WHAT IS THE MASS OF $\alpha$ CEN B ${\rm b}$?

The results of our simulation suggest that we cannot place any dynamical constraints on α Cen B b at its present location, with stable orbits found at all inclinations tested. However, the current inclination of α Cen B b with respect to the AB binary orbital plane is also tied to the formation mechanism of the exoplanet. α Cen B b could have formed in situ, migrated to its current location in a disk, migrated in resonance with another planet, or migrated via the Kozai mechanism or via planet–planet scattering, among other possibilities.

We first consider whether it is feasible that α Cen B b migrated via the Kozai mechanism to its present location. It is clear from Figure 12 that Kozai migration in this system requires a fine-tuned decreasing initial inclination as a function of increasing semimajor axis. Higher initial inclinations are dynamically unstable, and lower inclinations are stable. In our three simulations in which Kozai migration was successfully completed, the final inclinations are 46°–53° relative to the AB orbital plane and less than the initial inclinations. Only one of our three migration simulations resulted in a circularized orbit exterior to the Roche radius of the star at a position comparable to the present semimajor axis of α Cen B b. The migration in the other two simulations is so rapid that the planet may not survive the tidal dissipation of the orbital energy. For the planet with an initial inclination of 60° and semimajor axis of 1.3 AU (Figure 15), a steady inward Kozai migration does start. However, the process is clearly very fine-tuned to initial conditions, as the planet is destroyed halfway through the migration. Consequently, Kozai migration is likely not a robust planet formation outcome for α Cen B b in this system from initial semimajor axes larger than ∼1 AU.

While the fragility of the Kozai migration mechanism may seem unexpected, Wu (2003) does note that Kozai migration is not a common outcome of their simulations of HD 80606 b. Further, the binary companion to HD 80606 in Wu (2003) is located at 1100 AU, compared to ∼23 AU for α Cen A. The much shorter orbital timescale for α Cen A, when compared to the tidal circularization and GR precession timescales, may account for the fragility of the Kozai mechanism in our simulations.

Similarly, although not directly demonstrated in our simulations, planet–planet scattering events are unlikely to result in a significantly inclined orbit of α Cen B b with respect to the AB orbital plane at its present semimajor axis. If α Cen B b was initially formed at a larger semimajor axis and it received a significant orbital inclination boost from a planet scattering event, then the Kozai oscillation from α Cen A would dominate the dynamical evolution of α Cen B b thereafter. Thus, α Cen B b would still require a fine-tuned inclination as a function of its semimajor axis to avoid ejection or tidal disruption.

We next turn to consider the formation of α Cen B b in situ, or its formation and subsequent migration in a primordial disk with or without a hypothetical second planet in resonance.

Detailed numerical simulations of planet formation in binaries (Quintana et al. 2007; Wu et al. 2007; Fragner et al. 2011; Xie et al. 2011; Zhao et al. 2012), and in particular planet formation around α Cen A and B (Quintana et al. 2002; Lissauer et al. 2004; Quintana & Lissauer 2006; Guedes et al. 2008; Xie et al. 2010; Andrade-Ines & Mitchtchenko 2014; Rafikov & Silsbee 2014), give us the strongest constraints on the range of allowed inclinations for exoplanets in the α Cen system. The salient points from this extensive list of references can be summarized as follows.

• 1.  
  Planet formation is more efficient around α Cen A compared to α Cen B.
• 2.  
  Planet formation is less efficient for increasing misalignments, decreasing rapidly at inclinations of $\sim \gt 25{\rm --}45{}^\circ$ due to the Kozai mechanism.
• 3.  
  Mutual orbital inclination decreases with decreasing orbital semimajor axis.
• 4.  
  Planet formation of ∼2–5 exoplanets within 2 AU is feasible.
• 5.  
  Short orbital period planets within ∼0.05 AU are often discarded due to the short dynamical timescale with respect to numerical time steps for computational efficiency, which is one factor that motivated our analysis.
• 6.  
  Additional simulations of the α Cen system assume coplanarity in a 2D code, and are thus not relevant to our discussion (Kley & Nelson 2007; Muller & Kley 2012).

