aa r X i v : . [ a s t r o - ph . H E ] D ec The emission positions of kHz QPOs and Kerrspacetime influence
ZHANG Chengmin ∗ , WEI Yingchun YIN Hongxing ZHAO Yongheng LEI YaJuan SONG Liming ZHANG Fan YAN Yan
1. National Astronomical Observatories, Chinese Academy of Sciences, Beijing100012, China2. Institute of High Energy Physics, Chinese Academy of Sciences, 19B YuquanRoad, Beijing 100049, China3.Urumqi Observatory, National Astronomical Observatories, CAS, Urumqi830011, China
AbstractBased the Alfven wave oscillation model (AWOM) and relativistic pre-cession model (RPM) for twin kHz QPOs, we estimate the emissionpositions of most detected kHz QPOs to be at r = 18 ± km ( R/ km ) except Cir X − at r ∼ ± km ( R/ km ) . For the proposed Keplerianfrequency as an upper limit to kHz QPO, the spin effects in Kerr Space-time are discussed, which have about a 5% (2%) modification for thatof the Schwarzchild case for the spin frequency of 1000 (400) Hz.Theapplication to the four typical QPO sources, Cir X − , Sco X − , SAXJ1808.4-3658 and XTE 1807-294, is mentioned. Key words: kHz QPO, neutron star, low-mass X-ray binaries
In thirty more low-mass X-ray binaries (LMXBs), the kiloHertz quasi-periodicoscillations (kHz QPOs) have been found , where 2 / ∼
100 Hz - ∗ Corresponding author.
Email address: [email protected] (ZHANG Chengmin ). Preprint submitted to Phys Mech Astron 26 October 2018
300 Hz for the sources with the different spectrum states, e.g. Atoll and Z [2].The separations of twin kHz QPOs are not constant[1,3,4,5,6,7], which areinconsistent with the beat model[8,9]. The low frequency QPOs have also beenfound, which follow the tight correlations with the kHz QPOs [1,4]. Some kHzQPO models have been proposed,most of which are ascribed to the accretionflow[10], and the Alfven wave mode oscillation[11,12]. To account for the variedkHz QPO separation, the relativistic precession model (RPM) is proposed byStella and Vietri[15], which ascribes the upper frequency to the Keplerianfrequency of orbiting material in an accretion disk and the lower frequency tothe periastron precession of the same matter.However, for the detected twin kHz QPOs of neutron star (NS) in a LMXB,their average ratio value is also 3 : 2, but varies with the accretion, which mayindicate some distinctions between BHC and NS[1]. In this short letter, we willinvestigate the orbital positions of kHz QPO emissions, based on the AlfvenWave Oscillation Model (AWOM) [13,14] and RPM[15]. The Kerr spacetimemodification is discussed by considering the spin influence on the Keplerianfrequency. AWOM ascribes an upper frequency to the Keplerian frequency of orbitingmatter at radius r, and a lower frequency to the Alfven wave oscillation fre-quency at the same radius, as described in the following) [13,14], ν = ν k = 1850 AX (1)with the parameter X = R/r (ratio between star radius R and disk radius r )and A = ( m/R ) / with R = R/ ( cm ) and m the mass M in the units ofsolar masses. The ratio of twin kHz QPO frequencies can be obtained as, ν/ν = (1 + (1 − x ) / ) /X (2)which only depends on the position parameter X = R/r , and has nothing todo with the other parameters. The twin kHz QPO separation is obtained as, ν − ν = ν [1 − (1 − (1 − x ) ) ] ∗ X (3)In FIG.1, the upper kHz QPO frequency is plotted against the position pa-rameter X = R/r ( Y = 3 Rs/r , Rs is the Schwarzschild radius) for AWOM(RPM). For the detected twin kHz QPOs, the mass density parameter A is2 .0 0.2 0.4 0.6 0.8 1.0050010001500 m=3 X = . X = . U pp e r f r e qu e n cy X or Y
A=0.70A=0.451300 Hz830 Hz m=2Cir X-1Sco X-1XTE 1807-294 SAXJ 1808.4--3658
Fig. 1. Upper kHz QPO frequency vs. the position function( X = R/r, Y = 3
Rs/r, Rs = 2
GM/C A = 0 . A = 0 . m = 2 ( m = 3) solar masses, where the maximumfrequency is 2200/m (Hz). Sax J1808.4-3658XTE 1807-294 k H z Q P O s epa r a t i on X or Y (a) Cir X-1 Sco X-1
Fig. 2. Twin kHz QPO separation vs. the position function . Curve 1 and 2 representAWOM with mass density parameters A = 0 . A = 0 .
