10
Slowly rotating neutron stars with small differential rotation: Equilibrium models and oscillations in the Cowling approximation Cecilia Chirenti, 1, * Jozef Ska ´kala, 1,and Shin’ichirou Yoshida 2,1 Centro de Matema ´tica, Computac ¸a ˜o e Cognic ¸a ˜o, UFABC, 09210-170 Santo Andre ´, Sa ˜o Paulo, Brazil 2 Department of Earth Science and Astronomy, Graduate School of Arts and Sciences, University of Tokyo, Komaba, Meguro-ku 3-8-1, 153-8902 Tokyo, Japan (Received 22 January 2013; published 21 February 2013) Newly born neutron stars can present differential rotation, even if later it should be suppressed by viscosity or a sufficiently strong magnetic field. And in this early stage of its life, a neutron star is expected to have a strong emission of gravitational waves, which could be influenced by the differential rotation. We present here a new formalism for modelling differentially rotating neutron stars: working on the slow rotation approximation and assuming a small degree of differential rotation, we show that it is possible to separate variables in the Einstein field equations. The dragging of inertial frames is determined by solving three decoupled ordinary differential equations. After we establish our equilibrium model, we explore the influence of the differential rotation on the f- and r-modes of oscillation of the neutron star in the Cowling approximation, and we also analyze an effect of the differential rotation on the emission of gravitational radiation from the f-modes. We see that the gravitational radiation from the f-modes is slightly suppressed by introducing differential rotation to the equilibrium stars. DOI: 10.1103/PhysRevD.87.044043 PACS numbers: 04.40.Dg I. INTRODUCTION Differential rotation, until it becomes suppressed by viscosity or strong enough magnetic fields [1,2], might play an important role in the evolution of a newly born neutron star. (For a typical neutron star it takes between 10 and 100 years to become uniformly rotating [3].) The equilibrium stellar models representing a neutron star’s differential rotation were explored in some older papers [4,5], and the oscillation frequencies for some types of fluid modes were calculated later in Refs. [3,68]. In this work we explore the evolution of linear pertur- bations in a slowly rotating neutron star with a polytropic equation of state. Moreover, the rotation profile of the star represents a first order deviation from the uniform rotation. We generalize the old semianalytical results of Hartle for the uniformly rotating equilibrium model [9] to a small deviation from the uniform rotation following the relativ- istic j-constant law. In particular we show that under a first order deviation from the slow uniform rotation one can still separate spherical harmonics and obtain only a very small number of non-zero terms in their expansion. Furthermore, similar to the uniformly rotating case [9], one can find an exact analytic solution for the metric dragging function outside the star. In Sec. II of this paper we use the consistency conditions imposed by the equilibrium model to constrain the value of the parameter representing the differential rotation, (for the given equilibrium parameters of the star). Such a relatively simple equilibrium model with the constrained value of the differential rotation parameter is then used in Sec. III, in the Cowling approximation, to explore the various types of fluid modes. To obtain numerical results for the modes, in Sec. IV we evolve the initial perturbation in time using a two- dimensional Lax-Wendroff scheme [10]. Our choice of a time evolution treatment is motivated by the corotation problem that affects the usual eigenvalue approach for ob- taining the mode frequencies. The f-modes were computed for the slow rotation case in Ref. [6,7] and we compare our results with the results from those papers, further constrain- ing the domain of validity of the differential rotation parame- ter. In addition to the results of [6,7], we explore in more detail the behavior of the f-modes as a function of small values of the differential rotation parameter and we also find numerical values for some of the r-mode frequencies. (As can be seen in Ref. [11], the r-modes excite differential rotation in the stellar fluid.) Also, the f-mode eigenfunctions are extracted by using a pointwise discrete Fourier transform (DFT) on the evolution data. As the equilibrium stars have slow rotation with a low degree of differential rotation, the eigenmodes extracted show small change from their non- rotating counterpart. In Sec. V , by using a quadrupole esti- mate of gravitational emission timescale, we see a tendency that the differential rotation slightly suppresses the gravita- tional emission. These results are consistent with the results of Kruger et al.[12] for the rapidly rotating stars. Finally, we finish with our conclusions in Sec. VI. II. EQUILIBRIUM STELLAR MODEL Consider the background spacetime of a slowly rotat- ing star, * [email protected] [email protected] [email protected] PHYSICAL REVIEW D 87, 044043 (2013) 1550-7998= 2013=87(4)=044043(10) 044043-1 Ó 2013 American Physical Society

Slowly rotating neutron stars with small differential rotation: Equilibrium models and oscillations in the Cowling approximation

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Page 1: Slowly rotating neutron stars with small differential rotation: Equilibrium models and oscillations in the Cowling approximation

Slowly rotating neutron stars with small differential rotation: Equilibrium modelsand oscillations in the Cowling approximation

Cecilia Chirenti,1,* Jozef Skakala,1,† and Shin’ichirou Yoshida2,‡

1Centro de Matematica, Computacao e Cognicao, UFABC, 09210-170 Santo Andre, Sao Paulo, Brazil2Department of Earth Science and Astronomy, Graduate School of Arts and Sciences, University of Tokyo,

Komaba, Meguro-ku 3-8-1, 153-8902 Tokyo, Japan(Received 22 January 2013; published 21 February 2013)

Newly born neutron stars can present differential rotation, even if later it should be suppressed by

viscosity or a sufficiently strong magnetic field. And in this early stage of its life, a neutron star is expected

to have a strong emission of gravitational waves, which could be influenced by the differential rotation.

