Mass-angular-momentum relations implied by models of twin peak quasi-periodic oscillations
Gabriel Torok, Pavel Bakala, Eva Sramkova, Zdenek Stuchlik, Martin Urbanec, Katerina Goluchova
aa r X i v : . [ a s t r o - ph . H E ] A ug . Draft version March 1, 2018
Preprint typeset using L A TEX style emulateapj v. 12/16/11
MASS–ANGULAR-MOMENTUM RELATIONS IMPLIED BY MODELS OF TWIN PEAK QUASI-PERIODICOSCILLATIONS
Gabriel T¨or¨ok, Pavel Bakala, Eva ˇSr´amkov´a, Zdenˇek Stuchl´ık, Martin Urbanec,and Kateˇrina Goluchov´a (Dated:
Received 2010 November 30; accepted 2012 September 26; published 2012 November 16 ) Draft version March 1, 2018
ABSTRACTTwin peak quasi-periodic oscillations (QPOs) appear in the X-ray power-density spectra of several accreting low-mass neutron star (NS) binaries. Observations of the peculiar Z-source Circinus X-1 display unusually low QPOfrequencies. Using these observations, we have previously considered the relativistic precession (RP) twin peak QPOmodel to estimate the mass of central NS in Circinus X-1. We have shown that such an estimate results in a specificmass–angular-momentum ( M – j ) relation rather than a single preferred combination of M and j . Here we confrontour previous results with another binary, the atoll source 4U 1636–53 that displays the twin peak QPOs at very highfrequencies, and extend the consideration to various twin peak QPO models. In analogy to the RP model, we findthat these imply their own specific M – j relations. We explore these relations for both sources and note differencesin the χ behavior that represent a dichotomy between high- and low-frequency sources. Based on the RP model, wedemonstrate that this dichotomy is related to a strong variability of the model predictive power across the frequencyplane. This variability naturally comes from the radial dependence of characteristic frequencies of orbital motion. Asa consequence, the restrictions on the models resulting from observations of low-frequency sources are weaker thanthose in the case of high-frequency sources. Finally we also discuss the need for a correction to the RP model andconsider the removing of M – j degeneracies, based on the twin peak QPO-independent angular momentum estimates. Subject headings: stars: neutron — X-rays: binaries INTRODUCTIONSeveral low-mass neutron star binaries (NS LMXBs)exhibit in the high-frequency part of their X-ray power-density spectra (PDS) two distinct peaks, so-called twinpeak quasi-periodic oscillations (QPOs). The two peaksare referred to as the upper and lower QPO. Centroidfrequencies of these QPOs, ν L and ν U , vary over time,but follow frequency correlations specific to individualsources. However, these specific correlations are qualita-tively similar (see Psaltis et al. 1999; Stella et al. 1999;Abramowicz et al. 2005a,b, and references therein). Insome cases, the frequency ranges spanned by a singlesource are as large as a few hundreds of Hz. At present,there is no consensus on the QPO origin. Numerousmodels have been proposed, mostly assuming that thetwo twin QPOs carry important information about theinner accreting region dominated by the effects of strongEinstein’s gravity. In principle, several of these mod-els imply restrictions to netron star (NS) parameters (asystematic treatment of these restrictions through thefitting of twin peak QPO correlations was pioneered byPsaltis et al. 1998). A brief introduction to QPOs andtheir models can be found in van der Klis (2006).In our previous work, T¨or¨ok et al. (2010), hereafterPaper I, we focused on restrictions of a particular “rel-ativistic precession” (RP) QPO model and a peculiarbright Z-source Circinus X-1. The RP model introducedby Stella & Vietri (1999) and Stella et al. (1999) identi-fies the lower and upper kHz QPOs with the periastronprecession ν P and Keplerian ν K frequency of a perturbed Mailto: [email protected] circular geodesic motion at the given radii r , ν L ( r ) = ν P ( r ) = ν K ( r ) − ν r ( r ) , ν U ( r ) = ν K ( r ) , (1)where ν r is the radial epicyclic frequency of the Keplerianmotion. In Paper I we noticed that the RP model wellmatches the data points of Circinus X-1 for any dimen-sionless NS angular momentum, j ≡ cJ/GM , when theassumed NS mass reads M ∼ . M ⊙ [1+0 . j + j )]. Wehave shown that the existence of such a mass–angular-momentum ( M − j ) relation is generic for the model.Circinus X-1 that we discussed in Paper I is a rela-tively well-known source, since it displays twin QPOsat unusually low frequencies, ν L ∈ (50Hz , ν U ∈ (200Hz , ν L ∈ (550Hz , ν U ∈ (800Hz , MASS–ANGULAR-MOMENTUM RELATIONFROM RP MODELAs found in Paper I, the data points of Circinus X-1 arewell matched by the RP model when the combinations ofthe source mass and angular momentum are in the formof M ∼ . M ⊙ [1 + 0 . j + j )]. The existence of such an( M − j ) relation is generic to the model. Let us brieflyrecall the major points and implications of our previousresults.We found that due to the properties of the RP modeland the NS spacetime the quality of fit for a given sourceshould not differ much from the following general rela-tion: M ∼ M (cid:2) k (cid:0) j + j (cid:1)(cid:3) . (2)In this relation, M is the mass that provides the best fitassuming a non-rotating star ( j = 0). The coefficient k ,implied by the model, would read k = 0 . ν L ∈ ( ν ≪ ν ISCO , ν
ISCO ) , (3)where ν ISCO denotes the Keplerian orbital frequency atthe innermost stable circular orbit r ms (hereafter ISCO).The available data points are, however, unequally sam-pled and often cluster, either simply due to incompletesampling and weakness of the two QPOs outside thelimited frequency range, or due to the intrinsic sourceclustering (Abramowicz et al. 2003a; Belloni et al. 2005;Belloni et al. 2007b; T¨or¨ok et al. 2008a,b,c; Barret &Boutelier 2008; Boutelier et al. 2010).2.1. Importance of Frequency Ratio for the RP ModelPredictions in Different Sources
In the detailed analysis presented in Appendix A.2 ofPaper I, we elaborated the influence of unequal samplingof the frequency correlation ν U ( ν L ). It is important thatfrequencies predicted from the RP model scale as 1 /M for a fixed j , and in this sense the expected frequencyratio, R ≡ ν U /ν L , is mass independent. Moreover, in theRP model, it is R = ν K / ( ν K − ν r ) . (4)The frequency ν r vanishes when the radial coordinateapproaches ISCO, r → r ms , and therefore R →
1. Onthe other hand, when r → ∞ the spacetime becomesflat reaching Newtonian limit where ν r → ν K and R di-verges. The QPOs that are expected to arise close toISCO therefore always reveal a low R , while those ex-pected to arise in a large radial distance from the NSreveal a high R . For any NS parameters, the top part (relatively high frequencies) of a given frequency correla-tion ν U ( ν L ) predicted by the RP model then reveals a fre-quency ratio close to R = 1. The bottom part (relativelylow frequencies) of the frequency correlation reveals highfrequency ratio R & Based on the above-mentioned theoretical predictionof the RP model, in Paper I we found that the value of k in mass–angular-momentum relation (2) must tend to k ∼ .
75 when the range of the ratio of the lower andupper QPO frequencies in the sample falls to low valuesclose to R = 1. On the other hand, it is k ∼ . R has a high value ( R ∼ M ∼ M ⊙ . Assuming this mass, thelow frequency ratio R . . ν L ∼ . − R ∼ (2 −
5) cor-responds to low QPO frequencies, ν L ∼ − k = 0 . k ∼ . − .
75 foravailable data on high-frequency twin peak QPO sources.For the available data on low-frequency twin peak QPOsources, the effect of unequal sampling is more impor-tant, changing k to ∼ . − .
