Astron. Nachr. /AN 335, No. 6/7, 618 – 623 (2014) / DOI 10.1002/asna.201412082

f- and r-modes of slowly rotating stars: New results in the lineartreatment�

C. Chirenti1,��, J. Skakala2, and S. Yoshida3

1 Centro de Matematica, Computacao e Cognicao, UFABC, 09210-170 Santo Andre, Sao Paulo, Brazil2 School of Physics, Indian Institute of Science, Education and Research (IISER-TVM), Trivandrum 695016, India3 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 2014 Apr 30, accepted 2014 May 13Published online 2014 Aug 01

Key words gravitational waves – stars: neutron – stars: oscillations – stars: rotation

Newly born neutron stars can present differential rotation, even if later it should be suppressed by viscosity or a sufficientlystrong magnetic field. In this early stage of its life, a neutron star is expected to have a strong emission of gravitationalwaves, which could be influenced by the differential rotation. We present here a new formalism for modelling differentiallyrotating neutron stars, working on the slow rotation approximation and assuming a small degree of differential rotation.After we establish our equilibrium model, we explore the influence of the differential rotation on the f and r-modes ofoscillation of the neutron star in the Cowling approximation, and we also analyze an effect of the differential rotation onthe emission of gravitational radiation from the f-modes.

c© 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Isolated neutron stars can also become relevant sources ofgravitational radiation. Different families of oscillations canbe excited in the stellar fluid, each one corresponding to adifferent restoring force, and the non-radial oscillations arecoupled to the emission of gravitational waves (for moredetails, see Stergioulas 2003):

– Polar fluid modes:– f-modes (fundamental),– p-modes (pressure),– g-modes (gravity).

– Axial and hybrid modes:– inertial modes,– r-modes (rotation).

– Polar and axial space-time modes.

Differential rotation, until it becomes suppressed by vis-cosity or strong enough magnetic fields (see Etienne, Liu& Shapiro 2006; Duez et al. 2006), might play an impor-tant role in the evolution of a newly born neutron star. Fora typical neutron star it takes between 10–100 years to be-come uniformly rotating, according to Galeazzi, Yoshida &Eriguchi (2012). The equilibrium stellar models represent-ing a neutron star’s differential rotation were explored insome older studies (Hartle 1970; Will 1974), and the oscil-lation frequencies for some types of fluid modes were calcu-lated later in Galeazzi, Yoshida & Eriguchi (2012), Yoshida

� Based on results presented in Chirenti, C., Skakala, J., & Yoshida, S.2013, Phys. Rev.D, 87, 044043.�� Corresponding author: cecilia.chirenti@ufabc.edu.br

et al. (2002), Stavridis, Passamonti & Kokkotas (2007), andPassamonti, Stavridis & Kokkotas (2008).

2 Equilibrium stellar model

Consider the background spacetime of a slowly rotatingstar:

ds2 = −eνdt2 + eλdr2 + r2dθ2 + r2 sin2 θ (dφ− ωdt)2,

where ν and λ are functions of r and ω = ω(r, θ) is theframe dragging function. We will use a polytropic equationof state, p = Kε1+1/N , where p is the pressure and ε isthe rest-mass energy density of the star. The fluid rotation isdescribed by the 4-velocity

u(t,r,θ,φ) = (e−ν/2, 0, 0, Ω e−ν/2) .

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

Ω =Ωc + γ r2 sin2 θ e−νω

1 + γ r2 sin2 θ e−ν. (1)

The γ parameter in Eq. (1) describes the level of differentialrotation of the star. Furthermore, take the differential rota-tion to be also small, representing only a linear order per-turbation from the uniform rotation case: 0 < γ � 1. Thenone can expand ω up to linear order in the γ parameter as

ω = ω0(r) + γ [ ω11(r) + ω13(r){5 cos2 θ − 1}] , (2)

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Astron. Nachr. /AN 335, No. 6/7 (2014) 619

-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 = 0 θ = π/4θ = π/2

Fig. 1 Angular velocity Ω as a function of r for different anglesθ (γ = 10

−2).

