Issue 
A&A
Volume 522, November 2010



Article Number  A72  
Number of page(s)  7  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/201014731  
Published online  04 November 2010 
Discoscillation resonance and neutron star QPOs: 3:2 epicyclic orbital model
Institute of Physics,
Faculty of Philosophy and Science, Silesian University in Opava,
Bezručovo nám. 13,
74601
Opava, Czech
Republic
email: terek@volny.cz
Received: 3 April 2010
Accepted: 23 June 2010
The highfrequency quasiperiodic oscillations (HF QPOs) that appear in the Xray fluxes of lowmass Xray binaries remain an unexplained phenomenon. Among other ideas, it has been suggested that a nonlinear resonance between two oscillation modes in an accretion disc orbiting either a black hole or a neutron star plays a role in exciting the observed modulation. Several possible resonances have been discussed. A particular model assumes resonances in which the discoscillation modes have the eigenfrequencies equal to the radial and vertical epicyclic frequencies of geodesic orbital motion. This model has been discussed for black hole microquasar sources as well as for a group of neutron star sources. Assuming several neutron (strange) star equations of state and HartleThorne geometry of rotating stars, we briefly compare the frequencies expected from the model to those observed. Our comparison implies that the inferred neutron star radius R_{NS} is larger than the related radius of the marginally stable circular orbit r_{ms} for nuclear matter equations of state and spin frequencies up to 800 Hz. For the same range of spin and a strange star (MIT) equation of state, the inferrred radius is R_{NS} ~ r_{ms}. The “Paczyński modulation” mechanism considered within the model requires that R_{NS} < r_{ms}. However, we find this condition to be fulfilled only for the strange matter equation of state, masses below 1 M_{⊙}, and spin frequencies above 800 Hz. This result most likely falsifies the postulation of the neutron star 3:2 resonant eigenfrequencies being equal to the frequencies of geodesic radial and vertical epicyclic modes. We suggest that the 3:2 epicyclic modes could stay among the possible choices only if a fairly nongeodesic accretion flow is assumed, or if a different modulation mechanism operates.
Key words: Xrays: binaries / stars: neutron
© ESO, 2010
1. Introduction: HF QPOs and desire for stronggravity
Galactic low mass Xray binaries (LMXBs) display quasiperiodic oscillations (QPOs) in their observed Xray fluxes (i.e., peaks in the Xray power density spectra). Characteristic frequencies of these QPOs range from ~10^{2} Hz to ~10^{3} Hz. Of particular interest are the socalled highfrequency (HF) QPOs with frequencies typically in the range 50−1300Hz, which is roughly of the same order as the range of frequencies characteristic of orbital motion close to a low mass compact object. We briefly recall that there is a crucial difference between HF QPOs observed in black hole (BH) and neutron star (NS) systems. In BH systems, the HF QPO peaks are commonly detected at constant (or nearly constant) frequencies that are characteristic of a given source. When two or more QPO frequencies are detected, they usually come in smallnumber ratios, typically in a 3:2 ratio (Abramowicz & Kluźniak 2001; Kluźniak & Abramowicz 2001; McClintock & Remillard 2004; Török et al. 2005). For NS sources, on the other hand, HF (or kHz) QPOs often appear as twin QPOs. These features, on which we focus here, consist of two simultaneously observed peaks with distinct actual frequencies that substantially change over time. The two peaks forming twin QPO are then referred to as the lower and upper QPO in agreement with the inequality in their frequencies.
The amplitudes of twin QPOs in NS sources are typically much stronger and their coherence times much higher than those in BH sources (e.g. McClintock & Remillard 2004; Barret et al. 2005a,b, 2006; Méndez 2006). It is however interesting that most of the twin QPOs with high statistical significance have been detected at lower QPO frequencies 600−700 Hz and upper QPO frequencies 900−1200 Hz. Because of this the twin QPO frequency ratio clusters mostly around ≈ 3:2 value posing thus some analogy to BH case (see Abramowicz et al. 2003a; Belloni et al. 2007; Török et al. 2008a,b,c; Boutelier et al. 2009, for details and a related discussion). In several NS sources, the difference in the amplitudes of the two peaks changes sign as their frequency ratio passes through the (same) 3:2 value (Török 2009). A detailed review on the other similarities and differences in the HF QPOs features can be found in van der Klis (2006).
