Rotational parameters of strange stars in comparison with neutron stars

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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 whereas the subtraction doesthe reverse thing. On the other hand, for strange stars, Rgf3 Z2gis always much greater than Req and even after the addition or sub-

    traction of the smaller termRg 3 Z13 Z1f 2Z21=2g, it (rms) re-mains greater than Req.

    In Fig.8, I plot thevariationofmkrasa functionof r. Nosignicantdifference in mkr (at any chosen r) betweena strange star andaneu-tron star having the same values ofM andX is observed (specially athigher values of r). In comparison to thewhole range of mkr, the dif-ferences between mkr for the co-rotating and the counter-rotatingmotions (keeping all of the other parameters xed) and the variationof mkr with X (keeping all of the other parameters xed) are verysmall. jmk;cor mk;counterr=mk;counterrj varies around 0.20.001for r 0 to 500 km depending slightly upon the choice of the EoS,M and X; jmk;0:4r mk;0:2r=mk;0:2rj varies 0.10.001 for r 0 to500 km (where the third parameter in the subscript denotes the va-lue ofX in units of 104 s1) depending slightly upon the choice of theEoS,M and the direction of the motion. So as an example, the plot inFig. 8 is only for co-rotating motion with X 0:4 104 s1 and foronly two chosen values ofM.

    But these differences between mk;cor and mk;counterr become lar-ger for smaller values of r where mk;cor < mk;counterr, but at suf-ciently higher values of r; mk;corJ mk;counterr. This transitionoccurs at r 26 27 km for both strange stars and neutron starswith M 1:5M and X 0:4 104. Similar trends have been no-ticed for other values of M and X. The value of r at which this tran-sition occurs increases slightly with the increase of M but does notdepend signicantly with X.

    SUNM ( M )

    6 1.2 1.4 1.6 1.8 2 2.2 2.4

    Fig. 6. Variation of rms;co with the mass for strange stars (upper panel) and neutronstars (lower panel). The values of X in units of 104 s1 are shown. The EsoS used areEoS A for strange stars and EoS APR for neutron stars.

  • nomSUNM ( M )

    r ms,

    co

    un

    ter(k

    m)

    SS

    0.2

    0.3

    0.40.5

    10.5

    11

    11.5

    12

    12.5

    13

    13.5

    14

    14.5

    15

    15.5

    1.1 1.2 1.3 1.4 1.5 1.6

    r ms,

    co

    un

    ter(k

    m)

    NS

    0.20.30.40.5

    12

    14

    16

    18

    20

    22

    24

    M. Bagchi / New AstroWe will now study the variation of mkReq, i.e. mkr at r ReqwithM taking different values of X and both co-rotating and coun-ter-rotating motions. mk;coReq is the Keplerian frequency of a par-ticle orbiting the star at the stars equatorial surface and I obtain itby equating the gravitational force with the centrifugal force.

    In Figs. 9 and 10 I plot mkReq with the stellar mass for the co-rotating and the counter-rotating motions. mk;coReq > mk;counterReqalways. Let us concentrate mainly on Fig. 9 as mk;coReq can repre-sent the rotational frequency of both the constituent particles ofthe star and the accreting material at the equatorial surface ofthe star whereas mk;counterReq can represent only the accretingmaterial. I see that mk;coReq > mspin in the chosen range of M andmspin. Fig. 11 shows that the difference Dm mk;coReq mspin de-creases at higher mspin and/or lower M. The condition Dm 0 isthe mass shedding limit as for mk;coReq < mspin, matter from thestar would y way due to the centrifugal force. The spin frequencywhere the mass shedding starts (i.e. Dm 0) is the maximumpossible rotational frequency of the star and it is known as theKeplerian frequency of the star mk mspin mk;coReq

    and the cor-

    responding angular frequency is called as the Keplerian angularfrequency Xk of the star. For xed values of M andmspin;DmSSA > DmAPR. This implies that the mass shedding limit willbe reached in case of neutron stars at comparatively a lower valuefor mspin than for strange stars which means that strange stars aremore stable than neutron stars against rotations (we have checkedthat this fact remains true even if one uses BAG model EoS forstrange stars). The Keplerian angular frequency Xk for compactstars has been studied in the literature by different groups likeFriedman et al. (1986), Haensel and Zdunik (1989), Lattimer et al.(1990) and many other authors.

