Table Of Content(DOI: will be inserted by hand later)
The applications of the MHD Alfv´en wave oscillation model for
kHz quasi-periodic oscillations
C.M. Zhang1,⋆, H.X. Yin1, Y.H. Zhao1, H.K. Chang2, and L.M. Song3
1 National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China
2 Department of Physicsand Instituteof Astronomy,National Tsing HuaUniversity,Hsinchu30013, Taiwan
3 Astronomical Institute,InstituteofHighEnergy Physics,ChineseAcademyofSciences,Beijing 100039, China
thedate of receipt and acceptance should beinserted later
7
0 Abstract. Inthispaper,weimprovethepreviousworkontheMHDAlfv´enwaveoscillationmodelfortheneutron
0 star(NS)kHzquasi-periodicoscillations (QPOs),andcomparethemodelwiththeupdatedtwinkHzQPOdata.
2 For the 17 NS X-ray sources with the simultaneously detected twin kHz QPO frequencies, the stellar mass M
n and radius R constraints are given by means of the derived parameter A in the model, which is associated with
a the averaged mass density of star as hρi=3M/(4πR3)≃2.4×1014(g/cm3)(A/0.7)2, and we also compare the
J M-R constraints with thestar equations of states. Moreover, wealso discuss thetheoretical maximum kHzQPO
0 frequencyandmaximumtwinpeakseparation, andsomeexpectationsonSAXJ1808.4-3658 arementioned,such
3 as its highest kHz QPO frequency ∼ 870 (Hz), which is about 1.4-1.5 times less than those of the other known
kHzQPOsources. Theestimated magneticfieldsfor bothZ sources(about Eddington accretion rate M˙ ) and
Edd
4 Atoll sources (∼1%M˙ ) are approximately ∼109 G and ∼108 Grespectively.
Edd
v
3
Key words.accretion: accretion disks– stars:neutron – binaries: close – X-rays: stars
3
2
8
0 1. Introduction linearcorrelationsbetweenthe twinkHzQPOpeaks(e.g.
4 Zhangetal.2006a),andithasbeenimpliedbytheAlfv´en
0 The Rossi X-Ray Timing Explorer (RXTE) has observed oscillation model by Zhang (2004)and relativistic preces-
/ the kilo-Hertz quasi-periodic oscillations (kHz QPOs) in
h sion model by Stella & Vietri (1999). Moreover, the kHz
the X-ray flux in about 25 accretingneutron stars in low-
p QPO frequencies also follow the rather tight correlations
- mass X-ray binaries (LMXBs), and the twin peak kHz between themselves and with the other timing features of
o
r QPOs, the upper and lower frequencies (ν2 and ν1), are the X-ray emissions (see, e.g., Zhang et al. 2006a; Psaltis
t usuallyshownintheFourierpowerspectrum(e.g.,vander
s etal.1998;Psaltisetal.1999ab;Stellaetal.1999;Belloni
a Klis2000,2005,2006fortherecentreviews).Remarkably, et al. 2002; Titarchuk & Wood 2002), and the ratios be-
: the ranges of kHz QPO frequencies are almost homoge-
v tween twin kHz QPO peaks systematically decrease with
Xi neous at about ≃200Hz - 1300 Hz for both the less lumi- the frequency with the averaged ratio value of about 1.5
nousAtollsourcesandthebrightZsources(onthedefini-
(Zhang et al. 2006a).
r tionofAtollandZ,seeHasinger&vanderKlis1989),and
a
they increasewiththe inferredmassaccretionrate,which
implies that the kHz QPO mechanisms should be related In order to account for the mechanisms of twin kHz
to the common properties of both classes of sources (e.g., QPOphenomena,someviablemodelshavebeenproposed.
