BLACK HOLE SPIN–ORBIT MISALIGNMENT IN GALACTIC X-RAY BINARIES

The formation of Roche-lobe overflowing X-ray binaries (XRBs) with black hole (BH) accretors in galactic fields involves the evolution of a primordial isolated binary of which at least one of the component stars is massive enough to form a BH. Right before the supernova (SN) explosion that will create the BH, the binary orbit is assumed to be circular, due to tidal interactions and possible preceding mass-transfer episodes, and the spins of the two stars are assumed to be aligned with the orbital angular momentum at the formation of the binary. The subsequent SN explosion will change the orbital characteristics of the binary. If an asymmetric kick is imparted on the BH during its formation, the orientation of the orbit will change and this in turn will cause a misalignment between the spin axis of the BH and the orbital angular momentum.

Through studies of the position and spatial velocity of radio pulsars in the Galaxy, it has been established in the literature that neutron stars (NSs) can receive large asymmetric SN kicks (∼265 Km s⁻¹; Gunn & Ostriker 1970; Hansen & Phinney 1997; Hobbs et al. 2005). On the other hand, the possibility of an asymmetric kick imparted to BHs remains still an open question. Jonker & Nelemans (2004), using corrected distances of Galactic BH XRBs, suggested that BHs receive kicks from a similar distribution to NSs. However, indirect BH kick observations indicate that low-mass BHs receive high asymmetric kicks, while high-mass BHs low or no kicks (Mirabel et al. 2002, 2001; Mirabel & Rodrigues 2003; Dhawan et al. 2007). On the other hand, detailed theoretical modeling of the evolutionary history of two Galactic BH XRBs (GRO J1655−40 and XTE J1118+480) showed that for the case of GRO J1655−40 a moderate asymmetric kick is possible, while in the case of XTE J1118+480 an asymmetric kick larger than ∼100 Km s⁻¹ is necessary in order to explain the current properties of the system (Willems et al. 2005; Gualandris et al. 2005; Fragos et al. 2009).

Unless the direction of this asymmetric kick during the formation of the BH is finely tuned to be on the orbital plane, it will induce a misalignment of the BH spin and the orbital angular momentum. In fact, two microquasar-type Galactic BH XRBs (GRO J1655−40 and V4641 Sgr) have been observed with relativistic radio jets that are misaligned with respect to their orbital plane. This misalignment angle is estimated to be ≳15° for GRO J1655−40 (Hjellming & Rupen 1995; Orosz & Bailyn 1997) and ≳55° for V4641 Sgr (Orosz et al. 2001). Assuming that the radio jet has the same direction as the spin of the BH, this misalignment angle can be translated into a tilt of the BH spin axis with respect to the orbital plane.

In this Letter, we present population synthesis (PS) models of Roche-lobe overflowing, Galactic field BH XRBs, and study their BH spin–orbit misalignment and their dependence on model parameters.

We performed the simulations presented here with StarTrack, a current binary PS code that has been tested and calibrated using detailed binary star calculations and incorporates all the important physical processes of binary evolution (see Belczynski et al. 2008, for details).

2.1. Model Parameters and Initial Conditions for Milky Way Models

2.2. Asymmetric Black Hole Kicks

At the time of birth, BHs and NSs may receive an instantaneous asymmetric kick, due to asymmetries involved in the SN explosion mechanisms. Hobbs et al. (2005) studied the proper motion of 233 Galactic pulsars and found that their inferred kick velocities are consistent with a single Maxwellian distribution with σ = 265 Km s⁻¹.