We present here a new formalism for modelling differentially rotating neutron stars: working on the slow

rotation approximation and assuming a small degree of differential rotation, we show that it is possible to

separate variables in the Einstein field equations. The dragging of inertial frames is determined by solving

three decoupled ordinary differential equations. After we establish our equilibrium model, we explore the

influence of the differential rotation on the f- and r-modes of oscillation of the neutron star in the Cowling

approximation, and we also analyze an effect of the differential rotation on the emission of gravitational

radiation from the f-modes. We see that the gravitational radiation from the f-modes is slightly

suppressed by introducing differential rotation to the equilibrium stars.

DOI: 10.1103/PhysRevD.87.044043 PACS numbers: 04.40.Dg

I. INTRODUCTION

Differential rotation, until it becomes suppressed byviscosity or strong enough magnetic fields [1,2], mightplay an important role in the evolution of a newly bornneutron star. (For a typical neutron star it takes between 10and 100 years to become uniformly rotating [3].) Theequilibrium stellar models representing a neutron star’sdifferential rotation were explored in some older papers[4,5], and the oscillation frequencies for some types offluid modes were calculated later in Refs. [3,6–8].

In this work we explore the evolution of linear pertur-bations in a slowly rotating neutron star with a polytropicequation of state. Moreover, the rotation profile of the starrepresents a first order deviation from the uniform rotation.We generalize the old semianalytical results of Hartle forthe uniformly rotating equilibrium model [9] to a smalldeviation from the uniform rotation following the relativ-istic j-constant law. In particular we show that under a firstorder deviation from the slow uniform rotation one can stillseparate spherical harmonics and obtain only a very smallnumber of non-zero terms in their expansion. Furthermore,similar to the uniformly rotating case [9], one can find anexact analytic solution for the metric dragging functionoutside the star.

In Sec. II of this paper we use the consistency conditionsimposed by the equilibrium model to constrain the value ofthe parameter representing the differential rotation, (for thegiven equilibrium parameters of the star). Such a relatively

simple equilibrium model with the constrained value of thedifferential rotation parameter is then used in Sec. III, in theCowling approximation, to explore the various types of fluidmodes. To obtain numerical results for the modes, in Sec. IVwe evolve the initial perturbation in time using a two-dimensional Lax-Wendroff scheme [10]. Our choice of atime evolution treatment is motivated by the corotationproblem that affects the usual eigenvalue approach for ob-taining the mode frequencies. The f-modes were computedfor the slow rotation case in Ref. [6,7] and we compare ourresults with the results from those papers, further constrain-ing the domain of validity of the differential rotation parame-ter. In addition to the results of [6,7], we explore in moredetail the behavior of the f-modes as a function of smallvalues of the differential rotation parameter and we also findnumerical values for some of the r-mode frequencies. (Ascan be seen in Ref. [11], the r-modes excite differentialrotation in the stellar fluid.) Also, the f-mode eigenfunctionsare extracted by using a pointwise discrete Fourier transform(DFT) on the evolution data. As the equilibrium stars haveslow rotation with a low degree of differential rotation, theeigenmodes extracted show small change from their non-rotating counterpart. In Sec. V, by using a quadrupole esti-mate of gravitational emission timescale, we see a tendencythat the differential rotation slightly suppresses the gravita-tional emission. These results are consistent with the resultsofKrugeret al. [12] for the rapidly rotating stars. Finally, wefinish with our conclusions in Sec. VI.

II. EQUILIBRIUM STELLAR MODEL

Consider the background spacetime of a slowly rotat-ing star,

*[email protected][email protected][email protected]

PHYSICAL REVIEW D 87, 044043 (2013)

1550-7998=2013=87(4)=044043(10) 044043-1 2013 American Physical Society

Page 2: Slowly rotating neutron stars with small differential rotation: Equilibrium models and oscillations in the Cowling approximation

ds2 ¼ edt2 þ edr2 þ r2d2

þ r2sin 2ðÞ½d!dt2;where and are functions of r, and ! ¼ !ðr; Þ is theframe dragging function. We will use a polytropic equation

of state p ¼ K1þ1=N, where p is the pressure and is therest mass energy density of the star. The fluid rotation isdescribed by the four-velocity,

uðt;r;;Þ ¼ ðe=2; 0; 0; e=2Þ:

Further, we consider that the rotation of the fluid,, obeysthe j-constant law,

¼ c þ r2sin 2ðÞe!

1þ r2sin 2ðÞe: (1)

The parameter in Eq. (1) describes the level of thedifferential rotation of the star. Then the Tolman-Oppenheimer-Volkoff equations remain unchanged underthe slow rotation (in the linear order in ), and the framedragging parameter ! has to be a solution of the equation,

1

r4@

@r

r4eðþÞ=2 @!

@r

þ eðÞ=2

r2sin 3ðÞ@

@

sin 3ðÞ @!

@

16eðÞ=2 ðþ pÞ

!c þ r2sin 2ðÞe!

