Models of high-frequency quasi-periodic oscillations and black hole spin estimates in Galactic microquasars

For constant angular momentum tori, Abramowicz et al. (1978) (see also Kozlowski et al. 1978) have shown that equilibrium structures of the equipressure and equidensity surfaces coincide with those of the constant effective potential surfaces determined by

Here, $g^{\mu\nu}$ denote components of the spacetime metric expressed in the Boyer–Lindquist coordinates. The surface of the torus coincides with one of the equipotential surfaces where the pressure vanishes. The critical points (extrema and saddles) of the effective potential correspond to vanishing pressure gradients and thus to time-like circular geodesics. At these points, the rotation of the flow is Keplerian. The centre of the torus (i.e., the circle at $r=r_{\mathrm{c}}$ where the pressure has its maximum) corresponds to a stable time-like circular geodesic located at the local minimum of the effective potential in the equatorial plane.

For angular momenta inside a specific range, there can be another unstable circular geodesic that corresponds to the saddle point of the effective potential and puts a limit on the possible size of the torus located at a given $r_{\mathrm{c}}$. The related critical equipotential has a characteristic ‘cusp’, through which the matter may be accreted onto the central BH without the need for any viscous processes, similarly as for the well-known Roche-lobe overflow in binary systems (Kozlowski et al. 1978).

While the marginally stable circular orbit, $r_{\mathrm{ms}}$, is often called the innermost stable circular orbit (ISCO) of a thin accretion disk and forms its inner edge, the inner edge of a thick disk has a different location. Situated between $r_{\mathrm{ms}}$ and the marginally bound circular orbit, $r_{\mathrm{mb}}$, this location depends on the disk’s angular momentum, $\ell_{\mathrm{c}}$ (Abramowicz et al. 1978). For $\ell_{\mathrm{c}}>\ell_{\scriptscriptstyle\!\mathrm{K}}(r_{\mathrm{mb}})$, the disk is infinite with its inner edge located at $r_{\mathrm{mb}}$, but still well inside the cusp self-crossing equipotential. The $\ell_{\mathrm{c}}=\ell_{\scriptscriptstyle\!\mathrm{K}}(r_{\mathrm{mb}})$ case corresponds to an infinite torus terminated by the cusp at $r_{\mathrm{mb}}$. For $\ell_{\mathrm{c}}<\ell_{\scriptscriptstyle\!\mathrm{K}}(r_{\mathrm{mb}})$, the configuration corresponds to a finite, marginally overflowing torus. This torus has its inner edge located closer to BH than $r_{\mathrm{ms}}$, but not closer than $r_{\mathrm{mb}}$.

Examples of various tori configurations are illustrated in the left panel of Figure 1. While the equilibrium configuration of fluid that embodies the cusp equipotential represents a rather simplified stationary analytic model of a non-accreting disk, in accretion, the very existence of the cusp is of general importance. As long as the disk dynamical timescale is fairly shorter than the viscous timescale, which is true for most disk models, the cusp plays a key role in the accretion of matter onto the BH. Fluid elements above the cusp inside the torus stay at more or less the same radius for a large number of dynamical periods, whereas, below the cusp, accretion is more of a Bondi-like type. The presence of the cusp is indeed often seen in sophisticated numerical simulations of accretion flows (e.g. Qian et al. 2009; Fragile et al. 2007, 2009; Gimeno-Soler & Font 2017; Lančová et al. 2019). This is illustrated in the right panel of Figure 1.

To quantify the torus size, Abramowicz et al. (2006) have introduced the ‘thickness’ $\beta$ parameter,

where $n$ is the polytropic index and $c_{\mathrm{sc}}$, $u^{t}_{\mathrm{c}}$, $\Omega_{\mathrm{c}}$ are the polytropic sound speed, the contravariant time component of the four-velocity, and the angular velocity of the flow defined at the centre of the torus, $r=r_{\mathrm{c}}$. For angular momenta $\ell_{\scriptscriptstyle\!\mathrm{K}}(r_{\mathrm{ms}})<\ell_{\mathrm{c}}<\ell_{\scriptscriptstyle\!\mathrm{K}}(r_{\mathrm{mb}})$, this parameter is limited by

whereas, for $\ell_{\mathrm{c}}\geq\ell_{\scriptscriptstyle\!\mathrm{K}}(r_{\mathrm{mb}})$, the possible range of $\beta$ is given by

Here, $W_{\mathrm{c}}$ and $W_{\mathrm{cusp}}$ are the effective potential values corresponding to the centre and cusp equipotential.