vanderKlis2000,2005,2006;Bellonietal.2005;Zhanget At the early stage of discovery of kHz QPOs, the sonic-
al. 2006a). Furthermore, it is also found that the separa- point beat-frequency model is proposed (see, e.g., Miller
tionoftwinkHzQPOpeaks∆ν =ν2 ν1isnotaconstant at al. 1998), which predicts a constant ∆ν at the stellar
−
and usually decreases with the mass accretion rate (e.g., spinfrequency,however,withthefurtherobservations,the
M´endez & van der Klis 1999; van der Klis 2000, 2006). simple beat model is inadequate for the varied kHz QPO
However, it is found that the peak separations increase separations. Instead, in the developed sonic-point model
with the accretion rate when the kHz QPO frequencies by Lamb & Miller (2001), the authors consider the disk
are low in Cir X-1 (Boutloukos et al. 2006) and in 4U flow at the spin-resonantradius to be smoothor clumped
1728-34 (Migliari et al. 2003), which supports the non- to interpret why the occurrence of kHz QPO pair separa-
tion is close to spin frequency as detected in XTE J1807-
⋆ Corresponding author: e-mail: [email protected] 294 (see, e.g. Linares et al. 2005; Zhang et al. 2006b) or
halfspinfrequencyasdetectedinSAX J1808.4-3658(see, 2.1. MHD wave and its excitation in the accretion flow
e.g. Wijnands et al. 2003)
In MHD, the Alfv´en wave is a transverse wave and prop-
Later on, the relativistic precessionmodel is proposed
agates along the magnetic field line, the effects of which
by Stella and Vietri(1999),in whichthe upper kHz QPO
have been studied in solar physics to interpret the de-
frequency ν2 is identified with the Keplerian frequency of
tectedquasi-periodicoscillationofseveralminutesincoro-
an orbit in the disk and the lower kHz QPO frequency ν1
nal loops (e.g., Roberts 2000; Nakariakov et al. 1999;
with the periastron precession of that orbit, and the var-
Aschwandenetal.1999).Asknown,thecoronalloopsmay
ied ∆ν can be consistently explained in this model. More
be setintooscillationswith variousmodes,suchasa kink
recently, Rezania and Samson (2005) propose a model to
mode (Roberts 2000 for a review), thus we assume the
explainthe kHz QPOsbasedon the interactionof accret-
similar oscillations to occur in the accretion disks with
ing plasma with the neutron star magnetosphere, where
the loop length of circumference 2πr at the disk orbit ra-
the matter is accelerated by the gravitational pull of the
dius r. As pointed out, the MHD turbulence by the shear
compact object and hits the star magnetosphere with a
flow in the accretion disk (e.g., Ruediger & Pipin 2000)
sonicorsupersonicspeed.Basicallytheconclusionsofthis
will trigger the strong variation of plasma energy density
model are consistent with the observed data for the ap-
and ignite the shear Alfv´en wave motion along the or-
propriate choices of the parameters.
bit. In a wave length of circular perimeter, the intensely
Moreover, Titarchuk et al. (1998), Osherovich &
excited shear Alfv´en wave can dedicate to the observed
Titarchuk (1999ab) and Titarchuk & Osherovich (1999)
X-ray flux fluctuations, which is then responsible for the
have developed the alternative models. They require the
observed kHz QPO phenomena.
lowerkHz QPOfrequency to be due to the Keplerianfre-
However,thepropagationoftheAlfv´enwavewillarise
quencyofmatter inthe diskandthe upper kHz QPOfre-
its damping because ofthe dissipationby the X-rayemis-
quency to be the hybrid between the lower kHz QPO fre-
sion and the viscous interactions in MHD, which may ac-
quency and the stellar spin frequency. Nonetheless, there
count for why sometimes we only measure a single kHz
have not yet been any agreements on the origins of the
QPO or nothing. From the conventional accretion disk
QPOsinneutronstarandblackholebinaryX-raysources,
prescription, the accretion flow in the equatorial plane of
noronwhatphysicalparametersdetermine theirfrequen-
disk will drag the polar field lines into the azimuthal di-
cies, which have also been identified with various char-
rection, and it is often assumed that the field strength
acteristic frequencies in the inner accretion flows (e.g.
of the azimuthal component B is comparable to that
φ
Abramowicz et al. 2003ab; Abramowicz 2005; Kluzniak
of the unperturbed polar field in the equatorial plane.
et al 2004; Lee et al. 2004; Rebusco 2004; Rebusco &
i.e. B (r) B(r) (e.g., Ghosh & Lamb 1979; Shapiro
φ
Abramowicz 2006; Petri 2005;Horak & Karas 2006). ≃
& Teukolsky 1983).
This paper is an improved one of the previous work
on the MHD Alfv´en wave oscillation mechanism for the
2.2. On the preferred radius and the coherence
NS kHz QPOs (Zhang 2004), where we consider the ra-
dial dependence of the accretion flow velocity, and the
On the preferred radius r, it is supposed to be a “coher-
applications have been done by comparing the model’s
ence” location at where the characteristic Alfv´en velocity
predictions with the updated kHz QPO data.
v (r) (frequency) calculated by the plasma mass density
A
The paper is organized as follows: In section 2, the
ofquasi-sphericalflowcoincideswiththe Keplerianveloc-
overview and scenario of the model are described, and
ity vk(r) (frequency), i.e., vA(r)=vk(r). We assume that
the derivations of kHz QPO frequencies are given. The
atthisradiusthereisaresonancebetweentheAlfv´enwave
further applications of model are presented in section 3.
frequency and Keplerian orbital frequency.
