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ce. thor s

and improved future techniques for observations and data analysis. 2009 Elsevier B.V. All rights reserved.

efforts

reproduced as mentioned by Klahn et al. (2006). Moreover, to con-strain EsoS from QPO observations, one needs to believe in a partic-ular model of QPO which is again a subject of debate. Anotheralternative method might be the measurement of the moment ofinertia from the faster component of the double pulsar systemPSR J0737-3039 (Lattimer and Schutz, 2005; Bagchi et al., 2009).Some high mass stars like PSR J1903+0327, EXO 0748-676, etc. pre-fer stiff EoS and some other stars like 4U 1728-34 (Li et al., 1999),

and neutron stars. Although stellar structures for rotating neutronstars or strange stars have been already studied by a number ofgroups (some of which I discuss in Section 3), systematic studiesof all relevant stellar parameters as well as disk parameters werelacking. That is why here I report the variation of a number of dif-ferent stellar parameters as well as disk parameters for differentvalues of stars spin frequency and mass in Sections 2 and 3. AlsoI use one EoS for strange stars and another EoS for neutron starswhereas in the earlier works people discussed either only neutronstar rotations or only strange star rotations, there was no compari-son between the properties of rotating neutron stars and rotating

* Fax: +91 22 2280 4610.

New Astronomy 15 (2010) 126134

Contents lists availab

tr

.e lsE-mail address: manjari@tifr.res.inmatter Equations of State (EsoS) through astronomical observationsof compact stars. The usual approach is to determine the mass andthe radius of the stars with the help of various observational fea-tures like gravitational redshifts (z) from spectral lines, coolingcharacteristics, kHz quasi-periodic oscillations (QPO), etc. (Lattimerand Prakash, 2007; Li et al., 1999; zel, 2006; zel et al., 2009;Zhang et al., 2007). But these methods are not foolproof, e.g. the va-lue of z used in zels (2006) analysis of EXO 0748-676 can not be

of EsoS for neutron stars and also for strange stars. Until then, it isinteresting to compare the stellar properties for different EsoS. Forsufciently fast spinning stars, stellar structures depend upon thespin frequency (mspin). So the study of stellar structures for rotatingstars will help in better understanding of the characteristics of fastspinning compact stars like LMXBs and millisecond pulsars. That iswhy here I study stellar structures with rotations in Section 2. InSection 3, I study different rotational parameters for strange starsPACS:26.60.Kp97.10.Nf97.10.Pg97.60.Jd97.80.Jp97.10.Kc

Keywords:Dense matterEquation of stateX-rays: binariesStars: neutronStars: rotation

1. Introduction

Presently there are a number of1384-1076/$ - see front matter 2009 Elsevier B.V. Adoi:10.1016/j.newast.2009.07.003to constrain the dense

EXO 1745-248 (zel et al., 2009), prefer soft EoS. This fact hintsto the possibility of existence of both neutron stars and strangestars. But even then, I need some constrains as there are a numberAccepted 3 July 2009Available online 9 July 2009Communicated by E.P.J. van den Heuvel

values of the radii of the marginally stable orbits and Keplerian orbital frequencies. By equating kHz QPOfrequencies to Keplerian orbital frequencies, I nd corresponding orbital radii. Knowledge about theseparameters might be useful in further modeling of the observed features from LMXBs with advancedRotational parameters of strange stars in

Manjari Bagchi *

Tata Institute of Fundamental Research, Colaba, Mumbai 400 005, India

a r t i c l e i n f o

Article history:Received 22 June 2009Received in revised form 3 July 2009

a b s t r a c t

I study stellar structures, i.different spin frequencies f

New As

journal homepage: wwwll rights reserved.omparison with neutron stars

e mass, the radius, the moment of inertia and the oblateness parameter attrange stars and neutron stars in a comparative manner. I also calculate the

le at ScienceDirect

onomy

evier .com/locate /newast

SUN

M (

M

)

c (10 gm / cm )15 3

00.2 0.3

0.40.5

SS

1.1

1.2

1.3

1.4

1.5

1.6

1 1.5 2 2.5 3 3.5 4 4.5

SUN

M (

M

)

