The Orbital Architecture of Qatar-6: A Fully Aligned Three-body System?

The system's secondary star, Qatar-6 B, was first identified by Mugrauer (2019). Mugrauer (2019) used the Gaia DR2 astrometric data set (Gaia Collaboration et al. 2016; Brown et al. 2018) to demonstrate that Qatar-6 B is bound to the primary. In this section, we use the updated Gaia DR3 data set to confirm that Qatar-6 B is the only candidate stellar companion to Qatar-6 A.

We queried for all sources within $10^{\prime}$ of Qatar-6 A and then followed the methods of El-Badry et al. (2021) to vet potential companions. Our search included any sources with parallaxes ϖ > 1 mas, fractional parallax uncertainties σ_ϖ /ϖ < 0.2, and absolute parallax uncertainties σ_ϖ < 2 mas. For each source that passed this initial cut, we checked the following three requirements:

• 1.  
  The sky-projected separation s between the two stars must be less than 1 pc, that is,

  $$\begin{eqnarray}&&\left(\displaystyle \frac{{\theta }_{s}}{\mathrm{arcsec}}\right)\lt 206.265\left(\displaystyle \frac{\varpi }{\mathrm{mas}}\right)\end{eqnarray} \tag{ 2 }$$

  for projected angular separation θ_s calculated as

  $$\begin{eqnarray}&&\cos ({\theta }_{s})=(\sin {\delta }_{p}\sin {\delta }_{{cc}}+\cos {\delta }_{p}\cos {\delta }_{{cc}}\cos ({\alpha }_{p}-{\alpha }_{{cc}})).\end{eqnarray} \tag{ 3 }$$

  The subscript p refers to the primary (Qatar-6 A), whereas the subscript cc refers to a companion candidate. R.A. and decl. are given as α and δ, respectively.
• 2.  
  The parallax of the primary and the candidate companion must be consistent within b σ, such that

  $$\begin{eqnarray}&&| {\varpi }_{{cc}}-{\varpi }_{p}| \lt b\sqrt{{\sigma }_{{\varpi }_{{cc}}}^{2}+{\sigma }_{{\varpi }_{p}}^{2}}.\end{eqnarray} \tag{ 4 }$$

  Following El-Badry et al. (2021), we set b = 3 for pairs with angular separation θ_s > 40, or b = 6 for pairs with θ_s < 40. The weaker threshold for pairs at small angular separation is set to counteract the systematic underestimate of parallax uncertainties at close angular separations (El-Badry et al. 2021).
• 3.  
  The two stars must have relative proper-motion measurements consistent with a bound Keplerian orbit

  $$\begin{eqnarray}&&{\rm{\Delta }}\mu \lt {\rm{\Delta }}{\mu }_{\mathrm{orbit}}+2{\sigma }_{{\rm{\Delta }}\mu }.\end{eqnarray} \tag{ 5 }$$

  Here Δμ_orbit is given by

  $$\begin{eqnarray}&&{\rm{\Delta }}{\mu }_{\mathrm{orbit}}=(0.44\,\mathrm{mas}\,{\mathrm{yr}}^{-1}){\left(\displaystyle \frac{\varpi }{\mathrm{mas}}\right)}^{3/2}{\left(\displaystyle \frac{{\theta }_{s}}{\mathrm{arcsec}}\right)}^{1/2}.\end{eqnarray} \tag{ 6 }$$

This relation, which is drawn from El-Badry & Rix (2018), provides the maximum difference in proper motion for a circular binary orbit with total system mass 5 M_⊙. We use this as a conservative estimate to encapsulate a range of potential candidate stellar companion velocities.

The uncertainty σ_Δμ in the proper-motion difference between the two stellar components is given as

$$\begin{eqnarray}&&{\sigma }_{{\rm{\Delta }}\mu }=\displaystyle \frac{1}{{\rm{\Delta }}\mu }\sqrt{({\sigma }_{{\mu }_{\alpha ,1}^{* }}^{2}+{\sigma }_{{\mu }_{\alpha ,1}^{* }}^{2}){\rm{\Delta }}{{\mu }_{\alpha }^{* }}^{2}+({\sigma }_{{\mu }_{\delta ,1}}^{2}+{\sigma }_{{\mu }_{\delta ,2}}^{2}){\rm{\Delta }}{\mu }_{\delta }^{2}},\end{eqnarray} \tag{ 7 }$$

while the proper-motion difference Δμ is

$$\begin{eqnarray}&&{\rm{\Delta }}\mu =\sqrt{{\rm{\Delta }}{{\mu }_{\alpha }^{* }}^{2}+{\rm{\Delta }}{\mu }_{\delta }^{2}}.\end{eqnarray} \tag{ 8 }$$