In the SN explosions of more massive stars that can form BHs, the effects of material fall-back (ejected initially in the SN explosion) during the star's final collapse are included. For each core-collapse event we calculate the fall-back factor (f_fb), i.e., the fraction of the stellar envelope that falls back. The value of f_fb is calculated based on the mass of the carbon/oxygen core (M_CO) of the pre-SN star and is interpolated linearly between M_CO = 5 M_☉ (f_fb = 0) and M_CO = 7.6 M_☉ (f_fb = 1). The regimes of no fall-back, partial fall-back, and direct collapse are estimated from earlier studies by Fryer (1999) and Fryer & Kalogera (2001). In the formation of BHs, kicks are lowered proportionally to the amount of fall-back associated with core-collapse event, according to the relation

$$\begin{equation} V_{\rm kick} = (1-f_{\rm fb})V, \end{equation} \tag{ 1 }$$

where V is the kick magnitude drawn from the single Maxwellian. Furthermore, we consider the extreme case where the magnitude of the BH kick is drawn directly from the Maxwellian distribution and is not scaled to fall-back mass (model 19). Finally, the standard assumption is that the direction of the kick is isotropically distributed, but we also consider the case of polar kicks, where the kick direction is aligned with the spin axis of the BH progenitor.

2.3. Initial Black Hole Spin–Orbit Misalignment

During the XRB phase, the material that is being accreted onto the BH can be a significant fraction of the BH's initial mass, and the effects of the accretion on the BH spin cannot be considered negligible. We therefore want to investigate whether this mass accretion can significantly alter the post-SN spin–orbit misalignment. In this section, we describe how we model these effects.

Let us assume a BH of spin a and mass M_BH surrounded by an accretion disk which has an inclination i with respect to the BH's spin (the accretion disk is assumed to be parallel to the orbital plane), and that mass m_acc is gradually being accreted onto the BH. The mass accreted by the BH carries into the BH not only its mass but also its angular momentum l_acc, and as a result the orbital inclination is altered. It is reasonable to assume that l_acc is going to be the angular momentum right at the innermost stable circular orbit (ISCO) of the BH, since inside the ISCO the infalling mass is following a radial plunge. However, the position of the ISCO depends on the BH mass and spin and on the infalling particle's orbital inclination.

The equation that describes the radial geodesic motion of a mass particle in the Kerr spacetime in terms of the Boyer–Lindquist coordinates (t, r, θ) is given by Misner et al. (1973),

$$\begin{equation} \Sigma ^2 \left(\frac{dr}{d\tau } \right)^2=[E(r^2+a^2)-aL_z]^2-\Delta [r^2+(L_z-aE)^2+Q]\equiv R \end{equation} \tag{ 2 }$$

$$\begin{equation} \Sigma \left({dt\over d\tau }\right) = E\left[{(r^2+a^2)^2\over \Delta } - a^2\sin ^2\theta \right] +aL_z\left(1 - {r^2+a^2\over \Delta }\right), \end{equation} \tag{ 3 }$$

where Σ = r² + a²cos²θ and Δ = r² − 2Mr + a². The quantities E, Lz, and Q (the three constants of motion) are the specific energy, angular momentum, and Carter constant, with the Carter constant being defined as: $Q=\frac{L_z^2}{\cos ^2i}-L_z^2=L_z^2\tan ^2i$. Note that all equations make use of geometrized units (G = c = 1). A mass particle orbiting at the ISCO should fulfill three conditions: R = 0 (for the orbit to be circular), R' = 0 (for the orbit to remain circular, i.e., zero radial acceleration), and finally R'' = 0 (for the orbit to be stable). By solving this set of equations (R = R' = R'' = 0) we get the position of the ISCO r_ISCO, as well as the angular momentum (L_ISCO) and energy (E_ISCO) of the accreted mass. In our calculations, the accretion of mass m_acc is treated as a series of accretion episodes of infinitesimal mass parcels m_i (∑_im_i = m_acc). After every accretion episode, the mass and angular momentum of the BH is updated and so is the inclination. We proceed to the next accretion episode using these updated values until m_acc has been accreted.