45 respectively. Curve 3(4) represents RPM with mass parameter m = 2 ( m = 3), respectively. .0 0.2 0.4 0.6 0.8 1.0012345 XTE 1807-294 Ratio=1.95Ratio=1.0 U ppe r / Lo w e r X or Y
AWOMRPM X = . Y = . Ratio=1.5 X = . Cir X-1
Sco X-1
Sax J 1808.4-3658
Fig. 3. Twin kHz QPO ratio vs. the position function (Same meaning as shownin FIG.1). The ratio 1.5 (1) is the averaged (minimum limit) value of the detectedtwin kHz QPOs. found to be about 0.7 (e.g. Sco X −
1) [13,14]. In most cases (except Cir X − X = R/r is lies in the range from 0.7 to 0.92,or radius from r = 1 . R to r = 1 . R . This implies that the emission positionsof most kHz QPOs are close to the surface of the NS X = 1 for AWOM (forRPM the emission positions are close to 3 Rs ), which means that the max-imum kHz QPO frequency occurs at the surface (ISCO of star r = R forAWOM (or r = 3 Rs for RPM). In FIG.2, the twin kHz QPO separation vs.position parameter is plotted, where the maximum separation 375 (200) Hzis achieved for A = 0 . X = 0 . A = 0 .
45, whichpresents relatively low kHz QPO separations. For RPM, the maximum kHzQPO separations are 360 Hz (210 Hz) for the different choices of mass param-eter m = 2(3) solar masses, which occurs at Y = 0 .
76. For the two AMXPs,Sax J 1808.4-3658 and XTE 1807-294, RPM has to assume their star massesare close to the NS mass upper limit, 3 solar masses, if consistent theoreticalcurves with the detected data can be fitted. FIG.3 is the diagram of twinkHz QPO ratios vs. position parameter. It can be noticed that the averagedratio 1.5 of the detected kHz QPOs corresponds to the position X = 0 .
83 forAWOM ( Y = 0 .
89 for RPM). The ratios of all sources but Cir X − ratio = 1 and ratio = 2. The kHz QPO data of Cir X − . < X < . . R > r > . R , centered at about 2 R .4 Kerr spacetime effect on the kHz QPO
If the influence of Kerr spacetime on the Keplerian frequency is taken intoaccount, then the orbital frequency of a spinning point mass M with angularmomentum J is expressed as below[1] ν = νν k ξ ; ν k = ( GM/ r ) (4)with the Kerr modification parameter ξ = 1 + jR g ; R g = R s / j = J c/GM ; J = 2 πIν s (6)where I is the moment of inertia, with the maximum value for the homo-geneous sphere I = (2 / M R . In the Schwarzschild geometry, j = 0, Eq.4recovers the conventional Keplerian frequency; 0 < j < m = M/M ⊙ , radius and spin fre-quencyparameters, we have the following simplified expressions, j = 4Π ν R /R g C = (0 . /m ) R ( ν s / HZ ) (7) ξ = 1 + (0 . m ) R ( ν s / hz ) (8)If we set the conventional values M = 1 . M ⊙ , R = 15 km and s = 400 Hz ,then the Kerr modification parameter has about a 2% contribution to theKeplerian frequency, which cannot have too much influence on the kHz QPOmodel based on the Keplerian frequency. For the maximum spin fre-quency1122 Hz , the Kerr modification contributes about 5% to the Schwarzschildspacetime, so this influence should be considered when we estimate the NSparameters. The kHz QPO emission positions are analyzed by the models (AWOM andRPM), which shows that most kHz QPOs (e.g. Sco X −
1) come from theregimes of several kilometers away from the stellar surface. This may corre-spond to the condition of a spinning up NS, since the detected NS spin fre-quencies are averaging 400 Hz [16], less than the upper frequencies. In RPM,the star mass can be derived by the detected twin kHz QPOs, then it usually5ives a value of 2 solar masses, higher than the typical NS mass of 1.4 solarmasses. One reason for RPM’s prediction of high NS mass may be originatingfrom its assumption of the vacuum circumstance around the star in introduc-ing the perihelion precession term[15], but the accretion disk does not satisfythis clean condition. A value of about 3 solar masses for SAX J1808.4-3658 [17](e.g. XTE 1807-294) is obtained, which seems to suggest that RPM should bemodified. AWOM cannot predict a stellar mass by QPO but rather an aver-aged mass density ( A ∼ M / /R / , by which one can evaluate the equationof state (EOS) of the star. For the presently known kHz QPO frequencies,AWOM cannot give the prediction of quark matter[18] inside the star unlessthe QPO frequency over 1500 Hz is detected. In addition, the Kerr spacetimeinfluence is investigated, and a 5% modification factor in Keplerian frequencyexists for a high spin frequency of 1000 Hz, which will increase the estimationof the mass density parameter. Though the spectral properties of Cir X − X − Acknowledgements
This work is supported by National Basic Research Program of China-973 Pro-gram 2009CB824800; National Natural Science Foundation of China (10773017).
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