The proper-motion differences ${\rm{\Delta }}{\mu }_{\alpha }^{* }$ and Δμ_δ in the R.A. and decl. directions, respectively, are calculated as

$$\begin{eqnarray}&&{\rm{\Delta }}{{\mu }_{\alpha }^{* }}^{2}={\left({\mu }_{\alpha ,1}^{* }-{\mu }_{\alpha ,2}^{* },\right)}^{2}\end{eqnarray} \tag{ 9 }$$

and

$$\begin{eqnarray}&&{\rm{\Delta }}{\mu }_{\delta }^{2}={\left({\mu }_{\delta ,1}-{\mu }_{\delta ,2}\right)}^{2}.\end{eqnarray} \tag{ 10 }$$

We note that proper motions reported by Gaia DR3 in the RA direction already account for a $\cos \delta$ corrective factor, ¹⁴ such that ${\mu }_{\alpha }^{* }\equiv {\mu }_{\alpha }\cos \delta$. Our provided relations have implicitly included this correction within them. For clarity, we include a star superscript on each variable that includes this corrective factor.

A single source passed all three of the tests outlined above. We first compared the projected separation and orientation of the system to confirm that the recovered source is the previously identified companion Qatar-6 B. Then, we verified that the same binary companion was also identified in El-Badry et al. (2021), which predicts a low fractional chance alignment probability R = 1.92125073 × 10⁻⁶ for the binary pair. We conclude that the previously identified companion Qatar-6 B—the only candidate companion that we identify for Qatar-6 A—is likely not a chance alignment.

We also confirmed that the projected separation of the binary is s ≪ 30,000 au, a threshold above which El-Badry et al. (2021) find that chance alignments dominate over true bound companions. Using Gaia DR3, we measured a sky-projected separation s = 482 au between Qatar-6 A and Qatar-6 B. This separation is similar to, but slightly smaller than, the s = 486 au separation determined by Mugrauer (2019) using Gaia DR2.

Lastly, we checked each star's Renormalized Unit Weight Error (RUWE) parameter—a χ²-based metric provided by Gaia to quantify the robustness of the astrometric fit for a star. RUWE ∼ 1.0 typically corresponds to a high-quality single star fit, while RUWE > 1.4 generally indicates a poor astrometric fit that may result from the presence of an unresolved companion. For Qatar-6 A and B, respectively, Gaia DR3 reports RUWE = 1.22 and RUWE = 1.20. Although both Qatar-6 stars fall comfortably below the commonly adopted limit RUWE < 1.4, the astrometric fit for each star deviates substantially from RUWE = 1.0. With this caveat in mind, we proceed to further characterize the properties of the binary star orbits.

To examine the relative orbital configuration of the Qatar-6 AB binary star system, we first measured the angle γ between the position vector connecting the two binary star components and the relative proper-motion vector (Tokovinin 1998; Tokovinin & Kiyaeva 2015). We selected the convention that γ = 180° where the secondary star is moving directly in the opposite direction of the primary, whereas γ = 0° where the secondary star is moving directly toward the primary. We obtained γ = 17911 for Qatar-6 AB, and the geometry of the system is visualized in Figure 3 for reference. This nearly linear configuration indicates that the position and velocity vectors are well aligned within the sky plane, suggesting an edge-on orbit for the binary system.

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Next, we further constrained the highest-likelihood binary orbits for the Qatar-6 AB system using the lofti_gaia Python package (Pearce et al. 2020). The Linear OFTI (LOFTI) algorithm implemented by lofti_gaia was designed to constrain the orbital properties of binary star systems based on the linear sky-plane velocity vector of each star measured by the Gaia mission. LOFTI builds on the Orbits For The Impatient (OFTI) Bayesian rejection sampling method (Blunt et al. 2017) for orbit fitting to short orbital arcs.

We ran LOFTI up to 100,000 accepted orbits using astrometric constraints provided by Gaia DR3. Stellar masses ${M}_{A}={0.829}_{-0.022}^{+0.019}\,{M}_{\odot }$ (this work) and ${M}_{B}={0.244}_{-0.020}^{+0.013}\,{M}_{\odot }$ (Mugrauer 2019) were adopted for Qatar-6 A and Qatar-6 B, respectively. Because LOFTI requires a symmetric mass uncertainty, we used the larger uncertainties ${\sigma }_{{M}_{A}}=0.022\,{M}_{\odot }$ for Qatar-6 A and ${\sigma }_{{M}_{B}}=0.020\,{M}_{\odot }$ for Qatar-6 B.

A subsample of 1000 accepted orbits is shown in Figure 4, demonstrating a clear tendency toward aligned orbits in agreement with our γ analysis. The posteriors of our orbit fit are provided in Figure 5. We find that the current set of Gaia DR3 astrometry is unable to provide a strong constraint on the system's eccentricity. There is also a degeneracy between position angles PA = 180° and PA = 360° owing to the relatively small, 2.2σ difference in parallaxes between the two components, which produces a corresponding degeneracy in the argument of periapsis ω. Regardless, we find a strong preference for an edge-on binary orbit, with inclination ${i}_{B}={90.17}_{-1.06}^{+1.07\circ}$.