The conclusions and consequences are summarized in the
From the definition, at the Alfv´en radius RA, the
final section. As the conventional usage, the Newtonian
Alfv´en velocity is the free fall velocity, expressed as (e.g.,
gravitational constant G and the speed of light c=1 are
Shapiro & Teukolsky 1983),
exploited.
vA(RA)=B(RA)/ 4πρ(RA)=vff(RA), (1)
2. The overview of the improved model where B(r) 1/r3pis the dipole magnetic field and
∼
The previous model for the NS kHz QPOs proposed by
Zhang(2004)does not mentionthe excitationmechanism ρ(r)=M˙ /[Srvff(r)], Sr =4πr2 (2)
of Alfv´en wave oscillation to modulate the X-ray flux to
is the plasma mass density defined by the spherical ac-
produce the observed kHz QPOs. Therefore, in this pa-
perwetrytoanswerthesequestions.Tostraightforwardly cretion with the free fall velocity vff(r) or the Keplerian
graspourpointofviews,animaginedgeometricalillustra- velocity vk(r),
tion of the magnetosphere and accretion disk associated
with our model is plotted in FIGURE 1. vff(r)= 2GM/r=√2vk(r), vk(r)= GM/r . (3)
p p
Therefore, at the preferred radius r = φRA, where φ is The accretedmaterials accumulatedonthe polar capwill
an introduced parameter, the Alfv´en velocity equals the become denser by a factor of ratio of the spherical area
Keplerian velocity, thus, to the polar cap area. These denser materials will flow
equator-ward and pass through the perpendicular field
vA(r)=B(r)/ 4πρ(r)=√2φ−7/4vk(r)=vk(r), (4) lines by the instabilities, so this flow is influenced by the
the ‘coherence’pradius is defined by the condition of Lorentz force. The all denser materials will be expelled
√2φ−7/4 =1 or φ=22/7 1.2. out through the magnetic tunnel and enter into the tran-
≃ sitional zone near the ‘coherence’ radius because the flow
alongthefieldlinesisforcefreeandwillnotexperiencethe
2.3. The identifications of twin kHz QPO frequencies
Lorentzforce.Henceforth,weascribetheAlfv´enwavefre-
quency calculated by the denser mass density of plasma
Therefore, at the preferred radius, the Alfv´en wave fre-
formed on the polar cap to be the lower kHz QPO fre-
quency is
quency.Forthe mathematicalconvenience,itispresumed
vA(r) vk(r) that both the upper and lower kHz QPOs occur at the
νA(r)= = =νk(r), (5)
2πr 2πr same radius. Thus, the lower kHz QPO frequency ν1 is
which we identify to the upper kHz QPO frequency (see described below,
also Zhang 2004), or written as
v (r)
Ap
ν (r)= , (8)
ν2 =νk = 1850(ξA)X3/2(Hz) (6) Ap 2πr
= 1295(ξA)X3/2(Hz), where the Alfv´en velocity vAp(r) is defined by the mass
0.7 density expelled from the polar cap,
with the parameters X=R/r and A = (m/R3)1/2 where
6
6 M B(r) B(r) ρ(r) ρ(r)
Ran6d=mRas/s1M0 (cinmt)haenudnmits=ofM10⊙6a(rcemt)heansdteslloalrarramdaiussseRs, vAp(r)= 4πρp = 4πρ(r)s ρp =vk(r)s ρp , (9)
respectively. It is noted that the quantity A2 is a mea- with p p
surement of the averaged mass density of star, expressed
as ρp =M˙ /[Spvff(R)], (10)
ρ =3M/(4πR3) 2.4 1014(g/cm3)(A/0.7)2.
hThie reasons for in≃troduc×ing the “non-Keplerian factor” ξ where the magnetic polar cap area Sp is obtained as
in Eq.(6) come from the facts of the complicated physical (Zhang & Kojima 2006),
environments,whichdonotsatisfytheidealconditionsfor
2
S =4πR (1 cosθ), (11)
applying the Keplerian frequency, i.e. the point mass or- p −
biting around a central gravitationalsource in a vacuum.
with
ThemodificationstotheKeplerianfrequencywillbetaken
into accountif the followingsituations areconsidered:(a) sin2θ =R/r X, (12)
≡
the influence by rotation or the modification from Kerr
spacetime to Schwarzschild spacetime; (b) the considera- where θ is the open angle of the last field line to close
tionofthefactualenvironmentaroundstardeviatingfrom at radius r. As an approximation, the polar cap area is
the ideal situation of a test particle motion in a vacuum; usually written as Sp = 2πrR3 if R ≪ r (see e.g. Shapiro
(c) the plasma blob moving in a MHD with the strong & Teukolsky 1983, P.453). Henceforth, if we ascribe the
magnetic field. In principle, this “non-Keplerianfactor” ξ upper and lower kHz QPO frequencies to the two Alfv´en
should be not a multiplicative constant. For instance, it wave frequencies with different mass densities described
could depend upon the radius in the disk at which the in Eqs.(6) and (8), then we have
QPO is produced, and hence upon the frequency itself.
