Impact of updated Multipole Love numbers and f-Love Universal Relations in the context of Binary Neutron Stars Bikram Keshari Pradhan1Aditya Vijaykumar2 3and Debarati Chatterjee1

2025-05-08 0 0 6.57MB 21 页 10玖币

侵权投诉

Impact of updated Multipole Love numbers and f-Love Universal Relations in the

context of Binary Neutron Stars

Bikram Keshari Pradhan,1, ∗Aditya Vijaykumar,2, 3 and Debarati Chatterjee1

1Inter-University Centre for Astronomy and Astrophysics,Pune University Campus,Pune,411007, India

2International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India

3Department of Physics, The University of Chicago,

5640 South Ellis Avenue, Chicago, Illinois 60637, USA

(Dated: January 18, 2023)

Neutron star (NS) equation of state (EoS) insensitive relations or universal relations (UR) involv-

ing neutron star bulk properties play a crucial role in gravitational-wave astronomy. Considering a

wide range of equations of state originating from (i) phenomenological relativistic mean ﬁeld models,

(ii) realistic EoS models based on diﬀerent physical motivations, and (iii) polytropic EoSs described

by spectral decomposition method, we update the EoS-insensitive relations involving NS tidal de-

formability (Multipole Love relation) and the UR between f-mode frequency and tidal deformability

(f-Love relation). We analyze the binary neutron star (BNS) event GW170817 using the frequency

domain TaylorF2 waveform model with updated universal relations and ﬁnd that the additional

contribution of the octupolar electric tidal parameter and quadrupolar magnetic tidal parameter

or the change of multipole Love relation has no signiﬁcant impact on the inferred NS properties.

However, adding the f-mode dynamical phase lowers the 90% upper bound on ˜

Λ by 16-20% as well

as lowers the upper bound of NSs radii by ∼500m. The combined URs (multipole Love and f-Love)

developed in this work predict a higher median (also a higher 90% upper bound) for ˜

Λ by 6% and

also predict higher radii for the binary components of GW170817 by 200-300m compared to the

URs used previously in the literature. We further perform injection and recovery studies on simu-

lated events with diﬀerent EoSs in A+ detector conﬁguration as well as with third generation (3G)

Einstein telescope. In agreement with the literature, we ﬁnd that neglecting f-mode dynamical tides

can signiﬁcantly bias the inferred NS properties, especially for low mass NSs. However, we also ﬁnd

that the impact of the URs is within statistical errors.

I. INTRODUCTION

The density in the core of a neutron star (NS) surpasses

the nuclear saturation density (n0∼2×1014 g cm−3)

and even the highest density that can be achieved in

terrestrial experiments. It is argued that strangeness

in the form of hyperons, meson condensates, or even

deconﬁned quark matter may appear at such high

densities, aﬀecting several NS observable properties [1].

The behavior of ultra-dense NS matter is still largely

unknown, and therefore, NSs provide a natural labora-

tory to study nuclear matter under extreme conditions

such as high magnetic ﬁeld and rotation [2].

The NS macroscopic observables relate to the mi-

croscopic NS physics through the pressure density

relationship or the equation of state (EoS). The two

main approaches used to model nuclear EoS are (i)

microscopic or ab-initio [3–7] and (ii) phenomeno-

logical (eﬀective theories with parameters ﬁtted to

reproduce saturation nuclear properties) [8, 9]. Yet

another approach is to construct empirical ﬁts rather

than microphysics-based EoS. Where the high-density

behavior of EoS is described by polytropic models

(pressure is proportional to a power of density) either

using piece-wise polytropes [10] or using the spectral

∗bikramp@iucaa.in

decomposition method [11]. One can also infer the NS

EoS from observations using a nonparametric description

of NS EoS [12] or can even use a hybrid approach to

describe the NS EoS [13].

On the observational side, NSs are observed at mul-

tiple wavelengths in current generation electromagnetic

telescopes. The masses of these objects are typically

well measured in binary systems, whereas the measure-

ment of radius involves larger uncertainty. However,

the recently launched NICER (Neutron Star Interior

Composition Explorer) mission [14] has improved radius

measurement and can further improve the estimation

of the NS radius upto 5% [15, 16]. Additionally, binary

neutron star (BNS) mergers are sources of gravitational

waves (GW). In a BNS merger, tidal deformation of

the NSs carries information about NS radii and is

used to constrain the NS EoS in combination with

mass measurements from the same binary [17–21].

