New Astronomy 15 (2010) 126–134
Contents lists available at ScienceDirect
New Astronomy
journal homepage: www.elsevier .com/locate /newast
Rotational parameters of strange stars in comparison with neutron stars
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 2009Accepted 3 July 2009Available online 9 July 2009Communicated by E.P.J. van den Heuvel
PACS:26.60.Kp97.10.Nf97.10.Pg97.60.Jd97.80.Jp97.10.Kc
Keywords:Dense matterEquation of stateX-rays: binariesStars: neutronStars: rotation
1384-1076/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.newast.2009.07.003
* Fax: +91 22 2280 4610.E-mail address: [email protected]
a b s t r a c t
I study stellar structures, i.e. the mass, the radius, the moment of inertia and the oblateness parameter atdifferent spin frequencies for strange stars and neutron stars in a comparative manner. I also calculate thevalues of the radii of the marginally stable orbits and Keplerian orbital frequencies. By equating kHz QPOfrequencies to Keplerian orbital frequencies, I find corresponding orbital radii. Knowledge about theseparameters might be useful in further modeling of the observed features from LMXBs with advancedand improved future techniques for observations and data analysis.
� 2009 Elsevier B.V. All rights reserved.
1. Introduction
Presently there are a number of efforts to constrain the densematter 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; Özel, 2006; Özel et al., 2009;Zhang et al., 2007). But these methods are not foolproof, e.g. the va-lue of z used in Özel’s (2006) analysis of EXO 0748-676 can not bereproduced 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),
ll rights reserved.
EXO 1745-248 (Özel 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 numberof EsoS for neutron stars and also for strange stars. Until then, it isinteresting to compare the stellar properties for different EsoS. Forsufficiently 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 starsand 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 star’s 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
SUN
M (
M
)
ε 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
ε 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 s�1.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) 126–134
strange 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 find 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 fluid solutions of the Einstein fieldequations 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 ¼ 1122 Hz, although themeasurement has not been confirmed later. The second fastest one isJ1748-2446ad (Hessels et al., 2006) with mspin ¼ 716 Hz. In this work,I choose the angular frequency (X ¼ 2pmspinÞ as 2000, 3000, 4000 and5000 s�1 (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 sufficient for slow objects like EXO 0748-676(mspin ¼ 45 Hz). Throughout this work, I take the stellar mass (M) tobe always greater than 1.1 M� as observations usually hint 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 fixed value of �c;M increaseslittle bit with the increase of X. For all of the values of X;M first in-
creases with the increase of �c@M@�c> 0
� �and then after a certain value
of MðMmaxÞ starts to decrease @M@�c< 0
� �. The stars are unstable when
@M@�c< 0. This instability appears around �c ¼ 4:1� 1015 g cm�3 for
strange stars and around �c ¼ 2:8� 1015 g cm�3 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 ¼ 1:1M�
at �c � 1:7� 1015 g cm�3 for strange stars and at �c � 0:80�1015 g cm�3 for neutron stars.
Fig. 2 shows the mass–radius plots. With the increase of X, forboth strange stars and neutron stars, the maximum mass ðMmaxÞ in-creases and for any fixed 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-dius Rp. For a fixed X, the maximum mass for a neutron star is greaterthan that of a strange star. For fixed values of X and M;Req is larger fora neutron star than that of a strange star. The compactness factorðM=RÞ of strange stars is larger than that of neutron stars and the var-iation of M with Req follows an approximate R3
eq law for strange starsin contrast to neutron star’s approximate R�3
eq variation.Fig. 3 shows the variation of the moment of inertia ðIÞ with the
mass. For any fixed mass, the moment of inertia increases with theincrease of X both for strange stars and neutron stars. For fixed 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 ðRp=ReqÞ with the mass. It is clear
Fig. 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 s�1. TheEsoS used are EoS A for strange stars and EoS APR for neutron stars.1 <http://www.gravity.phys.uwm.edu/rns>.