However, we have to stress that it is just the reason of ν1 ρ(r) Spvff(R)
= = . (13)
simplicity to chose it as a constant. In practice, for the
ν2 s ρp sSrvff(r)
mathematical convenience, we still take ξ = 1 to process
the calculations, and finally the real parameter A can be
After considering Eq.(1) and Eq.(11) with the parameter
obtained through dividing it by the “non-Keplerian fac-
definition X R/r, Eq.(13) gives
tor” ξ, the implication of which is discussed in the last ≡
section. S
Equivalently, the NS radius can be expressed by the ν1 =ν2X−1/4 Sp =ν2X3/4[1−√1−X]1/2 . (14)
r r
parameters A and m,
For convenience, the ratio of the twin kHz QPO frequen-
R6 =1.27m1/3(A/0.7)−2/3(10 km). (7) cies is obtained to be,
Furthermore, on the formation scenario of the lower kHz ν2 =X−5/4 1+√1 X, (15)
QPO frequency (ν1), we have the following arguments. ν1 −
q
which only depends on the position parameter X R/r andradiusrelationsandtheirconstrainconditionsarecal-
≡
where X-ray flux responsible for kHz QPOs emits and is culatedforthe17knownsourceswhostwinkHzQPOsare
independent of the averaged mass density parameter A simultaneously detected, which are listed in TABLE I. In
and mass M. Furthermore, the twin kHz QPO separation Figure 3, the mass-radius relations inferred from the val-
is written as, ues of parameter A have been plotted, and it is found
that, for the conventionallyaccepted NS mass lower limit
∆ν ν2 ν1 =ν2[1 X3/4(1 √1 X)1/2], (16)
≡ − − − − M⊙,thereexistsdifficultyinaccordancewiththe many
∼
which is not a constant with the variation of ν1 or ν2. modern equations of states (EOSs) if the parameter A is
From Eq.(6) and Eq.(14), if the twin kHz QPO fre- too low, for instance A 0.47 for SAX J1808.4-3658(see
∼
quencies are known simultaneously, then the values of A TABLE I). In general,the lower the value of A, the more
andXcanbedetermined.Forthedetectedsamplesources difficult it is to reconcile the M-R relations with the re-
listedinTABLEI(seee.g.vanderKlis2000,2006;Belloni alistic EOSs of stars. The situation can be altered if the
et al. 2005; data are provided by T. Belloni, M. M´endez introduced ‘non-Keplerianfactor’ ξ < 1, which makes the
andD. Psaltis),suchas Sco X-1 (vander Klis etal. 1997; parameter A increased.
M´endez & van der Klis 2000),4U1608 52(M´endez et al.
−
1998ab) , 4U1735-44 (Ford et al. 1998) and 4U1728 34
3.2. The maximum twin kHz QPO frequencies and
−
(M´endez & van der Klis 1999).