In particular, the results of oligarchic growth simulations carried to 200 Myr in Guedes et al. (2008, Table 1) yield populations of planets with a standard deviation of inclinations of ∼±8.6°, orbital semimajor axes of ∼0.2–1.8 AU, and eccentricities of ∼0.02–0.35 from an initial set of co-aligned planetesimals. Additionally, Quintana et al. (2002) specifically investigated oligarchic growth simulations carried to 400 Myr for a planetesimal disk initially inclined with respect to the AB orbital plane. The outcome of the simulations for planet formation around α Cen A (Figures 4, 8, 9; Quintana et al. 2002) and α Cen B (Figure 10) generally produce planets inside of ∼0.5 AU with inclinations of $\lt 20{}^\circ$, regardless of the initial planetesimal disk inclination with respect to the AB binary orbital plane. For planets exterior to ∼1 AU, Quintana et al. (2002) find that they can retain their initial misalignment. Finally, Zhao et al. (2012) presented an analysis of the inclination evolution of an exoplanet in the presence of a primordial gas disk in a binary system and concluded that the inclination will also remain small.

From these literature simulations, we conclude that α Cen B b could have likely formed with an initial inclination of ±20° with respect to the AB binary orbital plane, which is less than the critical Kozai angle of 392. Regardless of whether α Cen B b migrated in a disk with or without a resonance planetary companion, or formed in situ at its present location, these literature simulations strongly suggest that α Cen B b is not misaligned with the AB orbital plane by more than 20°. However, Quintana et al. (2002) start their simulation with planetesimals exterior to 0.36 AU. Thus, for semimajor axes inside of this value, we are extrapolating our conclusion about the evolution of protoplanets inclinations. Future detailed studies of planetesimal evolution interior to 0.36 AU are warranted.

α Cen C is currently far enough away (∼15,000 AU) from the AB binary to be ignored as having any current dynamical influence on α Cen B b. However, at some point during the early formation of the α Cen system, assuming that the C component is bound, C may have had a closer approach to the AB system in a fashion sufficient to warp or disturb the circumsecondary disk or young protoplanets around the B component. This could have resulted in a misalignment, disruption, or migration of the B circumsecondary disk/ protoplanets, but such a misalignment would not have been likely to survive the dynamical influence of α Cen A in our simulations and literature simulations.

Next, we consider whether or not we can infer any constraints on the orbital inclination of α Cen B b from the stellar spin axis alignment of α Cen B. The spin of α Cen A is observed to be aligned with the AB orbital plane, as inferred from projected rotational velocity combined with the observed rotation period and radius measurements obtained from asteroseismology and interferometry (Kervella et al. 2003; Bazot et al. 2007). However, the slower rotation period of α Cen B (∼36–42 days; Jay et al. 1997; Buccino & Mauas 2008; Dewarf et al. 2010), combined with the low v sin i of 1.1 ± 0.8 km s⁻¹ (Saar & Osten 1997), precludes a useful constraint on the inclination of the stellar spin axis from R ∼ 10⁶ spectroscopy (Frutiger et al. 2005). Given the observed rotation period and stellar radius, the expected rotational velocity is ∼1 km s⁻¹ for α Cen B, which is consistent with the observed v sin i.

The best observational constraint on the inclination of the spin axis of α Cen B comes from Dumusque (2014). Through modeling of simultaneous radial velocity and photometric observations of α Cen B, Dumusque (2014) derives an inclination on the sky of the stellar spin axis of α Cen B of $45_{-19}^{+9}$ degrees. With an unknown orientation on the sky, this corresponds to a minimum misalignment of $\gt$20° at 2σ with the orbital plane of the AB binary.