SUNM ( M )
I (
10
gm
cm
)45
2
0.2 0.3 0.4
0.5
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
)45
2
SUNM ( M )
NS
0.2
0.50.4
0.3
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
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 s�1. 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
SUNM ( M )
NS
0.2
0.3
0.4
0.5
0.75
0.8
0.85
0.9
0.95
1
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 s�1. TheEsoS used are EoS A for strange stars and EoS APR for neutron stars.
M. Bagchi / New Astronomy 15 (2010) 126–134 128
that for a fixed 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 fixed 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 ¼ GM=c2 and a ¼ IX=Mc.As expected from their expressions, the plot shows that for anyfixed mass, a=Rg increases with the increase of X as expected andfor a fixed 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 beidentified as the specific 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 M–R curve will remain the same, i.e. M / R3
eq forstrange stars and M / R�3
eq for neutron stars. The nature of M � �c
curve 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 isdefined as (Bardeen et al., 1972):
rms ¼ Rg 3þ Z2 � ð3� Z1Þð3þ Z1 þ 2Z2Þ½ �1=2n o
ð1Þ
where
Z1 ¼ 1þ 1� ða=RgÞ2h i1=3
ð1þ a=RgÞ1=3 þ ð1� a=RgÞ1=3h i
ð2Þ
and
Z2 ¼ 3ða=RgÞ2 þ Z21
h i1=3ð3Þ
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
mkðrÞ ¼1
2pFmðrÞ
r
� �1=2
ð4Þ
where FmðrÞ is the force per unit mass. As an example, I take FmðrÞ asderived from a pseudo-Newtonian potential by Mukhopadhyay andMisra (2003)
FmðrÞ ¼Rgc2
r2 1� rms
r
� �þ rms
r
� �2� �
ð5Þ
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 anyfixed values of X and M. For a fixed X, both rms;co and rms;counter in-creases linearly with the increase of M. But for a fixed M; rms;co
decreases with the increase of X and rms;counter increases with the
SUNM ( 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
SUNM ( M )
0.2
0.3
0.4
0.5
NS
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
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 s�1. 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
SUNM ( M )
r ms,
co(k
m)
0.20.30.40.5
NS
6
8
10
12
14
16
18
20
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 s�1 are shown. The EsoS used areEoS A for strange stars and EoS APR for neutron stars.
129 M. Bagchi / New Astronomy 15 (2010) 126–134
increase 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 Rgf3þ Z2g and
Rg ½ð3� Z1Þð3þ Z1 þ 2Z2Þ�1=2n o
are positive quantities and the sec-
ond one is smaller than the first one, their sum ððrms;counterÞ mustbe greater than their difference (rms;co). As for a fixed value of X,
both Rgf3þ Z2g and Rg ½ð3� Z1Þð3þ Z1 þ 2Z2Þ�1=2n o
increases with
the increase of M with the first term having much steeper slope,both rms;co and rms;counter increase with increase of M. For a fixedmass, Rgf3þ Z2g remains almost constant but
Rg ½ð3� Z1Þð3þ Z1 þ 2Z2Þ�1=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), Rgf3þ Z2g � Req for any value of M. Soafter addition or subtraction of a comparatively small term
ðRg ½ð3� Z1Þð3þ Z1 þ 2Z2Þ�1=2n o
Þ with it, the expression (rms;counter
or rms;co) remain always greater than Req.For neutron stars (EoS APR), Rgf3þ Z2g < Req at lower values of
M where the values for Req are sufficiently larger, butRgf3þ Z2g > Req at larger values of M. Here Rg ½ð3� Z1Þð3þfZ1 þ 2Z2Þ�1=2g is very small in comparison to both Rgf3þ Z2g andReq for all values of M. So the addition or subtraction of
Rg ½ð3� Z1Þð3þ Z1 þ 2Z2Þ�1=2n o
with Rgf3þ Z2g (to get rms for coun-
ter-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, Rgf3þ Z2gis always much greater than Req and even after the addition or sub-
traction of the smaller term Rg ½ð3� Z1Þð3þ Z1þf 2Z2Þ�1=2g, it (rms) re-mains greater than Req.