1þ r2sin 2ðÞe

¼ 0:

(2)

Furthermore, take the differential rotation to be also small, representing only a linear-order perturbation from theuniform rotation case: 0< 1. Then one can expand ! in the parameter as

!ðr; Þ ¼ !0ðr; Þ þ !1ðr; Þ þOð2Þ: (3)

Here !0 corresponds to the case of uniform rotation and as we know from Ref. [9], it depends only on r. (In the zeroth-order expansion of (2) in , we obtain Hartle’s equation for the uniform rotation problem [9].) Now, let us write the first-order equation (in ), which represents a correction due to a small differential rotation modifying the uniform rotationproblem. This will be

1

r4@

@r

r4eðþÞ=2 @!1

@r

þ eðÞ=2

r2sin 3ðÞ@

@

sin 3ðÞ@!1

@

16eðÞ=2 ðþ pÞ½!1 ð!0 cÞr2sin 2ðÞe ¼ 0:

(4)

Due to the fact that !0 does not depend on , one cansimplify the problem by decomposing the terms in Eq. (4)into the vector spherical harmonics and one obtains (sym-bol ‘‘0’’ means r-derivative),

½r4j!01‘0 ¼ ejr2½f‘ð‘þ 1Þ 2g!1‘ þ 16r2ðþ pÞ

f!1‘ r2eC‘½!0 cg: (5)

Here jðrÞ ¼ eðþÞ=2 and C‘ is a ‘ th coefficient of thedecomposition of sin 2ðÞ into vector spherical harmonics.One can express the decomposition as

sin 2ðÞ ¼ 4

5 2

15

15

2cos 2ðÞ 3

2

; (6)

and thus the only two nonzero coefficients C‘ of thedecomposition are C1 ¼ 4=5 and C3 ¼ 2=15.

The two linearly independent solutions of Eq. (5) behavefor ‘ > 1, both close to zero and at infinity, as

Cþrð2þ‘Þ þ Cr‘1: (7)

The ‘ ¼ 1 case shows the same behavior (7) at the infinity,but the close to zero one has to be more careful. One can tryto Taylor expand the solutions at the origin, proving thatonly one of the solutions is analytic around zero. Then onecan naturally expect, that also in the case ‘ ¼ 1, one of thesolutions is singular at zero. (For more details see (A1).)

It can be easily shown that in the case C‘ ¼ 0, ‘ > 1, theregular behavior of the solution at the infinity cannot bematched with the regular behavior at zero (see again theAppendix). This means for C‘ ¼ 0 no nontrivial relevantsolutions exist. On the other hand, for C‘ 0 we can,through the Green function, construct everywhere regularnontrivial solutions. This means the frame dragging func-tion ! can be expressed in the linear order of as

! ¼ !0ðrÞ þ ½!11ðrÞ þ!13ðrÞf5cos 2ðÞ 1g: (8)

The fact that !0ðrÞ can be analytically solved outsidethe star is a known result [9], and the solution is given as(r > R),

!0ðrÞ ¼ B1

r3: (9)

Moreover, similar to the uniformly rotating case, one canalso find analytic solutions for !11, !13 outside the star.(This is because Eq. (5) can be rewritten outside the starinto the form of the hypergeometric equation.) The analyticsolutions are (r > R),

!11ðrÞ ¼ B2

r3(10)

and

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!13ðrÞ ¼ B3 1

z3 5

z2 30

zþ 210 180z

þ ln

z

z 1

ð120 300zþ 180z2Þ

; (11)

with z ¼ r=2M (M being the mass of the star). Althoughthis is maybe not obvious, the solution for !13 can beshown to behave as r5 when approaching infinity. (Allthe terms with powers higher than r5 cancel out. Thederivation of the solutions and their asymptotic behavior isleft for the Appendix.) Let us also add that the physicalmeaning of the constants B1;2 is the following:

B1 þ B2

2¼ J; (12)

where J is the angular momentum of the star.The procedure for numerically computing the frame

dragging ! is the following: First we numerically deter-mine !0 by solving Eq. (2) inside the star for ¼ 0 (theusual Hartle equation from Ref. [9]) with !0

0ð0Þ ¼ 0 and

!0ð0Þ finite. The value !0ðRÞ then fixes the constant B1

from Eq. (9), determining the behavior of !0 outside thestar. The other components !11 and !13 are obtained fromthe numerical integration of Eq. (5) with ‘ ¼ 1 and 3,respectively. In order to pick the regular solutions, we write!11 and !13 up to the second order in r with regular seriesexpansions near the center as

!11ðrÞ ¼ b0

1þ 8

5ð0 þ p0Þr2

; !13ðrÞ ¼ c0r

2;

and the constants b0, c0 and B2 and B3 [see Eqs. (10)and (11)] are determined with a shooting method, byrequiring that both !11 and !13 and their first derivativesbe continuous on the stellar surface. Finally, the totaldragging !ðr; Þ is computed with Eq. (8).In Figs. 1 and 2, we present some plots of the angular

velocity as a function of the radial coordinate, for differ-ent values of angle and different values of the inverted parameter. We also present (Figs. 3 and 4) plots of theframe dragging ! as a function of the radial coordinate,for different values of angle and different values ofthe inverted parameter. (The star is taken in the units

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 1 2 3 4 5 6 7 8 9

Ω(r

, θ)/

Ωk

r

θ = 0θ = π/4θ = π/2

FIG. 1 (color online). Angular velocity as a function of r fordifferent angles ( ¼ 102).