We can perform a related analysis by comparing the stellar jitter activity level of α Cen B to the Sun. D12 reported the observation of differential rotation for α Cen B from the radial velocity jitter induced by the rotational modulation of starspots. The 2008-2011 radial velocity observations in D12 yield periods of 39.76, 37.80, and 36.71 days in 2009, 2010, and 2011, respectively. These observations span the minimum to maximum in the ∼8.8 yr stellar activity cycle reported for α Cen B (Ayres 2009; Dewarf et al. 2010). The rotation period evolution is consistent with α Cen B exhibiting a Sun-like "butterfly diagram" evolution of starspots from latitudes of ∼±30° to the stellar equator from epochs of minimum to maximum activity. Thus, we can constrain the spin axis of α Cen B to be $\sim \gt 30{}^\circ$ deviant from an axis normal to the sky. Otherwise, the spots would be visible at all rotational phases in 2009, and the projected radial velocity rotational modulation would likely not be observed at the detected amplitude. This is consistent with the measurement in Dumusque (2014), but as noted in Dumusque (2014) the rotation periods reported are susceptible to error because of the harmonic fitting to the radial velocity time series.

We can take this line of inquiry one step further. D12 derives radial velocity rms in the 2008-2011 seasons of 1.18, 1.50, 2.19, and 2.15 m s⁻¹ respectively, corresponding to an increase in quadrature of ∼1.8 m s⁻¹ in projected radial velocity jitter from the increased stellar activity from 2008–2011. The Sun's expected radial velocity jitter due to rotational modulation is ∼0.4 m s⁻¹ (Makarov 2009). This value is a factor of ∼4 below the observed jitter for α Cen B despite the slower rotational velocity of α Cen B compared to the Sun—∼1.1 km s⁻¹ for α Cen B versus ∼1.6 km s⁻¹ for the Sun (Pavlenko et al. 2012). Further, D12 measures activity levels of ${\rm log} R_{{\rm HK}}^{\prime }$ = −4.99, −4.94, −4.89, and −4.90 in 2008-2011, respectively, compared to the solar mean activity level of −4.90 (Mamajek & Hillenbrand 2008). In other words, the activity level of α Cen B is comparable to that of the Sun. If we assume α Cen B to have a spot frequency, size, and temperature contrast similar to that of the Sun, then the projected radial velocity jitter of α Cen B is difficult to reconcile with the estimated jitter of the Sun without the spin axis of α Cen B being nearly perpendicular to normal to the plane of the sky (e.g., $\sim \gt 60{}^\circ$). Instead, α Cen B must have larger spots or larger spot temperature contrast to account for the factor of ∼3 needed to reconcile the observed RV jitter and rotation period relative to the Sun, which likely can be attributed to the differences in spectral type.

Finally, if the axis of the stellar spin was perpendicular to the normal to the sky and aligned with the direction of orbital motion rather than the orbital plane, then this would be consistent with the arguments presented thus far. However, a noteable cycle in the activity of α Cen B could be apparent over the course of the 80 yr orbital period of the AB binary, and this is not observed with detailed studies of activity dating back a few decades (Flannery & Ayers 1978, Dewarf et al. 2010). While this scenario is not expressly discounted, we conclude from the above arguments that the stellar spin axis of α Cen B is likely $\sim \gt 30{}^\circ$ deviant from normal to the plane of the sky, with the most likely value of 45° coming from Dumusque (2014) implying a misalignment of $\gt$20° with the AB orbital plane. However, the misalignment is likely not significantly larger—e.g., $\sim \lt$45° misalignment—lest we run into difficulty accounting for the observed differential rotation and jitter amplitude, and the lack of activity modulation over the ∼80 yr binary orbit. Thus, unlike α Cen A, α Cen B is likely ∼20–45° misaligned with the AB orbital plane.