65, which also correspondsto the case of Circinus X-1 elaborated in Paper I. De-tailed quantification of restrictions on k can be found inTable 1 of Paper I.Next we justify our result comparing the case of Circi-nus X-1 to the case of high-frequency source 4U 1636 − k ∼ . − .
75. Then we explorewhether several other QPO models imply their own M − j relations or not. DATA AND MODELSFigure 1(a) shows several twin peak QPO data pointscoming from the works of Barret et al. (2005b,c); Boirinet al. (2000); Di Salvo et al. (2003); Homan et al. (2002);Jonker et al. (2002a,b); M´endez & van der Klis (2000);M´endez et al. (2001); van Straaten et al. (2000, 2002);Zhang et al. (1998), and Boutloukos et al. (2006). Forthe analysis presented in this paper we use the twin peakQPO data of 4U 1636-53 (from Barret et al. 2005b,c)and Circinus X-1 (from Boutloukos et al. 2006). Thesedata points are denoted in the figure by the color-codedsymbols. Each of them corresponds to an individual con-tinuous segment of the source observation. One can seethat our choice of the two representative NSs allows usto demonstrate the confrontation between the low- andhigh-frequency sources, as mentioned in the previous sec-tion. Details of the observations, data analysis tech-niques, and properties of the twin peak QPOs in the two In Paper I we have shown that more than 60% of the lengthof the expected curve ν U ( ν L ) correspond to R < ass–angular-momentum Relations Implied by NS kHz QPO Models 3sources discussed can be found in Barret et al. (2005b,c);Barret et al. (2006); Boutloukos et al. (2006); M´endez(2006), and van der Klis (2006).Each of the many QPO models proposed (e.g., Alpar& Shaham 1985; Lamb et al. 1985; Miller at al. 1998;Psaltis et al. 1999; Wagoner 1999; Wagoner et al. 2001;Abramowicz & Klu´zniak 2001; Titarchuk & Kent 2002;Rezzolla et al. 2003; P´etri 2005; Zhang 2005; Kato 2007;Stuchl´ık et al. 2008; Mukhopadhyay 2009) still faces sev-eral difficulties and, at present, none of them is favored.In such a situation, we expect that the estimations ofmass and angular momentum based on the individualmodels could be helpful for the further development orfalsification of an appropriate model. In the next sectionwe therefore consider several of these models in addi-tion to the RP model investigated in Paper I, and exam-ine what mass–angular-momentum relations they imply.Since we do not attempt to describe the individual mod-els and resolve all their specific issues in detail, in whatfollows we just give a short summary of the models exam-ined and highlight some of their distinctions along withthe related references. Individual Models
The RP model has been proposed in a series of papersby Stella & Vietri (1998a,b, 1999, 2002) and Morsink& Stella (1999) and explains the kHz QPOs as a directmanifestation of modes of relativistic epicyclic motion ofblobs at various radii r in the inner parts of the accre-tion disk. Within the model, the twin peak QPO fre-quency correlation arises due to periastron precession ofthe relativistic orbits. Because of the existence of an-other so-called Lense–Thirring RP the model also pre-dicts another frequency correlation extending to highertimescales. The kHz QPO frequencies are indeed corre-lated with the low-frequency QPO features observed farbelow 100 Hz, which was first noticed and discussed inthe works of Psaltis et al. (1999), Stella & Vietri (1999),and Stella et al. (1999). Here we restrict our attentionmostly to kHz features but the low-frequency QPO in-terpretation within the RP model is briefly considered inSection 6 and Appendix B.1.Recently, ˇCadeˇz et al. (2008), Kosti´c et al. (2009), andGermana et al. (2009) have introduced a similar con-cept in which the QPOs were generated by a “tidal dis-ruption” (TD) of large accreting inhomogeneities. Itis assumed–and is supported by some hydrodynamicsimulations–that blobs orbiting the central compact ob-ject are stretched by tidal forces forming a “ring-section”features that are responsible for the observed modula-tion. The model has been proposed for black hole (BH)sources (both supermassive and stellar mass) but, inprinciple, it should work for compact NS sources as well.In some cases at least, the PDS produced within themodel seem to well reproduce those observed.It is often argued that QPOs arise due to “disk os-cillations” (in contrast to the above models considering“hot-spot motion”) and that some resonances can be in-volved. The disk-oscillation concept has a good poten-tial for explaining the high QPO coherence times ob- Some more details on these models and a discussion of theirrelevance to the black-hole QPOs can be found in T¨or¨ok et al.(2011). served in some NS systems (see Barret et al. 2005a, whofirst recognized the importance of the high QPO qualityfactor measured in 4U 1636–53, Q ∼ R ≡ ν U /ν L clustering (see Abramowicz et al. 2003a;Belloni et al. 2005; Belloni et al. 2007b; T¨or¨ok et al.2008a,b,c; Boutelier et al. 2010, for details and relateddiscussion). As found recently, in the six atoll NS sys-tems including 4U 1636 −
53, the difference between therms amplitudes of the upper and lower QPOs changesits sign for resonant frequency ratios R = 3 : 2 (T¨or¨ok2009). This interesting effect still requires some furtherinvestigation, since the rms amplitudes of kHz QPOs areenergy dependent and this must be taken into account.Nevertheless, we note that it was suggested by Hor´ak etal. (2009) that the “energy switch” effect could be nat-urally explained in terms of the theory of the nonlinearresonance.Two examples of the often quoted resonant disk-oscillation models are the epicyclic resonance (ER)model (Klu´zniak & Abramowicz 2001; Abramowicz etal. 2003b,c; Klu´zniak et al. 2004) assuming axisymmet-ric modes and the “warped disk” (WD) oscillation modelsuggested by Kato (2001, 2007, 2008) that assumes non-axisymmetric modes. We consider these and also anothertwo QPO resonance models dealing with different combi-nations of non-axisymmetric disk-oscillation modes. Thelatter two models are of particular interest because theyinvolve oscillation modes whose frequencies almost co-incide with the frequencies predicted by the RP modelwhen the NS rotates slowly. We denote them as RP1(Bursa 2005) and RP2 (T¨or¨ok et al. 2007; T¨or¨ok et al.2010) models and assume that the resonant correctionsto the eigenfrequencies are negligible.3.1.1. Frequency Relations
The relations that define the upper and lower QPOfrequencies in terms of the orbital frequencies are givenfor each of the above models in the first column of Ta-ble 1. We include these terms for the case of the Kerrspacetimes in Appendix A.1. The applicability of anapproach assuming the Kerr spacetimes for high-massNSs was elaborated in Paper I. The relevance and limi-tations of the same approach within the work and resultspresented here are discussed more in Section 6 and Ap-pendix A.3.For the RP model, one can easily solve the definitionrelations to arrive at the explicit formula which relatesthe upper and lower QPO frequencies. A similar sim-ple evaluation of an explicit relation between the two ob-served QPO frequencies is also possible for the TD model.For the RP and TD models, we give the explicit formulasin Equations (A3) and (A4). For the WD, RP1, and RP2models the definition relations lead to high-order poly-nomial equations that relate the lower and upper QPOfrequencies. In these cases, in Appendix A.1 we give onlythe implicit form of the ν U ( ν L ) function which has to betreated numerically.For the version of ER model assumed here, we expect T¨or¨ok et al.the ν U ( ν L ) function in the form of a linear relation. Thisapproach follows the work of Abramowicz et al. (2005a,b)and related details are briefly recalled in Section 4.4. DATA MATCHINGIn this section we fit the data points of 4U 