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0 1 2 3 4 5 6 7 8 9

Ω(r

,θ =

π/2

)/Ω

k

r

uniform rotationA = 30A = 20A = 10A = 8 A = 6 A = 5

FIG. 2 (color online). Angular velocity as a function of r fordifferent values of A ¼ 1=2.

0

0.001

0.002

0.003

0.004

0.005

0.006

0 2 4 6 8 10 12 14 16 18

ω(r

,θ)

r

r = R

θ = 0θ = π/4θ = π/2

FIG. 3 (color online). Frame dragging function as a function ofr for different values of angle ( ¼ 102).

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0 2 4 6 8 10 12 14 16 18

ω(r

,θ =

π/2

)

r

r = R

uniform rotationA = 30A = 20A = 10A = 8 A = 6 A = 5

FIG. 4 (color online). Frame dragging function as a function ofr for different values of A ¼ 1=2.

SLOWLY ROTATING NEUTRON STARS WITH SMALL . . . PHYSICAL REVIEW D 87, 044043 (2013)

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c ¼ G ¼ M ¼ 1 with the compactness M=R ¼ 0:15 andwith the equation of state parameters N ¼ 1, K ¼ 100.This gives the stellar mass to be M ¼ 1:4. Note also thatunless explicitly stated otherwise, we will use everywherein the paper the units c ¼ G ¼ M ¼ 1.) In Figs. 5–7 weshow the dependence of the central and the surface angularvelocities on the parameter. (All the plots are taken at theequatorial plane.) The angular velocities are normalized bythe Keplerian mass shedding limit, K. The minimalbounds on the parameter are given by the equilibriummodel, when either the central angular velocity reaches thevalue 0:8K, or when the surface angular velocity

reaches zero. In this sense the minimal bounds on areobtained naturally in the equatorial plane, as one can easilyanalytically observe that the second bound on , given bythe surface angular velocity, has lowest value in the equa-torial plane. (The first bound given by the central angularvelocity is independent on .) In the next section, aftercomputing the f-modes we further restrict the value of by comparing our results for the f-mode frequencies withthe results of Passamonti et al. [7]. We confirm there is avery good agreement (less than 3% error) up to the value 302, but for 202 the error is already 25%, so thebound (B) on can be put as B & 202.

III. LINEARIZED PERTURBATION EQUATIONSFOR THE FLUID

We work in the Cowling approximation, thus we haveonly fluid perturbation variables, in particular: , p,u. There are two more principles one uses to reducethe number of the variables to four: the four-velocitynormalization condition ðuuÞ ¼ 0, and the fact that

the perturbed fluid is barotropic,

¼ þ p

pp: (13)

The remaining variables are ur, u, u, Q, withQ ¼ p=ðpþ Þ. The dynamical equations are obtainedfrom the three independent components of the perturbedEuler equation ðð

þ uuÞT; Þ ¼ 0 and the perturbed

energy conservation equation ðuT; Þ ¼ 0. The final four

equations for the linearized dynamics of the fluid can bewritten as

-0.2

0

0.2

0.4

0.6

0.8

0 0.001 0.002 0.003 0.004

Ω/Ω

k

γ

FIG. 5 (color online). The solid lines represent the centralangular velocities and the dashed lines the angular velocitiesat the surface as functions of , for different values of angularmomenta. We use the star with the compactness M=R ¼ 0:1 andthe equation of state with N ¼ 1:5, K ¼ 10:86. (The total massof the star is M ¼ 1:47.)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.005 0.01 0.015 0.02

Ω/Ω

k

γ

N = 1.5N = 1.0N = 0.5

FIG. 6 (color online). The solid lines represent the centralangular velocity and the dashed lines the angular velocity atthe surface as functions of , for different stars with differentangular momenta. The mass of the star decreases for differentlines representing different cases from left to right. The compact-ness of the star is in all the three cases M=R ¼ 0:1 and theparameters of the equation of state are (from right to left in theplot) N ¼ 0:5, 1, 1.5, K ¼ 78106, 100, 10.86.

-0.2

0

0.2

0.4

0.6

0.8

0 0.002 0.004 0.006 0.008 0.01 0.012

Ω/Ω

k

γ

M/R = 0.10M/R = 0.15M/R = 0.20

FIG. 7 (color online). The solid line represents the centralangular velocity and the dashed line the angular velocity at thesurface as functions of , for different stars with differentangular momenta. The compactness of the stars grows for differ-ent lines representing different cases from left to right and takesthe values M=R ¼ 0:1, 0.15, 0.2. The star has the equation ofstate with N ¼ 1 and K ¼ 100. (The corresponding stellarmasses are M ¼ 1:06, 1.4, 1.62.)