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Ignoring selection effects (see, e.g., El-Badry & Rix 2018; Ferrer-Chávez et al. 2021), the probability distribution function for inclinations i_B drawn from an isotropically distributed set of orbits follows a uniform distribution in $\sin ({i}_{B})$, where i_B = 90° is defined as an edge-on orbit. In the case that stellar binary orbits are randomly oriented, there would be a ∼2% occurrence rate for chance alignments in i_B within 12. Correspondingly, there is a ∼2% chance that the observed alignment results from a chance alignment among a randomly distributed set of orbits.

Selection biases favor relatively edge-on orbits, such that even an isotropically distributed set of orbits (uniform in $\sin ({i}_{B})$) should include an overdensity toward i_B ∼ 90°. Furthermore, orbit fitting with little to no orbital coverage suffers from known degeneracies between inclination and eccentricity (Ferrer-Chávez et al. 2021). To examine the robustness of our edge-on orbit fit, we produced a comparison sample of 10 systems with similar binary separation, parallax, and magnitude properties to that of Qatar-6 AB. This sample includes the 10 systems within the El-Badry et al. (2021) catalog with the most similar properties to that of Qatar-6 AB based on the metric adopted in Christian et al. (2022). Masses were extracted using the isoclassify Python package in the same configuration described in Section 3, but with an uncertainty floor of 0.5 mag to facilitate convergence. We integrated each comparison system to 1000 accepted orbits using LOFTI, with results that are shown in Figure 6. This exercise reaffirms the edge-on nature of Qatar-6 AB, which has an inclination distribution that is restricted to a much narrower range of edge-on values than the comparison sample.

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Recent population studies have demonstrated that hosts of transiting planets tend to have stellar binary orbits closer to an edge-on configuration than field binary star systems with no known planets, suggesting a systematic trend toward alignment between the binary plane and the planetary orbits (Christian et al. 2022; Dupuy et al. 2022). Our results are consistent with this finding; that is, we show that the Qatar-6 AB binary system, which includes one confirmed transiting planet, lies in a precisely edge-on configuration. Combined with previous findings, this suggests that the planet's orbital plane may be closely aligned with the stellar binary orbital plane (a configuration that we refer to as "orbit–orbit alignment").

However, we emphasize that the only well-constrained angle in our binary system is inclination. That is, the sky-plane direction in which the planet is transiting is not constrained relative to the plane of the stellar orbit. While the line-of-sight orientation of the system is consistent with alignment, it is not currently possible to measure the sky-plane angle between the planetary orbit and the stellar orbit in most planetary systems. Additional population-wide studies examining the prevalence of orbit–orbit alignment in hot and warm Jupiter systems may further inform the role of stellar multiplicity in the evolution of planetary systems.

6.1.1. Angular Momentum of the System

To evaluate the feasibility of tidal alignment within the system, we first compared the orbital angular momentum of Qatar-6 A b with that of the convective layer of Qatar-6 A. In the case that tidal alignment occurs prior to the completion of the orbital decay process (and the subsequent disruption of the Jovian planet), the companion planet's orbit should host more angular momentum than the host star's convective layer. We calculated the angular momentum L_CZ of the star's convective zone using the relation

$$\begin{eqnarray}&&{L}_{\mathrm{CZ}}=\omega \int {l}^{2}{dm},\end{eqnarray} \tag{ 11 }$$

for distance l from each mass element dm to the spin axis, with $l=r\sin \phi$ and

$$\begin{eqnarray}&&{dm}=\rho {dV}=\rho {r}^{2}\sin \phi {drd}\phi d\theta .\end{eqnarray} \tag{ 12 }$$

Here ω = v/R_* is the angular velocity of the convective layer's rotation, where we assume no shear between layers. We integrated the density ρ over each volume element dV of the convective layer, using spherical coordinates to integrate from the radius of the convective zone boundary (r = R_CZ) to the full radius of the star (r = R_*). The radius of the convective zone boundary was set as R_CZ/R_* = 0.69 based on the models of Van Saders & Pinsonneault (2012) for the mass and age of Qatar-6 A. The density of the convective layer was approximated as uniform, with a total mass M_CZ = 10^−1.35 M_⊙ drawn from the stellar interior models of Pinsonneault et al. (2001).

Comparing this with the planet's orbital angular momentum, which is given by

$$\begin{eqnarray}&&{L}_{p,\mathrm{orb}}={M}_{b}{v}_{b}{r}_{b},\end{eqnarray} \tag{ 13 }$$

at a given orbital distance r_b and momentary velocity v_b , we find that L_p,orb/L_CZ ∼ 15. This excess of angular momentum in the planet's orbit indicates that tidal alignment could feasibly occur within this system.