-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 Angular velocity Ω as a function of r for different valuesof A = γ−1/2.

and the equations for the radial functions ω0, ω11 and ω13

are given in Chirenti, Skakala, & Yoshida (2013). Outsidethe star (r > R) we have the following analytic solutions:

ω0 =B1

r3, (3)

ω11 =B2

r3, (4)

ω13 = B3

{−

1

z3−

5

z2−

30

z+ 210− 180z +

ln

[z

z − 1

](120− 300z + 180z2)

}, (5)

with z = r/2M (M being the total mass of the star) and ω13

behaves as r−5 for very large r. Also, the physical meaningof the constants B1,2 above is given by the relation

B1 + γB2

2= J, (6)

where J is the total angular momentum of the star.In Figs. 1 and 2, we present some numerical solutions

obtained for the behavior of the angular velocity Ω given by

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 = 0 θ = π/4θ = π/2

Fig. 3 Frame dragging function as a function of r for differentvalues of angle θ (γ = 10

−2).

-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 Frame dragging function as a function of r for differentvalues of A = γ−1/2.

Eq. (1). Some representative solutions for the frame drag-ging ω given by Eq. (2) are shown in Figs. 3 and 4.

In Figs. 5, 6, and 7, we show the dependence of the cen-tral and the surface angular velocities on the γ parameter,all taken at the equatorial plane. The angular velocities arenormalized by the Keplerian mass shedding limit, ΩK. Theminimal bounds on the γ parameter are given by the equilib-rium model, when either the central angular velocity reachesthe value ∼0.8 ΩK, or when the surface angular velocityreaches zero. In this sense the minimal bounds on γ are ob-tained naturally in the equatorial plane, as one can easilyanalytically observe that the second bound on γ, given bythe surface angular velocity, has lowest value in the equa-torial plane. (The first bound given by the central angularvelocity is independent on θ.)

3 Linearized perturbation equations for thefluid

We work in the Cowling approximation, thus we have onlyfluid perturbation variables, in particular: δε, δp, δuμ. There

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620 C. Chirenti, J. Skakala & S. Yoshida: f and r-modes of slowly rotating stars

-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 The solid lines represent the central angular velocities andthe dashed lines the angular velocities at the surface as functions ofγ, for different values of angular momenta. We use the star with thecompactness M/R = 0.1 and the equation of state with N = 1.5,K = 10.86 (the total mass of the star is M = 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 The solid lines represent the central angular velocity andthe dashed lines the angular velocity at the surface as functions ofγ, for different stars with different angular momenta. The mass ofthe star decreases for different lines representing different casesfrom left to right. The compactness of the star is in all the threecases M/R = 0.1, and the parameters of the equation of state are(from right to left) N = 0.5, 1, 1.5, K = 78106, 100, 10.86.

are two more principles one uses to reduce the number ofthe variables to four: the four-velocity normalization con-dition δ(uμuμ) = 0, and the fact that the perturbed fluid isbarotropic:

δε =ε + p

Γpδp . (7)

The remaining variables are δur, δuθ, δuφ, δQ, with δQ =δp/(p + ε). The dynamical equations are obtained from thethree independent components of the perturbed Euler equa-tion δ((δμ

κ + uμuκ)T κν;ν ) = 0 and the perturbed energy

conservation equation δ(uκT κν;ν ) = 0. The final four equa-

-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 The solid line represents the central angular velocity andthe dashed line the angular velocity at the surface as functions ofγ, for different stars with different angular momenta. The com-pactness of the stars grows for different lines representing differentcases from left to right and takes the values M/R = 0.1, 0.15, 0.2.The star has the equation of state with N = 1 and K = 100. (Thecorresponding stellar masses are M = 1.06, 1.4, 1.62.)

tions for the linearized dynamics of the fluid can be foundin Chirenti, Skakala & Yoshida (2013).

4 Numerical results for the modes

We used a 2D Lax-Wandroff scheme for solving the per-turbation equations and the frequencies for the fluid modeswere obtained through the Fourier transform of the timeevolution of δp at a given point inside the star. For thenumerical integration we used the form of the perturba-tion equations rewritten in the variables {δp, f i}, wheref i is a momentum-like variable defined as f i = (p + ε)δui

(similarly to what was done in Jones, Andersson & Ster-gioulas 2002 for Newtonian polytropes). We used symmet-ric boundary conditions for δp (at the equatorial plane) forthe f-modes and antisymmetric boundary conditions for δpfor the r-modes. Also we used the regularity condition atboth the radial center and the rotational axis. For the r-modes we used the initial data from Owen et al. (1998) andfor the f-modes we used the initial value conditions fromJones, Andersson & Stergioulas (2002).

In Fig. 8, we have a representative power spectrum ob-tained from our evolution data. One can see in this plot thecorrection to the rotational split added by the differentialrotation. The relative heights of the peaks are rather arbi-trary, and depend only on the initial data used. The widthof the peaks is caused by the numerical dissipation of thealgorithm used: combining that and our comparisons withvalues from the literature, we estimate that our numericalerror in the determination of the frequencies (see below) iswithin 3 %.