In Fig. 8, I plot the variation ofmkðrÞas a function of r. No significantdifference in mkðrÞ (at any chosen r) between a strange star and a neu-tron star having the same values of M and X is observed (specially athigher values of r). In comparison to the whole range of mkðrÞ, the dif-ferences between mkðrÞ for the co-rotating and the counter-rotatingmotions (keeping all of the other parameters fixed) and the variationof mkðrÞ with X (keeping all of the other parameters fixed) are verysmall. j½mk;coðrÞ � mk;counterðrÞ�=½mk;counterðrÞ�j varies around 0.2–0.001for r ¼ 0 to 500 km depending slightly upon the choice of the EoS,M and X; j½mk;0:4ðrÞ � mk;0:2ðrÞ�=½mk;0:2ðrÞ�j varies 0.1–0.001 for r ¼ 0 to500 km (where the third parameter in the subscript denotes the va-lue of X in units of 104 s�1) 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 ¼ 0:4� 104 s�1 and foronly two chosen values of M.
But these differences between mk;coðrÞ and mk;counterðrÞ become lar-ger for smaller values of r where mk;coðrÞ < mk;counterðrÞ, but at suffi-ciently higher values of r; mk;coðrÞJ mk;counterðrÞ. This transitionoccurs at r � 26� 27 km for both strange stars and neutron starswith M ¼ 1:5M� and X ¼ 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 significantly with X.
SUNM ( M )
r ms,
cou
nter
(km
)
SS
0.2
0.3
0.4
0.5
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
SUNM ( M )
r ms,
cou
nter
(km
)
NS
0.2
0.3
0.4
0.5
10
12
14
16
18
20
22
24
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 s�1. TheEsoS used are EoS A for strange stars and EoS APR for neutron stars.
1.1 MSUN
1.5 MSUN
k, c
oν
(r)
Hz
r (km)
SS
1
10
100
1000
10000
100000
1e+06
0 50 100 150 200 250 300 350 400 450 500
1.5 MSUN
1.1 MSUN
r (km)
k, c
oν
(r)
Hz
NS
1
10
100
1000
10000
100000
1e+06
0 50 100 150 200 250 300 350 400 450 500
Fig. 8. Variation of mkðrÞ with the mass for strange stars (upper panel) and neutronstars (lower panel) for X ¼ 0:4� 104 s�1. Stellar masses are taken as 1:1M� and1:5M� . The EsoS used are EoS A for strange stars and EoS APR for neutron stars.
M. Bagchi / New Astronomy 15 (2010) 126–134 130
We will now study the variation of mkðReqÞ, i.e. mkðrÞ at r ¼ Req
with M taking different values of X and both co-rotating and coun-ter-rotating motions. mk;coðReqÞ is the Keplerian frequency of a par-ticle orbiting the star at the star’s equatorial surface and I obtain itby equating the gravitational force with the centrifugal force.
In Figs. 9 and 10 I plot mkðReqÞ with the stellar mass for the co-rotating and the counter-rotating motions. mk;coðReqÞ > mk;counterðReqÞalways. Let us concentrate mainly on Fig. 9 as mk;coðReqÞ 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;counterðReqÞ can represent only the accretingmaterial. I see that mk;coðReqÞ > mspin in the chosen range of M andmspin. Fig. 11 shows that the difference Dm ¼ mk;coðReqÞ � mspin de-creases at higher mspin and/or lower M. The condition Dm ¼ 0 isthe ‘‘mass shedding limit” as for mk;coðReqÞ < mspin, matter from thestar would fly way due to the centrifugal force. The spin frequencywhere the ‘‘mass shedding” starts (i.e. Dm ¼ 0) is the maximumpossible rotational frequency of the star and it is known as theKeplerian frequency of the star mk ¼ mspin ¼ mk;coðReqÞ
� �and the cor-
responding angular frequency is called as the Keplerian angularfrequency ðXkÞ of the star. For fixed 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 ðXkÞ 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.