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Page 5: Slowly rotating neutron stars with small differential rotation: Equilibrium models and oscillations in the Cowling approximation

u;tþ u;þ sin2ðÞ ½!;2cot ðÞ ð!Þ u ¼e=2

r2Q;; (14)

u;t þ u; þð!Þ;r þ

2

r ;r

ð!Þ

ur þ ½ð!Þ; þ 2cot ðÞ ð!Þ u

¼

e=2

r2sin 2ðÞ Q; þ e=2ð!Þ Q;t

; (15)

e ur;t þ e ur; þ r sin 2ðÞ ½r !;r þ ðr ;r 2Þ ð!Þ u ¼ e=2 Q;r; (16)

Q;t þ Q; þ e=2 p

þ p½er2sin 2ðÞ ð!Þ u;t þ ur;r þ u; þ u;

¼ e=2

p

þ p

;r þ ;r

2þ 2

r

;r

2

ur e=2 p

þ p cot ðÞ u: (17)

IV. NUMERICAL RESULTS FOR THE MODES

We used a two-dimensional Lax-Wandroff scheme forsolving the perturbation equations, and the frequencies forthe fluid modes were obtained through the Fourier trans-form of the time evolution of p at a given point inside thestar. For the numerical integration we used the form of theEqs. (14)–(17) rewritten in the variables fp; fig, where fiis a momentumlike variable defined as fi ¼ ðpþ Þui(similarly to what was done in Ref. [13] for Newtonianpolytropes). We used symmetric boundary conditions forp (at the equatorial plane) for the f-modes and antisym-metric boundary conditions for p for the r-modes. Alsowe used the regularity condition at both the radial centerand the rotational axis. For the r-modes we used the initialdata from Ref. [14], and for the f-modes we used the initialvalue conditions from Ref. [13].

In Fig. 8 we have a representative power spectrumobtained from our evolution data. One can see in thisplot the correction to the rotational split added by thedifferential rotation. The relative heights of the peaks arerather arbitrary, and depend only on the initial data used.The width of the peaks is caused by the numerical dissi-pation of the algorithm used: combining that and ourcomparisons with values from the literature, we estimatethat our numerical error in the determination of the fre-quencies (see below) is within 3%.We present in Fig. 9 detailed results for the f-modes in

the appropriate range of validity of . (As previouslymentioned, for < B & 202 the agreement with the

1000 2000 3000 4000 5000f (kHz)

0

1e-06

2e-06

3e-06

4e-06

5e-06

6e-06

7e-06

Pow

er s

pect

rum

uniform rotationdifferential rotation

FIG. 8 (color online). The power spectrum obtained from ourtime evolution data for two stars with the same angular momen-tum J, with uniform (solid line) and differential rotation with ¼ 0:003 (dashed line). Both stars have J ¼ 0:2 and equationof state with N ¼ 1:5, K ¼ 10:86 (compactness M=R ¼ 0:14and mass M ¼ 1:5).

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

10 100 1000

-(σ-

σ 0)/

(Ωe/

ΩK)

(kH

z)

A (km)

N = 1.0N = 1.5

FIG. 9 (color online). The rotational correction for thef-mode frequencies within the range of reliability of A ¼1=2 for the stars with the equation of state with N ¼ 1, 1.5,K ¼ 100, 10.86, and with compactness M=R ¼ 0:15, 0.14, (andmasses M ¼ 1:4, 1.5) in sequences with constant J ¼ 0:2. Thequantity plotted is analogous to the quantities defined inRef. [17] for the uniformly rotating case, with the frequencyof the f for the given value of A, and 0 the frequency of f forthe correspondent nonrotating star.

SLOWLY ROTATING NEUTRON STARS WITH SMALL . . . PHYSICAL REVIEW D 87, 044043 (2013)

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results of Passamonti et al. [7] is within less than 3% error.)We present in this figure an equivalent to the correction tothe frequency given by the rotational splitting of thef-modes in the uniformly rotating case, normalized nowby the surface angular velocity of the star at the equatorialplane,e, in units ofK. The two different data sequencescorrespond to constant J sequences of two polytropic starswith different polytropic indexes N, but approximately thesame compactness. We can see that, for larger value of N,the correction starts with lower values, but grows fasterwith increasing differential rotation. This very fast growthshows a limitation of our first order treatment of thedifferential rotation. As seen in Ref. [7], when second-order terms are taken into account, this growth becomesmuch less steep and much more ‘‘well behaved.’’

We also present in Tables I and II results for the r-modefrequencies, which were not computed in Ref. [7]. Wecomputed r-modes also for the uniformly rotating case,compared our results with Ref. [15], and they agreed againwith less than 3% error. In Ref. [15] they used decompo-sition of the perturbations into spherical harmonics andafter truncating the coupled equations at ‘max , they timeevolved the one-dimensional wave function to obtain thefrequencies. Table I presents the values of the r-modesfrequencies for a sequence of stars with constant angularvelocity at the center c and increasing differential rota-tion, while Table II presents the same results for a sequenceof stars with constant angular momentum J.

In Fig. 10 we present an equivalent to the correctiongiven in Fig. 9, but using now as central value the r-modefrequency for a uniformly rotating star, r0. We used theresults from both Tables I and II in order to calculate theseresults. Note that the effect of the differential rotationseems to be much weaker for the r-modes than for thef-modes (the scale of the vertical axis is now in Hz, and notin kHz, as it was in Fig. 9).