In general, the value of $\beta$ that corresponds to marginally overflowing tori, $\beta=\beta_{\mathrm{cusp}}(a)$, lies in the range

where $\beta_{\mathrm{cusp}}(a)=0$ describes a torus located at the marginally stable circular orbit, $r_{\mathrm{ms}}=r_{\mathrm{ms}}(a)$. For the purpose of the present work, it is useful to introduce the ‘effective’ $\beta$ parameter,

For $\ell_{\mathrm{c}}\geq\ell_{\scriptscriptstyle\!\mathrm{K}}(r_{\mathrm{mb}})$, the possible equilibrium configurations correspond to $0\leq\beta_{\mathrm{eff}}\leq 1$, while, for $\ell_{\scriptscriptstyle\!\mathrm{K}}(r_{\mathrm{ms}})<\ell_{\mathrm{c}}<\ell_{\scriptscriptstyle\!\mathrm{K}}(r_{\mathrm{mb}})$, the allowed range is $0\leq\beta_{\mathrm{eff}}\leq\beta_{\mathrm{cusp}}/\beta_{\infty}$. Based on the value of $\beta_{\mathrm{eff}}$, one may determine whether a given cusp configuration corresponds to a small torus, or to a large torus with $\beta\approx\beta_{\mathrm{\infty}}$.

No fully self-consistent concept of resonant models that would incorporate some of the epicyclic modes has been proposed so far. Despite it being a necessary requirement, the frequency commensurability is not a sufficient condition for the resonance to occur. Other important requirements follow from the symmetry properties of the involved oscillatory modes, such as their parities with respect to the equatorial plane or the azimuthal wavenumber. Out of all oscillatory modes combinations discussed in this work, only the axisymmetric modes seem to fully satisfy these conditions (Horák 2008).

In this context, we examine the “epicyclic” (Ep) model of Abramowicz and Kluźniak, which attributes the HF QPOs to the axisymmetric radial and vertical epicyclic oscillation modes in the accretion disk. Alternatively, in the so-called Kep model, the two QPO frequencies are associated with the axisymmetric radial mode frequency and the Keplerian orbital frequency (see Török et al. 2005, for details). Both models have been extensively studied by Abramowicz & Kluźniak (2001); Abramowicz et al. (2003); Kluźniak et al. (2004); Horák & Karas (2006); Horák et al. (2009).

There are two combinations of modes that we denote as the “RP1” model (Bursa 2005) and the “RP2” model (Török et al. 2010). In the slender torus limit, these two predict for a non-rotating BH the same observable frequencies as the RP model. In this limit, for any BH spin, the same observable frequencies as those predicted by the RP model are also predicted by another combination of modes that we in next denote as the RP0 model. This model deals with the possibility of the observed QPO frequencies being associated to the $m=-1$ non-axisymmetric radial mode frequency and the Keplerian orbital frequency. We consider the RP0 model in addition to the set of models discussed by Török et al. (2011). The RP0 model is of special importance in the context of the recent findings on NS QPOs presented in the studies of Török et al. (2016, 2018, 2019), which is further discussed in Section 7.2.

Finally, we also assume a combination of modes that in the slender torus limit leads to the same observable frequencies as the “warped disk” model proposed by Kato (2001, 2004) in the context of diskoseismology (the study of oscillations of thin disks). We denote it as the WD model.

In non-slender tori, the frequencies of oscillation modes are modified by the pressure forces. As a result, the non-geodesic radial and vertical epicyclic frequencies will differ from those corresponding to the perturbed circular geodesic motion.

The study of oscillation and stability properties of fluid tori has been initiated by Papaloizou & Pringle (1984), who have explored global linear stability of Newtonian fluid tori with respect to non-axisymmetric perturbations. Considering small linear perturbations to the torus equilibrium, they have derived a single partial differential equation governing the linear dynamics of oscillations of a Newtonian constant specific angular momentum torus (Papaloizou & Pringle 1984). Later on, general relativistic form of the Papaloizou–Pringle equation has been introduced by Abramowicz et al. (2006) and Blaes et al. (2006). In general, this equation cannot be fully solved analytically. Using a perturbation method, Straub & Šrámková (2009) and Fragile et al. (2016) have derived fully general relativistic formulae determining the frequencies of axisymmetric and non-axisymmetric radial and vertical epicyclic modes in a slightly non-slender constant specific angular momentum torus within a second-order accuracy in the torus thickness.