Barring direct observational constraints of the stellar spin alignment of α Cen B, we can invoke spin–orbit alignment measurements of young binaries of comparable separations to α Cen AB in Hale (1994), Howe & Clarke (2009), and at larger separations in Jensen et al. (2004) and Monin et al. (2006), to estimate the likely spin–orbit alignment of α Cen B. These studies identify that at separations of $\sim \lt$ 100 AU, the spins of stars in a young binary are typically aligned to within ∼10–30° of the orbital plane. Additionally, observations of young low-mass stellar binaries indicate that circumprimary and circumsecondary disks are usually aligned with the orbit of the binary as well (Monin et al. 2006; Prato & Weinberger 2007; Watson et al. 2011; Wheelwright et al. 2011). These observations are supported by binary disk modeling of systems including eccentric systems like α Cen (Pichardo et al. 2005). Thus, this suggests on an ensemble basis, although it is not conclusively demonstrated through direct observation, that the spin of α Cen B is aligned with the AB orbital plane to within ∼30°. Such an alignment is consistent with the observed rotational modulation and radial velocity jitter amplitude of α Cen B presented in D12 and marginally consistent with Dumusque (2014) given the arguments presented thus far.

On the other hand, however, Skemer et al. (2008), Jensen & Akeson (2014), and Roccatagliata et al. (2011) present evidence for misaligned disks in the triple system T Tauri, the binary system HK Tauri, and the binary Haro 6-10, respectively. Jensen et al. (2004) notes that compact triples can increase the odds of spin–orbit misalignment. Thus, if Proxima Centauri were closer to α Cen AB earlier in the evolution of this triple system, it could have warped the circumsecondary disk around α Cen B into misalignment.

In Sections 4.3 and 4.4, we have made the argument that the spin of α Cen B is aligned with the orbital plane of the AB binary with an angle of ∼20–45° from direct observational constraints, and from less conclusive comparative studies of binary stars with and without circumstellar disks. We next consider if the orbital plane of the exoplanet α Cen B b is aligned with the stellar spin of α Cen B. This might be expected for an exoplanet forming in situ or migrating in an aligned protoplanetary disk. Observations of solitary transiting exoplanets in short orbital periods around single stars indeed demonstrate that the exoplanet orbit and stellar spin are well-aligned for older systems with effective temperatures $\lt$6000 K and exoplanet masses $\gt$0.2 M_J (Winn et al. 2010; Albrecht et al. 2012). Additionally, for multiple, compact terrestrial exoplanet systems, co-alignment within a few degrees is the norm (Fang & Margot 2012). However, there are a number of short-period systems that are misaligned, and this is thought to be due to migration of the exoplanets via planet–planet scattering or the Kozai mechanism. Furthermore, a number of studies have predicted that exoplanets in binary systems are more likely to be misaligned (Wu et al. 2007; Parker & Goodwin 2009; Xie et al. 2011).

Tidal interactions of a close-in exoplanet, even if initially misaligned, can be re-aligned with the spin axis within the tidal circularization timescale (Winn et al. 2010). Given the age of α Cen of ∼5 Gyr, one could assume that the system has had adequate time to re-align α Cen B b to the stellar spin axis of α Ceb B via the action of stellar tides, regardless of any initial misalignment. However, the mass of α Cen B b is orders of magnitude smaller than 0.2 M_J, and thus the re-alignment timescale is orders of magnitude longer (Albrecht et al. 2012). It is likely that α Cen B b is not massive enough to excite stellar tides to induce the re-alignment on a timescale that is shorter than the age of the α Cen system. Indeed, HAT-P-11 b is one of the lowest mass exoplanets with a measured Rossiter–McLaughlin effect at ∼26 ${{M}_{\oplus }}$, and it is an outlier that is misaligned with the stellar spin axis of its host star (Winn et al. 2010). More observations of the Rossiter–McLaughlin effect for single, short-period terrestrial mass planets are needed to determine if orbital re-alignment is likely or rare. Thus, we cannot conclusively argue that an initially misaligned α Cen B b was re-aligned with its host stellar spin axis. However, we can assert that if the initial planet-forming disk around α Cen B was aligned with the spin axis of α Cen B and not the AB orbital plane, and if α Cen B b did not undergo Kozai migration, then α Cen B b will have likely retained that primordial spin–orbit alignment (Fang & Margot 2012).