V. AN EFFECT OF DIFFERENTIAL ROTATIONON GRAVITATIONALWAVE EMISSION

FROM THE f-MODE

We here study an effect of differential rotation on gravi-tational radiation by using a simple analysis. We use theNewtonian mass quadrupole formula to evaluate the gravi-tational wave emission. Luminosity of an eigenmode iscomputed by using the eigenfunction extracted by DFT.The luminosity is the quadratic functional of the eigen-function. On the other hand, we compute the kinetic energyof the eigenmode, which is also a quadratic functional ofthe eigenfunction. Taking the ratio of the luminosity andthe energy, we obtain an inverse of the damping timescaleof the eigenmode due to gravitational radiation. By com-paring the timescale for different degrees of differentialrotation, we evaluate how differential rotation affectsgravitational emission from the eigenmode oscillation.

A. Extracting eigenfunctions

We extract the eigenfunction of the f-modes using theprocedure described as follows. On each spatial grid point,we performed a DFT of the physical variables to extracttheir power spectra. An eigenmode corresponds to a peakin the spectrum whose frequency is constant in space and isshared by different physical variables. We approximate thepeak with a Lorentzian profile and extract the central

TABLE II. The table of r-mode frequencies r with constantJ and changing A ¼ 1=2. The star is taken with N ¼ 1,K ¼ 100, M=R ¼ 0:15, J ¼ 0:2135, M ¼ 1:4.

A rðHzÞ cðHzÞ eðHzÞ2262 491.55 347 347

126.14 492.47 349 346

68.03 495.19 354 346

39.26 511.02 371 343

29.24 549.59 393 339

22.06 589.43 436 331

19.91 602.76 463 327

-300

-200

-100

0

100

200

300

400

500

0 0.0005 0.001 0.0015 0.002 0.0025

(σr-

σ r0)

/(Ω

e/Ω

K)

(Hz)

γ

Ωc const.

J const.

FIG. 10 (color online). The equivalent of the rotational cor-rection for the r-modes for sequences with constant J andsequences with constant angular velocity at the center c.This are the values from Tables I and II, so the star is takenwith N ¼ 1, K ¼ 100, M=R ¼ 0:15, (and M ¼ 1:4).

TABLE I. The table of r-mode frequencies r with constantangular velocity at the center (c) and changing A ¼ 1=2. (e

is surface angular velocity at the equatorial plane.) The star is takenwith N ¼ 1, K ¼ 100, M=R ¼ 0:15, c ¼ 347 Hz, M ¼ 1:4.

A rðHzÞ eðHzÞ J

1000 491.55 347 0.2135

500 491.49 347 0.2134

100 490.78 343 0.2114

50 489.92 331 0.2050

40 489.57 321 0.2002

30 486.92 302 0.1898

20 446.90 245 0.1603

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frequency, the peak amplitude, and its width. The collec-tion of the amplitudes on each grid point gives the absolutevalue of the eigenmode excited in the simulation. Weobtained sufficiently smooth eigenfunction profiles forthe f-modes, but we failed to extract higher orderp-modes. It might be that we need to prepare initial datathat contains the p-mode component with a larger ampli-tude than our current cases. In Fig. 11 we present sometypical results obtained for p, fr, f, and f for the

fþ-mode which limits to the ‘ ¼ m ¼ 2 f-mode in thenonrotating limit.

We also obtain the eigenfunctions for the f-mode,which shows qualitatively very similar functional profiles.Although we tried to extract p-mode eigenfunctions, wewere hindered by numerical noise.

B. Radiation timescale

Formulas for the gravitational radiation from stellar os-cillations are found in Ref. [16] in the Newtonian limit. Theenergy of linear perturbation is defined as Eq. (15) there.Since our model is in the Cowling approximation, we haveno gravitational perturbation (which is U in Ref. [16]).Therefore, we approximate the linear perturbation energy as

E ¼Z

vavadV; (18)

where we assume the potential energy is regarded asthe same size as the kinetic energy (i.e., we assume‘‘equipartition’’ as in a simple oscillator).

Energy loss by gravitational radiation is given by

dE

dt¼ X

‘m

ð1Þ‘N‘<d2‘þ1

dt2‘þ1

Dm‘

d

dtDm

‘ imDm‘

; (19)

where

N‘ ¼ 4G

c2‘þ1

ð‘þ 1Þð‘þ 2Þ‘ð‘ 1Þ½ð2‘þ 1Þ!!2 (20)

and

Dm‘ ¼

Zr‘Ym

‘ dV: (21)

Since the amplitude of the eigenmodes extracted fromthe linear evolution is arbitrary, we should study the damp-ing timescale g to characterize the efficiency of gravita-

tional radiation,

1g ¼ 1

E

dE

dt(22)

instead of an absolute amount of energy radiated.To compute g numerically, we need to define and com-

pute the perturbed quantities appearing in the equationsabove. We need to have vi (perturbed three-velocity), (perturbed mass density) and dV (three-volume element).As for the density we used the perturbed rest mass density.The volume element is defined in the spatial hypersurfacewith t ¼ const, where t is Schwarzschild time. The three-metric ij is naturally chosen as ¼ diagðe2; r2; r2sin 2Þ.Thus the corresponding volume element is dV ¼er2 sindrdd’.As for the three-velocity perturbation, we adopted the

definition below. The three-velocity perturbation is ex-pressed by the perturbed four-velocity components as

vi ¼

ui

ut

¼ ui

ut ui

ðutÞ2 ut; (23)

where u is the four-velocity. Then the components of theperturbed velocity (in coordinate basis) are expressed byour basic variables fiði ¼ r; ; ’Þ as

vr ¼ ur

ut¼ fr

ðþ pÞut ; (24)

v ¼ u

ut¼ f

ðþ pÞut ; (25)

v’ ¼ u’

ut u’