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Astron. Nachr. /AN 335, No. 6/7 (2014) 621

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 The power spectrum obtainedfrom our time evolution data for twostars with the same angular momentumJ , with uniform (solid line) and differ-ential rotation with γ = 0.003 (dashedline). Both stars have J = 0.2 and equa-tion of state with N = 1.5, K = 10.86(compactness M/R = 0.14 and massM = 1.5).

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

10 100 1000

-(σ

-σ0)

/(Ω

e/Ω

K)

(kH

z)

A (km)

N = 1.0N = 1.5

Fig. 9 The rotational correction for the f−-mode frequencieswithin the range of reliability of A = γ−1/2 for stars with theequation of state with N = 1, 1.5, K = 100, 10.86, and withcompactness M/R = 0.15, 0.14, (and masses M = 1.4, 1.5) insequences with constant J = 0.2. The quantity plotted is analo-gous to the quantities defined in Yoshida & Kojima (1967) for theuniformly rotating case, with σ the frequency of the f− for thegiven value of A, and σ0 the frequency of f− for the correspon-dent non-rotating star.

We present in Fig. 9 detailed results for the f-modes inthe appropriate range of validity of γ.1 We present in thisfigure an equivalent to the correction to the frequency givenby the rotational splitting of the f-modes in the uniformlyrotating case, normalized now by the surface angular veloc-ity of the star at the equatorial plane, Ωe, in units of ΩK.The two different data sequences correspond to constant Jsequences of two polytropic stars with different polytropic

1 As previously mentioned, for γ < γB � 20−2 the agreement with

the results of Passamonti, Stavridis & Kokkotas (2008) is within less than3 % error.

-300

-200

-100

0

100

200

300

400

500

0 0.0005 0.001 0.0015 0.002 0.0025

(σr-

σr0

)/(Ω

e/Ω

K)

(Hz)

γ

Ωc const.J const.

Fig. 10 The equivalent of the rotational correction for the r-modes for sequences with constant J and sequences with constantangular velocity at the center Ωc. The star is taken with N = 1,K = 100, M/R = 0.15 (and M = 1.4).

indexes N , but approximately the same compactness. Wecan see that, for larger value of N , the correction starts withlower values, but grows faster with increasing differentialrotation. This very fast growth shows a limitation of our firstorder treatment of the differential rotation. As seen in Pas-samonti, Stavridis & Kokkotas (2008), when second orderterms are taken into account, this growth becomes much lesssteep and much more “well-behaved”.

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, σr0. Note that theeffect of the differential rotation seems to be much weakerfor the r-modes than for the f-modes (the scale of the verticalaxis is now in Hz, and not in kHz as it is in Fig. 9).

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622 C. Chirenti, J. Skakala & S. Yoshida: f and r-modes of slowly rotating stars

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

fθ

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

fφ

R

z

0 1e-08 2e-08Fig. 11 Eigenfunction of � = |m| = 2 f+ mode for a differentially rotating star with A = 25.89. The upper half of the meridionalsection of the star is shown, with R the equatorial coordinate distance and z the coordinate distance parallel to the rotational axis. topleft: δp; top right: fr; bottom left: fθ; bottom right: fφ. The coordinate distance is normalized to the stellar radius.

5 The effect of differential rotation ongravitational wave emission from the f-mode

We here study the effect of differential rotation on gravi-tational radiation by using a simple analysis. We use theNewtonian mass quadrupole formula to evaluate the grav-itational wave emission. Luminosity of an eigenmode iscomputed by using the eigenfunction extracted by DFT.The luminosity is the quadratic functional of the eigenfunc-tion. On the other hand, we compute the kinetic energy ofthe eigenmode, which is also a quadratic functional of theeigenfunction. Taking the ratio of the luminosity and the en-ergy, we obtain an inverse of the damping timescale of theeigenmode due to gravitational radiation. By comparing thetimescale for different degrees of differential rotation, weevaluate how differential rotation affects gravitational emis-sion from the eigenmode oscillation.

In Fig. 11 we present some typical results obtained forδp, fr, fθ , and fφ for the f+-mode which limits to the � =m = 2 f-mode in the non-rotating limit.

We compare the gravitational damping timescale τGW

for different degrees of differential rotation parametrized byγ for a fixed value of total angular momentum of the equi-librium star. In Fig. 12, we plot sequences of τGW for the� = |m| = 2 f-modes with the angular momentum J = 0.2.