    SUNM ( M )

    10 1.2 1.4 1.6 1.8 2 2.2 2.4

    Fig. 7. Variation of rms;counter with the mass 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.1.1 MSUN

    1.5 MSUN

    k, c

    o

    (r) H

    z

    r (km)

    SS

    1

    10

    100

    1000

    10000

    100000

    1e+06

    0 50 100 150 200 250 300 350 400 450 500

    1.5 MSUN1.1 MSUN

    k, c

    o

    (r) H

    z

    NS

    10

    100

    1000

    10000

    100000

    1e+06

    y 15 (2010) 126134 130Haensel and Zdunik (1989) proposed an analytic relation be-tween Xk and maximum allowable static mass Mmax and the cor-responding radius Rmax

    Xk 7:7 103MmaxM

    0:5 Rmax10 km

    1:56

    With their choice of neutron star EsoS, Lattimer et al. (1990) foundXk to lie between 0:76 104 1:6 104 s1 from Eq. (6). Using Bagmodel EsoS for strange stars, Prakash et al. (1990) foundXk 6 1:0 104 s1 for M > 1:44M.

    Afterwards, Lattimer and Prakash (2004) gave a more usefulexpression of Xk of a star of mass M and non-rotating radius R.

    Xk 6:6 103MM

    0:5 R10 km

    1:57

    To test these two simple analytical expressions (Eqs. (6) and(7)), I run the task kepler in RNS code which produces stellar con-gurations for stars rotating with Xk. Using the RNS outputs (i.e.M;R; I;Xk), I calculate mk;coReq from Eq. (4). At Xk, I should getXk 2pmk;coReq (by the denition of Xk).

    In Fig. 12, I plot the variation of Xk with mass. The line labeled(1) is the maximum limit of Xk given by Haensel and Zdunik(1989), i.e. Eq. (6). The curve labeled (2) is from the analyticalexpression given by Lattimer and Prakash (2004), i.e. Eq. (7). Thecurve labeled (3) is the output of the RNS code and the curve la-beled (4) is 2pmk;coReq.

    For neutron stars (APR EoS) the curves (24) are very close toeach other which supports the correctness of both the analytical

    r (km)

    1 0 50 100 150 200 250 300 350 400 450 500

    Fig. 8. Variation of mkr with the mass for strange stars (upper panel) and neutronstars (lower panel) for X 0:4 104 s1. Stellar masses are taken as 1:1M and1:5M . The EsoS used are EoS A for strange stars and EoS APR for neutron stars.

  • (6) which is 10;695 s1. The strange stars are very oblate havingRp=Req 0:38 0:42.

    The probable fastest spin frequency of a neutron star (XTEJ1739-285) is mspin 1122 Hz or X 7049:734 s1, which is lessthan the value of Xk of both strange stars and neutron stars as de-rived from Eq. (6) or even less than as derived from Eq. (7) for acanonical value of the stellar mass as M 1:5M (see Fig. 12). Sothere is no problem of this star being either a strange star or a neu-tron star. One needs to conclude about its nature by other observa-tional evidences.