ðutÞ2 ut

¼ ½1þð!Þr2sin 21 f’

ðþ pÞut ; (26)

where fiðr; ; ’Þ are defined as before as fi ¼ ðþ pÞui.We have used here

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

δp

R

z

0 1e-08 2e-08

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

fr

R

z

0 2e-08 4e-08 6e-08

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

R

z

0 5e-09 1e-08

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

R

z

0 1e-08 2e-08

FIG. 11 (color online). Eigenfunction of ‘ ¼ jmj ¼ 2 fþmode for a differentially rotating star with A ¼ 25:89. The upperhalf of the meridional section of the star is shown, with R theequatorial coordinate distance and z the coordinate distanceparallel to the rotational axis. Upper left: p. Upper right: fr.Bottom left: f. Bottom right: f. The coordinate distance is

normalized by the stellar radius.

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ut ¼ ½e2 r2sin 2ð!Þ212 (27)

and

ut ¼ 1

utr2sin 2ð!Þ

e2 r2sin 2ð!Þ2 v’: (28)

Together with the equilibrium values of ,, and metriccoefficients, these perturbed variables are used to compute

E, dE=dt, and GW.

C. Results

We compare the gravitational damping timescale GWfor different degrees of differential rotation parametrizedby for a fixed value of total angular momentum of theequilibrium star. In Fig. 12 we plot sequences of GW for the‘¼jmj¼2 f-modes with the angular momentum J ¼ 0:2.

In Fig. 12, GW is normalized by the correspondingtimescale for a uniformly rotating star GW;0. ¼ 0corresponds to the uniformly rotating model. A largervalue of GW means smaller amount of gravitational radia-tion from the eigenmode. We see that the emissivity ofgravitational radiation from each mode is reduced by in-troducing differential rotation. For the counterrotatingf-mode, this may be partly because the eigenfrequency isdecreasing as we increase the degree of differential rota-tion. However it does not explain the increase of GW forthe prograde f-mode, whose frequency increases as weincrease the degree of differential rotation. For the fþmode with differential rotation, the emissivity enhance-ment due to the increase of the frequency may be canceledby a modification of the eigenfunction from that of theuniformly rotating case which reduces mass multipoles.

VI. CONCLUSIONS

In this work we dealt with a slowly rotating relativisticpolytrope, which has a nearly uniform rotation profile. Wegeneralized the old result of Hartle [9] (for the equilibriummodel) for the first-order deviations from the uniformrotation. Similar to Ref. [9] we are able to also providean analytical solution for the metric in the exterior of thestar. Furthermore, we used our equilibrium model to nu-merically compute (in the Cowling approximation) both fand r-mode frequencies. We also estimated the range ofvalidity of our first-order approach in the differential rota-tion parameter . We used the consistency conditions ofthe equilibrium model to constrain the domain of the parameter, and some further restrictions were obtained bycomparing our results for the f-modes with the knownresults in the literature. We provided detailed plots of thef-modes for different polytropes with different compact-ness/angular momenta and also provided some new resultsfor the r-mode frequencies. By using a DFT we extractedthe low order f-mode eigencfunctions from the evolutiondata. With the eignfrequencies and their eigenfunctions,the damping time of the oscillation due to gravitationalradiation was estimated. Along the stellar models with aconstant value of total angular momentum, we see a largerdamping time as we increase the parameter to character-ize the degree of differential rotation. This suggests that theinclusion of the differential rotation with our functionalform tends to suppress the emission of gravitational wavefor f-modes.

ACKNOWLEDGMENTS

This research was supported by FAPESP and the MaxPlanck Society. S. Y. thanks the Center for Mathematics,Computation and Cognition at UFABC for the financialsupport on his stay at UFABC. The authors wish to thankLuciano Rezzolla for useful discussions on r-modesand invaluable help on the development of the time evolu-tion code.

APPENDIX: THE EQUILIBRIUM MODEL

1. Analysis of the regularity of solutions

Let us show that if C‘ ¼ 0, ‘ > 1, there does not exist aneverywhere regular solution. In such case we are lookingfor a solution of the equation,

½r4j!01‘0 ¼ ejr2½‘ð‘þ 1Þ 2þ 16r2ðþ pÞ!1‘:

(A1)

However, if a solution goes to zero at r ! 0 and in thesame time at r ! 1, it must have at some point a localmaximum/local minimum, depending on whether it isapproaching the zero at infinity from the negative values(close to the infinity the solution is negative) or fromthe positive values (close to the infinity the solution is

1

1.1

1.2

1.3

1.4

0 0.001 0.002 0.003 0.004

τ GW

/ τ G

W,0

γ

FIG. 12 (color online). Damping timescale of the eigenmodedue to gravitational radiation, GW, for ‘ ¼ jmj ¼ 2 f-modes.The sequences are obtained by fixing the angular momentum andincreasing , the degree of differential rotation. The timescale isnormalized by that of the same eigenmode in a uniformlyrotating star with the same angular momentum. The solid curveis for the f mode (counterrotating mode) and the dashed one isfor the fþ mode (prograde mode).

