIDAL LOVE NUMBERS OF NEUTRON STARS 119 TABLE 1 Relativistic Love Numbers k We now solve for k in terms of H and its derivative at the star s surface r ¼ R using equations () and (0), and use equation (5) to obtain the expression k ¼ 8C5 1 C ½ þ Cðy 1Þ yš C½6 3y þ 3C 5 ; (5y 8)Šþ4C y þ C(3y ) þ C (1 þ y) 1 þ 3(1 C ) ½ y þ C( y 1) Šlogð1 CÞ ; ð3þ where we have defined the star s compactness parameter C M/R and the quantity y RH 0 (R)/H(R), which is obtained by integrating equation (15) outward in the region 0 < r < R Newtonian Limit The first term in the expansion of equation (3) in M/R reproduces the Newtonian result k N ¼ 1 n M/R k y y þ 3 ; ð4þ TABLE Estimated Neutron Star Parameters from X-Ray Observations Cluster/Object where the superscript N denotes Newtonian. In the Newtonian limit, the differential equation (15) inside the star becomes H 00 þ r H 0 þ M (M ) 4 dp=d 6 r H ¼ 0: R ( km) M/R! Cen a... 1:61 0:15 10:99 0:71 0:18 0:04 M13 a... 1:36 0:04 9:89 0:08 0. NGC 808 a... 0:84 0:1 7:34 0:96 0: 0:01 EXO b... :1 0:8 13:8 1: Note. Parameters used to generate Fig. 3. a The parameters for these stars are the averages from the best-fit values of the data in Webb & Barret (007) for their three different spectral fits. The errors given here reflect only the deviations among the best-fit values for the fits. b The values are taken from Ozel (006). ð5þ For a polytropic index of n ¼ 1, equation (5) can be transformed to a Bessel equation with the solution that is regular at r ¼ 0given by H ¼ Ar/R ð Þ 1/ J 5/ (r/r), where A is a constant. At r ¼ R, we thus have y ¼ RH 0 /H ¼ ( 9)/3, and from equation (3) it follows that k N (n ¼ 1) ¼ 1 þ 15 0:5991; ð6þ which agrees with the known result of Brooker & Olle (1955). 4. RESULTS AND DISCUSSION The range of dimensionless Love numbers k obtained by numerical integration of equation (3) is shown in Figure 1 as a function of M/R and n for a variety of different NS models, and representative values are given in Table 1. These values can be Fig.. Difference in percent between the relativistic dimensionless Love numbers k and the Newtonian values k N. Fig. 3. Range of Love numbers for the estimated NS parameters from X-ray observations. Top to bottom sheets: EXO ,!Cen, M13, and NGC 808. For an inspiral of two 1:4 M NSs at a distance of 50 Mpc, LIGO II detectors will be able to constrain k to k 0:1 ; gcm s with 90% confidence (Flanagan & Hinderer 008).

5 10 HINDERER approximated to an accuracy of 6% in the range 0:5 n 1:0 and0:1 (M/R) 0:4 by the fitting formula k 3 0:56 M 0:003 0:41 þ n 0:33 : ð7þ R Both Figure 1 and Table 1 illustrate that the dimensionless Love numbers k depend more strongly on the polytropic index n than on the compactness C ¼ M/R. 3 This is expected since the weak field, Newtonian values k N given by equation (4) just depend on n (through the dependence on y). The additional dependence on the compactness for the Love numbers k in equation (3) is a relativistic correction to this. For M/R 10 5 our results for k agree well with the Newtonian results of Brooker & Olle (1955). Figure shows the percent difference (k N k )/k between the relativistic and Newtonian dimensionless Love numbers. As can be seen from the figure, the relativistic values are lower than the Newtonian ones for higher values of n. This can be explained by the fact that the Love number encodes information about the degree of central condensation of the star. Stars with a higher polytropic index n are more centrally condensed and, therefore, have a smaller response to a