The ground-breaking BNS merger GW170817 [21–23]

also observed electromagnetic (EM) counterparts in

addition to GWs, and hence opened a new window in

multi-messenger astronomy.

The tidal deformation of the stars in a BNS contributes

to phase of the GW signal starting at the ﬁfth Post

Newtonian (PN) order. In general, tidal deformability

(for electric type) of order ‘l’ (Λl) appears at 2l+ 1 Post

Newtonian (PN) order. The major contribution comes

arXiv:2210.09425v2 [astro-ph.HE] 17 Jan 2023

from the quadrupolar tides, followed by higher-order

octupolar (Λ3) and hexadecapolar (Λ4) terms [18, 24].

Recent waveform models have been updated to consider

the eﬀect of the higher order tidal parameters Λ3, Λ4

and also the magnetic type deformation (quadrupolar

magnetic Σ2in the GW phase [24, 25]. In most

analyses, the tides are considered adiabatic, ie. the GW

frequency is much lower than the NS resonant oscillation

mode frequency. However, recent eﬀorts largely support

the excitation of NS fundametal modes (f-modes) in the

late inspiral [26–31]. In a frequency domain model, the

dynamical tidal contribution to the GW phase due to the

excitation of f-modes appears at 8 PN order [18, 32] and

depends upon the f-mode frequency. Since the additional

tidal deformability parameters and the f-mode dynam-

ical phase contribute at high PN orders, they are often

dropped while constructing waveform approximants.

However, the f-mode dynamical tidal correction has been

shown to signiﬁcantly aﬀect the inference of NS proper-

ties from a binary due to its resonance behaviour [33, 34].

Previous works have suggested that there exist EoS-

insensitive universal relations (UR) between Λ2and each

of Λ3, Λ4, and Σ3( hereafter referred as ‘multipole Love’

relation) [35–37]. Similarly, URs exist between the

f-mode frequency and tidal parameters [38] ( hereafter

referred as ‘f-Love’ relation). This allows the additional

contribution due to higher order tidal parameters or

f-mode parameters to be considered without having to

actually sample over them in a parameter estimation

run, reducing the search parameter space. Additionally,

there exists UR involving stellar compactness and

quadrupole tidal parameters, which can be used to infer

NS radius from the measured mass and quadrupole tidal

parameters from a BNS event [39]. In this work, we will

only be interested in the URs mentioned above. There

also exist URs involving the combination of Λ2and mass

ratio qof the binaries, known as ‘binary Love relations,’

as well as URs involving NS spin which reduces eﬀort in

measuring the NS tidal parameter (and NS radius) in a

BNS event [40–46] 1.

The existing URs are employed by considering a few

theoretical EoS models and some of the selective EoSs

are now incompatible with the current astrophysical

constraints [35, 37–39]. Also, the existing URs do not

consider the EoS uncertainties resulting from the nuclear

parameters. In this work, we improve the multipole

Love and f-Love URs relevant for GW astronomy and

investigate the impact of the updated URs by analyzing

the BNS event GW170817 and performing several

injection and recovery studies with the future GW

detector conﬁgurations. This work is organized in the

1Recently a new EoS insensitive approach involving the NS mass

and tidal deformability is proposed to constrain NS properties

from a GW event [47].

following way. In section II we discuss the choices of

Eos, followed by the description of the methodology to

solve for multipole tidal parameters and to solve for the

NS f-mode characteristics. We compile our results in

Section III and summarise our conclusions in Section IV .

II. METHOD

A. Choices of Equations of State

The EoS is essentially the relation between the pressure

(p) and density (ρ) (or energy density ), i.e., p=p().

As described in Section I, diﬀerent physical motivations

develop a diverse family of EoSs depending upon the

physical descriptions of the NS matter. In this work,

we consider a wide range of EoSs based upon diﬀerent

physical descriptions. They are discussed below:

a. Relativistic Mean Field (RMF) Models: RMF

models are phenomenological models where baryon-

baryon interaction is mediated via exchange of mesons.