1636–53and Circinus X-1 with frequency relations predicted fromeach of the individual models (i.e., by functions (A3)and (A4) for the RP and TD model, respectively, bya straight line for the ER model, and by the numer-ically given solutions of Equations (A5)–(A7) for theother models). As in Paper I, we restrict the rangeof mass and angular momentum considered to [ M ∈ (1 , M ⊙ ] × [ j ∈ (0 , . j = 0) for a single freeparameter M using the least-squares fitting procedure(e.g., Press et al. 2007). Then we also inspect the two-dimensional χ behavior for the free M and j .Within the numerical approach adopted the modelfrequency curve is parameterized along its full lengththrough a parameter p which ranges from p ∞ to p ISCO .The exact definition of χ that we use here is then givenas χ ≡ m X n =1 ∆ n , with ∆ n = Min (cid:18) l n, p σ n ,p (cid:19) p ISCO p ∞ , (5)where l n, p is the length of a line between the n th mea-sured data point [ ν L ( n ) , ν U ( n )] and a point [ ν L ( p ) , ν U ( p )]belonging to the model frequency curve. The quantity σ n ,p equals the length of the part of this line locatedwithin the error ellipse around the data point.4.1. Results for the RP, RP1, and RP2 Models
Considering j = 0 for fitting the data of 4U 1636–53 with the RP model, we find a narrow χ mini-mum for M ∼ . M ⊙ but its value is rather high, χ . = 350 / j . Thus, assuming thatthe model is valid, we can only speculate that there isan unknown systematic uncertainty. Then it followsfrom Equation (5) that the χ of the best fit for j = 0drops to an acceptable value χ = 1dof when the uncer-tainties in the measured QPO frequencies are multiplied(underestimated) by factor ξ ≡ p χ / dof . = 4. Underthis consideration we find the NS mass from the best-fitreading M = 1 . M ⊙ . We express the correspondingscatter in the estimated mass as δM = [ ± . M ⊙ , as-suming the 2 σ confidence level which we henceforth useas the reference one.On the other hand, the best match to the data of Circi-nus X-1 for the RP model and j = 0 already reveals anacceptable value of χ . = 12 . / M inferred from the RP modelfor both sources as M = 1 . ± . M ⊙ At this point we should also note that our choice of modelsrepresents a subset of those recently discussed by Lin et al. (2011)for the two sources 4U 1636–53 and Sco X-1. An overlap with theirwork is discussed in Section 6. in 4U 1636 −
53 ( χ = 1dof ⇔ ξ . = 4) (6)and M = 2 . ± . M ⊙ in Circinus X − χ = 12 . / . (7)As found in Paper I and briefly recalled here in Sec-tion 2, for the RP model and a given source the χ should not differ much along the M − j relation M ∼ M [1 + k ( j + j )] where k ∼ . − .
75 for high-frequencysources and k ∼ . − . M and j agree well with this finding. The χ be-havior for 4U 1636–53 is depicted and compared to thecase of Circinus X-1 in the form of color-coded maps inFigure 1(b). Clearly, the best fits are reached when M and j are related through the specific relations denotedby the dashed green lines. We approximate these rela-tions in the form M = M × [1 + k ( j + j )] arriving atthe following terms: M = 1 . ± . M ⊙ × [1 + 0 . j + j )]in 4U 1636 −
53 (8)and M = 2 . ± . M ⊙ × [1 + 0 . j + j )]in Circinus X − . (9)4.1.1. Results for the RP1 Model
The frequencies predicted by the RP and RP1 modelsare very similar for slowly rotating NSs. The two modelscommonly define the lower observable QPO frequency as ν L = ν K − ν r . (10)The upper observable QPO frequencies differ, reading ν RP U = ν K , ν RP1 U = ν θ . (11)In the Schwarzschild limit j = 0, ν θ = ν K and ν U iscommon to both RP and RP1. Consequently, M RP1 = M RP , (12)where M RP is given in Equation (6) and (7) for 4U 1636–53 and Circinus X-1, respectively. For 4U 1636-53, thequality of the fits does not differ much between j = 0 and j = 0 and the same conclusions on the possible unknownsystematic uncertainty as in the case of RP model arevalid.One can expect that fits to the data based on the RP1model for j = 0 should exhibit M − j degeneracy quali-tatively similar to the case of the RP model. We do notrepeat for RP1 model the full analysis of M − j degener-acy presented in the Paper I for the RP model. Instead,we just inspect the behavior of χ for free M and j tocheck whether such degeneracy is present and evaluateit. The χ behavior resulting for free M and j is depictedin the form of color-coded maps in Figure 2(a). The two χ maps displayed clearly reveal M − j degeneracy qual-itatively similar to that of the RP model. Related M − j relations (best χ for a fixed M ) are denoted by dashedgreen lines in Figure 2(a). We approximate these rela-tions in the form M = M × [1 + k ( j + j )] arriving atthe following terms: M = 1 . ± . M ⊙ × [1 + 0 . j + j )]ass–angular-momentum Relations Implied by NS kHz QPO Models 5 (a) (b) Fig. 1.— (a) Twin peak QPO frequencies in the atoll source 4U 1636–53 (22 data points in purple), Z-source Circinus X-1 (11 data pointsin red/yellow), and several other atoll- and Z-sources (data points in black). (b) The χ dependence on M and j for the RP model. Thetop panel corresponds to 4U 1636–53 while the bottom panel corresponds to Circinus X-1. For 4U 1636–53 ξ = 4 is assumed. The dashedgreen line indicates the best χ for a fixed M . The continuous green line denotes its quadratic approximation. The white lines indicatecorresponding 1 σ and 2 σ confidence levels. The white cross-marker denotes the mass and angular momentum reported for 4U 1636–53 andthe RP model by (Lin et al. (2011); see Section 6). The dashed yellow line in the top panel indicates a simplified estimate on the upperlimits on M and j assuming that the highest observed upper QPO frequency in 4U 1636–53 is associated with the ISCO. This estimateis not included for Circinus X-1 because the observed frequencies clearly points to the radii far away from ISCO which can be seen fromFigure 6. in 4U 1636 −
53 (13)and M = 2 . ± . M ⊙ × [1 + 0 . j + j )]in Circinus X − . (14)4.1.2. Results for the RP2 Model
As in the previous case, the frequencies predicted bythe RP2 model are very similar to those of the RP modelfor a slowly rotating NS. The lower observable QPO fre-quency is commonly defined by Equation (10). The up-per observable QPO frequency differs from the RP modeland reads ν RP2 U = 2 ν K − ν θ . (15)However, in the Schwarzschild limit j = 0, ν θ = ν K andthe expression for the upper observable QPO frequency ν U = ν K is common for all the three models RP, RP1,and RP2. For j = 0, therefore, the frequency relationsimplied by these models merge (although the expectedmechanisms generating QPOs are different). Thus wecan write M RP2 = M RP1 = M RP , (16)where M RP is given in Equation (6) for 4U 1636–53 andEquation (7) for Circinus X-1. For 4U 1636–53, the qual-ity of the fits is again not much different between j = 0and j = 0 and the same conclusions are valid on the pos-sible unknown systematic uncertainty as in the case ofthe RP1 and RP2 models.The χ behavior resulting from fitting the data pointsfor free M and j is depicted in the form of color-codedmaps in Figure 2(b). These χ maps again clearly reveal M − j degeneracy qualitatively similar to that in thecase of RP and RP1 models. The best χ for a fixed M ( M − j relation) is in each case denoted by the dashedgreen line. The corresponding approximate relations inthe form M = M × [1 + k ( j + j )] read M = 1 . ± . M ⊙ × [1 + 0 . j + j )]in 4U 1636 −
53 (17)and M = 2 . ± . M ⊙ × [1 + 0 . j + j )]in Circinus X − . (18)4.2. Results for the WD Model
Considering j = 0 for fitting the data of 4U 1636–53we find a narrow χ minimum for M ∼ . M ⊙ but itsabsolute value is somewhat higher than in the case ofthe RP model, χ . = 450 / j . Thus, we canagain only speculate that there is an unknown system-atic uncertainty. The χ of the best fit for j = 0 dropsto an acceptable value χ = 1dof for ξ . = 4 .