6.1.2. Equilibrium Tides

Qatar-6 A is a cool star (T_eff ∼ 5063 ± 42 K) that lies well below the Kraft break rotational discontinuity at roughly T_eff ∼ 6100 K, below which stars typically have convective envelopes (Kraft 1967). The tidal alignment timescale τ_CE for stars with convective envelopes can be approximated as

$$\begin{eqnarray}&&{\tau }_{\mathrm{CE}}=\displaystyle \frac{{10}^{10}\mathrm{yr}}{{\left({M}_{b}/{M}_{* }\right)}^{2}}{\left(\displaystyle \frac{{a}_{b}/{R}_{* }}{40}\right)}^{6},\end{eqnarray} \tag{ 14 }$$

where M_b /M_* is the planet-to-star mass ratio (Zahn 1977; Albrecht et al. 2012). This scaling is calibrated based on the observed synchronization of stellar binaries under the framework of equilibrium tides. Consequently, it includes an implicit assumption that the tidal realignment timescale for planetary systems scales similarly to that of stellar systems.

Based on Equation (14), Qatar-6 A b has τ_CE ∼ 1.6 × 10¹³ yr, which is longer than the age of the universe. Therefore, under the assumption that equilibrium tides well approximate the dynamical behavior of this system, the companion planet's spin–orbit angle likely has not changed significantly since its initial formation.

6.1.3. Dynamical Tides

Alternatively, the system may have been aligned through the dissipation of inertial waves, which are driven by the Coriolis force in a rotating star. The tidal disturbances produced through this mechanism are collectively known as "dynamical tides." We focus on a specific mode of the dynamical tide—known as the "obliquity tide," with m = ± 1 and $m^{\prime} =0$—that damps only the stellar obliquity without altering the orbital semimajor axis of the companion planet (Lai 2012). In this case, the obliquity of a planetary system evolves as

$$\begin{eqnarray}&&\begin{array}{l}{\left.\frac{d\psi }{{dt}}\right|}_{10}=-\frac{3}{4}\frac{{k}_{2}}{{Q}_{10}}\left(\frac{{M}_{b}}{{M}_{* }}\right){\left(\frac{{R}_{* }}{{a}_{b}}\right)}^{5}{{\rm{\Omega }}}_{{\rm{K}}}\sin (\psi ){\cos }^{2}(\psi )\\ \times \left[1+\frac{{L}_{p,\mathrm{orb}}}{{L}_{* ,\mathrm{spin}}}\cos (\psi )\right],\end{array}\end{eqnarray} \tag{ 15 }$$

where L_*,spin is the stellar spin angular momentum, Ω_* is the star's spin frequency, k₂ is the planet's Love number, Q₁₀ is the tidal quality factor for the obliquity tide, and Ω_K is the Keplerian orbital angular frequency (Lai 2012; Spalding & Winn 2022). We use L_*,spin = L_CZ to consider the most conservative case in which only the convective zone realigns with the companion orbit. If the convective zone is not decoupled from the stellar core, then the timescale for realignment would be longer owing to the larger value of L_*,spin.

We numerically integrate this expression from t = 30 Myr (a rough starting age for a K-type star to enter the main sequence) to t = 1 Gyr (the age of the system) to determine the maximum initial ψ such that the system would today be observed to be aligned at ψ = 1°. The ratio k₂/Q₁₀ remains poorly constrained and provides the limiting uncertainty within our model.

We consider a range of k₂/Q₁₀ values in Figure 7, where we include both the case in which Qatar-6 A b has a true obliquity ψ = 22° and the case in which the true obliquity is within 1° of alignment (ψ = 1°). As demonstrated in Figure 7, a relatively large value of k₂/Q₁₀ ≳ 10⁻⁴ would be required to push Qatar-6 A b from a highly misaligned state to its currently observed spin–orbit angle over the system lifetime.

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Empirical measurements of tidal dissipation in hot Jupiter hosts typically range from k₂/Q = 10⁻⁵ to 10⁻⁷ (Penev et al. 2018), with larger values for wider-orbiting planets. The solar system's much wider orbiting Jupiter, for comparison, has been measured with k₂/Q = (1.102 ± 0.203) × 10⁻⁵ (Lainey et al. 2009). While the effective tidal quality factor Q differs for each mode of tidal dissipation, Q values for hot and warm Jupiters are typically expected to be high, leading to correspondingly low k₂/Q values. As a result, unless tidal dissipation in warm Jupiter systems is more efficient than previous estimates have suggested, it is unlikely that Qatar-6 A b has been realigned from a large previous misalignment over the post-disk system lifetime.