The Lagrangian density describes the interaction be-

tween baryons through the exchange of mesons. The

complete description of the Lagrangian density for

nucleonic (npeµ) and nucleon-hyperon (npeµY ) mat-

tered NS, and hence the EoS within the RMF model

considered in this work can be found in [48]. The

parameters of the RMF model are calibrated to the

nuclear and hypernuclear parameters at saturation:

nuclear saturation density (n0), the binding energy

per nucleon (E/A or Esat), incompressibility (K), the

eﬀective nucleon mass (m∗), symmetry energy (J) and

slope of symmetry energy (L) at saturation. Hyperon

coupling constants are ﬁxed using hyperon nucleon

potential depths (UY) or using symmetry properties. We

consider the total uncertainty ranges in the saturation

parameters resulting from nuclear and hypernuclear

experiments, summarized in Table I [49].

b. Selective EoSs: Along with the RMF mod-

els with nucleonic and hyperonic mattered EoSs, we

consider many realistic EoSs. The realistic EoSs are

taken either from CompOSE database [51–53] or

from LALSimulation [54]. The considered realistic

EoSs are APR4, APR3 [6, 55], SLy4 [56], BL [57],

DD2(GPPVA) [58], SRO-APR [59], BSk22 [60, 61],

WFF1 [62], MPA1 [63]. We also consider the Soft and

Stiﬀ EoS from [5]. In Figure 1a, though the Stiﬀ-EoS

terminates at earlier energy density, it is suﬃcient to

reach the maximum stable NS mass the EoS model

can reproduce [5]. To capture the hypothesis of a

deconﬁned quark phase in the interior of the NS core,

we account for some realistic hybrid EoSs containing the

quark phase in the interior. The selective hybrid EoS

models are BFH-D [64, 65], KBH (QHC21 −AT) [66],

JJ-VQCD [67] and OOS-DD2(FRG) [68]. We do not

consider the EoSs regarding quark stars only, as they

Model n0Esat K J L m∗/mNUΣUΞ

(fm−3) (MeV) (MeV) (MeV) (MeV) MeV MeV

RMF [48] [0.14, 0.17] [-16.5, -15.5] [200, 300] [28, 34] [40, 70] [0.55, 0.75] [0, +40] [-40,0]

TABLE I: Range of nuclear and hypernuclear saturation parameters considered in this work. Meson and nucleon

masses are ﬁxed at mσ= 550 MeV, mω= 783 MeV, mρ= 770 MeV, mσ∗= 975 MeV, mφ= 1020 MeV and

mN= 939 MeV. Masses of the hyperons are ﬁxed from [50]. Note that we ﬁx the Λ hyperon potential depth

(UΛ=-30 MeV).

might deviate from the universal behavior and leave

them for a separate investigation [35, 69].

c. Spectral Decomposition: To span the EoSs with

empirical ﬁt formalism, we consider the four parameters

spectral decomposition method developed in [11]. In

spectral decomposition, the adiabatic index (Γ) of the

EoS is spectrally decomposed onto a set of polynomial

basis functions and expressed as,

Γ(p) = exp X

γk[ln(p/p0)]k!(1)

where γkis the expansion coeﬃcient and p0is the ref-

erence pressure where the high-density EoS is stitched to

the low-density crustal EoS. The EoS can then be gener-

ated by integrating the relation,

d

dp =+p

pΓ(p)(2)

which can be reduced to,

(p) = (p0) + 1

µ(p)Zp

µ(p0)

Γ(p0)dp0(3)

where,

µ(p) = exp −Zp

dp0

p0Γ(p0)

We ﬁx the low-density EoS to SLy EoS [70] and stitch

the high-density EoS at a density below half of the sat-

uration density such that the NS macroscopic properties

will not be aﬀected signiﬁcantly. We generate spectral

decomposed EoSs as implemented in LALSimulation [54]

and consider the ranges for spectral indices (γk) from

[20, 71].

Before proceeding with any further calculations, we

ensure that each EoS satisﬁes the required physical con-

ditions, such as thermodynamic stability (dp/d > 0),

causality (pdp/d ≤1) and the monotonic behaviour of

pressure (dp/dρ > 0 and d/dρ > 0). We additionally im-

pose the constraint that the EoS must be able to produce

a 2Mstable NS and the tidal deformability of a 1.4M

is less than 800 (i.e, Λ1.4M≤800) [20, 21]. EoSs and

the corresponding mass-radius relations used in this work

are displayed in Figure 1a and Figure 1b respectively.