1.1. HF QPO interpretation
There is strong evidence supporting the origin of the twin QPOs inside 100 gravitational radii, r_{g} = GMc^{2}, around the accreting compact objects (e.g., van der Klis 2006). At present, there is no commonly accepted QPO theory. It is even unclear whether this theory could involve the same phenomena for both BH and NS sources. Several models have been proposed to explain the HF QPOs, most of which involve orbital motion in the inner regions of an accretion disc. When describing the orbital motion, the Newtonian approach necessarily fails close to the compact object. Two of the most striking differences arise from the relevant general relativistic description: Einstein’s strong gravity cancels the equality between the Keplerian and epicyclic frequencies, and (due to the existence of the marginally stable circular orbit r_{ms}) it applies a limit to the maximal allowed orbital frequency. Several effects such as the relativistic precessions of orbits then pop up in the inner accretion region. Finding a proper QPO model may thus help us to test the strong field regime predictions of general relativity and, in the case of NS sources, also the models of highly dense matter (see van der Klis 2006; Lamb & Boutloukos 2007, for a review).
1.2. Nonlinear resonances between “geodesic and nongeodesic” discoscillations
Numerous explanations of the observed lower and upper HF modulation of the Xray flux have been proposed while hypothetical resonances between the two QPO oscillatory modes are often assumed. Specific ideas considering nonlinear resonances between discoscillation modes have been introduced and extensively investigated by Abramowicz, Kluźniak and collaborators (Kluźniak & Abramowicz 2001; Abramowicz & Kluźniak 2001; Abramowicz et al. 2003a,b; Rebusco 2004; Török et al. 2005; Horák 2008; Stuchlík et al. 2008; Horák et al. 2009, and others; see also Aliev & Galtsov 1981 and Aliev 2007). These ideas have been widely discussed and adapted into numerous individual discoscillation models.
The subject of disc oscillations and their propagation has been extensively studied analytically for thin disc (i.e., nearly geodesic, radiatively efficient) configurations (Okazaki et al. 1987; Kato et al. 1998; Wagoner 1999, 2008; Wagoner et al. 2001; Silbergleit et al. 2001; OrtegaRodríguez et al. 2002). The derived results have been compared to those for “thick” (radiatively inefficient, slimdisc or toroidal) configurations foe which both analytical (Blaes 1985; Šrámková 2005; Abramowicz et al. 2006; Blaes et al. 2006, 2007; Straub & Šrámková 2009) and numerical (Rezzolla et al. 2003a,b; Rezzolla 2004; Montero et al. 2004; Zanotti et al. 2005; Šrámková et al. 2007) studies have been performed. Several consequences of discoscillation QPO models have been sketched, some having direct relevance to nonlinear resonance hypotheses. In particular, it has been found that, due to pressure effects, the values of the frequencies at radii fixed by a certain frequency ratio condition can differ between the geodesic and fairly nongeodesic flow of factors such as 15% (Blaes et al. 2007).
1.3. Aims and scope of this paper
GondekRosińska & Kluźniak (2002) suggested that the resonance theory of kHz QPOs can help us to discrimine between quark (strange matter) stars and neutron stars. In the spirit of this suggestion, we examine a particular, often quoted “3:2 epicyclic resonance model” (or rather a class of these models). The paper is arranged as follows.
In Sect. 2, we briefly highlight some important aspects of nonlinear resonance models specific to neutron stars and a 3:2 epicyclic resonance model. In Sect. 3, we compare the model to the HF QPO observations of a group of NS sources displaying the 3:2 ratio. The restrictions to the mass and radius implied by the equations of state for nonrotating NS are included. In Sect. 4, we explore the corrections required for NS rotation and again consider the equations of state. In Sect. 5, we assign some consequences and discuss possible falsification of the model, whereas the nearly geodesic and fairly nongeodesic cases are considered separately.
Throughout the paper, we use the standard notation where ν_{L}, ν_{U} represent the observed lower and upper QPO frequencies, while ν_{K}, ν_{r}, ν_{θ} represent the Keplerian, radial epicyclic and vertical epicyclic frequencies for the considered spacetime and its parameters.