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positive). In case !1‘ is close to infinity positive, it musthave a local maximum where!1‘ is positive, in case!1‘ isclose to infinity negative, it must have somewhere a localminimum where!1‘ is negative. Take Eq. (A1) at the pointof such a local minimum/maximum (call it re). Then,due to the fact that the first derivative of !1‘ vanishes atre, Eq. (A1) can be written as

r4ej½!001‘re ¼ ejr2e½‘ð‘þ 1Þ 2þ 16r2eðþ pÞ!1‘:

(A2)

But this equation cannot be fulfilled for a very simplereason: In the case of a local maximum f!00

1‘gre < 0, and

thus the left side of the equation must be negative, whereasthe right side of the equation has always the same sign as!1‘, and!1‘ is at the local maximum positive. (As we saidbefore, the local maximum is taken to be such that!1‘ is atthe local maximum positive.) In the case of local mini-mum, the opposite holds: f!00

1‘gre > 0 and thus the left side

of the equation must be positive, whereas the right side ofthe equation must be negative, because !1‘ is at the localminimum negative. This argumentation means that!1‘ forany ‘ 1, 3 must be represented by a trivial, zero solution.But what about !11 and !13? (With !11, regularity is notan issue, but one still requires that it shall have zero atinfinity.) Here the situation is very different, since on theright side of the Eq. (A2) appears another term propor-tional to C3, and this prevents us from determining the signof the right side of the equation at the extremum of !11,!13. This means both!11,!13 are in the game, and one hasto proceed further in the analysis.

a. Convergence of solutions near zero for ‘ ¼ 1

Let us analyze the solutions in case ‘ ¼ 1 near zero. For‘ ¼ 1 Eq. (5) turns close to 0 to a more complicatedequation,

r!0011 þ 4!0

11 K:r!11 ¼ 0 (A3)

with K ¼ 16½ð0Þ þ pð0Þ. This equation seems not tohave an analytic solution, but one can verify that one andonly one of the solutions is regular at 0 by decomposing!11 into McLaurin series (or Taylor series at zero)

!11 ¼X1n¼0

anrn

and after doing this, one obtains(i) a1 ¼ 0,(ii) anþ2 ¼ K an=ðn2 þ 5nþ 4Þ.

This means there is only one solution that can be decom-posed close to 0 to McLaurin series and that stronglyindicates that the other solution is singular at 0. Thus,qualitatively, the case ‘ ¼ 1 is the same as the other‘ > 1 cases.

2. The frame dragging function solutionsoutside the star

Take the Eq. (5) outside the star. Consider that outsidethe star holds (M being the mass of the star)

e ¼1 2M

r

1: (A4)

Then the Eq. (5) can be rewritten outside the star in theform,

rðr 2MÞ!001‘ þ 4ðr 2MÞ!0

1‘ ½‘ð‘þ 1Þ 2!1‘ ¼ 0:

(A5)

After redefining the variable r ¼: 2M z and some algebra,one can rewrite Eq. (A5) in the following form:

zð1 zÞ!001‘ þ 4ð1 zÞ!0

1‘ þ ½‘ð‘þ 1Þ 2!1‘ ¼ 0:

(A6)

By ‘‘0’’ we mean here a z derivative. Now consider that(A6) is a hypergeometric equation with coefficients thatcan be chosen as(i) a ¼ 2þ ‘,(ii) b ¼ 1 ‘,(iii) c ¼ 4.

[Note that a, b are in fact minus exponents in the asymp-totic formula (7).] Unfortunately, due to the fact that c is aninteger, (A6) cannot be solved by a linear combination of2-1-type hypergeometric functions (multiplied by powersof z), which is a generic solution for hypergeometricequation. Although this cannot be done, let us proceedfurther. The only relevant ‘ are ‘ ¼ 1, 3, for ‘ ¼ 1 wealready know the general solution and this is

D1 z3 þD2: (A7)

For ‘ ¼ 3, Eq. (A6) becomes

zð1 zÞ!0013 þ 4ð1 zÞ!0

13 þ 10!13 ¼ 0: (A8)

The software MATHEMATICA found for Eq. (A8) the follow-ing analytic solution:

!13 ¼D1

2

3 5

3zþ z2

þD2

1

z3 5

z2 30

zþ 210

180zþ ln

z

z 1

ð120 300zþ 180z2Þ

: (A9)

We can see that the first term is the z2 divergent term.It is slightly less obvious that the other term is actuallythe convergent part of the solution behaving as z5. Onehas to substitute z ¼ 1 and take the McLaurin seriesexpansion of

ln

z

z 1

¼ ln

1

1

¼ þ 1

2 2 þ 1

3 3 þ . . .

Then if one expresses the second solution term in ,

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D2 ð3 52 30þ 210 1801 þþ 1

2 2 þ 1

3 3 þ . . .

f120 3001 þ 1802gÞ; (A10)

one can observe (after some computation) that all the terms up to the fifth power cancel. This means!13 term is outside thestar given as

!13 ¼ D2

1

z3 5

z2 30

zþ 210 180zþ ln

z

z 1

ð120 300zþ 180z2Þ

:

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