In Fig. 12, τGW is normalized by the correspondingtimescale for a uniformly rotating star τGW,0. γ = 0 cor-responds to the uniformly rotating model. A larger value of

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 Damping timescale of the eigenmode due to gravita-tional radiation, τGW, for � = |m| = 2 f-modes. The sequencesare obtained by fixing the angular momentum and increasing γ,the degree of differential rotation. The timescale is normalized bythat of the same eigenmode in a uniformly rotating star with thesame angular momentum. The solid curve is for the f− mode (ret-rograde mode) and the dashed one is for the f+ mode (progrademode).

τGW means smaller amount of gravitational radiation fromthe eigenmode. We see that the emissivity of gravitationalradiation from each mode is reduced by introducing differ-ential rotation. For the counter-rotating f-mode, this may bepartly because the eigenfrequency is decreasing as we in-

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Astron. Nachr. /AN 335, No. 6/7 (2014) 623

crease the degree of differential rotation. However it doesnot explain the increase of τGW for the prograde f-mode,whose frequency increases as we increase the degree of dif-ferential rotation. For the f+ mode with differential rota-tion, the emissivity enhancement due to the increase of thefrequency may be cancelled by a modification of the eigen-function from that of the uniformly rotating case which re-duces mass multipoles.

6 Conclusions

In this work we dealt with a slowly rotating relativistic poly-trope, such that has a nearly uniform rotation profile. Wegeneralized the old result of Hartle (1967), (for the equi-librium model), for the first order deviations from the uni-form rotation. Similar to Hartle (1967) we are able to alsoprovide an analytical solution for the metric in the exteriorof the star. Furthermore, we used our equilibrium model tonumerically compute (in the Cowling approximation) bothf- and r-mode frequencies. We also estimated the range ofvalidity of our first order approach in the differential rota-tion parameter γ: We used the consistency conditions of theequilibrium model to constrain the domain of the γ param-eter, and some further restrictions were obtained by com-paring our results for the f-modes with the known resultsin the literature. We provided detailed plots of the f-modesfor different polytropes with different compactness/angularmomenta and also provided some new results for the r-mode frequencies. By using a DFT we extracted the loworder f-mode eigenfunctions from the evolution data. Withthe eigenfrequencies and their eigenfunctions, the dampingtime of the oscillation due to gravitational radiation was es-timated. Along the stellar models with a constant value oftotal angular momentum, we see a larger damping time aswe increase the γ parameter to characterize the degree ofdifferential rotation. This suggests that the inclusion of thedifferential rotation with our functional form tends to sup-press the emission of gravitational wave for f-modes.

Acknowledgements. This research was supported by FAPESP andthe Max Planck Society. SY thanks the Center for Mathematics,Computation and Cognition at UFABC for the financial supporton his stay at UFABC. The authors wish to thank Luciano Rez-zolla for useful discussions on r-modes and invaluable help on thedevelopment of the time evolution code, and also the organizers ofthe STARS2013 and SMFNS2013 conference for their support.

References

Chirenti, C., Skakala, J., & Yoshida, S. 2013, Phys. Rev. D, 87,044043

Jones, D.I., Andersson, & N.,Stergioulas, N. 2002, MNRAS, 334,933

Hartle, J.B. 1967, ApJ, 150, 1005Hartle, J.B. 1970, ApJ, 161, 111Yoshida, S., & Kojima, Y. 1997, MNRAS, 289, 117Stavridis, A., Passamonti, A., & Kokkotas, K. 2007, Phys. Rev. D,

75, 064019Passamonti, A., Stavridis, A., & Kokkotas, K. 2008, Phys. Rev. D,

77, 024029Owen, B.J., Lindblom, L.,Cutler, C., et al. 1998, Phys. Rev. D, 58,

084020Etienne, Z.B., Liu, Y.T., & Shapiro, S.L. 2006, Phys. Rev. D,

74,044030Duez, M.D., Liu, Y.T., Shapiro, S.L., Shibata, M., & Stephens,

B.C. 2006, Phys. Rev. D, 73, 104015Stergioulas,N. 2003, Living Rev. Relativity, 6,3Galeazzi, F., Yoshida, S., & Eriguchi, Y. 2012, A&A, 541, A156Yoshida, S., Rezzolla, L., Karino, S., & Eriguchi, Y. 2002, ApJ,

568, L41Will, C.M. 1974, ApJ, 190, 403

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