One caveat of this analysis is the assumption that the accretion disk around the BH is parallel to the orbital plane all the way down to the ISCO. Scheuer & Feiler (1996) showed that an accretion disc around a BH will be warped by the Lense–Thirring effect. The inner parts of the disc align with the BH spin, but the torque from the outer parts of the warped disc will eventually align the BH spin with the orbital plane. Scheuer & Feiler (1996) found that the timescale for the alignment of the BH spin with the outer parts of the disk is only a factor of a few longer than just assuming that angular momentum is transferred to the BH from the warping radius of the disk. However, for standard Shakura–Sunyaev accretion disk models this warping radius is located at 100–1000 Schwarzschild radii, in other words much further out than the ISCO. The exact position of the warping radius can vary by more than an order of magnitude depending on the adopted viscosity model and the exact geometry of the disk (see also Natarajan & Pringle 1998; Fragile & Anninos 2005; King et al. 2005; Lodato & Pringle 2006; Fragile et al. 2007; Martin et al. 2007, 2008; Fragile 2009), nevertheless, the overall effect is to accelerate the alignment of the BH spin compared to our calculations. In this sense, our calculation of BH spin evolution due to accretion poses an upper limit on the current BH spin–orbit misalignment in Galactic BH XRBs.

We considered a large library of PS models (over 200), where we varied a number of PS model parameters (see Table 1). For each of the models, we examined the probability density function (PDF) of post-SN BH spin–orbit misalignment angles in BH XRBs and its dependence on the masses of the BH and the donor star. We note here that in all of the results we present here, we considered only the transient BH XRB population. This selection was made in order to facilitate a direct comparison with the currently observed sample of Galactic BH XRBs. The dynamical measurement of the BH mass, which is required in order to identify the compact object accretor of an XRB as a BH, can only be done when the XRB is in quiescence. The same limitation holds for the two currently available observational techniques (continuum-fitting and Fe K methods) for the measurement of the BH spin magnitude.

In Figures 1 and 2, we present the PDF of the post-SN BH spin–orbit misalignment angle and the two-dimensional PDF of the post-SN misalignment angle versus BH mass, split into three donor mass ranges, for our "standard" model 1i and for model 1p, respectively. Model 1p has the same PS parameters as the "standard" model, except that the direction of the SN kicks is always perpendicular to the orbital plane and not isotropic. One general characteristic of the PDFs is that they are heavily skewed toward small misalignment angles. This characteristic is the result of the combination of two factors. BHs are generally believed to receive asymmetric kicks of smaller magnitude compared to that of NS. On the other hand, large kick magnitudes, which would result in a large misalignment angle, usually lead to the disruption of the binary, and thus these systems are effectively filtered out of the population. The latter effect becomes more prominent in the case of polar kicks (model 1p). One would initially expect that kicks always directed perpendicular to the orbital plane would result in larger misalignment angles. However, a large kick perpendicular to the orbital plane is also more efficient at disrupting the binary. Figures 1 and 2 show that the binary disruption effect is dominant and that only BHs that received a small kick manage to remain bound in a binary, and thus polar kicks lead to overall smaller misalignment angles.

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The two-dimensional PDFs of the post-SN misalignment angle versus BH mass in Figure 1 show that massive BHs tend to have smaller misalignment kicks. This trend is common in all of our models, and is a consequence of our prescription for the asymmetric kicks imparted to BHs. As described in Section 2.3, massive BHs receive on average smaller kicks, which in turn lead to smaller misalignment angles. Moreover, the higher the total mass of the binary is, the less the orbit will change from a given SN kick. The same physical mechanism is also associated with our finding that XRBs with massive donor stars have smaller BH spin tilts.