their separations
Thecomparisonsofthemodel’sconclusionstothewell
detected kHz QPOsample sourcesareshowninFIGURE The theoretical maximum frequency separation ∆νmax
2,andtheagreementbetweenthemodelandtheobserved canbe calculatedfromEq.(16)by vanishing the variation
QPO data is quite good for the selected ranges of NS of ∆ν respect to X, d[∆νmax]/dX = 0, where we obtain
parameters A=0.6, 0.7 and 0.8. In FIGURE 2b, we find X 0.69, ν2 = 750(A/0.7)(Hz), ν1 = 382(A/0.7)(Hz)
≃
that∆ν increaseswithν2ifν2 < 750(A/0.7)Hzand∆ν and∆νmax =368(A/0.7)(Hz). Ifν2 is lower(higher)than
∼
decreaseswithν2ifν2 > 750(A/0.7)Hz.Inaddition,the 750 (A/0.7) (Hz), then ∆ν will increase (decrease) with
∼
twin kHz QPO ratio (FIGURE 2c) follows the decreasing ν2. In the recent observations of 4U 1728-34 (Migliari et
tendency with ν2. al 2003), the twin simultaneously detected kHz QPO fre-
quencies are found at the central frequency ν1 = 308
3. The applications of the model (Hz) and ν2 = 582 (Hz) ( ∆ν = 274 (Hz)), so this is
the first detected event for a significant decrease of kHz
To inspect the model’s predictions, we demonstrate some QPO peak separation towards low frequencies, however,
applications of the model in the following. which is qualitatively consistent with our model’s predic-
tion. Nonetheless, the maximum values of twin kHz QPO
frequenciescoincideandoccuratX=1wherethepreferred
3.1. The constrain conditions of NS mass and radius
radius equals the stellar radius in case the ISCO is inside
The NS mass constrain condition by the kHz QPOs has thestar,ν1max =ν2max =1850A(Hz)=1295(A/0.7)(Hz),
been given by Miller et al (1998) and Zhang et al (1997) namely
through assuming that the accretion disk radius of show-
νk
ingkHzQPOsisbiggerthantheinnermoststablecircular A . (20)
≥ 1850(Hz)
orbit(ISCO:threeSchwarzschildradii),i.e.,themaximum
observed frequency is presumed to be the Keplerian fre- InTABLEI,thetheoreticalmaximumkHzQPOfrequen-
quencyatISCOwiththeconditionthatthestellarsurface cies of 17 kHz QPO sources are listed, and the averaged
is enclosed by ISCO, values of A and the maximum position parameter Xmax
are given for both the Z and Atoll sources, which are, re-
m≤2.2/ν2k. (17) spectively, A =0.66 and Xmax =0.87 (r 1.15R) for Z
h i h i ∼
From Eq.(7), the new radius constrain condition is ob- sources,and A =0.74andXmax=0.9(r 1.1R)forAtoll
h i ∼
tained if the mass constraincondition is known, therefore sources. However, we currently cannot figure out what
we exploit the generally assumed NS mass lower limit 1.0 physical mechanisms arise these systematical differences
M⊙ from the astrophysical argument and its upper limit betweentheZandAtollsources.TheinequalityXmax <1
condition Eq.(17) to set the radius constrain conditions, tells us the factthatthe accretiondisk doesnotreachthe
respectively, stellar surface, so it is likely that the ISCO prohibits the
disk fromarrivingatthe stellar surface.If this scenariois
R6 1.27(A/0.7)−2/3, (18) plausible, we can conclude that the ISCOs of almost all
≥
sources in TABLE I locate outside the stars and the esti-
and
mated positions of their masses and radii should appear
R6 ≤(ν22k.A22)1/3 =1.65ν2−k1/3(0A.7)−2/3. (19) aInbtoevreesItSinCgOly,ltinhee oinneMu-nRusduiaalgrcaamse isshothwenAintolFlIsGouUrRceE43U.
InEq.(19),theradiusisconstrainedbyAandν2,orequiv- 0614+09(e.g.,vanStraatenet al2000;vanderKlis 2000,
alentlybythetwinkHzQPOs.Furthermore,theNSmass 2006), and its parameter A 0.76 is inferred from the si-
≃
multaneously detected twin kHz QPOs, which implies its So, the magnetic fields are proportionally related to the
maximum upper (lower)kHz QPO frequency to be about accretion rates, and for both Z sources (Eddington accre-
1406(Hz). However,from the observation,one single kHz tion rate M˙ ) and Atoll sources ( 1%M˙ ) they are
Edd Edd
QPOpeakfrequency1330(Hz)wasdetectedinthissource about 109 G and 108 G respec∼tively if m 1 and
∼ ∼ ∼
(van Straaten et al 2000; van der Klis 2000, 2006), show- R6 1, which are consistent with the originally hinted
∼
ing X 0.97, thus this detected single frequency may be values from the X-ray spectra of both sources (Hasinger
∼
nearthe maximumkHz QPOsaturationfrequency,show- & van der Klis 1989).