2. Resonances in discs around neutron stars
Miscellaneous variations in the nonlinear, discoscillation resonances have been discussed in the past (see, e.g., Abramowicz & Kluźniak 2001, 2004; Török 2005a; Horák & Karas 2006). While the basic approaches have been common to both blackhole and neutronstar models, several differences between the two classes of sources have been considered. In particular, it has been suggested that, in a turbulent NS accretion flow, the resonant eigenfrequencies are not fixed (e.g., when oscillations of a tori changing its position are assumed; Zanotti et al. 2003; RubioHerrera & Lee 2005; Abramowicz et al. 2006; Török et al. 2007; Kluźniak et al. 2007), or that the resonant corrections to eigenfrequencies reach high values (Abramowicz et al. 2003b, 2005a,b). Both possibilities are taken into account in Sect. 3.
2.1. Modulation
One more important difference between the two aforementioned source types concerns the QPO modulation mechanism (Bursa et al. 2004; Horák 2005a; Abramowicz et al. 2007; Bursa 2008). In the black hole case, the weak modulation is assumed to be primarily connected to radiation of the oscillating disc and the related relativistic lensing, lightbending, and Doppler effects. In the neutron star case, the expected modulation is connected to the flux emitted from a hot spot on the NS surface causing a strong QPO amplitude.
We briefly describe the “Paczyńskimodulation” mechanism (Paczyński 1987), which was investigated by Horák (2005a) and Abramowicz et al. (2007). The schematic Fig. 1 displays the considered situation. The expected mass flow is described by the Bernoulli equation, while surfaces of constant enthalpy, pressure, and density coincide with surfaces of constant effective potential U(r,z) = constant (Abramowicz 1971). The disc equilibrium can exist if the disc surface corresponds to one of the equipotentials inside the socalled Roche lobe (region indicated by the yellow colour). No equilibrium is possible in the region of r < r_{in}. For a given accretion rate the dynamical mass loss occures when the fluid distribution overflows the surface of the disc for U_{0} = U(r_{in}). When the accretion disc oscillates, it slightly changes its position with respect to the equipotencial surfaces. At a particular location corresponding to the crossing of the equipotentials, the socalled cusp, even a small displacement of the disc causes a large change in the accretion rate. The change in the accretion rate is then nearly instantly reflected by the hotspot temperature leading to an enhanced Xray emission (Paczyński 1987; Horák 2005a; Abramowicz et al. 2007).
Fig. 1
Massflow leaving the disc and crossing the relativistic accretion gap (after Abramowicz et al. 2007). Top: Keplerian angular momentum versus the angular momentum in the flow. Bottom: The equipotential surfaces and the distribution of fluid in a meridional crosssection of the discconfiguration. The yellow area denotes the fluid in the disc, while the orange area corresponds to the overflow modulated by the oscillations. Enhanced luminosity arises as the flow enters the boundary layer (lightblue colour). 
The existence of the surface U_{0} above the neutron star is crucial to the model. Therefore, as a necessary condition for its applicability, it is required that
where R_{NS} denotes the neutron (strange) star radius (“accretion gap paradigm”, Kluźniak & Wagoner 1985; Kluźniak et al. 1990). We note that this is a necessary but insufficient condition, since the inner radius r_{in} is located between the marginally stable and marginally bound circular orbit (Kozlowski et al. 1978).
2.2. Epicyclic resonance
A particular example of the nonlinear resonance between discoscillation modes is represented by the concept of the “3:2 epicyclic internal resonance”. This hypothesis is widely discussed (e.g., Abramowicz et al. 2002; Kluźniak & Abramowicz 2002, 2005; Horák 2004, 2005b; van der Klis 2005; Török & Stuchlík 2005; Vio et al. 2006; Rebusco 2008; Reynolds & Miller 2009, among the other references in this paper). It assumes that the resonant modes have eigenfrequencies equal to radial and vertical epicyclic frequency of geodesic orbital motion given by
associated with the orbital radius r_{3:2}, where ν_{θ}/ν_{r} = 3/2. We emphasize that models consider oscillations of fluid configurations rather than test particle motion (see, e.g., Kluźniak 2008, for some details and related references). In the following sections we consider the 3:2 epicyclic model and whether the Paczyński modulation mechanism may be at work.