Haensel and Zdunik (1989) proposed an analytic relation be-tween Xk and maximum allowable static mass ðMmaxÞ and the cor-responding radius ðRmaxÞ
Xk ¼ 7:7� 103 Mmax
M�
� 0:5 Rmax
10 km
� �1:5
ð6Þ
With their choice of neutron star EsoS, Lattimer et al. (1990) foundXk to lie between 0:76� 104 � 1:6� 104 s�1 from Eq. (6). Using Bagmodel EsoS for strange stars, Prakash et al. (1990) foundXk 6 1:0� 104 s�1 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 ¼ 6:6� 103 MM�
� 0:5 R10 km
� �1:5
ð7Þ
To test these two simple analytical expressions (Eqs. (6) and(7)), I run the task ‘‘kepler” in RNS code which produces stellar con-figurations for stars rotating with Xk. Using the RNS outputs (i.e.M;R; I;Xk), I calculate mk;coðReqÞ from Eq. (4). At Xk, I should getXk ¼ 2pmk;coðReqÞ (by the definition 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;coðReqÞ.
For neutron stars (APR EoS) the curves (2–4) are very close toeach other which supports the correctness of both the analytical
k, c
oν
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
oν
eq(R
)
Hz
SUNM ( M )
NS
0.30.4
0.5
0.2
1000
1500
2000
2500
3000
3500
4000
4500
5000
1.2 1.4 1.6 1.8 2 2.2 2.4
Fig. 9. Variation of mk;coðReqÞ with the mass for strange stars (upper panel) andneutron stars (lower panel). The parameter is the value of X in units of 104 s�1
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
zν 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
zν k,
cou
nter
SUNM ( M )
0.2
0.30.40.5
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;counterðReqÞ with the mass for strange stars (upper panel) andneutron stars (lower panel). The parameter is the value of X in units of 104 s�1
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.
131 M. Bagchi / New Astronomy 15 (2010) 126–134
expression 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 s�1. 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” for�c < 1:51� 1015 g cm�3 and at �c ¼ 1:51� 1015 g cm�3, I getM ¼ 1:61M� which is greater than Mmax for static configuration.Moreover, at this mass, the value of the Kerr parameter a=Rg
is greater than 1, which is unphysical. At �c P 1:58�1015 g cm�3 ðM P 1:72M�Þ, a=Rg becomes less than 1, but veryhigh ð� 0:9Þ. 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 ¼ 13;086 s�1 at �c ¼ 1:51� 1015 g cm�3 and increases with theincrease of �c (or M). 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 ¼ 60:0 MeV=fm3
;ms ¼ 150:0 MeV;mu ¼ md ¼ 0;ac ¼ 0:17 whereB is the Bag parameter, ms;mu;md are masses of s;u and d quarks,respectively, giving static Mmax ¼ 1:82M�. Here also I get a=Rg > 1at lower �c; a=Rg < 1 when �c P 0:82� 1015 g cm�3 ðM P2:33M�). Using Eq. (7), I get Xk ¼ 8756 s�1 at �c ¼ 0:82�1015 g cm�3 which increases with increase of �c (or M). This valueof Xk is less than the maximum value of Xk obtained using Eq.
(6) which is 10;695 s�1. 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 ¼ 1122 Hz or X ¼ 7049:734 s�1, 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 ¼ 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 confined 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 distanceðr � ReqÞ and I report quantities like rms;co and rms;counter; mk;coðrÞ;mk;counterðrÞ 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;coðReqÞ; mk;counterðReqÞ, etc.There is no previous work where all this parameters were reportedtogether. 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 )
SS
ΔνH
z
0.50.4
0.3
0.2
2400
2600
2800
3000
3200
3400
3600
3800
4000
4200
4400
1.1 1.2 1.3 1.4 1.5 1.6
ΔνH
z
SUNM ( M )
0.2
0.30.4
0.5NS
0
500
1000
1500
2000
2500
3000
3500
1.2 1.4 1.6 1.8 2 2.2 2.4
Fig. 11. Variation of Dm ¼ mk;coðReqÞ � mspin with the mass for strange stars (upperpanel) and neutron stars (lower panel). The parameter is the value of X in units of104 s�1. The EsoS used are EoS A for strange stars and EoS APR for neutron stars.