NS

c (10 gm / cm )15 3

00.20.30.4

0.5

1.2

1.4

1.6

1.8

2

2.2

2.4

0.5 1 1.5 2 2.5 3 3.5

Fig. 1. Variation of the mass with the central density for strange stars (upper panel)and neutron stars (lower panel). The parameter is the value of X in units of 104 s1.The EsoS used are EoS A for strange stars and EoS APR for neutron stars.

SUN

M (

M

)

R (km)eq

00.2

0.30.4

0.5

SS

1.1

1.2

1.3

1.4

1.5

1.6

7 7.2 7.4 7.6 7.8 8

SUN

M (

M

)

R (km)eq

NS

0.20 0.3 0.4 0.5 1.2

1.4

1.6

1.8

2

2.2

2.4

9.5 10 10.5 11 11.5 12 12.5 13 13.5

127 M. Bagchi / New Astronomy 15 (2010) 126134strange stars. In addition I compare my results obtained by using apseudo-Newtonian potential with full general relativistic calcula-tions by other people like Haensel and Zdunik (1989), Lattimerand Prakash (2004) and the close matching found implies the cor-rectness of my approach and the validity of the pseudo-Newtonianpotential. In Section 4 I discuss a possible application of the knowl-edge of the rotational parameters in modeling kHz QPOs. I end witha discussion in Section 5.

2. Stellar structures with rotation

I use two sample EsoS of the dense matter among the numerousEsoS available in literature, one for the strange quark matter (EoS Aor SSA, Bagchi et al., 2006) and the other for the nuclear matter(EoS APR Akmal et al., 1998). To nd stellar structures with rota-tions, I use the RNS code.1 Following Komatsu et al. (1989), this codeconstructs the compact star models by solving stationary, axisym-metric, uniformly rotating perfect uid solutions of the Einstein eldequations with tabulated EsoS (supplied by the users).

The fastest rotating compact star known till date is probably XTEJ1739-285 (Kaaret et al., 2007) having mspin 1122 Hz, although themeasurementhasnotbeenconrmed later. The second fastest one isJ1748-2446ad (Hessels et al., 2006)with mspin 716 Hz. In thiswork,I choose the angular frequency (X 2pmspin as 2000, 3000, 4000 and5000 s1 (which correspond to mspin as 318Hz, 477 Hz, 637 Hz and796 Hz, respectively). All of the fast rotating compact stars exceptXTE J1739-285 have mspin in that range. I have also computed non-rotating, spherically symmetric stellar structures by solving TOVequations which are sufcient for slow objects like EXO 0748-676(mspin 45 Hz). Throughout this work, I take the stellar mass (M) tobealways greater than1.1M asobservationsusuallyhint the stellarmass to be greater than that value.

In Fig. 1 I plot the mass against the central density (c) both forstrange stars and neutron stars. For a xed value of c;M increaseslittle bit with the increase of X. For all of the values of X;M rst in-

creaseswith the increase of c @M@c > 0

and then after a certain value

ofMMmax starts to decrease @M@c < 0

. The stars are unstable when

@M@c

< 0. This instability appears around c 4:1 1015 g cm3 forstrange stars and around c 2:8 1015 g cm3 for neutron stars;these values does not change more than 5% with the change of X inthe chosen range. For any X in the chosen range, I get M 1:1Mat c 1:7 1015 g cm3 for strange stars and at c 0:801015 g cm3 for neutron stars.