6. Therelated mass corresponding to the best fit then reads M = 2 . ± . M ⊙ .In analogy to the RP model, the best match to thedata of Circinus X-1 for j = 0 reveals an acceptablevalue of χ . = 10 . / M for both sources as M = 2 . ± . M ⊙ in 4U 1636 −
53 ( χ = 1dof ⇔ ξ = 4 .
6) (19)and T¨or¨ok et al. (a)
RP1 model (b)
RP2 model (c) WD model (d)
TD model Fig. 2.—
Same as Figure 1(b), but for the other models. In 4U 1636–53 ξ = 4 is assumed for the RP1 and RP2 models, ξ = 4 . ξ = 2 . M and j from the highestobserved QPO frequency in 4U 1636–53 is not included since the model does not associate this frequency to the ISCO but to the radiuswhere the term ν K ( r ) + ν r ( r ) reaches its maximum. M = 1 . . , − . M ⊙ in Circinus X − χ = 10 . / . (20)The χ behavior resulting from fitting the data pointsfor free M and j that again exhibits the M − j degeneracyis depicted in Figure 2(c). The exact M − j relations inthis figure are denoted by the dashed green lines. Theirapproximations in the form M = M × [1 + k ( j + j )]are, as in the previous cases, marked by the continuousgreen lines and read M = 2 . ± . M ⊙ × [1 + 0 . j + j )]in 4U 1636 −
53 (21)and M = 1 . . , − . M ⊙ × [1 + 0 . j + j )]in Circinus X − . (22) 4.3. Results for the TD Model
Considering j = 0 for fitting the data of 4U 1636–53we find a narrow χ minimum for M ∼ . M ⊙ whileits value χ . = 137 / × lower than in the case of the RPmodel. Moreover, there is also no sufficient improvementalong the whole given range of mass, even up to the upperlimit of j . Thus, again we can only speculate that there isan unknown systematic uncertainty. The χ of the bestfit for j = 0 drops to the acceptable value χ = 1dof for ξ . = 2 .
5. The related mass corresponding to the best fitthen reads: M = 2 . ± . M ⊙ ( χ = 1dof ⇔ ξ = 2 . . (23)For the Circinus X-1 data we find no clear χ minimum.It is roughly χ ∼ / χ is only slowly decreasing with M de-ass–angular-momentum Relations Implied by NS kHz QPO Models 7 (a) (b) Fig. 3.— (a) Profiles of the lowest χ for a given M plotted for various models. As in the previous figures, the case of 4U 1636–53 isshown in the top panel and Circinus X-1 in the bottom panel. The schematic drawing in the inset indicates the relation between the χ behavior and j common to all the plotted curves. (b) The mass–angular-momentum combinations allowed by the ER model. The colorsymbols indicate different equations of state (after Urbanec et al. 2010a,b, see these papers for details). The lightened subset of thesesymbols is compatible with the 4U 1636–53 data. The black line denotes its quadratic approximation (Equation(25)). creasing (or j increasing).Color-coded maps of χ resulting for free M and j areshown in Figure 2(d). In the case of 4U 1636–53 there isclearly an M − j degeneracy. The M − j relation is wellapproximated in the form M = M × [1 + k ( j + j )] as M = 2 . ± . M ⊙ × [1 + 0 . j + j )] . (24)On the other hand, the χ distribution for Circinus X-1 is rather flat, exhibiting roughly χ ∼ / M and in-creasing j .For the case of 4U 1636–53 the detailed profile of χ along the relation (24) is shown and compared to the RP,RP1, RP2, and WD models in Figure 3(a). In the samefigure we also show an analogous comparison for CircinusX-1. The absence of an M − j relation and behavior of χ for the TD model in the case of Circinus X-1 is thendiscussed in Section 6.4.4. Results for the ER Model
Adopting the assumption that the observed frequen-cies are nearly equal to the resonant eigenfrequencies, ν U = ν θ ( r ) and ν L = ν r ( r ), the ER model does not fitthe NS data (e.g., Belloni et al. 2005; Urbanec et al.2010b; Lin et al. 2011). A somewhat more complicatedcase in which this assumption is not fulfilled has beenrecently elaborated by Urbanec et al. (2010b), who as-sumed data of 12 NS sources, including 4U 1636-53.They investigated the suggestion made by Abramowiczet al. (2005a,b) that the resonant eigenfrequencies in 12NS sources roughly read ν L = 600Hz versus ν U = 900Hzand the observed correlations follow from the resonantcorrections to the eigenfrequencies, ν L = ν L + ∆ ν L versus ν U = ν U + ∆ ν U . In this concept the resonance occursat the fixed radius r and the data of the individualsources are expected as a linear correlation. Intersec-tion of this correlation with the ν U /ν L = 3 / ν L = ∆ ν U = 0 when R = 3 /
2. More details and references to the model can be found in Urbanec et al.(2010b).For the sake of the comparison with the RP and othermodels examined here, we plot Figure 3(b) based on theresults of Urbanec et al. (2010b). The figure displayscombinations of mass and angular momentum requiredby the model. The color-coded symbols indicate solu-tions for different equations of state (EoS). We denotethe subset of these solutions compatible with the dataof 4U 1636–53 by lighter symbols. The determination ofthis subset comes from the fit of 4U 1636–53 data by astraight line ( χ = 37 / / dof). It is clear from the figurethat, as in the previous cases, for the ER model there is apreferred mass–angular-momentum relation. In contrastto the other models examined, it tends to a positive cor-relation between M and j only for low values of the angu-lar momentum, j . .
2, while for a higher j the requiredmass decreases with increasing j . This trend is connectedto a high influence of the NS quadrupole momentum andlarge deviation from the Kerr geometry that arise for thelow-mass NS configurations (see Urbanec et al. 2010b,for details). We find that the mass–angular-momentumrelation implied by the ER model for 4U 1636–53 canbe approximated by a quadratic term roughly as (blackcurve in Figure 3(b)) M = 0 . M ⊙ × (cid:2) . j − j (cid:3) ± . (25)For Circinus X-1, the observed frequency ratio is far awayfrom R = 3 / × lessthan the typical twin peak QPO frequencies observed in4U 1636–53. The related non-rotating mass would thenbe approximately 3 × higher than that corresponding to T¨or¨ok et al. TABLE 1The main definition relations for the models considered and the mass–angular-momentum relationsfound for 4U 1636–53 and Circinus X-1.
Model atoll source 4U 1636-53 Z-source Circinus X-1 χ / dof ∼ ξ ∼ ( M /M ⊙ ) × f ( j ) χ / dof ∼ ( M /M ⊙ ) × f ( j )RP ν L = ν K − ν r , 16 4.0 1 . ± . × [1 + 0 . j + j )] 1 . . ± . × [1 + 0 . j + j )] ν U = ν K TD ν L = ν K , 7 2.5 2 . ± . × [1 + 0 . j + j )] 30 X ν U = ν K + ν r WD ν L = 2( ν K − ν r ), 21 4.6 2 . ± . × [1 + 0 . j + j )] 1 . . a ν U = 2 ν K − ν r RP1 ν L = ν K − ν r , 16 4.0 1 . ± . × [1 + 0 . j + j )] 1 . . ± . × [1 + 0 . j + j )] ν U = ν θ RP2 ν L = ν K − ν r , 16 4.0 1 . ± . × [1 + 1 . j + j )] 1 . . ± . × [1 + 0 . j + j )] ν U = 2 ν K − ν θ ER ν L = ν r + ∆ ν L b , 3 1.7 0 . ± . × [1 + 0 . j − j ] 1 . . ± . × [1 + 1 . j + j )] c ν U = ν θ + ∆ ν U Note . — Symbols ν K , ν r , and ν θ denote the orbital Keplerian, radial epicyclic, and vertical epicyclic frequencies(see Appendix A.1 for the explicit terms). For both sources, except for the ER model, the errors in the estimatedmass corresponds to the 2 σ confidence level. For the ER model, the errors are given by the scatter in the estimatedresonant eigenfrequencies (see Urbanec et al. 2010b). a The mass–angular-momentum relation that we found reads M = 1 . . , − . M ⊙ × [1 + 0 . j + j )]. Due tothe low M , the M ( j ) dependence cannot be taken seriously (see Section 6 for a comment on this). b See Section 4.4 for details. c The possibility that the observed frequencies are the combinational frequencies is taken into account.