tidal field, resulting in a smaller Love number. Some estimates of the masses and radii of NSs, given in Table, have been inferred from X-ray observations (Ozel 006; Webb & Barret 007) using the information from three measured quantities: the Eddington luminosity, the surface redshift of spectral lines, and the quiescent X-ray flux. The range of the numbers k for these stars is shown in Figure 3. LIGO II detectors will be able to establish a 90% confidence upper limit of k :01 ; gcm s for an inspiral of two nonspinning 1:4 M NSs at a distance of 50 Mpc in the case that no tidal phase shift is observed (Flanagan & Hinderer 008). 3 Note, however, that LIGO measurements will yield the combination k R 5 and, therefore, will be more sensitive to the compactness than the polytropic index. The author thanks Éanna Flanagan for valuable discussions and comments. Abbott, B., et al. 008, Phys. Rev. D, 77, 0600 Abramowitz, M., & Stegun, I. A. 1964, Handbook of Mathematical Functions ( New York, Dover) Baumgarte, T. W., & Shapiro, S. L. 003, Phys. Rep., 376, 41 Berti, E., et al. 00, Phys. Rev. D, 66, Bildsten, L., & Cutler, C. 199, ApJ, 400, 175 Brooker, R. A., & Olle, T. W. 1955, MNRAS, 115, 101 Cutler, C., & Thorne, K. S. 00, preprint (gr-qc/004090) Cutler, C., et al. 1993, Phys. Rev. Lett., 70, 984 Faber, J. A., et al. 00, Phys. Rev. Lett., 89, 3110 Flanagan, É. E., & Hinderer, T. 008, Phys. Rev. D, 77, 0150 Geroch, R. 1970, J. Math. Phys., 11, 580 Gualtieri, L., et al. 001, Phys. Rev. D, 64, Hansen, R. O. 1974, J. Math. Phys., 15, 46 Kalogera, V., et al. 004, ApJ, 601, L179 Kochanek, C. S. 199, ApJ, 398, 34 Kokkotas, K. D., & Schafer, G. 1995, MNRAS, 75, 301 Lattimer, M., & Prakash, J. M. 007, Phys. Rep., 44, 109 REFERENCES Misner, C. W., Thorne, K. S., & Wheeler, J. A. 1973, Gravitation (San Francisco: W. H. Freeman and Co.) Mora, T., & Will, C. M. 004, Phys. Rev. D, 69, Ozel, F. 006, Nature, 441, 1115 Pons, J. A., et al. 00, Phys. Rev. D, 65, Regge, T., & Wheeler, J. A. 1957, Phys. Rev., 108, 1063 Shibata, M. 1994, Prog. Theor. Phys., 91, 871 Taniguchi, K., & Shibata, M. 1998, Phys. Rev. D, 58, Thorne, K. S. 1980, Rev. Mod. Phys., 5, 85 Thorne, K. S. 1998, Phys. Rev. D, 58, Thorne, K. S., & Campolattaro, A. 1967, ApJ, 149, 591 Thorne, K. S., & Hartle, J. B. 1985, Phys. Rev. D, 31, 1815 Vallisneri, M. 00, Phys. Rev. Lett., 84, 3519 Wald, R. M. 1984, General Relativity (Chicago: Univ. Chicago Press) Webb, N. A., & Barret, D. 007, ApJ, 671, 77 Zdunik, J. L., Haensel, P., Bejger, M., & Gourgoulhon, E. 008, in Proc. Int. Symp. on Exotic States of Nuclear Matter, in press (arxiv: v1)

6 The Astrophysical Journal, 697:964, 009 May 0 C 009. The American Astronomical Society. All rights reserved. Printed in the U.S.A. doi: / x/697/1/964 ERRATUM: TIDAL LOVE NUMBERS OF NEUTRON STARS (008, ApJ, 677, 116) Tanja Hinderer Center for Radiophysics and Space Research, Cornell University, Ithaca, NY 14853, USA; tph5@caltech.edu In the original paper, there are typographical errors in Equations (0) and (3), and some incorrect entries in Table 1. I thank Ryan Lang for pointing these out. Equation (0) should read as follows: H = c 1 ( r M +3c ( r M ) ( 1 M r ) ( 1 M r Equation (3) should be replaced by the following: )[ M(M r)(m +6Mr 3r ) r (M r) + 3 ( log r r M ). (0) k = 8C5 5 (1 C) [+C(y 1) y] { C (6 3y +3C(5y 8)) +4C 3 [ 13 11y + C(3y ) + C (1 + y) ] )] +3(1 C) [ y +C(y 1)] log (1 C) } 1. (3) The corrected values for the Love numbers in Table 1 are given in the table below. Table 1 Relativistic Love Numbers k n M/R k