Using relations (9) – (14), the calculated frequencies of the radial and vertical epicyclic modes can be written in the following form:

Here, $m$ is the azimuthal wavenumber, and $C_{\mathrm{r},\,m}(r_{\rm{c}},a)$ and $C_{\theta,\,m}(r_{\rm{c}},a)$ denote the negative second-order pressure corrections evaluated at the centre of the torus, $r=r_{\rm{c}}$. Since $C_{\mathrm{r},\,m}$ and $C_{\theta,\,m}$ are given by fairly long expressions, we provide their explicit form in the electronic Appendix.

It is necessary to determine the range of the $\beta$ parameter relevant for our study. Formulae (4.1) and (4.1) should provide reasonable results for tori of moderate thickness, but they are not fully applicable for tori of a substantial width. To specify this statement quantitatively, a concrete physical situation needs to be taken into account. It is useful to express the torus thickness in terms of $\beta_{\mathrm{eff}}$. One can expect that $\beta_{\mathrm{eff}}\sim 0.3$ will likely yield a solid approximation of a real situation, while considering $\beta_{\mathrm{eff}}\sim 0.7$ could lead to incorrect results. In next, we use $\beta_{\mathrm{eff}}=0.3$ and $\beta_{\mathrm{eff}}=0.7$ as the referential values.

Although we mostly focus on the 3:2 frequency ratio, other frequency ratios are explored as well. Figure 3 shows the results of such investigation for the Schwarzschild spacetime and tori whose thickness ranges from an infinitely slender torus to a torus with its outer edge extending to infinity. The QPO frequency ratio in this Figure is not fixed and takes values of up to $R=10$.

The enlarged area in Figure 3 illustrates the QPO frequency behaviour relevant for the Ep model close to $R=3:2$. It is apparent that for ratios from this area, the QPO frequency’s extremal value does not always correspond to a torus with a cusp. Such phenomenon has been discussed in more detail in our previous paper (Šrámková et al. 2015).

The position of the orbit that gives the 3:2 QPO frequency ratio within the Ep model changes in a non-trivial way as the torus size increases. As a consequence, the relation between $\beta$ and the QPO frequency is not always a function (Blaes et al. 2006; Šrámková et al. 2015) – see the loops on the Ep model curves illustrated in Figure 4. Nevertheless, for $a=0$, these frequencies only display a small variation across the whole range of the allowed torus thickness. The small sensitivity of the predicted QPO frequency towards the torus thickness persists up to $a\sim 0.9$.

For the RP0 model and the 3:2 frequency ratio, the predicted QPO frequency is a monotonic function of $\beta$. This is illustrated in Figure 4 that shows a comparison between the Ep and RP0 models. Note that, for $a\in[0,\,0.9]$, the RP0 model is associated with a much higher quantitative impact of the torus thickness on the QPO frequency.

Figure 5 shows a comparison between the overall behaviour of the QPO frequencies implied by the Ep and RP0 model as it changes with increasing BH spin and torus thickness. Different curves in the Figure indicate the referential values of the relative torus thickness, $\beta_{\mathrm{eff}}\in{0,0.3,0.7,1}$. There is only a small spread of the resonant frequency predicted by the Ep model for any but a very high spin ($a\gtrsim 0.9$). The lower limit on the spin given by this model, $a\approx 0.69$ for $\beta_{\mathrm{eff}}=0$, reaches the value of $a\approx 0.62$ for the maximal allowed torus thickness. The impact of the non-geodesic effects is clearly more important within the RP0 model. For a geodesic limit of this model (which corresponds to the RP model), the maximal value of the spin is about $a\sim 0.55$. Regarding larger tori, it can grow up to $a\sim 0.75$ for tori with a cusp, or even to $a\sim 0.83$ when tori with no cusp are assumed. A detailed quantification of the impact for each microquasar is presented in Table 2.