Fig. 2 shows the massradius plots. With the increase of X, forboth strange stars and neutron stars, the maximummass Mmax in-creases and for any xed mass, the radius also increases due to thelarger value of the centrifugal force. Note that here radius meansthe equatorial radius Req which is always greater than the polar ra-diusRp. For a xedX, themaximummass for a neutron star is greaterthan that of a strange star. ForxedvaluesofX andM;Req is larger fora neutron star than that of a strange star. The compactness factorM=R of strange stars is larger than that of neutron stars and the var-iation ofMwith Req follows an approximate R

3eq law for strange stars

in contrast to neutron stars approximate R3eq variation.Fig. 3 shows the variation of the moment of inertia I with the

mass. For any xed mass, the moment of inertia increases with theincrease of X both for strange stars and neutron stars. For xed val-ues of M and X, a neutron star possess much higher value of I thana strange star because of its larger value of Req.

In Fig. 4, I plot the oblateness parameter, i.e. the ratio of the po-lar radius to the equatorial radius Rp=Req with the mass. It is clearFig. 2. Variation of the mass with the radius for strange stars (upper panel) andneutron stars (lower panel). The parameter is the value of X in units of 104 s1. TheEsoS used are EoS A for strange stars and EoS APR for neutron stars.1 .

nomSUNM ( M )

I ( 10

gm

cm )

452

0.2 0.3 0.40.5

SS

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1.1 1.2 1.3 1.4 1.5 1.6

I ( 10

gm

cm )

452

NS

0.2

0.50.4

0.3

1

1.2

1.4

1.6

1.8

2

2.2

2.4

M. Bagchi / New Astrothat for a xed mass, this ratio decreases with the increase of Xboth for strange stars and neutron stars, i.e. the star becomes moreand more oblate due to the larger values of the centrifugal force.Moreover, for a xed value of X, this ratio decreases with the de-crease of the mass as there the centrifugal force becomes increas-ingly more dominant over the gravitational force. Note that thevariation of Rp=Req with M is steeper for neutron stars than thatfor strange stars, but for both of them, the steepness increases withthe increase of X.

In Fig. 5, I plot a=Rg with M where Rg GM=c2 and a IX=Mc.As expected from their expressions, the plot shows that for anyxed mass, a=Rg increases with the increase of X as expected andfor a xed X; a=Rg decreases with the increase of the mass. Forthe same value of M and X, a neutron star has larger value ofa=Rg than that of a strange star because of its larger value of I.a=Rg is an important parameter of the compact stars as it can beidentied as the specic angular momentum of the star and its va-lue determines many other properties of the star.

For any other EoS, the value of Mmax and corresponding radiuswill change depending upon the stiffness of that EoS. But the gen-eral trend of the MR curve will remain the same, i.e. M / R3eq forstrange stars and M / R3eq for neutron stars. The nature of M ccurve will also remain the same.

3. Rotational parameters

With the output of the RNS code, i:e: M;Req and I, I calculatesome rotational parameters for strange stars and neutron stars.First I calculate the radius of the marginally stable orbit which isdened as (Bardeen et al., 1972):

SUNM ( M ) 0.8

1.2 1.4 1.6 1.8 2 2.2 2.4

Fig. 3. Variation of the moment of inertia with the mass for strange stars (upperpanel) and neutron stars (lower panel). The parameter is the value of X in units of104 s1. The EsoS used are EoS A for strange stars and EoS APR for neutron stars.SUNM ( M )

R

/ Rp

eq

0.5

0.4

0.3

0.2

SS

0.96

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1.1 1.2 1.3 1.4 1.5 1.6

R /

Rp

eq

NS

0.2

0.3

0.4

0.5 0.8

0.85

0.9

0.95

1

y 15 (2010) 126134 128rms Rg 3 Z2 3 Z13 Z1 2Z2 1=2n o

1

where

Z1 1 1 a=Rg2h i1=3

1 a=Rg1=3 1 a=Rg1=3h i

2

and

Z2 3a=Rg2 Z21h i1=3

3

The sign in the expression of rms implies the co-rotating motionand the + sign implies the counter-rotating motion which I call asrms;co and rms;counter , respectively. As the values of a=Rg are alwaysvery small, both Z1 and Z2 have their values 3.