3. NS mass and radius implied by the 3:2 epicyclic resonant model
In resonance models of BH QPOs, the observed constant frequencies are expected to coincide with the resonant eigenfrequencies. Assuming a particular resonance, one may then relate the black hole spin or mass to the observed frequencies. This procedure was followed by Abramowicz & Kluźniak (2001) and later by Török et al. (2005) and Török (2005b) for various resonances and sets of sources. In principle, similar calculations can also be made for resonance models of NS QPOs. For neutron stars, the observed frequencies, however, change over time and, moreover, monotonic positive frequency correlations are similar, but specific to the individual sources. Within the framework of the resonance models we can consider two distinct simplifications to the observed frequency correlations when inferring the neutron star mass:
 a)
The observed frequencies are roughly equal to the resonanteigenfrequencies and the observed frequency correlationfollows from the changes in eigenfrequencies
implying for the 3:2 epicyclic model that
 b)
The eigenfrequencies are constant and the observed correlation is caused by the resonant corrections
implying for the 3:2 epicyclic model that
3.1. Mass
In this section, we neglect the effects of neutron star spin and assume the 3:2 epicyclic resonance model in the Schwarzschild spacetime^{1}. Introducing a relativistic factor ℱ ≡ c^{3}/(2πGM), Eq. (4) reads
implying that
It has been previously discussed in terms of a correlation between the QPO frequency (ν_{L} or ν_{U}) and frequency difference Δν = ν_{U} − ν_{L} that the correlation given by Eq. (8) clearly disagrees with the observations of NS sources (e.g., Belloni et al. 2005). The 3:2 epicyclic resonance model that is fully based on Eq. (3) is therefore excluded. Hence, in the following we focus on the option represented by Eq. (5).
The relation of Eq. (5) to the observation of several NS sources was considered by Abramowicz et al. (2005a,b). They assumed that the corrections Δν in Eq. (6) vanish when the observed frequency ratio ν_{U}/ν_{L} reaches the 3/2 value. They suggested that the resonant eigenfrequencies in a group of twelve NS sources are roughly equal to [ 600 Hz, 900 Hz ] . For the 3:2 epicyclic model we then find that
which, from terms given in Eq. (7), implies that the relevant mass must be around M = 1 M_{⊙} (as first noticed by Bursa 2004 unpublished).
Fig. 2
Massradius relations for several EoS assuming a nonrotating star. The shadow area indicates the region with NS radii higher than the radius of the marginally stable circular orbit (no accretion gap). The mass M = 1 ± 0.1M_{⊙} is denoted by the dashed and dotted horizontal lines. 
3.2. Radius
Modelling of NS equations of state (EoS) have been extensively developed by numerous published methods and codes (see, Lattimer & Prakash2001; Lattimer & Prakash2007 for a review). Here we calculate NS radii following the approach of Hartle (1967), Hartle & Thorne (1968), Chandrasekhar & Miller (1974), and Miller (1977). In Fig. 2, we plot the massradius relations for several EoS.
Skyrme represents nine different EoS (namely SkT5, SkO’, SkO, SLy4, Gs, SkI2, SkI5, SGI, and SV) given by the different parameterizations of the effective Skyrme potentials (see Říkovská Stone et al. 2003, and references therein). DBHF represents four different parameterizations, chosen to describe matter in the framework of DiracBruecknerHartreeFock theory. In particular, we choose the parameterizations labeled HA, HB, LA, and MA in Kotulič Bunta & Gmuca (2003) used by Urbanec et al. (2010) to describe the properties of static neutron stars. The EoS labeled APR has often been used. We chose the model labeled A18 + δv + UIX ∗ in the original paper (Akmal et al. 1998). Remaining pure neutronstar equations of state are FPS (Pandharipande & Ravenhall 1989) and BBB2 (Baldo et al. 1997). The model labeled GLENDNH3 also includes hyperons (Glendenning 1985).
The MIT model represents strange stars calculated using the socalled MIT bag model (Chodos et al. 1974), where we used the standard values B = 10^{14}g.cm^{3} for the bag constant and α_{c} = 0.15 for strong interaction coupling constant.
From Fig. 2 we can see that for M ~ 1M_{⊙} the NS radii are in all cases above r_{ms}. Thus, assuming the Schwarzschild metric the condition presented in Eq. (1) is not fulfilled for the Xray modulation given by the 3:2 epicyclic model.