Our model demonstrates that Qatar-6 A b may lie on a slightly eccentric orbit, with ${e}_{b}={0.051}_{-0.030}^{+0.032}$ (Section 4). Over time, the orbit will evolve along a path of constant orbital angular momentum characterized by

$$\begin{eqnarray}&&{a}_{b,\mathrm{final}}={a}_{b}(1-{e}_{b}^{2}),\end{eqnarray} \tag{ 16 }$$

toward e → 0 as energy is removed from the system through tidal dissipation within the planet. As a result, the orbit will ultimately settle to a slightly smaller separation if it currently has a true nonzero eccentricity.

We follow the formulation of Rice et al. (2022a) to calculate the timescale τ_circ for orbital circularization, with methods summarized here for convenience. τ_circ can be characterized as

$$\begin{eqnarray}&&{\tau }_{\mathrm{circ}}\sim {e}_{b}/({{de}}_{b}/{dt}),\end{eqnarray} \tag{ 17 }$$

where

$$\begin{eqnarray}&&\displaystyle \frac{{{de}}_{b}}{{dt}}=\displaystyle \frac{{dE}}{{dt}}\displaystyle \frac{{a}_{b}(1-{e}_{b}^{2})}{{{GM}}_{* }{M}_{b}{e}_{b}}\end{eqnarray} \tag{ 18 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{{dE}}{{dt}}=\displaystyle \frac{21{k}_{2}{{GM}}_{* }^{2}{\rm{\Omega }}{R}_{b}^{5}}{2{{Qa}}_{b}^{6}}\zeta ({e}_{b}),\end{eqnarray} \tag{ 19 }$$

for an incompressible, synchronously rotating planet. Here G is the gravitational constant, E is the orbital energy of the planet, Q is the planet's effective tidal dissipation parameter, k₂ is the planet's Love number, and Ω is the pseudo-synchronous rotation rate, given by

$$\begin{eqnarray}&&{\rm{\Omega }}=\displaystyle \frac{1+\tfrac{15}{2}{e}_{b}^{2}+\tfrac{45}{8}{e}_{b}^{4}+\tfrac{5}{16}{e}_{b}^{6}}{(1+3{e}_{b}^{2}+\tfrac{3}{8}{e}_{b}^{4}){\left(1-{e}_{b}^{2}\right)}^{3/2}}{n}_{b},\end{eqnarray} \tag{ 20 }$$

for planetary mean motion n_b . The corrective factor ζ(e) in Equation (18) was derived in Wisdom (2008) and is defined as

$$\begin{eqnarray}&&\zeta ({e}_{b})=\displaystyle \frac{2}{7}\left[\displaystyle \frac{{f}_{0}({e}_{b})}{{\beta }^{15}}-\displaystyle \frac{2{f}_{1}({e}_{b})}{{\beta }^{12}}+\displaystyle \frac{{f}_{2}({e}_{b})}{{\beta }^{9}}\right],\end{eqnarray} \tag{ 21 }$$

where

$$\begin{eqnarray}&&\begin{array}{l}{f}_{0}({e}_{b})=1+\displaystyle \frac{31}{2}{e}_{b}^{2}+\displaystyle \frac{255}{8}{e}_{b}^{4}\\ \,+\,\displaystyle \frac{185}{16}{e}^{6}+\displaystyle \frac{25}{64}{e}_{b}^{8}\end{array}\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{f}_{1}({e}_{b})=1+\displaystyle \frac{15}{2}{e}_{b}^{2}+\displaystyle \frac{45}{8}{e}_{b}^{4}+\displaystyle \frac{5}{16}{e}_{b}^{6}\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{f}_{2}({e}_{b})=1+3{e}_{b}^{2}+\displaystyle \frac{3}{8}{e}_{b}^{4}\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&\beta =\sqrt{1-{e}_{b}^{2}}.\end{eqnarray} \tag{ 25 }$$

Considering typical expected ranges of k₂/Q ∼ 10⁻⁵ to 10⁻⁷ for close-in giant planets (Penev et al. 2018), we obtain τ_circ ∼ 10⁷–10⁹ yr. Based on these relatively short timescales, we cannot exclude the possibility that Qatar-6 A b, with an age τ = (1.0 ± 0.5) × 10⁹ yr (Alsubai et al. 2018), began its dynamical evolution at a higher eccentricity that has been damped over time. However, because the planet's orbit is consistent with e_b = 0 within 2σ, we find no strong evidence requiring that the planet's orbit must have had a higher eccentricity in the past.