SUNM ( M )
(1)
(2)
ΩΚ*
SS
0
2000
4000
6000
8000
10000
12000
14000
16000
0 0.5 1 1.5 2 2.5 3
SUNM ( M )
NS
ΩΚ*
(1)
(2)
(3)
(4)
0
2000
4000
6000
8000
10000
12000
14000
16000
0 0.5 1 1.5 2 2.5 3
Fig. 12. Variation of Xk with mass. The line labeled (1) is the maximum limit of Xkgiven by Haensel and Zdunik (1989). The curve labeled (2) is from the analyticalexpression given by Lattimer and Prakash (2004). The curve labeled (3) is theoutput of RNS code and the curve labeled (4) is 2pmK;coðReqÞ. The EsoS used are EoS Afor strange stars and EoS APR for neutron stars.
M. Bagchi / New Astronomy 15 (2010) 126–134 132
4. Applications in kHz QPO models
kHz QPOs in the LMXBs are very interesting phenomena. Themost popular model to explain kHz QPOs is the beat frequencymodel (Strohmayer et al., 1996) which assumes the upper QPO(mup) as the Keplerian orbital frequency mkðrÞ of the innermost orbitin the accretion disk around the star, the separation between thetwo peaks, i.e. Dmpeak ¼ mup � mlow as mspin and the lower QPO (mlow)as the beat frequency of mkðrÞ and mspin. This model suggest thatfor a particular star Dmpeak ¼ mspin will remain constant.
However, it has been observed that in many sources, Dmpeak
changes with time. This phenomenon discards the reliability ofthe beat frequency model for the QPOs. Two examples of alternatemodels can be found in Titarchuk and Osherovich (1999) andMukhopadhyay et al. (2003). Both of these later two models sug-gest mlow ¼ mkðrÞ. But in the present work, I don’t prefer any partic-ular QPO model over the others.
We take two sample LMXB for which kHz QPOs have been ob-served, namely KS 1731-260 with mspin ¼ 524 Hz and 4U 1636-53mspin ¼ 581 Hz. As none of the above mentioned models for kHzQPOs is established beyond doubt, I fit both mup and mlow to mkðrÞto find the corresponding radial distance r ¼ rk (from Eq. (4))which I call rk;up and rk;low, respectively. Mukhopadhyay et al.(2003) calculated rk;low for these two sources taking mass–radiusvalues for non-rotating strange stars and using a=Rg (J in their nota-tion) as an parameter. Here I get rk;up and rk;low taking exact valuesof Req and a=Rg as generated by the RNS code with the specific val-ues of mspin and chosen values of M (as masses of these two stars arenot well determined). QPO frequencies for KS 1731-260 has beentaken from Wijnands and van der Klis (1997) and those for 4U
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 mkðrÞwith QPO frequencies. I choosethe stellar mass to be 1:1M�;1:5M� and 1:9M�. For a given star(i:e: mspin is fixed), the values of rms depend significantly 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 significantly only on M when I keep mspin
and mk fixed.For KS 1731-260, rk;low � rk;up � 2 km for both strange stars and
neutron 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 � 4—7 for KS 1731-260 and �7–10 km for 4U 1636-53 for co-rotating motions, �2–9 km for KS 1731-260 and �8–10 km for 4U 1636-53 for counter-rotating motions; whereasrk;up � rms � 1—5 km for KS 1731-260 and �2.5–6 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 ¼ 1:9M�; rk;up;counter < rms. Remember, Mmax <
1:9M� for EoS A for the values of X of KS 1731-260 and 4U1636-53.
Table 1Rotational parameters for two LMXBs.