Fig. 3
Solution of the 3:2 frequency Eq. (9) projected onto M − j plane and colourscaled in terms of . 
4. Effects related to NS spin
We have restricted attention to the implications of Eq. (9) for nonrotating NS. The spin of the astrophysical compact objects and related oblateness however introduce some modifications of the Schwarzschild spacetime geometry. Without the inclusion of magnetic field effects, it has been found that the rotating spacetimes induced by most of the uptodate neutron star equations of state (EoS) are well approximated with the solution of Hartle & Thorne (1968) (see Berti et al. 2005, for details). We use this solution (in next HT) to discuss the spin corrections to the above results.
The HT solution reflects three parameters, the neutron star mass M, angular momentum J, and quadrupole moment Q. We note that the Kerr geometry represents the “limit” to the HT geometry for up to the second order in J. The formulae for Keplerian and epicyclic frequencies in the HT spacetime were derived by Abramowicz et al. (2003c). We applied these formulae to solve Eq. (9). Figure 3 displays the resulting surface colourscaled in terms of M/M_{⊙}, j = cJ/GM^{2}, and . We can see that for low values of and any j the implied M increases with increasing j, while exactly the opposite dependence M(j) occurs for high values of and j ≥ 0.2.
Fig. 4
a) NS configurations fulfilling the 3:2 frequency condition (10). b) Related relationships between NS spin and radii evaluated in terms of ISCO radii r_{ms}(M, j, q). Only subset of configurations obeying the condition (10) are depicted in the figures for clearness. Lines tending to appear on both figures correspond to configurations with same central parameters and different rotational spin. 
4.1. EoS and radii
For a given EoS, the parameter decreases with increasing M/M_{max}. In more detail, it is usually for 1M_{⊙}, while ) for the maximal allowed mass (e.g. Török et al. 2010, Fig. 3 in their paper). Since the nonrotating mass inferred from the model is about 1M_{⊙}, one can expect that realistic NS configurations will be related to M − j solutions associated with high . These are denoted in Fig. 3 by colours of the yellowred spectrum.
We checked this expectation using the same set of EoS as in Sect. 3.2. We calculated the configurations for each EoS covering the range of the central density ρ_{c} implying that M ∈ (≈ 0.5 M_{⊙}, M_{max}) and the spin frequency ν_{s} ∈ (0, ν_{max}), using thousand bins in each of both independent quantities. The mass M_{max} is the maximal mass allowed for a given EoS and spin frequency ν_{s}. The frequency ν_{max} is the maximal frequency of a given neutron star and is equal to the Keplerian frequency at the surface of the neutron star at the equator, corresponding to the socalled massshedding limit. In this way, we obtained a group of 15 × 1000^{2} ≈ 10^{7} configurations. From these we retained only those fulfilling the condition in Eq. (9) for the epicyclic frequencies (ν = ν(M,j,q), Abramowicz et al. 2003c) extended to
We note that this range of considered eigenfrequencies is based roughly on the range of the observed 3:2 frequencies (Abramowicz et al. 2005a,b). The combinations of mass and angular momentum selected in this way are displayed in Fig. 4a. Inspecting the figure, one can see that the mass decreases with increasing j above j ~ 0.3.
Figure 4b indicates the ratio R_{NS}/r_{ms} for the selected configurations (shown in Fig. 4a). The modulationcondition presented in Eq. (1) appears only to be fulfilled for MITEoS and high spins above 800 Hz.
5. Discussion and conclusions
The neutron star masses inferred for the 3:2 epicyclic resonance model by the considered EoS (Fig. 4a) are very low compared to the "canonical" value of 1.4 M_{⊙}. For the nonrotating case, the implied NS configurations are in addition insufficiently compact to fulfill the modulation condition in Eq. (1). We find that this condition is satisfied only for high spin values, above 800 Hz, and strange matter EoS (MIT) (see the shaded region in Fig. 4b).
Searching through the region, we find the highest mass satisfying Eq. (1) to be M = 0.97 M_{⊙}. The related NS spin is 960 Hz. This mass and spin correspond to Hz. For higher frequencies , the required mass is even lower. For Hz, it is M = 0.85 M_{⊙}, whereas the related NS spin is 900 Hz.