4U 1636–53, i.e., M ∼ M ⊙ . The related fit of theCircinus X-1 data by a straight line has χ = 16 / j < .
5, we can express the formula forthe mass of Circinus X-1 implied by the ER model ap-proximately as M = 3 M ⊙ × (cid:2) . (cid:0) j + j (cid:1)(cid:3) ± . (26)While for 4U 1636–53 the mass decreases with increas-ing j (Equation (25)), for Circinus X-1 the trend is op-posite. This behavior is associated with the choice of thespacetime geometry. The low mass M ∼ M ⊙ inferredfrom the model for 4U 1636–53 implies high deviationsfrom the Kerr geometry due to the NS oblateness (Ur-banec et al. 2010a,b). In such situation orbital frequen-cies can decrease with increasing j . For Circinus X-1,the high mass M = 3 M ⊙ justifies the applicability of theKerr geometry chosen. For this geometry, the orbital fre-quencies must increase with increasing j (provided that j < CHI-SQUARED DICHOTOMY ANDCORRECTIONS TO THE RP OR OTHERMODELSIt has been noticed by Stella & Vietri (1999) and laterby a number of other authors that data of sources with QPOs sampled mostly on low frequencies are better fit-ted by the RP model than data for sources with QPOssampled mostly on high frequencies. Inspecting χ -squaremaps (Figures 1 and 2) and Table 1, we can see thatthe comparison between Circinus X-1 (good χ ) and 4U1636–53 (bad χ ) well demonstrates such a “dichotomy”.The χ maps and profiles for the RP model are quali-tatively similar for both 4U 1636–53 and Circinus X-1.Both sources also exhibit a decrease of χ with increas-ing j (see Figure 3(a)). The χ values reached for 4U1636–53 are, however, much worse than those in the caseof Circinus X-1 ( ≈
10 versus 1 dof), and their spread with M is much narrower. Moreover, we find that a similar di-chotomy also arises for all the other models considered as-suming that the observed twin peak QPO frequency cor-relation arises directly from a correlation between char-acteristic frequencies of the orbital motion. Below webriefly discuss the relation between this dichotomy, thepredictive power of the model, and possible non-geodesiccorrections. We restrict our attention mostly to the RPmodel but argue that there is a straightforward general-ization to the other models.5.1. Data versus Predictive Power of the RP Model
Figure 4(a) shows the frequency relations predicted bythe RP model for a non-rotating NS and several valuesof mass M . These curves run from the common point[ ν L , ν U ] = [0Hz, 0Hz] corresponding to infinite r . Theyterminate at specific points [ ν ISCO , ν
ISCO ] correspondingass–angular-momentum Relations Implied by NS kHz QPO Models 9 (a) (b) (c)
Fig. 4.— (a) Frequency relations predicted by the (geodesic) RP model for j = 0 vs. data of 4U 1636-53 and Circinus X-1. (b) Quantity P illustrating the variability of the predictive power of the RP model across the frequency plane. (c) Profiles of the orbital, radial epicyclic,and periastron frequencies of the perturbed circular motion. Solid curves correspond to the geodesic case ( β = 0). The dashed and dottedcurves correspond to the case of non-geodesic radial oscillations ( β > to r = r ms = r ISCO . This behavior follows the fact thatfor low excitation radii close to ISCO, a certain change in M leads to a modification of the orbital frequency thatis much higher than those for radii far away from ISCO.In other words, the predictive power of the RP model ismuch weaker for radii far away from ISCO than for radiiclose to ISCO.As recalled in Section 1, in the RP model the radius r is proportional to R (e.g., T¨or¨ok et al. 2008c). Because ofthis, the predictive power of the RP model is strongly de-creasing with increasing R . In Appendix A.2 we discussthis in terms of the quantity P ∝ R − determining thesquared distance ds measured in the frequency planebetween data points related to different masses. Thisquantity has a direct impact on the spread of χ . For acertain variation of the mass, δ ≡ ∆ M/M , it is ds ∝ δ (1 + δ ) P . (27)Detailed formulae are given in Equations (A11) and(A12). Figure 4(b) shows behavior of P in the frequencyplane.Taking into account the data points included in Fig-ures 4(a) and (b) and the behavior of P we can deducethat the difference in the spread of χ in 4U 1636–53 andCircinus X-1, as well as the very different values of the χ minima in these sources, can be related to both thesize of the error bars (affected by a low significance ofkHz QPOs on low frequencies) and the location of datapoints. In Circinus X-1, the data points lie in the re-gion of relatively low frequencies related to high R . Forthese, the predictive power of the model is low, since thecurves ν U ( ν L ) expected for various parameters M and j converge. On the other hand, in 4U 1636–53, the datapoints lie in the region of relatively high frequencies re-lated to low R . These correspond to the strong gravityzone where different correlations are much more distin-guished and the predictive power of the model is high.Similar consideration is also valid for several other mod-els that predict frequency curves converging at low R .Clearly, from Figures 1 and 2 we can see that the uncer-tainties of the inferred mass expressed at 2 σ confidence levels in 4U 1636–53 are ∼ − × smaller comparedto Circinus X-1 for each of the RP, RP1, RP2, and WDmodels.5.2. Toy Non-geodesic Modification of the RP Model
Based on the above findings, we can speculate thatthe same systematic deviation from the particular modelconsidered may be involved in both sources. We justifythis speculation using an arbitrary example of a toy non-geodesic version of the RP model. We attempt to use amodification that would mimic the behavior of real data.In the vicinity of the inner edge of an accretion disk itis natural to expect a modification of the radial epicyclicfrequency rather than a modification of the Keplerianfrequency. The orbital motion in this region is highlysensitive to radial perturbations and even very small de-viations from the geodesic idealization can strongly affectthe radial oscillations (see in this context Stuchl´ık et al.2011). In our example we therefore assume that the fre-quency of the hot-spot radial oscillations is somewhatlowered due to pressure or magnetic field effects (e.g.,Straub & ˇSr´amkov´a 2009; Bakala et al. 2010, 2012). Forsimplicity, we postulate that the effective frequency ofthe radial oscillations is˜ ν r = ν r (1 − β ) , (28)where β is a small constant. The related lower QPOfrequency actually observed is then given by˜ ν L = ν L + β ( ν U − ν L ) , (29)where ν L ( ν U ) is the frequency relation of the geodesic RPmodel given in Equation (A3). Assuming Equation (29), β = 0 . j = 0, and M = 2 M ⊙ we produce 20 datapoints uniformly distributed along the frequency corre-lation. We then fit the simulated data by the geodesicmodel. Figure 5(a) shows the resulting χ profile calcu-lated in the same way as those in Figure 3(a). Clearly, χ decreases with growing j similarly to the results ob-tained for real data points in both sources discussed. Forcomparison, we also present the fit of data simulated for β = 0, where, in contrast, χ increases with growing j .Having this boost we use Equation (29) for the fitting of0 T¨or¨ok et al. (a) (b) (c) Fig. 5.— (a) Profile of the best χ M calculated when the simulated data are matched by the geodesic RP model. Thecontinuous line is plotted for M = 2 M ⊙ , j = 0, and β = 0 .