Regarding the QPO models’ falsifiability, we conclude that much like in the geodesic case the Kep model is fully incompatible with the GRO J1655$-$40 and XTE 1550$-$564 data. Provided that the mechanism responsible for the QPO phenomenon is the same in all the three microquasars, this model would be ruled out. Although there is currently no final agreement on the spectral spin estimates, the overall sample of the iron line and continuum based studies suggests that at least one of the microquasars should exceed the value of $a=0.65$ (McClintock et al. 2006; Middleton et al. 2006; Blum et al. 2009; McClintock et al. 2014; Miller & Miller 2015). If this was confirmed, our study would in addition to the Kep model rule out also the WD and RP2 model. Moreover, if two very different values of the spin, such as $a\approx 0.65$ in GRO J1655$-$40 vs. $a\approx 1$ in GRS 1915+105, were confirmed, all the models except the RP1 model would remain unsupported by our results.

As briefly mentioned in Sections 1–3, two of the models considered in this work are of special importance. The first one is the Ep model, which is prominent within the class of the resonance models proposed by Abramowicz & Kluźniak (2001) and Török et al. (2005). This model, which involves the axisymmetric modes, is by far the most developed resonance model of QPOs. Using the solution of the improved relativistic Papaloizou–Pringle equation, we confirm the previous result of Šrámková et al. (2015) that indicates small sensitivity of the resonant frequency to the torus thickness and the requirement of a high BH spin. The second one is the RP0 model, which provides outstanding results regarding the overall context of matching the BH and NS HF QPOs.

In the slender torus limit, the RP0 model predicts the same QPO frequencies as the RP model. For the marginally overflowing torus ($\beta=\beta_{\mathrm{cusp}}$), it merges with a model recently discussed by Török et al. (2016) in the context of NS QPOs – hereafter the CT model. This model has been suggested as the RP model alternative which deals with torus oscillations. It is based on the expectation that cusp configurations are likely to appear in real accretion flows, in which case the actual overall accretion rate through the inner edge of the disk can be strongly modulated by the disk oscillations (Paczynski & Abramowicz 1982; Abramowicz et al. 2006). The CT model provides generally better fits of the NS data than the RP model. It also predicts a lower NS mass than the RP model which, in some cases, implies a mass estimate that is too high (Török et al. 2018, 2019).

The spin predicted by the CT model is given by the upper solid red curve in Figure 5 (The RP0 model and $\beta=\beta_{\mathrm{cusp}}$). We conclude that the upper limit on the spin implied by this model is significantly higher than in the RP model’s case, namely $a\sim 0.75$ vs. $a\sim 0.55$. This is presumably in better agreement with the spectral spin estimates.

One should be aware of the limitations associated with the adopted perturbative approach. The results of our calculations that stem from the consideration of $\beta_{\mathrm{eff}}\gtrsim 0.7$ should be confronted with exact numerical treatment of the Papaloizou–Pringle equation.⁴⁴4The relation between $\beta$ and $\beta_{\mathrm{eff}}$ depends on the specific torus configuration underlying a given QPO model. For all tori configurations considered in this paper, we have $\beta=0.3\beta_{\infty}\approx 0.2$ and $\beta=0.7\beta_{\infty}\in[0.4,\,0.6]\approx 0.5$. A similar reservation applies to the consideration of $a\gg 0.9$, since, for high spins, the epicyclic mode frequencies are very sensitive to even small changes of the torus thickness.

It is not fully clear to what degree our assumptions match the real situation. We compare the observed 3:2 QPO frequencies with frequencies of the oscillation modes calculated for one particular torus configuration. There are not many observations of HF QPOs available for BH binaries, and even fewer of those that are available display the two peaks simultaneously. The integration time required for the QPO identification is a few orders of magnitude longer than the characteristic QPO period. Furthermore, there are observations suggesting that the BH HF QPO frequencies may vary in time and form continuous correlations similar to those observed in NSs that reach into (but are not necessarily constrained to) the 3:2 frequency ratio range (Belloni et al. 2012; Belloni & Altamirano 2013; Motta et al. 2014; Varniere & Rodriguez 2018). Despite these uncertainties, our assumptions are sufficient for the presented simplified analysis that provides a brief comparison of BH spin estimates associated to models that deal with different combinations of disk oscillation epicyclic modes.