B. Macroscopic Structure and Tidal

Deformabilities

For a given EoS, NS mass (M) and radius (R) are

obtained by solving the Tolman-Oppenheimer-Volkoﬀ

(TOV) equations. The vanishing of pressure at the sur-

face of NS (p(R) = 0) provides the stellar radius Rand

the stellar mass M=m(R). TOV equations correspond-

ing to a static and spherically symmetric metric (4) are

summarised in Eq. (5) [74, 75].

ds2=−e2Φ(r)dt2+e2λ(r)dr2+r2dθ2+r2sin2θdφ2(4)

dm(r)

dr = 4πr2(r),

dp(r)

dr =−[p(r) + (r)] m(r)+4πr3p(r)

r(r−2m(r)) ,

dΦ(r)

dr =−1

(r) + p(r)

dr ,

e2λ(r)=r

r−2m(r). . (5)

The NS can be tidally deformed in a binary system due

to mutual gravitational interaction. The tidal ﬁeld can

be decomposed to an electric (Eij ) and magnetic (Mij )

component, leading to the induction of mass multipole

moment (Qij ) and current multipole moment (Sij ). The

gravitoelectric tidal deformability (λl) and gravitomag-

netic tidal deformability (σl) of order lcan be deﬁned

as [76],

Qij =λlEij ,

Sij =σlMij (6)

kl=(2l−1)!!

2λl,

jl= 4(2l−1)!! σl.(7)

The electric Love number (kl) and magnetic Love num-

ber (jl) relate to the tidal deformability parameters as

given in Eq. (7). The valuable parameters that can be

determined from a GW signal are dimensionless tidal de-

formability. The dimensionless gravitoelectric tidal de-

formability parameter (Λl) and gravitomagnetic tidal de-

formability parameter (Σl) can be expressed using the

corresponding tidal Love numbers and stellar compact-

ness (C=M/R) as [77]:

500 1000 1500 2000

( MeV fm−3)

200

400

600

800

1000

p( MeV fm−3)

RMF(npeµ)

RMF(npeµY)

Spectral

APR3

APR4

BFH-D

DD2

JJ-VQCD

KBH

MPA1

OOS-DD2

SLy4

SRO-APR

Stiﬀ-EoS

WFF1

BSk22

Soft-EoS

(a)

10 12 14 16

R(km)

0.5

1.0

1.5

2.0

2.5

3.0

M(M)

PSR J0740+6620

PSR J0348+0432

GW170817 M1

GW170817 M2

Causality

Rotation

RMF(npeµ)

RMF(npeµY)

Spectral

APR3

APR4

BFH-D

DD2

JJ-VQCD

KBH

MPA1

OOS-DD2

SLy4

SRO-APR

Soft-EoS

Stiﬀ-EoS

WFF1

BSk22

(b)

FIG. 1: (a) EoSs used in this work. (b) Mass radius relation corresponding to the EoSs used in this work. For each

EoS, the corresponding M-R relationship is shown up to the maximum possible stable mass. Horizontal bands

correspond to masses M= 2.072+0.067

−0.066Mof PSR J0740+6620 [72] and M= 2.01+0.04

−0.04Mof PSR J0348+0432

[73]. The mass radius estimates of the two companion neutron stars in the merger event GW170817 [20] are shown

by the shaded area labeled with GW170817 M1 (M2) a.

ahttps://dcc.ligo.org/LIGO-P1800115/public

Λl=2

(2l−1)!!

C2l+1 ,

Σl=1

4(2l−1)!!

C2l+1 .(8)

For computing the electric and magnetic Love numbers,

one needs to integrate additional set of diﬀerential equa-

tions [77] along with the TOV equations. We follow the

methodology developed in [77] to solve for the Love num-

bers. We test our numerical scheme by reproducing the

Love numbers corresponding to selective EoSs used in

[77]. We provide the electric tidal deformability up to

order ‘`= 4’ and magnetic tidal deformability up to or-

der ‘`= 3’.