While, in the line-of-sight direction, the orbit of Qatar-6 A b appears to be aligned with the Qatar-6 AB binary system orbit (i_b ∼ i_B ∼ 90°), we cannot rule out the possibility that its orbit may be misaligned in the sky plane and therefore inclined relative to the Qatar-6 AB binary system. If this is the case, then Qatar-6 A b could be located along a low-eccentricity trough of a Kozai–Lidov cycle, where the z-component of the angular momentum vector

$$\begin{eqnarray}&&{L}_{z}=\sqrt{1-{e}_{b}^{2}}\cos {i}_{\mathrm{tot}},\end{eqnarray} \tag{ 26 }$$

is conserved at the quadrupole level. Here i_tot refers to the true inclination between the Qatar-6 A b warm Jupiter orbit and the Qatar-6 AB stellar binary orbit.

At the quadrupole level of approximation, the Kozai–Lidov timescale in a hierarchical three-body system is given by

$$\begin{eqnarray}&&{\tau }_{\mathrm{KL}}=\displaystyle \frac{16}{30\pi }\displaystyle \frac{{P}_{2}^{2}}{{P}_{1}}{\left(1-{e}_{2}^{2}\right)}^{3/2}\left(\displaystyle \frac{{m}_{1}+{m}_{2}+{m}_{3}}{{m}_{3}}\right),\end{eqnarray} \tag{ 27 }$$

where subscripts 1 and 2 refer to the inner and outer orbits, respectively (Naoz 2016). In our case, m₁ ≪ m₂, m₃ such that we can approximate (m₁ + m₂ + m₃) ∼ (m₂ + m₃). Simplifying our general expression and reconfiguring it for our system, we obtain

$$\begin{eqnarray}&&{\tau }_{\mathrm{KL}}\approx \displaystyle \frac{16}{30\pi }\displaystyle \frac{{P}_{B}^{2}}{{P}_{b}}{\left(1-{e}_{B}^{2}\right)}^{3/2}\left(\displaystyle \frac{{M}_{A}+{M}_{B}}{{M}_{B}}\right).\end{eqnarray} \tag{ 28 }$$

The primary star Qatar-6 A is referred to here with the subscript A, whereas the orbital properties of the stellar binary companion Qatar-6 B are denoted by the subscript B.

From our LOFTI orbital fitting results, we adopt a semimajor axis of a_B = 536, which translates to a_B = 541 au at a distance of 100.95 pc measured through the parallax reported in Gaia DR3. Given our poor constraint on the orbital eccentricity of the stellar binary system, we approximate this timescale for a range of possible eccentricities from e = 0 to e = 0.9. We find a timescale ranging from τ_KL ∼ 10⁹ yr for e = 0.9 to τ_KL ∼ 10¹⁰ yr for e = 0. These timescales are comparable to the estimated age of the system ((1.0 ± 0.5) × 10⁹ yr; Alsubai et al. 2018), indicating that Kozai–Lidov migration likely has not played a major role in the evolution of this system if Qatar-6 A b formed near its current location. The same planet would have a much shorter Kozai–Lidov timescale if it instead formed on a wider orbit (ranging from τ_KL ∼ 10⁷ yr for e = 0.9 to τ_KL ∼ 10⁸ yr for e = 0 if the planet began with an orbital period P_b = 1 yr), such that migration through Kozai–Lidov orbital evolution may have occurred in the past.

Kozai–Lidov oscillations can be suppressed by additional perturbations that produce apsidal precession at a rate more rapid than that of the Kozai–Lidov mechanism, reducing the orbit-averaged torque induced by the companion star (Holman et al. 1997; Wu & Murray 2003). We consider, in particular, the timescale τ_GR for apsidal precession due to general relativity

$$\begin{eqnarray}&&{\tau }_{\mathrm{GR}}=\frac{2\pi {c}^{2}(1-{e}_{b}^{2}){a}_{b}^{5/2}}{3{\left({{GM}}_{* }\right)}^{3/2}},\end{eqnarray} \tag{ 29 }$$

which is relatively short for short-period giant planets. In this expression, c is the speed of light.

For Qatar-6 A b, the timescale for precession induced by general relativistic effects is only τ_GR = 2 × 10⁴ yr—much shorter than the system's current Kozai–Lidov timescale τ_KL ∼ 10⁹–10¹⁰ yr (Section 6.3). We can, therefore, rule out the possibility that the Qatar-6 system is currently undergoing Kozai–Lidov oscillations.

A past mutual inclination of i_tot ≥ 392 would have been required between the planetary orbit and the binary star orbit to initiate Kozai–Lidov oscillations. In the case of the Qatar-6 system, a significant mutual inclination could remain undetected within the sky-plane direction if Kozai–Lidov oscillations occurred in the past. Alternatively, the system may have settled at a low final mutual inclination after undergoing Kozai–Lidov cycles (as in, e.g., some of the systems simulated by Naoz et al. 2012).