1. The dashed line is plotted for β = 0. The arrows indicate increasing j. (b)Profiles of the best χ M in the case when Equation (29) is assumed for fitting of the real data. The arrows in each panelindicate increasing j. The vertical arrow denotes the improvement ∆ χ . (c) Comparison of the geodesic ( β = 0, thick blue line) andnon-geodesic ( β >
0, red line) fits is included in the “zoom” from Figure 4(a). The top panel is plotted for 4U 1636-53 while the bottompanel is plotted for Circinus X-1. Both panels have the same scaling of the axes. the real data points. The resulting ”best χ ” improvesfor both sources, although in the case of Circinus X-1the improvement is only marginal. More specifically, for4U 1636–53 the best χ improves up to β ∼ . χ ∼ β ∼ . χ ∼
4. The representative χ profiles areillustrated in Figure 5(b), which also shows the relatedimpact on mass restrictions. The strong improvementin 4U 1636–53 data corresponds to only a marginal ef-fect on the mass restriction (∆ M . . M ⊙ ). On theother hand, the small improvement of χ in CircinusX-1 causes a large modification of the mass restriction(∆ M ∼ . M ⊙ ). The related fits to the data are shownin Figure 5(c).The toy model (29) naturally does not represent anelaborate attempt to describe the QPO mechanisms, butit demonstrates well that, in spite of the good qualityof fit, in both 4U 1636–53 and Circinus X-1 sources,the same physical correction to the RP model could beinvolved. A similar consideration should also be validfor several other models discussed. In this context, wenote that sophisticated implementations of non-geodesiccorrections have been developed in the past within theframework of various models of accretion flow dynam-ics and QPOs (see, e.g., Wagoner et al. 1999, 2001;Kato 2001; Alpar & Psaltis 2008, and references therein). We also note that some corrections to the orbital fre-quencies can arise directly due to corrections to theKerr or Hartle–Thorne (HT) spacetimes that we assumehere (see, e.g., Kotrlov´a et al. 2008; Psaltis et al. 2008;Stuchl´ık & Kotrlov´a 2009; Johannsen & Psaltis 2011). DISCUSSION AND CONCLUSIONSExcept the TD model applied to Circinus X-1 data,all applications of the models examined to the 4U 1636–53 and Circinus X-1 data result in the preferred mass–angular-momentum relations. These are summarized inTable 1.Comparing the χ map of the TD model and Circi-nus X-1 (Figure 2(d)) to the other χ maps we can seethat it is very different with its flat χ behavior. More-over, the TD model is the only model of those consideredhere giving very bad χ for Circinus X-1 ( χ ∼ χ ∼ R impliedby the model. The TD model states ν L = ν K , ν U = ν K + ν r , (30)where ν r ≤ ν K . In more detail, ν r vanishes at r = r ISCO and, in a flat spacetime limit ( r = ∞ ), ν r = ν K . Conse-quently, the TD model allows only R ∈ (1 , R ∼ . (a) (b) Fig. 6.—
Best fits to the data by individual models for j = 0. (a) Frequency relations. Error bars corresponding to ξ = 4 for RP models, ξ = 4 . ξ = 2 . and R ∼ . R = 2. This disfavors the TD model.6.1. Quality of Fits and Inferred Masses: Models with ν ( r )Table 1 provides a summary of results of fits to the datafor both sources by individual models. The comparisonbetween fits by individual models is illustrated in Fig-ure 6 which also indicates the inferred QPO excitationradii. Within the RP, RP1, RP2, and WD models, thequality of fits is rather comparable (bad for 4U 1636 andgood for Circinus X-1). The mass–angular-momentumrelations are similar for the RP, RP1, and RP2 modelswhile for the WD model they differ (see Table 1). In moredetail, the RP, RP1, and RP2 models require relativelysimilar masses for both sources, namely M ∼ . M ⊙ for 4U 1636–53 versus 2 . M ⊙ for Circinus X-1. On theother hand, the required masses differ quite a lot whenthe WD model is assumed. We then have M ∼ . M ⊙ for 4U 1636–53 versus 1 . M ⊙ for Circinus X-1. We notethat the QPO excitation radii inferred for each modelin 4U 1636–53 lie within the innermost part of the ac-cretion disk. This is depicted in detail in Figure 6(b)assuming a non-rotating star. We can see that the radiispan the interval r ∈ (6 M − M ) for the RP model, r ∈ (7 M − M ) for the WD model, and the largest inter-val r ∈ (6 M − M ) for the TD model. On the contrary,the radii inferred in Circinus X-1 are above r = 10 M ,belonging to the interval r ∈ (10 M − M ) for the RPmodel and r ∈ (15 M − M ) for the WD model.The above models have, along with few others, recentlybeen considered for 4U 1636–53 by Lin et al. (2011).They reported mass and angular momentum correspond-ing to χ minima for each of the models. The data pointsthey investigated especially for this purpose come froma sophisticated, careful application of a so-called shift-add procedure over a whole set of the available RXT E observations (see their paper for details and references).The data we use here for 4U 1636–53 come from thepreviously well-investigated individual continuous obser-vations of the source (see Barret et al. 2005b,c; T¨or¨ok2009). While the two sets of the applied data comefrom different methods, the values of mass and angu-lar momentum reported by Lin et al. (2011) agree withthe mass–angular-momentum relations that we find here(see Figures 1(b) and 2).One should note that, in con-trast to M − j relations, the single M − j combinationcorresponding to the χ minimum of a given model is notvery informative as the (bad) χ is comparable along alarge range of mass. Moreover, in each case examined2 T¨or¨ok et al. (a) (b) Fig. 7.— (a) The ambiguities of parameters of RP model frequency relations illustrated for the range j ∈ (0 , .
3) and ˜ q ∈ (1 , M in the Schwarzschild spacetime. The dark blue set of curves marked as “Kerr”represent the degeneracy in the Kerr spacetimes given by Equation (A14). The light blue set marked as “Hartle-Thorne” includes curvesresulting from the generalized degeneracy in HT spacetimes given by Equation (A15). The shadow cone denotes the range of frequencyratio R corresponding to the data of 4U 1636-53. (b) Removing the M − j degeneracy in the case of 4U 1636-53 and the RP model. The χ map displayed is calculated for β = 0 while the best fits correspond to β = 0 . − .