ing the X-ray spectrum information near the stellar sur- On the correlation between QPO frequency and the
face,theconfirmationofwhichneedsthefurtherproposal accretion rate, we can write it in the following by means
and analysis of this source. of the definition of magnetosphere (Shapiro & Teukolsky
As for the level-off or frequency saturation of kHz 1983),
QPO, it has been paid much attention since the early
discovery of kHz QPOs, which is ascribed to the occur- ν2 ∼ (R/r)3/2 ∼(R/RM)3/2(RM/r)3/2 (26)
rence of ISCO or stellar surface (e.g., Zhang et al 1998; (hM˙ i)3/7( M˙ )3/7 (hM˙ i)3/7M˙ 3/7 , (27)
Kaaret et al 1999; Miller 2004; Swank 2004). However, ∼ B2 M˙ ∼ B2 x
from our model, if RISCO > R, the saturation frequency h i
wouldoccuratISCOwiththemaximumupper kHzQPO whereM˙x =M˙ / M˙ isaratiooftheinstantaneousaccre-
h i
frequency ν2max = 2200/m (Hz) (see also, e.g. Miller et tion rate to the long-term accretion rate. If the neutron
al 1998; Miller 2004); if RISCO < R, the saturation fre- star accreted 0.01M⊙, the NS magnetic field will enter
∼
quency would occur at the stellar surface R with ν2max = into a ‘bottom state’ where the B-field is proportionally
1295(A/0.7) (Hz). Therefore, the model inferred maxi- related to the accretionrate as B M˙ 1/2 by the accre-
∝h i
mumpositionsofshowingthekHzQPOssatisfyXmax <1 tion induced magnetic evolution model (Zhang & Kojima
(see TABLE I) for the simultaneously detected kHz QPO 2006). Thus, we obtain a unified expression of QPO fre-
sources,whichimpliesthateithertheirISCOsappearout- quency vs.the accretionrateforbothAtollandZsources
side stars or the unclear mechanisms diminish the kHz tobeν2 M˙x3/7,althoughbothsourcesshareadiversified
∼
QPO X-ray spectra just above the stellar surface, which luminosity of even more than two magnitude orders.
needs the further investigations.
4. Summaries and conclusions
3.3. The estimations of magnetic field strengths of
In the paper, we compare the observations with the im-
NSs in Z and Atoll sources
proved model for kHz QPOs, and the main conclusions
The estimation of the NS magnetic field strength B can are summarized in the following.
be given by the definition of its magnetosphere (see, e.g. (1)The theoreticalmaximumtwin kHz QPOfrequen-
Shapiro & Teukolsky 1983), or described by the accre- cies coincide and occur at about 1295(A/0.7) (Hz) when
tioninducedmagneticevolutionmodel(see,e.g.Zhang& the accreted matters to show these QPOs clash on the
Kojima 2006; Cheng & Zhang 1998), NS surface, and the maximum twin kHz QPO separation
is 368 (A/0.7) (Hz). (2) For SAX J1808.4-3658, we find
B =(RM)7/4B , (21) that its stellar mass density parameter A=0.47 is about
f
R
1.5 times less than the typical values of other kHz QPO
B 4.3 108 (G)( M˙ /M˙ )1/2m1/4R−5/4 , (22) sources with A 0.7, and we also obtain its highest kHz
f Edd 6 ∼
≃ × h i QPO frequency to be 870 (Hz), which needs the proof
∼
whereRMisthemagnetosphereradiusdefinedbythelong- of future detections. (3)The averagedmass density of NS
term accretion rate M˙ and M˙ is Eddington limited can be described by the defined parameter A as
Edd
accretion rate. kHzhQPiOs are assumed to be produced ρ =3M/(4πR3) 2.4 1014(g/cm3)(A/0.7)2, and we
aroundRM with the variationof the instantaneous accre- ohbitain ρ 2.4 ≃1014(×g/cm3) for most of NS kHz QPO
h i≃ ×
tion rate, thus R/RM should be comparable to the aver- sources with A 0.7. (4) With the derived parameter A
≃
agedpositionparameter X ,i.e. X R/RM,whichcan listedinTABLEIfortheidealcaseofthe“non-Keplerian
h i h i∼
be estimated by Eq.(6) as factor” ξ = 1, the mass-radius relation curves are plot-
ted in FIGURE 3, and we find that the EOSs of strange
hν2i≃1850AhXi3/2 , (23) matters (CS1 and CS2) seems to be not favorite except
where ν2 istheaveragedvalueofthedetectedupperkHz A 1,resultinginanextremelyhighkHzQPOfrequency
h i ∼
QPOofZ( 870Hz)orAtoll( 980Hz)sourcesandboth ν2max 1800 (Hz). Moreover, for the generally assumed
∼ ∼ ∼
sources share the homogeneous kHz QPO distributions NSmasslowerlimit, M⊙ forinstance,EOSsofthe nor-
∼
(see, e.g. Zhang et al. 2006a),therefore, mal neutron matters (CN1 and CN2) do not fit for the
stars in the detected kHz QPO sources unless A 0.88,
B ≃ (1850A/hν2i)7/6Bf (24) corresponding to ν2max 1600 (Hz). If the EOSs o∼f CPC
109(G)( hν2i )−7/6( hM˙ i )1/2m5/6R−3 . (25) (the star core becomes ∼a Bose-Einstein condensate of pi-
≃ 900Hz M˙ 6 ons) are the possible choices, the mass and radius ranges
Edd
ofthesestarsarefrom1.0M⊙to1.7M⊙andfrom15kmto ofabovecomplicationswillimprovethe presentversionof
18km,respectively,forA=0.45–0.79.Inaddition,ifthe the model.
introduced“non-Keplerianfactor”ξ islessthanunity,for
instance ξ 0.7, then the derived values of A in TABLE Acknowledgements. We thank T. Belloni, M. M´endez, D.