The Keplerian frequency of a particle orbiting around the star ata radial distance r can be expressed as

mkr 12pFmrr

1=24

where Fmr is the force per unit mass. As an example, I take Fmr asderived from a pseudo-Newtonian potential by Mukhopadhyay andMisra (2003)

Fmr Rgc2

r21 rms

r

rms

r

2 5

In Figs. 6 and 7, I plot rms;co and rms;counter , respectively, with themass. For any EoS, rms;co is always smaller than rms;counter for anyxed values of X and M. For a xed X, both rms;co and rms;counter in-creases linearly with the increase of M. But for a xed M; rms;codecreases with the increase of X and rms;counter increases with the

SUNM ( M )

0.75 1.2 1.4 1.6 1.8 2 2.2 2.4

Fig. 4. Variation of the oblateness parameter, i.e. the ratio of the polar radius to theequatorial radius with the central density for strange stars (upper panel) andneutron stars (lower panel). The parameter is the value of X in units of 104 s1. TheEsoS used are EoS A for strange stars and EoS APR for neutron stars.

omSUNM ( M )

a / R

g

0.3

0.2

0.4

0.5

SS

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

1.1 1.2 1.3 1.4 1.5 1.6

a / R

g

0.2

0.3

0.4

0.5

NS

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

129 M. Bagchi / New Astronincrease of X. For strange stars, both rms;co and rms;counter are alwaysgreater than Req. For neutron stars, rms;co and rms;counter are smallerthan Req for low masses and they become greater than Req for high-er masses; rms;co becomes equal to Req around 1:5 1:7M andrms;counter becomes equal to Req around 1:15 1:2M.

All these happen because of the nature of the terms in theexpression of rms (Eq. (1)). As both Rgf3 Z2g andRg 3 Z13 Z1 2Z21=2n o

are positive quantities and the sec-

ond one is smaller than the rst one, their sum rms;counter mustbe greater than their difference (rms;co). As for a xed value of X,

both Rgf3 Z2g and Rg 3 Z13 Z1 2Z21=2n o

increases with

the increase of M with the rst term having much steeper slope,both rms;co and rms;counter increase with increase of M. For a xedmass, Rgf3 Z2g remains almost constant butRg 3 Z13 Z1 2Z21=2n o

increases with the increase of X,

so their sum (rms;counter) increases with the increase of X and the dif-ference (rms;co) decreases with the increase of X.

For strange stars (EoS A), Rgf3 Z2g Req for any value ofM. Soafter addition or subtraction of a comparatively small term

Rg 3 Z13 Z1 2Z21=2n o

with it, the expression (rms;counteror rms;co) remain always greater than Req.

For neutron stars (EoS APR), Rgf3 Z2g < Req at lower values ofM where the values for Req are sufciently larger, butRgf3 Z2g > Req at larger values of M. Here Rg 3 Z13fZ1 2Z21=2g is very small in comparison to both Rgf3 Z2g andReq for all values of M. So the addition or subtraction of

Rg 3 Z13 Z1 2Z21=2n o

with Rgf3 Z2g (to get rms for coun-

SUNM ( M )

0.1 1.2 1.4 1.6 1.8 2 2.2 2.4

Fig. 5. Variation of a=Rg with the mass for strange stars (upper panel) and neutronstars (lower panel). The parameter is the value of X in units of 104 s1. The EsoSused are EoS A for strange stars and EoS APR for neutron stars.SUNM ( M )

SS

r ms,

co(k

m)

0.20.3

0.40.5

8.5

9

9.5

10

10.5

11

11.5

12

12.5

13

1.1 1.2 1.3 1.4 1.5 1.6

r ms,

co(k

m)

0.20.30.40.5

NS 8

10

12

14

16

18

20

y 15 (2010) 126134ter-rotating or co-rotating motions, respectively) does not changethe overall trend, only the addition (for counter-rotation) shifts thetransition towards lower values of M...