For compact objects in the NS kHz QPO sources, there are at present no clear QPO independent mass estimates. In contrast, there is convincing evidence of the spin of several sources from the Xray burst measurements (see, e.g, Strohmayer & Bildsten 2006). In the group of sources discussed by Abramowicz et al. (2005a,b) considered in this paper, there are several that have spins in the range ~250–650Hz. The NS parameters implied by the 3:2 epicyclic model therefore include not only very low masses, but also spins excluded by the QPO independent methods.
The results obtained that falsify the epicyclic hypothesis are doubtless as far as:

i)
The Paczyński modulation mechanism is involved, implying that the inequality R_{NS} < r_{ms} is valid.
The eigenfrequencies considered within the model are equal to (nearly) geodesic frequencies.
As quoted in Sect. 1, for the 3:2 epicyclic resonance the values of the resonant eigenfrequencies for a nongeodesic flow are higher than those calculated for a nearly geodesic motion (Blaes et al. 2007; Straub & Šrámková 2009). It has been shown that in a certain case the difference can reach about 15%. This would change the nonrotating mass to a higher value, ~1.2 M_{⊙}. From Fig. 2 we can see that this value rather barely fits the modulation condition for the MIT EoS. It thus cannot be fully excluded that the model is compatible with observations if the flow is fairly nongeodesic. However, needless to say that a serious treatment of this possibility will require investigation of the related pressure effects on the disc structure in the HartleThorne geometry, since the aforementioned studies only consider Schwarzschild (in a pseudoNewtonian approximation) and Kerr geometry. Moreover, they assume a constant specific angular momentum distribution within the disc, while there is evidence from numerical simulations of the evolution of accreting tori that real accretion flows tend to have rather nearKeplerian distributions (e.g. Hawley 2000; De Villiers & Hawley 2003). In this case, one expects the pressure corrections to be considerably smaller than those calculated for the marginal case of a constant angular momentum torus. However, there is no clear guarantee of this expectation and further investigation will be neccessary to resolve this issue.
We can conclude that the resonance model for NS kHz QPOs should involve a combination of discoscillation modes that differ from the geodesic radial and vertical epicyclic modes, or a modulation mechanism that differs from the Paczyński modulation. The results also suggest that the two modes together with the considered modulation may operate as long as a fairly nongeodesic accretion flow is assumed for strange or some nuclear matter EoS.
We consider here this standard spacetime description for nonrotating neutron stars, although some alternatives have been discussed in a similar context (see Kotrlová et al. 2008; Stuchlík & Kotrlová 2009).
Acknowledgments
This work has developed from several debates initiated and richly contributed to by Marek Abramowicz and Wlodek Kluźniak in the past few years. We appreciate useful suggestions and comments of an anonymous referee, which helped to improve it. The paper has been supported by the Czech grants MSM 4781305903, LC 06014, and GAČR 202/09/0772. The authors also acknowledge the internal student grant of the Silesian University in Opava, SGS/1/2010. Part of the work reported here was carried out during the stay of GT and ZS at the Göteborg University, cosupported by The Swedish Research Council grant (VR) to M. Abramowicz.
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All Figures
Fig. 1
Massflow leaving the disc and crossing the relativistic accretion gap (after Abramowicz et al. 2007). Top: Keplerian angular momentum versus the angular momentum in the flow. Bottom: The equipotential surfaces and the distribution of fluid in a meridional crosssection of the discconfiguration. The yellow area denotes the fluid in the disc, while the orange area corresponds to the overflow modulated by the oscillations. Enhanced luminosity arises as the flow enters the boundary layer (lightblue colour). 

In the text 
Fig. 2
Massradius relations for several EoS assuming a nonrotating star. The shadow area indicates the region with NS radii higher than the radius of the marginally stable circular orbit (no accretion gap). The mass M = 1 ± 0.1M_{⊙} is denoted by the dashed and dotted horizontal lines. 

In the text 
Fig. 3
Solution of the 3:2 frequency Eq. (9) projected onto M − j plane and colourscaled in terms of . 

In the text 
Fig. 4
a) NS configurations fulfilling the 3:2 frequency condition (10). b) Related relationships between NS spin and radii evaluated in terms of ISCO radii r_{ms}(M, j, q). Only subset of configurations obeying the condition (10) are depicted in the figures for clearness. Lines tending to appear on both figures correspond to configurations with same central parameters and different rotational spin. 

In the text 
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