The stellar binary companion, Qatar-6 B, provides a small disturbing force

$$\begin{eqnarray}&&dF=\bar{R}\hat{r}+\bar{T}\hat{\theta }+\bar{N}\hat{z},\end{eqnarray} \tag{ 30 }$$

which perturbs the orbit of Qatar-6 A b, altering the observed orbital elements of the system. Here $\bar{R}$, $\bar{T}$, and $\bar{N}$ are the radial, tangential, and normal components of the force induced by the perturber, respectively. In particular, a nonzero mutual inclination between the planet's and the stellar binary's orbits should induce nodal and inclination precession in the orbit of Qatar-6 A b that may manifest as observable transit duration variations (TDVs) in the system. Our best-fitting solution includes a ∼5° mutual inclination between the planetary orbit (i_b = 8595 ± 024) and the stellar binary orbit (${i}_{B}={90.17}_{-1.06}^{+1.07\circ}$).

The rates of nodal (dΩ_b /dt) and inclination (di_b /dt) precession under the influence of a perturbing force dF are given by (Murray & Dermott 1999)

$$\begin{eqnarray}&&\displaystyle \frac{{{di}}_{b}}{{dt}}=\sqrt{\displaystyle \frac{{a}_{b}(1-{e}_{b}^{2})}{G({M}_{A}+{M}_{b})}}\displaystyle \frac{\bar{N}\cos ({\omega }_{b}+{f}_{b})}{1+{e}_{b}\cos {f}_{b}}\end{eqnarray} \tag{ 31 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{d{{\rm{\Omega }}}_{b}}{{dt}}=\sqrt{\displaystyle \frac{{a}_{b}(1-{e}_{b}^{2})}{G({M}_{A}+{M}_{b})}}\displaystyle \frac{\bar{N}\sin ({\omega }_{b}+{f}_{b})}{\sin {i}_{b}(1+{e}_{b}\cos {f}_{b})},\end{eqnarray} \tag{ 32 }$$

where f_b is the true anomaly of the planetary orbit, and both precession timescales are driven by the normal component of the perturbing force. We consider the range f_b ∈ (0, 2π) and find the largest possible nodal and inclination precession rates across this range, adopting a mutual inclination 5° between the planetary orbital plane and the stellar binary orbital plane and using the values derived in this work (Table 2). We obtain precession rates dΩ_b /dt = 9.6 × 10⁻⁴ deg yr⁻¹ and di_b /dt = 8.5 × 10⁻⁵ deg yr⁻¹, corresponding to a projected transit duration change of <1 minute over the course of a decade. Thus, we do not expect to observe significant TDVs caused by the binary perturber Qatar-6 B.

Together, the tidal alignment, tidal circularization, Kozai–Lidov, and apsidal precession timescales jointly indicate that the warm Jupiter Qatar-6 A b likely formed quiescently. The system's long tidal alignment timescale suggests that Qatar-6 A b formed within a protoplanetary disk that was primordially aligned with the Qatar-6 A host star, and the system was not dramatically altered from that point to push the system out of alignment.

The planet's short tidal circularization timescale prevents us from ruling out an initially eccentric orbit within the plane of that protoplanetary disk. The planet's current low-eccentricity orbit is consistent with either an initially circular orbit or a previously higher-eccentricity orbit that was not pushed to a high inclination relative to the stellar spin axis. A higher eccentricity could have been previously excited within the plane of the protoplanetary disk through planet–disk interactions (Goldreich & Sari 2003), planet–planet scattering (Rasio & Ford 1996; Chatterjee et al. 2008), or resonance crossings with a planetary companion (Chiang et al. 2003).

Lastly, the Kozai–Lidov timescale for the Qatar-6 system is comparable to or longer than the age of the system, depending on the true Qatar-6 AB stellar binary orbital properties. If the planet formed near its currently observed orbit, this indicates that the Kozai–Lidov mechanism has likely not played a major role in the evolutionary past of the Qatar-6 AB system. The possibility that the planet is currently undergoing Kozai–Lidov oscillations can also be ruled out based on the short timescale for general-relativistic-induced precession within the system. If Qatar-6 A b instead formed on a wider orbit and migrated inward over time, it is possible that the system may have experienced Kozai–Lidov oscillations in the past. To trigger Kozai–Lidov oscillations, however, the system would have needed a past mutual inclination of i_tot ≥ 392 between the planetary orbit and the binary star orbit.

The line-of-sight orbit–orbit alignment demonstrated in this work, together with the observed evidence for spin–orbit alignment, suggests a fully quiescent formation mechanism: if the system formed with no large mutual inclinations, the requirement of i_tot ≥ 392 to initialize Kozai–Lidov oscillations would have never been met, even in the case in which Qatar-6 A b began as a much wider orbiting planet that migrated inward over time. We conclude that Qatar-6 A b most likely reached its current orbit quiescently, either in situ or through disk migration.