≃ Psaltis and M.C. Miller for providing the data files, and help-
I will be increased by a factor of about 1.4, for instance
ful discussions with T.P. Li, X.D.Li,J.L. Qu,S.N.Zhang,M.
from A 0.7 to A 1.0, which in turn, as shown in
≃ ≃ Abramowicz, S. Boutloukos, J. Horak, J. Homan, V. Karas,
FIGURE 3, results in a fact that many sources listed in
P. Rebusco and J. Petri are highly appreciated. This research
TABLEImaybethecandidatesofstrangestars(see,e.g.,
hasbeensupportedbytheinnovativeprojectofCASofChina.
Cheng et al1998;Lattimer & Prakash2004).However,in
C.M. Z. thanks MPE-Garching and TIARA-NTHU for visit-
our model it seems to be difficult to take the star in SAX ing supports. H.-K. C. is supported by the National Science
J1808.4-3658 (its A=0.47) as a candidate of strange star Council through grants NSC 94-2112-M-007-002 and NSC 94-
as expected by Li et al. (1999), as seen in FIGURE 3. 2752-M-007-002-PAE.Weareverygratefulforthecriticalcom-
Nevertheless, the EOSs of stars in kHz QPO sources are ments and helpful suggestions from the anonymous referee,
still the open issues before the physical influences of the whichadviseustothoroughlyimprovethequalityofthepaper.
“non-Keplerianfactor”ξ arethoroughlysettled.(5)From
the homogeneous kHz QPO distributions for both Atoll
and Z sources (see, e.g. Zhang et al. 2006a), we conclude
theNSmagnetospherescalesofbothsourcestobesimilar,
whichwillarisetheNSmagneticfieldtobeproportionally
related to the long-term averaged accretion rate, i.e. NS
in Z source possesses a stronger magnetic field than that
of Atoll source.
Furthermore, the parameter A = 0.78 is implied by
the model for KS1731-260, corresponding to the max-
imum kHz QPO frequency 1443 (Hz), which is bigger
than the known detected maximum value ν2 1330 (Hz)
∼
(4U 0614+091, van Straaten et al. 2000). Moreover, the
proposal for the detection of 1500 (Hz) and 1800 (Hz)
QPOfrequencyissuggestedbyMiller(2004),correspond-
ing to A 0.8 and A 1.0 in our model, respectively, so
∼ ∼
these measurements of QPOs above 1500 (Hz) therefore
have excellent prospects for stronger constraints on the
mass and radius relations, as well as on the models. It is
claimed by Miller (2004) that a QPO frequency as high
as 1800 (Hz) would be large enough to argue against all
standard nucleonic or hybrid quark matter EOS, leaving
onlystrangestars(see alsoFIGURE 3).Nevertheless,the
proposalofdetectingtheQPOdataatlowfrequencyν1
∼
100 (Hz) is also meaningful, by which the model is tested
throughinspecting∆ν vs.ν2relationasshowninthemid-
dle panel of FIGURE 2. It is also predicted by Stella and
Vietri (1999) ∆ν will increase with the accretion rate if
ν2 < 700Hz,andtheevidenceforthishasbeenrecently
∼
detected in Cir X-1 by Boutloukos et al. (2006).