20. The blue spot roughly indicates the combination ofmass and spin restricted when the spin frequency 290Hz and several concrete equations of state are assumed. The red spot indicates thesame but for the spin frequency 580Hz. The shaded region around the dashed horizontal line indicates the angular momentum j = 0 . ± . here the χ minima correspond only to the end of theangular momentum interval considered since the qualityof fit is a monotonic function of j . Thus, we can concludethat the differences between the M coefficients in Table1 provide the main information about the differences be-tween predictions of the individual QPO models.In relation to the quality of fits by the RP, RP1, RP2,and WD models, we can also note that these modelsneed some correction, as has also been noted by Lin etal. (2011). As demonstrated in Section 5, differences inthe χ behavior between low- and high-frequency sourcescan be related to the variability of the model’s predictivepower across the frequency plane. This variability nat-urally comes from the radial dependence of the charac-teristic frequencies of orbital motion. As a consequence,the restrictions to the models resulting from the obser-vations of low-frequency sources are weaker than thosein case of high-frequency sources. A small required cor-rection is then likely to be common to both classes ofsources, which has been demonstrated using the non-geodesic modification of the RP model based on Equa-tion (29).6.1.1. Applicability of Results Based on the SpacetimeDescription Adopted
Both the poor quality of fits to the data by geodesicmodels and the mass–angular-momentum relations asso-ciated with these models have been obtained assumingthe Kerr spacetimes. This approximate description ofthe exterior of rotating NS neglects the NS oblateness.As argued in Paper I, the uncertainty in NS oblatenesscauses only small inaccuracies in the modeling of kHzQPOs for the compact high-mass NSs. In the case ofthe WD model applied to the Circinus X-1 data, a con-sequent application of a more sophisticated approach isstill needed. The Kerr approximation suggested in Pa-per I is clearly not valid here due to low M ∼ . M ⊙ .Such a low mass can imply high deviations from the Kerrgeometry due to the strong influence of the NS oblate- ness. In principle, the related mass–angular-momentumrelation can be very different in this case from that qual-itatively implied, e.g., by the RP model, and correctionsto the quadrupole moment should be included in anal-ogy to the ER model and 4U 1636–53. For the otherapplications of the WD, RP, RP1, and RP2 models re-ported here we can trust the M − j trends following fromthe Kerr approximation since the inferred masses M arerather high.We justify the applicability of our results in Ap-pendix A.3. In general, the differences between geodesicfrequencies associated with Kerr spacetimes and thosegiven for realistic NSs due to their oblateness are roughlyof the same order as the corrections required to obtaina good match between the predicted and observed QPOfrequencies (e.g., Morsink & Stella 1999). We illustratehowever, that these differences cannot improve the fitssufficiently for NSs with j . . M & . M ⊙ . Weshow that in HT spacetimes describing the exterior ofoblate NSs there is a degeneracy not only between theNS mass and angular momentum but also between thesequantities and the NS quadrupole moment q . Withinsuch “generalized degeneracy” the frequency curves pre-dicted by QPO models scale with the quantities M, j ,and q but the related qualitative change in their shapeis only small. Thus, our results obtained for the Kerrspacetimes have a more general relevance except for thecase of high values of j (see Appendix A.3 for details).6.1.2. Prospects of Eliminating the M − j Degeneracy
The M − j degeneracies implied by individual kHz QPOmodels can in principle be eliminated using angular mo-mentum estimates independent of the kHz QPOs. In Ap-pendix B we subsequently focus on the RP model anddiscuss such possible elimination. Based on the X-rayburst observations of Strohmayer & Markwardt (2002)we assume that the rotational frequency (spin) of the NSin 4U 1636–53 is around 290Hz or 580Hz. Applying fewconcrete NS EoS we show that the modified RP modelass–angular-momentum Relations Implied by NS kHz QPO Models 13well matches these spins for j ∼ . j ∼ .
2. We alsoshow that a further consideration of low-frequency QPOsand the Lense–Thirring precession mechanism within themodel can be finally crucial for fixing the value of j andchallenging for application of the concrete EoS (see illus-tration in Figure 7). We note that this issue as well asmodeling of kHz QPO correlations for rapidly rotatingNS require an additional detailed treatment.6.2. Resonance between m = 0 AxisymmetricDisk-oscillation Modes
Last but not least, we can draw conclusions about theversion of the ER model examined assuming the fixedradius r = r . It well fits the data of both the sourcesdiscussed here with a χ / dof of the order of unity. Thegood fits, however, arise because the present model pre-dicts a linear correlation which has slope and interceptgiven by unspecified (free) parameters. One should alsonote that for Circinus X-1 the model requires additionalconsideration of the resonant combinational frequencies. Moreover, application of the ER model leads to a ques-tionably low mass for 4U 1636–53, M ≤ M ⊙ , while forCircinus X-1 the implied mass is on the contrary ques-tionably high, M ≥ M ⊙ . All these along with the re-sults of Urbanec et al. (2010b) suggest that if a resonanceis involved in the process of generating the NS QPOs,modes other than those corresponding to the radial andvertical axisymmetric oscillations should be considered.We thank Marek Abramowicz, Wlodek Klu´zniak, Mi-lan ˇSenk´yˇr, and Yong-Feng Lin for discussions. We alsothank to the anonymous referee for his/her commentsand suggestions that helped greatly to improve the pa-per. This work has been supported by the Czech grantsMSM 4781305903, LC 06014, GA ˇCR 202/09/0772, andGA ˇCR 209/12/P740. The authors further acknowledgethe project CZ.1.07/2.3.00/20.0071 ”Synergy” support-ing the international collaboration of IF Opava and alsothe internal student grants of the Silesian University inOpava, SGS/1/2010 and SGS/2/2010.APPENDIX APPROXIMATIONS, FORMULAS, AND EXPECTATIONS
Relations for the Upper and Lower QPO Frequencies in the RP, TD, WD, RP1, and RP2 Models
Formulas for the Keplerian, radial, and vertical epicyclic frequency were first derived by Aliev & Galtsov (1981). Ina commonly used form (e.g., T¨or¨ok & Stuchl´ık 2005) they readΩ K = F j + x / , ν r = ΓΩ K , ν θ = ∆Ω K , (A1)where Γ = r − j + 8 j √ x + ( − x ) xx , ∆ = r j (3 j − √ x ) x , (A2) x ≡ r/M , and the ”relativistic factor” F reads F ≡ c / (2 πGM ).Relations defining the upper and lower QPO frequencies in terms of the orbital frequencies are given for each of themodels considered in the first column of Table 1. For the RP model, one can easily solve these relations to arrive atan explicit formula which relates the upper and lower QPO frequencies in the units of Hertz as (Paper I) ν L = ν U − " jν U F − jν U − (cid:18) ν U F − jν U (cid:19) / − j (cid:18) ν U F − jν U (cid:19) / / . (A3)A similar simple evaluation of the explicit relation between the two observed QPO frequencies is also possible forthe TD model, where we find ν U = ν L " jν L F − jν L − (cid:18) ν L F − jν L (cid:19) / − j (cid:18) ν L F − jν L (cid:19) / / . (A4)An apparent “asymmetry” between relations (A3) and (A4) arises from an analogical asymmetry in the model definitionof the observable frequencies (see Table 1). We note that in both models, one of the two observable frequencies simplyequals to the Keplerian orbital frequency, which makes the evaluation of the explicit formula very straightforward.For the WD, RP1, and RP2 models, the definition relations lead to high-order polynomial equations that relate thelower and upper QPO. In these cases we can give only parametric form relating ν U and ν L . The upper and lower QPOfrequencies for the WD model can be then expressed as ν U = 2 (1 − Γ) Ω K , ν L = (2 − Γ) Ω K . (A5)For the RP1 model they can be written as ν U = Ω K ∆ , ν L = (1 − Γ) Ω K , (A6)and for the RP2 model as ν U = (2 − ∆) Ω K , ν L = (1 − Γ) Ω K . (A7)4 T¨or¨ok et al. Predictive Power of the RP Model
Let us assume a non-rotating star. The radial epicyclic frequency vanishes at ISCO, x = 6, where the orbitalfrequency takes the value of ν K = ν ISCO = c √ GM π (A8)and within the RP model it is ν U = ν L = ν ISCO . (A9)When a certain variation of the mass, δ ≡ ∆ M/M , is assumed, the point in the frequency plane given by Equation(A9) changes its position. The corresponding square of the distance ds (important for the fitting of data) reads ds = ν ISCO δ (1 + δ ) . (A10)For any other specific orbit inside the accretion disk (e.g., x = 8, where the radial epicyclic frequency takes its maximalvalue), the analogous change of the related data point position in the frequency plane is always smaller, ds < ds .It is useful to utilize the fact that each specific orbit can be related to a certain frequency ratio R higher than R = 1corresponding to ISCO (e.g., for x = 8 it is R = 2). Using the relation between x and R (e.g., T¨or¨ok et al. 2008c), onecan find that ds = ds × P , (A11)where P = (cid:0) R + 1 (cid:1) (2 R − R . (A12)The quantity P = P ( R ) reads P = 1 for R = 1 and strongly decreases with increasing R . This naturally illustratesthat the predictive power of the model is high only for orbits close to ISCO. For instance, for the maximum of theradial epicyclic frequency where R = 2, it is roughly P = 0 . P for a non-zero j is less straightforward and does not bring any new interesting information. Generalized Degeneracy
As recalled in Section 2, the frequency curves predicted by the model (and other kHz QPO models) scale with theNS mass and angular momentum, but do not change their shape much when j . .