C. Finding f-mode oscillation characteristics

A neutron star can have several quasi-normal modes

depending upon the restoring force. Mode characteris-

tics (frequency and damping time) contain information

about the NS interior. Hence, the determination of mode

parameters can be employed to constrain the NS inte-

rior or EoS. In our previous work [78], we had shown

that depending upon the EoS, the Cowling approxima-

tion can overestimate the quadrupolar f-mode frequency

up to about 30%. As recent eﬀorts are going on to im-

prove the gravitational waveform models with considera-

tion of excitation of f-modes which depends on the mode

frequency [32], one should consider the general relativis-

tic formalism to ﬁnd the mode characteristics. We use the

direct integration method developed in [78–81] to ﬁnd

the NS f-mode frequency. In short, the coupled equations

for perturbed metric and ﬂuid variables are integrated

within the NS interior with appropriate boundary con-

ditions [80]. Outside of the NS, ﬂuid variables are set

to zero, and then the Zerilli’s wave equation [82] is inte-

grated to far away from the star. Then a search is carried

out for the complex f-mode frequency (ω= 2πf +i

τf) for

which one has only outgoing wave solution to the Zer-

illi’s equation at inﬁnity. The real part of ωrepresents

the f-mode angular frequency, and the imaginary part

represents the damping time. For ﬁnding the mode char-

acteristics, we use the numerical methods developed in

our previous work [78].

III. RESULTS

A. Multipole Love and f-Love Relations

The universality among higher-order electric tidal de-

formability parameters (Λ`≥3) or the magnetic de-

formability parameter (Σ`≥2) with the quadrupolar tidal

deformability parameter (Λ`=2) were ﬁrst discussed in

[35, 40] with few selective EoSs. The universality of com-

pactness (C) with Λ2initially introduced in [39] and the

UR involving f-mode frequency and tidal deformability

parameters are introduced in [38]. Original URs from

[35, 37–39] are expressed using a polynomial ﬁt of the

following form (the order of the polynomial diﬀers in dif-

ferent works).

k=0

ak[ln (Λ2)]k,(9)

where,

P={ln(Λ3),ln(Λ4),ln(|Σ2|),ln(|Σ3|), C,

Mω2, Mω3, M ω4}

In recent works [43, 83], the universal relations for elec-

tric type deformability were updated by considering the

phenomenological piece-wise polytropic EoSs. In diﬀer-

ent waveform models, the correction on the tidal phase

includes the electric tidal parameter up to order `≤4

and for magnetic deformation up to order `≤3. Hence,

we provide the universal relations for Λ3and Λ4with

Λ2for electric type, whereas for magnetic type, we pro-

vide the universal relation of Σ2and Σ3with Λ2. From

the tidal deformability and mass obtained from a bi-

nary system, one can infer the NS radius (R) by using

the EoS-independent relation that exists between stellar

compactness (C=M/R), and tidal deformability param-

eter [39, 43, 47, 84–86]2. There are EoS-independent

relations, which involve the tidal deformability of both

binary NSs and binary parameters ( like q=m1/m2).

The universal relation involving the binary parameters

and the C−Λ2relation is also used to infer the NS

radii [40, 41, 43, 87]. Though binary relations reduce the

parameter space of BNS search parameters, the inferred

NS parameters are model dependent [88].

As discussed in Section II A, we consider tentatively

6000 EoSs originating from diﬀerent physical motiva-

tions. We ﬁx the lower mass for an EoS by imposing the

constraint resulting from the maximum rotation pulsar

PSR J1748-2446 [89, 90] (also notice the rotation lim-

iting curve in Figure 1b). The maximum mass for an

EoS is ﬁxed at the maximum stable NS mass the EoS

can produce. We then use the method described in Sec-

tion II B to obtain the multipole tidal parameters (Λ`and

Σ`). We obtain the multipole Love relations by solving

the NS properties (with tidal parameters) for ∼1.2×106

neutron stars. We produce the universal relation for mul-

tipole Love relations as a polynomial ﬁt (9) as introduced

in [35]. Note that the original ﬁt from Yagi [35] was a

4th order polynomial which was then updated to a 6th or-

der polynomial for a larger data set in [43]. In our data

set, we notice that by updating the polynomial from a

quartic polynomial to a 5th order polynomial, the result-

ing goodness of ﬁt improved signiﬁcantly and then did

2The systematic errors that could be occurred in the inferred NS

properties due to the choice of binary Love relation or C−Λ2

UR are recently discussed in [43, 86].