Figure 3 shows the PDF of the current BH spin–orbit misalignment angle, split into three donor mass ranges, for models 1i^M0.2 and 1i^M0.9. In this case, we take into account the evolution of the BH spin's direction and magnitude due to accretion during the XRB phase. The accretion of matter onto the BH always results in a gradual alignment of its spin with the orbital angular momentum. The efficiency of this process depends on the initial spin magnitude of the BH—the higher the initial BH spin is, the more difficult it is to change its direction via accretion. In Figure 3, we show two models with different distributions of initial BH spin magnitude, one with a Maxwellian distribution with a maximum at 0.2 and truncated at 1, and one where the maximum of the Maxwellian is at 0.9. Comparing these PDFs with Figure 1, which shows the PDFs of the post-SN misalignment angle of the same model, we see that for both models (1i^M0.2 and 1i^M0.9) the PDFs of Figure 3 are shifted toward lower misalignment angles and that indeed this effect is more prominent for smaller initial BH spin magnitudes (model 1i^M0.2).

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Table 2 contains the 67%, 95.4%, and 99.7% confidence levels of BH spin–orbit misalignment angle PDFs for a selected representative list of models (for the complete list of PS models see the online supplemental material). After accounting for the partial alignment due to accretion onto the BH, we see that the majority of BH XRBs (at a 67% confidence level) have BH spin–orbit misalignment angles below (and some times well below) 10°. This is a robust conclusion as it is true for all of our models. At the same time, we find that in most of our models there is a small part of the XRB population (<5%) with misalignment angles above 20°.

The main effect of varying the IMF or the stellar wind strength is on the BH mass spectrum. A flatter IMF or a weaker stellar wind leads to more massive BHs. This in turn, as we have explained earlier, leads to smaller SN kicks and overall more massive binaries, and thus smaller BH spin–orbit misalignment angles. On the other hand, a variation of the CE efficiency does not seem to affect spin misalignment of our BH XRB population. Variations of the σ parameter of the Maxwellian distribution for the SN kicks affect, as expected, the BH tilts, with higher kicks resulting in higher misalignment angles. However this effect saturates for very high kicks, as they lead to disruption rather than to very high misalignment angles.

Finally, we examine the extreme case where BHs receive the same kicks as NSs, directly drawn from a Maxwellian distribution and not scaled to the fall-back mass (Model 19). Although this model predicts unrealistically large BH kicks, it serves as an absolute upper limit on the possible BH spin–orbit misalignment. We find that even in this extreme case, the majority of the BH XRBs have misalignment angles below ∼15°.

We performed PS simulations of a Galactic field population of BH XRBs and studied the distribution of BH spin–orbit misalignment angles of transient, Roche-lobe overflowing BH XRBs. In the determination of the current misalignment angle, we took into account the evolution of the BH spin magnitude and direction due to accretion of matter during the XRB phase. We examined over 200 PS models, varying PS parameters such as IMF, stellar wind strength, CE efficiency, SN kick's magnitude and direction distributions, and BH's initial spin magnitude distribution. We found that for the majority of BH XRBs in our model populations, the misalignment angle is below 10°, while at the same time there is a small part of the population (<5%) that has misalignment angles higher than 20° which in some cases can even exceed 90°. However this sub-population of highly misaligned BH XRBs account for less than 0.3% of the total population. These results are robust among all our models and all combinations of PS parameters.

In recent years, several research groups have attempted to measure the BH spin magnitude in BH XRBs. Both currently available methods for the measurement of BH spins depend on the BH spin–orbit misalignment angle (McClintok & Remillard 2009; Miller et al. 2009). The X-ray continuum spectral fitting method (Shafee et al. 2006; Miller 2007) assumes a full alignment between the BH spin and the orbital angular momentum. The Fe K spectral line method (Brenneman & Reynolds 2006) does not make any explicit assumption, but its accuracy could be improved from additional constraints on the BH spin tilts. This Letter is the first study that provides theoretical constraints on the BH spin–orbit misalignment in BH XRBs.

The authors thank Professor Kalogera, Dr. McClintock, and Dr. Mandel for their comments on the manuscript and other helpful discussions that greatly improved this work. T.F. acknowledges support from the Northwestern Presidential Fellowship.