5. This was explored in detailassuming the Kerr spacetimes. The exterior of a rotating NS is in general well described by the HT spacetimes whichare determined by the NS mass M , angular momentum j , and a quadrupole moment q reflecting the NS oblateness.One can ask whether there can be a “generalized degeneracy” related to all these three quantities similar to thoserelated just to M and j in the Kerr spacetimes. We briefly attempt to resolve this issue using formulas for epicyclicfrequencies in HT spacetimes derived by Abramowicz et al. (2003a).The orbital frequency at a marginally stable circular orbit increases with increasing angular momentum j while itdecreases with increasing quadrupole moment q . Thus, following Appendix A.2 of Paper I, we can expect that theeventual generalized degeneracy can, to first order in q and second order in j , be expressed as M ∼ M (cid:0) k j + k j − k q (cid:1) . (A13)In the limit of ˜ q = 1, where ˜ q ≡ q/j is the so-called “Kerr parameter”, relation (A13) has to merge with the massspin relation derived for the Kerr spacetimes. This relation is represented by Equation (2) which, assuming wholefrequency curves, reads M ∼ M (cid:2) . (cid:0) j + j (cid:1)(cid:3) . (A14)Therefore we choose k = 0 . k = k + k . Then only k remains as a “tunable” parameter.We searched for a value of k providing the eventual generalized degeneracy. For a particular choice of k = 0 . M = M (cid:0) . j + 1 . j − . q (cid:1) , (A15)we found results in full analogy to those that we had previously obtained for the Kerr spacetimes. This finding isillustrated in Figure 7(a). The figure is plotted for j ∈ (0 , .
3) and ˜ q ∈ (1 , M , j , and q there is a nearly identical curve drawn for the Schwarzschild spacetime given byEquation (A15). Thus, consideration of NS oblateness cannot improve the poor quality of fits of models to the datawithin the limits of j and q assumed for the figure. These limits correspond to almost any NS modeled using the usualEoS for the mass M > . M ⊙ and spin frequencies up to 600Hz (Lattimer & Prakash 2001, 2007).Considering the above facts, we can summarize the findings as follows: the results on M − j relations obtained forthe Kerr spacetimes have rather general validity and NS oblateness could only cause some correction to the slope of aparticular M − j relation. The only exceptions exceeding the framework of the work presented are represented by thecases of j ≫ . M < . M ⊙ , or some unusual NS models that have to be treated in detail assuming concrete EoS.ass–angular-momentum Relations Implied by NS kHz QPO Models 15 REMOVING DEGENERACY IN THE CASE OF THE RP MODEL AND 4U 1636–53
For the atoll source 4U 1636–53 there is good evidence on the NS spin frequency based on X-ray burst measurements.Depending on the (two- or one-) hot-spot model consideration, the spin frequency ν S reads either ν S ∼ Hz or ν S ∼ Hz (Strohmayer & Markwardt 2002). Thus, one can, in principle, infer the angular momentum j and removethe M − j degeneracies related to the individual twin peak QPO models. In Figure 7 we illustrate the potential of such an approach requiring a complex usage of various versions of a detailedultra-dense matter description. The figure is made for the non-geodesic version of the RP model based on Equation (29)with β = 0. It includes a χ map resulting from the fitting of 4U 1636–53 data with the model together with the M − j relations inferred from the equalities ν S = 290 Hz or ν S = 580 Hz. These M − j relations that depend on ultra-densematter properties were calculated using the approach of Hartle (1967), Hartle & Thorne (1968), Chandrasekhar &Miller (1974), Miller (1977), and Urbanec et al. (2010a). They assume the same set of several EoS as we used in Paper I,namely SLy 4 (Rikovska Stone et al. 2003), APR (Akmal et al. 1998), AU-WFF1, UU-WFF2, and WS-WFF3 (Wiringaet al. 1988; Stergioulas & Friedman 1995).Comparing the χ map to the M − j relations based on our choice of EoS we can conclude that the parameters ofthe NS implied by the model must be either j ∼ .
11 and M ∼ . M ⊙ , or j ∼ .
22 and M ∼ M ⊙ . In panel (b) ofFigure 7 we can check that in both cases the quality of fit to twin peak QPO data is acceptable (the best fits wereobtained for the value of β ∼ . − . Adding Low-frequency QPOs
The RP model associates the observed low-frequency QPOs to the Lense–Thirring precession that occurs at thesame radii as the periastron precession crucial for the high-frequency part of the model. It is then expected that theirfrequencies ν ℓ equal the Lense–Thirring precession frequency, ν ℓ = ν LT . (B1)Naturally, the value of ν LT depends more strongly on the angular momentum j than on the concrete radius r , since itvanishes for j → S = { ν ℓ , ν L , ν U } . For instance, Jonker et al. (2005) reported clear measurements oflow-frequency QPOs in 4U 1636–53 as well as their relation to the high-frequency part of PDS. For the PDS relatedto the middle part of the frequency correlation,[ ν L , ν U ] = [700 − , − , (B2)the frequencies ν ℓ were approximately around ν ℓ . = 42Hz . (B3)For the PDS related to the upper part of the frequency correlation,[ ν L , ν U ] = [800 − , − , (B4)the frequencies ν ℓ were around ν ℓ . = 43 . . (B5)Assuming these frequency intervals we can apply the equalities ν U = ν K , ν L = ν RP = ν K − ν r and ν ℓ = ν LT = ν K − ν θ . (B6)For the application we consider Equation (29) with β = 0 .
17 which provides acceptable fits to the twin peak QPOs.The spin j is then fixed just by the ratio between the observed frequencies (B6). Consequently we find that j mustbe about j = 0 . − .
3. Moreover, when using the measured frequency values, the relations (B6) determine both M and j just for a single point in the 3D frequency space S . Using this fact and the values of Jonker et al. (2005) wefind that M = (2 . − . M ⊙ for j = 0 . − . M and j are marked in Figure 7 by the green box. Note, however, that the considerationneeds to be further expanded for a larger set of data and some χ mapping in the 3D frequency space S should bedone. This can be somewhat complicated by the fact that low-frequency QPOs are, in general, broader than the kHzfeatures. In addition, the quadrupole momentum influence on ν LT could be overestimated due to the Kerr geometryapproximation considered here. Nevertheless, assuming all these uncertainties we can still expect from the abovenumbers that a further detailed consideration should confirm the value of j roughly inside the interval j LT = 0 . ± . . (B7)Figure 7 finally integrates both the implications of X-ray burst measurements and the Lense–Thirring precessionmodel for low-frequency QPOs. We can see that an EoS relatively distant from those which we consider here could be6 T¨or¨ok et al.needed in order to match both phenomena and fix the NS spin. This challenging issue clearly requires further futurework joining data analysis in the field of 3D frequency space and modeling the detailed influence of the NS EoS.(B7)Figure 7 finally integrates both the implications of X-ray burst measurements and the Lense–Thirring precessionmodel for low-frequency QPOs. We can see that an EoS relatively distant from those which we consider here could be6 T¨or¨ok et al.needed in order to match both phenomena and fix the NS spin. This challenging issue clearly requires further futurework joining data analysis in the field of 3D frequency space and modeling the detailed influence of the NS EoS.