    Several other people performed numerical studies on structuresof rapidly rotating compact stars like Friedman and Ipser (1992),Weber and Glendenning (1992), Cook et al. (1994), Eriguchi et al.(1994), Salgado et al. (1994a,b), Gourgoulhon et al. (1999), Bhatta-charyya et al. (2001), Bombaci et al. (2000), Gondek-Rosinska et al.(2000), Bhattacharyya and Ghosh (2005), Haensel et al. (2008,2009). In these, people usually kept themselves conned in study-ing the properties of stars like M; c;Req; T=W , etc. or at most prop-erties particles rotating at the stellar surface. But the use of thepseudo-Newtonian potential enables us to study the propertiesof particles orbiting around the star at a much higher distancer Req and I report quantities like rms;co and rms;counter; mk;cor;mk;counterr which are useful to study accretion onto rotating neu-tron stars or strange stars (see next section). I also report variousstellar parameters M;R; I;Rp=Req; a=Rg ; mk;coReq; mk;counterReq, etc.There is no previous work where all this parameters were reported

    omy 15 (2010) 126134

    k,

    co

    eq(R )

    Hz

    SUNM ( M )

    SS

    0.3

    0.4

    0.5

    0.2

    3400

    3600

    3800

    4000

    4200

    4400

    4600

    4800

    5000

    5200

    5400

    1.1 1.2 1.3 1.4 1.5 1.6

    k, c

    o

    eq(R )

    Hz

    NS

    0.30.4

    0.5

    0.2

    2000

    2500

    3000

    3500

    4000

    4500

    5000

    131 M. Bagchi / New Astronexpression of Lattimer and Prakash (2004) (Eq. (7)) and the pseu-do-Newtonian potential of Mukhopadhyay and Misra (2003). Themaximum value of Xk obtained using Eq. (6) is 11;519 s1. Herethe Kerr parameter a=Rg lies around 0.66 which is much greaterthan the values at lower frequencies (Fig. 5) and the neutron starsare very oblate having Rp=Req 0:59 0:56.

    For EoS A, the RNS code failed to perform the task kepler forc < 1:51 1015 g cm3 and at c 1:51 1015 g cm3, I getM 1:61M which is greater than Mmax for static conguration.Moreover, at this mass, the value of the Kerr parameter a=Rgis greater than 1, which is unphysical. At c P 1:581015 g cm3 M P 1:72M, a=Rg becomes less than 1, but veryhigh 0:9. So no direct comparison of Xk obtained with theRNS code with the analytical expressions are possible and inFig. 1 only plot the analytical expressions. Using Eq. (7), I getXk 13;086 s1 at c 1:51 1015 g cm3 and increases with theincrease of c (orM). This value of Xk is less than the maximum va-lue of Xk obtained using Eq. (6) which is 14;940 s

    1. Here thestrange stars are very oblate with Rp=Req 0:38 0:47. To checkwhether all these facts are intrinsic to strange star properties or de-pend upon the particular model of strange stars, I have evenchecked with the BAG model having model parameters as:B 60:0 MeV=fm3;ms 150:0 MeV;mu md 0;ac 0:17 whereB is the Bag parameter, ms;mu;md are masses of s;u and d quarks,respectively, giving static Mmax 1:82M. Here also I get a=Rg > 1at lower c; a=Rg < 1 when c P 0:82 1015 g cm3 M P2:33M). Using Eq. (7), I get X

    k 8756 s1 at c 0:82

    1015 g cm3 which increases with increase of c (or M). This valueof Xk is less than the maximum value of X

    k obtained using Eq.

    together. Also I use one EoS for strange stars and another EoS forneutron stars whereas in the earlier works people discussed eitheronly neutron star rotations or only strange star rotations. I also dis-cuss about kHz QPOs within the scenario of Mukhopadhyay et al.(2003) in the next section.

    SUNM ( M ) 1000

    1500

    1.2 1.4 1.6 1.8 2 2.2 2.4

    Fig. 9. Variation of mk;coReq with the mass for strange stars (upper panel) andneutron stars (lower panel). The parameter is the value of X in units of 104 s1

    corresponding to mspin 318 Hz, 477 Hz, 637 Hz and 796 Hz, respectively. The EsoSused are EoS A for strange stars and EoS APR for neutron stars.SUNM ( M )

    eq(R

    ) H

    z k

    , cou

    nter

    0.5

    0.2

    0.3

    0.4

    SS

    3500

    4000

    4500

    5000

    5500

    6000

    6500

    1.1 1.2 1.3 1.4 1.5 1.6

    eq(R

    ) H

    z k

    , cou

    nter

    SUNM ( M )