As a summary, it is remarkedthat our model is still a
simple one,andthe further improvementsarestill needed
through considering the details of accretion flow or disk
structure,wheretheplausibleexistenceofthetransitional
layer (e.g., Titarchuk et al 1998; Titarchuk & Osherovich
2000) or the nonlinear disk resonances (e.g., Abramowicz
etal.2003ab)maybeimportant.Inaddition,wehavenot
yetconsideredtherotationaleffectofKerrspacetime(see,
e.g., Miller 2004; van der Klis 2006), the factual MHD
flow velocity around star but perhaps not a Keplerian
velocity of test particle, the accretion flow in a strong
gravity regime, the plasma instabilities and turbulence in
strong magnetic field, etc. Therefore, the considerations
Magnetic Polar Cap Spherical Accretion
1000 (a)
Sco X-1
Z exclude Sco X-1
Magnetosphere Accretion Disk O (Hz) 800 A4AUt=o0 l1l. 67e02x8cl-u3d4e 4U 1728-34
QP 600 A=0.70
Hz A=0.80
Outflow wer k 400
Lo
200
0
(b)
400
Hz)
Peak Separation (230000
100
Fig.1. An imagined schematic illustration of an accret- 0
3.5
ingneutronstarmagnetosphereanddiskforthekHzQPO (c)
productions. θ is the open angle between the magnetic 3.0
axis and the closed field line of magnetosphere, defined 2.5
icbtrysetsmiionan2gθnfre=otmoRspa/hlrle,rowev,ehrreerssetpaRerctfaainvlldeslyri.naTraehnedthqetuhraeansdi-icsiupomhfeusrltiacaatrelsaanacdt- Upper/Lower12..50
the magnetic polar cap, where the blobs or patches of ac- 1.0
creting plasmas are piled up and condensed into the high
0.5
mass density, and some of materials will diffuse onto the
0.0
whole stellar areathroughthe side-flow onaccountof the 0 200 400 600 800 1000 1200 1400
plasma instabilities and the others sprout out along the Upper kHz QPO (Hz)
closedfieldlinesthroughtheout-flowtoenterintotheor-
bital flow. The preferred radius is supposed at where the
shearAlf´enwavefrequencywiththequasi-sphericalaccre-
tion propagating in the orbit equals the orbital Keplerian
frequency, namely the “coherence” is presumed to occur
there.
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TABLE I. Neutron star parameters of detected kHzQPO sources
Source∗ A(1) R(2) m(3) R(4) ν (5) ν (6) X (7)
2max 2obs max
m1/3 (km) (M⊙) (km) (Hz) (Hz)
Millisecond pulsar
SAXJ1808.4-3658a 0.47 16.9 1.0-2.99 16.8-24.4 870 725 0.90
Z source
Sco X-1 0.66 13.3 1.0-2.05 13.3-17.0 1221 1075 0.93
GX340+0 0.64 13.5 1.0-2.62 13.5-19.0 1184 840 0.80
GX349+2 0.68 13.0 1.0-2.23 13.1-17.2 1258 985 0.86
GX5–1 0.63 13.8 1.0-2.49 13.8-18.8 1166 890 0.84
GX17+2 0.67 13.2 1.0-2.02 13.3-16.8 1240 1087 0.93
CygX-2 0.70 12.8 1.0-2.19 12.8-16.7 1295 1005 0.85
Atoll source
4U0614+09 0.76 12.2 1.0-1.65 12.1-14.4 1406 1330 0.97
4U1636–53 0.74 12.4 1.0-1.79 12.1-15.1 1369 1230 0.94
4U1608−52 0.69 12.9 1.0-2.00 12.8-16.2 1277 1099 0.90
4U1702–43 0.75 12.3 1.0-2.03 12.2-15.6 1388 1085 0.85
4U1728−34 0.77 12.0 1.0-1.88 11.9-14.8 1425 1173 0.88
KS1731–260 0.78 11.9 1.0-1.83 11.9-14.6 1443 1205 0.89
4U1735−44 0.73 12.5 1.0-1.91 12.4-15.5 1351 1150 0.90
4U1820–30 0.73 12.5 1.0-2.00 12.4-15.8 1351 1100 0.88
4U 1916–053 0.73 12.5 1.0-2.08 12.4-16.0 1351 1058 0.85
XTEJ2123–058 0.72 12.6 1.0-1.93 12.5-15.8 1332 1140 0.91
*: The sources are chosen from those that the twin kHz QPOs are detected simultaneously (van der Klis 2000,
2006; the original references therein), then 4U1915–05 is not included because its two incompatible values of ∆ν
are reported (van derKlis 2006). a: Wijnands et al. 2003. (1):Calculated by thesimultaneously detected twin kHz
QPOdata.(2):ObtainedbyEq.(7).(3):EstimatedbythegenerallyassumedNSmasslowerlimit1.0M⊙ andbythe
constrainconditionm≤2.2/ν (Milleretal.1998).(4):ObtainedbyEq.(18)andEq.(19).(5):ν =1295(A/0.7)
2k 2max
(Hz)isthepossiblemaximumkHzQPOfrequency.(6):ν isthedetectedmaximumkHzQPOfrequency.(7):the
2obs
detection inferred maximum X position, X =(ν /ν )2/3.
max 2obs 2max