Source mspin (Hz) M ðM�Þ EoS Req (km) I ð1045 g cm2Þ Rp=Req a=Rg rms;co (km) rms;counter (km)
KS 1731-260 524 1.1 SSA 7.344 0.564 0.982 0.174 8.801 10.6561.1 NS 11.874 0.975 0.918 0.301 8.081 11.297
4U 1636-53 581 1.1 SSA 7.348 0.565 0.978 0.194 8.694 10.7541.1 NS 12.015 0.990 0.897 0.339 7.860 11.485
KS 1731-260 524 1.5 SSA 7.688 0.908 0.984 0.151 12.179 14.3661.5 NS 11.634 1.494 0.940 0.248 11.435 15.042
4U 1636-53 581 1.5 SSA 7.699 0.911 0.981 0.168 12.051 14.4851.5 NS 11.718 1.508 0.926 0.278 11.205 15.244
KS 1731-260 524 1.9 NS 11.255 2.032 0.959 0.210 14.894 18.7234U 1636-53 581 1.9 NS 11.310 2.045 0.949 0.235 14.617 18.9357
Table 2Keplerian radius (rk) by fitting kHz QPOs for two LMXBs.
Source mlow (Hz) mup (Hz) M ðM�Þ EoS rk;low;co (km) rk;low;counter (km) rk;up;co (km) rk;up;counter (km)
KS 1731-260 898.3 1158.6 1.1 SSA 15.135 15.341 12.923 13.2451.1 NS 15.099 15.442 12.835 13.379
4U 1636-53 688 1013 1.1 SSA 18.030 18.098 14.018 14.3101.1 NS 18.061 18.166 13.953 14.448
KS 1731-260 898.3 1158.6 1.5 SSA 17.071 17.477 14.758 15.2551.5 NS 16.958 17.618 14.609 15.420
4U 1636-53 688 1013 1.5 SSA 20.080 20.378 15.891 16.3961.5 NS 20.023 20.504 15.748 16.572
KS 1731-260 898.3 1158.6 1.9 NS 18.766 19.631 16.346 17.3074U 1636-53 688 1013 1.9 NS 21.896 22.662 17.507 18.530
133 M. Bagchi / New Astronomy 15 (2010) 126–134
5. Discussion
We study the rotational parameters for strange stars and neu-tron stars using the RNS code which consider the generalized axi-symmetric metric for rotating stars. I find that the rotationalparameters like Mmax;Req; I;Rp=Req; a=Rg depend on mspin and thechoice of the EoS. The value of rms depends on the stellar mass,the choice of the EoS and whether the motion is co-rotating orcounter-rotating. The dependence of mkðrÞ on the choice of EoS, Xand the direction of motion is prominent only at low rð6 ReqÞ. Thatis why, the values of rk as obtained by fitting the kHz QPO frequen-cies do not depend much on the choice of the EoS and on the direc-tion of motion as here rk > Req.
The beat frequency model suggests that rk;up is the radius ofthe innermost Keplerian orbit of the ðrinÞ of the accretion diskwhereas the models by Titarchuk and Osherovich (1999) andMukhopadhyay et al. (2003) suggest that rk;low ¼ 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 fixed set of M and mspin; rin is different for different EsoS,but the difference is always <1%, whereas presently rin is mea-sured only accuracy up to 20% (Reis et al., 2009; Di Salvo et al.,2009). Moreover, one should remember that this determinationof rin depends upon the QPO model which is not beyond doubtat the present moment. Indeed there is a strong need of betterexplanation of kHz QPOs to understand various features like theshift of the peaks, their correlations (Belloni et al., 2007; Yinet al., 2007), side-bands (Jonker et al., 2005), etc. Considering allthese facts, I conclude that, to constrain dense matter Eos bymeasuring rin using X-ray spectral analysis, I need much betteraccuracy and hopefully future advanced technology like ASTRO-SAT will provide us such ultra-high accuracy.
In this work I have primarily confined myself to a maximum va-lue of mspin as 796 Hz. But then I have also studied the properties ofthe compact stars if they rotate with the maximum spin frequency,i.e. the Keplerian spin frequency.
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