    0.2

    0.30.4 0.5

    NS

    1000

    1500

    2000

    2500

    3000

    3500

    4000

    4500

    5000

    5500

    1.2 1.4 1.6 1.8 2 2.2 2.4

    Fig. 10. Variation of mk;counterReq with the mass for strange stars (upper panel) and

    neutron stars (lower panel). The parameter is the value of X in units of 104 s1

    corresponding to mspin 318 Hz, 477 Hz, 637 Hz and 796 Hz, respectively. The EsoSused are EoS A for strange stars and EoS APR for neutron stars.

  • 1636-53 has been taken from Jonker et al. (2002). Note that theQPO frequencies of 4U 1636-53 shift with time and hence I havetaken only one set of values as an example.

    Table 1 shows the rotational parameters for KS 1731-260 and4U 1636-53. Table 2 shows the values of the radius of the Keplerianorbit (rk) obtained by equating mkrwith QPO frequencies. I choosethe stellar mass to be 1:1M;1:5M and 1:9M. For a given star(i:e: mspin is xed), the values of rms depend signicantly on (a)whether I consider the co-rotating or the counter-rotating motion,(b) on the choice of the EoS and (c) on the chosen value of M. Butthe values of rk depend signicantly only on M when I keep mspinand mk xed.

    For KS 1731-260, rk;low rk;up 2 km for both strange stars andneutron stars (independent of the choice of mass) and for both co-rotating and counter-rotating motions whereas for 4U 1636-53this value is 4 km.

    rk;low rms 47 for KS 1731-260 and 710 km for 4U 1636-53 for co-rotating motions, 29 km for KS 1731-260 and 810 km for 4U 1636-53 for counter-rotating motions; whereasrk;up rms 15 km for KS 1731-260 and 2.56 km for 4U1636-53 for co-rotating motions rk;up rms 1:5 to 2:5 km forKS 1731-260 and 0.5 to 3.5 km for 4U 1636-53 for counterrotating motions. These values decrease with the increase of Mand note that at M 1:9M; rk;up;counter < rms. Remember, Mmax Req.

    The beat frequency model suggests that rk;up is the radius ofthe innermost Keplerian orbit of the rin of the accretion diskwhereas the models by Titarchuk and Osherovich (1999) andMukhopadhyay et al. (2003) suggest that rk;low rin. The diskparameter rin can be estimated by X-ray spectral analysis. As rel-ativistic broadening is more dominant at the innermost edge ofthe disk, the existence of a relativistically broadened iron K a linehelps one to determine the value of rin (Reis et al., 2009; Di Salvoet al., 2009 and references therein). It is clear from Table 2 thatfor a xed set of M and mspin; rin is different for different EsoS,but the difference is always

  • Salgado, M., Bonazzola, S., Gourgoulhon, E., Haensel, P., 1994a. A&A 291, 155.Salgado, M., Bonazzola, S., Gourgoulhon, E., Haensel, P., 1994b. A&A 108 (Suppl.),

    455.Strohmayer, T.E., Zhang, W., Swank, J.H., et al., 1996. Astrophys. J. 469, L9.Titarchuk, L., Osherovich, V., 1999. Astrophys. J. 518, L95.

    Weber, F., Glendenning, N., 1992. Astrophys. J. 390, 541.Wijnands, R.A.D., van der Klis, M., 1997. Astrophys. J. 482, L65.Yin, H.X., Zhang, C.M., Zhao, Y.H., et al., 2007. Astron. Astrophys. 471, 381.Zhang, C.M., Yin, H.X., Kojima, Y., et al., 2007. Mon. Not. Royal Astron. Soc. 374, 232.

    M. Bagchi / New Astronomy 15 (2010) 126134 134

    Rotational parameters of strange stars in comparison with neutron starsIntroductionStellar structures with rotationRotational parametersApplications in kHz QPO modelsDiscussionReferences

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