There are a few potential avenues through which a binary star system could form, some of which are more or less likely to produce the observed line-of-sight orbit–orbit alignment. Three key mechanisms for binary star formation are dynamical capture, disk fragmentation, and turbulent fragmentation. In this section, we examine the likelihood of each of these scenarios in the context of the Qatar-6 AB binary system.

In the dynamical capture scenario, the binary companion would have been captured by the gravitational potential of the primary star and its protoplanetary disk (Tokovinin 2017). This formation mechanism produces a relatively high rate of highly misaligned systems, with the orientation of the final system set by the impact parameter of nearby passing stars. Consequently, it is unlikely that the observed line-of-sight orbit–orbit alignment would arise directly from dynamical capture. Furthermore, dynamical capture tends to produce much wider binaries (a > 10⁴ au) and requires a series of specific conditions that must be satisfied: (1) the presence of a third companion or a highly dissipative disk/envelope system to remove kinetic energy from the binary such that the binding energy of the system becomes negative, (2) a nearby star that falls into a specific range of appropriate relative velocities, and (3) an impact angle that is conducive to capture. Because of these conditions, dynamical capture is thought to be relatively rare in systems with a sub-solar-mass primary star (Clarke & Pringle 1991; Heller 1995; Moeckel & Bally 2007). Therefore, it is plausible but unlikely that the Qatar-6 AB system was produced by the capture of a companion star.

Alternatively, Qatar-6 AB may have formed through disk fragmentation, which can naturally produce binary star systems with primordially aligned protoplanetary disks. In this framework, a massive, gravitationally unstable circumprimary disk produces a stellar companion that inherits its angular momentum vector (Adams et al. 1989; Bonnell & Bate 1994). However, disk fragmentation is expected to produce relatively close-in companions (a ≲ 200 au; e.g., Krumholz et al. 2007; Tobin et al. 2016) with separation bounded by the extent of the circumprimary disk. A distant stellar companion with sky-projected separation s = 482 au would, accordingly, be unexpected for the relatively low mass (M_A = 0.829 M_⊙) Qatar-6 A host star within the disk fragmentation framework.

A third possibility is that the binary system formed through turbulent fragmentation (Offner et al. 2010, 2016; Lee et al. 2017), where a gravitationally unstable overdensity already in the contraction phase further fragments into two separate stars. While the turbulent environment can imprint potentially misaligned angular momentum vectors onto the stars, the overall angular momentum of the contracting overdensity is expected to preferentially produce relatively aligned systems with i_tot ≲ 45° (Bate 2018). In this case, the Qatar-6 A b planet could have quiescently formed within the relatively aligned protoplanetary disk, producing either a primordial orbit–orbit alignment or a relatively small primordial misalignment.

If a primordial misalignment existed between the circumprimary disk and the companion star's orbit, the binary potential would drive disk precession about the binary orbit normal (Bate et al. 2000; Batygin 2012; Zanazzi & Lai 2018). The gas-rich disk can dissipate the energy available from precession into heat, and in the process the disk can be pushed toward alignment with the binary orbit (Papaloizou & Terquem 1995; Bate et al. 2000; Lubow & Ogilvie 2000). As the associated timescale of this mechanism is well within the typical lifetime of protoplanetary disks for binary separations of order 500 au (Christian et al. 2022), dynamical alignment via energy dissipation during binary-driven disk precession could robustly explain the sky-projected inclination match between the binary orbit and transiting warm Jupiter orbit in the Qatar-6 system.

On the other hand, binary-driven disk precession has been invoked numerous times to explain observed spin–orbit misalignments (Batygin 2012). Indeed, the alignment scenario described above would, naively, leave the angular momentum vector of the star unchanged and thus, given the long tidal realignment timescale in Equation (14), produce a spin–orbit misalignment of order the initial binary–disk misalignment. Our constraint on the true spin–orbit angle $\psi ={21.82}_{-18.36}^{+8.86\circ}$ leaves room for a nonnegligible spin–orbit misalignment that could have resulted from a dynamical forcing of an initially misaligned protoplanetary disk into alignment with the binary orbit.

However, a true joint spin–orbit and orbit–orbit alignment could naturally arise from dynamical orbit–orbit alignment paired with additional gravitational and magnetic processes that occur over the disk lifetime. During the embedded phase of star formation, accretion and gravitational star–disk coupling can efficiently transfer angular momentum between the star and the disk, suppressing the production of large spin–orbit misalignments (Spalding et al. 2014). Furthermore, strong magnetic torques in young systems with relatively low mass stars (M_* ≲ 1.2 M_⊙) can push systems with primordial spin–orbit misalignments back to alignment within the protoplanetary disk's lifetime (Spalding & Batygin 2015). As a result, dynamical orbit–orbit alignment during the protoplanetary disk phase does not necessarily preclude spin–orbit alignment.