not improve signiﬁcantly after increasing the order of the

polynomial. However, we provide the universal relations

with a 6th order polynomial (9) to be consistent with the

updated relations. Universal relations Λ3−Λ2, Λ4−Λ2,

Σ2−Λ2and Σ3−Λ2are displayed in Figures 2a, 2b, 3a

and 3b respectively. The ﬁt parameters for the multipole

Love relations (9) involving tidal parameters obtained

from this work and other works are tabulated in Ta-

ble II. Similarly, for C−Λ2relation we provide the ﬁt

parameters in Table III and displayed the relation in

Figure 4. Our multipole universal relations are valid for

2.3≤Λ2≤4×104which is the essential range for GW

astronomy ( mostly in the range ≤104) and for C−Λ2,

the relation is valid for 0.08 ≤C≤0.33.

Although the detection of f-modes from BNS would

only be possible with third generation detectors [34],

the impact on the inferred NS parameters can be seen

in the A+ detector conﬁgurations [31, 33]. In the fre-

quency domain, the phase correction due to the dynami-

cal excitation of f-modes depends solely on the mode fre-

quency [18, 26, 32, 91]. The universality of mass scaled

f-mode angular frequency (ie.,Mω) with the tidal pa-

rameter (Λ) was ﬁrst studied in the work of Chan et al.

[38], where the universality was explained by using the

universal behavior of f-mode frequency with the moment

of inertia (I), and the I−Love−Qrelations [38, 44]. To

avoid the numerical instabilities, we ﬁx the lower mass

∼1Mfor each EoS despite the rotation limit while solv-

ing for the f-mode frequency. The proposed quartic poly-

nomial ﬁt of [38] is updated to a 5th order ﬁt in [78, 92].

We provide the polynomial ﬁt (9) up to 6th order to be

consistent with other multipole universal relations. Our

f-Love relations are valid for 2.3≤Λ2≤4×104.

B. Error Analysis

We analyze the errors and compare the URs in the

range Λ2≤104as required for GW astronomy. Our

Λ3−Λ2relation holds a maximum error of 11%, with 90%

of the errors are below 7%. In comparison, the original ﬁt

from Yagi [35] holds a maximum error of 16% with 90%

of the errors below 10%, and the updated Λ3−Λ2UR

from Godzieba et al. [43] holds a maximum error of 23%

with 90% of the errors below 16%. On similar lines, the

Λ4−Λ2relation developed in this work holds a maximum

error of 24% with 90% of the errors below 13%, whereas,

the original ﬁt from Yagi holds a maximum error of 35%

with 90% of the errors below 20%, and the UR from [43]

holds a maximum error ∼44% with 90% of the errors

below 30%. For 50 ≤Λ2≤104, the URs developed in

this work behave quite similarly to the URs developed

in [43] and also in this range, the updated URs, as well

as the URs from [43] account less error compared to the

original ﬁts from Yagi [35]. However, in the complete

range of Λ2, our relation have lower error bounds ( see

Figure 9 in Appendix A or Table II).

For Λ2≤104, our Σ2−Λ2ﬁt holds a maximum error

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

ImpactofupdatedMultipoleLovenumbersandf-LoveUniversalRelationsinthecontextofBinaryNeutronStarsBikramKeshariPradhan,1,AdityaVijaykumar,2,3andDebaratiChatterjee11Inter-UniversityCentreforAstronomyandAstrophysics,PuneUniversityCampus,Pune,411007,India2InternationalCentreforTheoreticalSciences,TataInst...

展开>> 收起<<

Impact of updated Multipole Love numbers and f-Love Universal Relations in the context of Binary Neutron Stars Bikram Keshari Pradhan1Aditya Vijaykumar2 3and Debarati Chatterjee1.pdf

共21页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Impact of updated Multipole Love numbers and f-Love Universal Relations in the context of Binary Neutron Stars Bikram Keshari Pradhan1Aditya Vijaykumar2 3and Debarati Chatterjee1

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: