Slowly rotating close binary stars in Cassini states

ABSTRACT

Recent asteroseismic measurements have revealed a small population of stars in close binaries, containing primaries with extremely slow rotation rates. Such stars defy the standard expectation of tidal synchronization in such systems, but they can potentially be explained if they are trapped in a spin-orbit equilibrium known as Cassini state 2 (CS2). This state is maintained by orbital precession due to an outer tertiary star, and it typically results in a very sub-synchronous rotation rate and high degree of spin-orbit misalignment. We examine how CS2 is affected by magnetic braking and different types of tidal dissipation. Magnetic braking results in a slower equilibrium rotation rate, while tidal dissipation via gravity waves can result in a slightly higher rotation rate than predicted by equilibrium tidal theory, and dissipation via inertial waves can result in much slower rotation rates. For seven binary systems with slowly rotating primaries, we predict the location of the outer tertiary predicted by the CS2 theory. In five of these systems, a tertiary companion has already been detected, although it is closer than expected in three of these, potentially indicating tidal dissipation via inertial waves. We also identify a few new candidate systems among a population of eclipsing binaries with rotation measurements via spot modulation.

1 INTRODUCTION

Stars in close binary systems undergo tidal dissipation, which usually brings the system to a low-energy state characterized by a circular orbit and the synchronization and alignment of each star’s spin. Indeed, stars in sufficiently short-period binaries (with orbital periods less than ∼3-10 d for main sequence stars) are usually observed to have circular orbits (Duquennoy & Mayor 1991; Meibom & Mathieu 2005; Meibom, Mathieu & Stassun 2006; Geller et al. 2009; Milliman et al. 2014; Leiner et al. 2015; Shporer et al. 2016; Windemuth et al. 2019; Justesen & Albrecht 2021), synchronized rotation rates (Giuricin, Mardirossian & Mezzetti 1984a, b; Claret & Cunha 1997; Lurie et al. 2017), and spins aligned with their orbits (Marcussen & Albrecht 2022). However, there are notable exceptions (discussed below), indicating more complex dynamics are sometimes at play.

The spin and orbital dynamics of stars in triple systems are much more complicated than two-body systems. While the orbital dynamics of such systems has been studied extensively (e.g. Naoz 2016), the spin evolution has received less attention. Storch, Anderson & Lai (2014) (see also Storch & Lai 2015; Anderson, Storch & Lai 2016; Storch, Lai & Anderson 2017) examined the spin evolution of hot Jupiter host stars during high-eccentricity migration, showing that spin-orbit precession combined with Kozai cycles and tidal circularization can produce a wide range of spin-orbit misalignment angles. Anderson, Lai & Storch (2017) demonstrated that spin-orbit misalignment can also be created by this process for stellar binaries.

Even for systems with circular orbits, complicated spin-orbit evolution can occur due to the interaction between spin precession of the inner primary and orbital precession of the inner binary induced by the external third body. Anderson & Lai (2018) showed how a resonance between the spin and orbital precession frequencies can excite the obliquity of exoplanet host stars. In a similar manner, spin-orbit misalignment can also be generated due to the precession induced by a protoplanetary disc (Spalding & Batygin 2014, 2015; Millholland & Batygin 2019; Su & Lai 2020; Anderson & Lai 2021).

Cassini states (Colombo 1966; Peale 1969; Ward 1975) are configurations in which the spin-orbit misalignment angle remains constant due to the combined action of spin and orbital precession. In many cases, there not one but two stable Cassini states, and systems undergoing any sort of dissipation (e.g. tidal dissipation) will become trapped in one of these states. The first Cassini state typically has nearly synchronous and aligned rotation, similar to the expectation in the absence of the tertiary companion. However, the second Cassini state typically exhibits large spin-orbit misalignment and a sub-synchronous stellar rotation rate.

In exoplanet systems, planets trapped in Cassini state 2 (CS2) would continue to experience tidal dissipation via obliquity tides, which could contribute to tidal heating and orbital decay of hot Jupiters (Winn & Holman 2005; Fabrycky, Johnson & Goodman 2007; Millholland & Laughlin 2018) and super-Earth systems (Millholland & Laughlin 2019; Millholland & Spalding 2020). Su & Lai (2021) studied this process in great detail, incorporating tidal dissipation torques into the dynamics to derive the modified Cassini states and capture probabilities. There is a substantial parameter space in which a star/planet can be captured into CS2 with large spin-orbit misalignment and slow rotation.

The same dynamics can occur in close stellar binaries, where stellar spin rates can be measured much more easily. Indeed, recent observations have uncovered a small but growing population of stars in short-period binaries with very slow stellar rotation rates and/or spin-orbit misalignment (Borkovits et al. 2015; Kallinger et al. 2017; Sowicka et al. 2017; Lampens et al. 2018; Fekel, Henry & Sowell 2019; Guo et al. 2019; Li et al. 2020; Zhang et al. 2020). These stars not only spin sub-synchronously, but also spin much slower than single stars of similar mass and temperature. We suggest that such systems may, in fact, be triple systems, with the spin of one (or both) stars in the inner binary trapped in CS2.

In this paper, we apply the pioneering analysis of Su & Lai (2021) to the case of stellar triples. We extend their work to include the effects of magnetic braking, which can modify or destabilize the Cassini states. We also derive results for a few different tidal dissipation mechanisms, which can similarly affect the spin rates and stability of the Cassini states. We analyse several of the stellar systems discussed above to determine whether they could be trapped in CS2. Several of these systems are already known to have tertiary companions that we show are indeed consistent with the Cassini state hypothesis. In the others, we predict the properties of a putative tertiary star. Finally, we examine the eclipsing binary catalogue of Lurie et al. (2017) to identify a few more candidates for CS2, although we show that many of the sub-synchronous rotators in their sample are false positives (FPs).

2 CASSINI EQUILIBRIA

2.1 Systems under consideration

We consider binary star systems where the rotation of the primary star can be measured. The semimajor axis of this star around the secondary star is a_in, the orbital frequency is n, the primary’s spin magnitude is Ω_s, and its spin direction is |$\hat{\boldsymbol{s}}$|⁠. The orbital angular momentum (AM) of the primary star is |$\hat{{\boldsymbol L}}_{\rm in}$|⁠, and the angle between the spin and orbital AM vectors is the obliquity θ. The masses of the primary and secondary stars are denoted M₁ and M₂, respectively, and the radius of the primary star is R. See Fuller & Felce (2023) for a diagram of the spin-orbital configuration.

We then consider a third star, of mass M_out, orbiting the inner binary system at a semimajor axis of a_out, with orbital AM |$\hat{{\boldsymbol L}}_{\rm out}$|⁠. The angle between |$\hat{{\boldsymbol L}}_{\rm out}$| and |$\hat{{\boldsymbol L}}_{\rm in}$| is denoted by I, the mutual orbital inclination.

2.2 Full equations and defining parameters of the system

We follow Su & Lai (2021), and assume that the spin AM, S, is much smaller than the orbital AM, L. The spin-orbit evolution equations are given by

where θ is the obliquity, and the precession rates α and g are defined via

Here, k = I_s/M₁R² is the primary’s gyration radius, and k₂ is the Love number.

Following Su & Lai (2021), using the weak theory of equilibrium tides, the alignment rate |$t_{al}^{-1}$| is related to the spin-up rate |$t_{\rm s}^{-1}$| via

In this case, the spin and orbital elements evolve as

where the tidal spin-up time is

The tidal Cassini equilibrium state has |$\dot{\Omega }_s = 0$|⁠, |$\dot{\theta }_s = 0$| and |$\dot{\phi }_s = 0$|⁠. This requires

and therefore Ω_s approaches zero as θ approaches |$\pi$|/2. The other requirement is more complicated and depends on g and α. Defining

Su & Lai (2021) show that the equilibrium of CS2 is

as long as η_sync ≪ 1. Hence, very slow spins can be achieved when η_sync ≪ 1.

3 MAGNETIC BRAKING

We now extend our analysis to determine the Cassini equilibria in the presence of both tides and magnetic braking. Magnetic braking slows a star’s spin and thus modifies the Cassini equilibria.

We model this effect by adding a frictional term to the star’s spin evolution equation,

Here, the constant t_mag is determined by the AM braking prescription (see e.g. El-Badry et al. 2022). The maximum (saturated) magnetic braking AM loss rate is (see also Sills, Pinsonneault & Terndrup 2000):

with |$\tau _1 \sim 10^{35}\, {\rm erg}$| and |$P_{\rm crit} \simeq 3 \, {\rm d}$| for Sun-like stars. This leads to a magnetic braking time-scale of t_mag ∼ 150 Myr for Sun-like stars at spin periods of 5 d.

For simplicity, we consider only saturated magnetic braking prescriptions where the spin-down torque is linearly proportional to Ω_s. Standard non-saturated braking descriptions have torques proportional to |$\Omega _{\rm s}^3$| and could exhibit different Cassini equilibria than those derived below.

Rewriting equation (8) to include magnetic braking yields

where r_mag is the ratio of the magnetic braking and tidal synchronization time-scales and is defined as

3.1 Condition for existence of CS1

There is a minimum value of r_mag below which CS1 does not exist. For CS1 to exist, we require that (e.g. Su & Lai 2021):

where η = −g/α = (n/Ω_s)η_sync, and η_crit is defined to be

This implies

Solving equation (16) for the equilibrium value of cos θ, we get

For this to be real, we need

giving a lower bound on r_mag for CS1 to exist:

where in the last approximation we have assumed η_sync ≪ 1.

3.2 Modified Cassini equilibrium

Solving equation (16) for equilibrium gives

Following appendix A of Su & Lai (2021), we consider slow tidal dissipation such that (|g|t_s)⁻¹ ≪ 1, such that CS2 has ϕ ≃ |$\pi$|⁠. In the limit of η ≪ 1, θ ≃ |$\pi$|/2 − ηcos I. Then solving equation (16) for Ω_{s, eq} again gives

and therefore:

In the presence of strong magnetic braking, the CS2 rotation rate is thus decreased by a factor of |$\sqrt{t_{\rm mag}/t_{\rm s}}$|⁠. This will be especially important in more widely separated systems where t_s is larger.

The CS2 equilibrium obliquity angle is still given by cos θ_eq  ≃ ηcos I, and hence

Stronger magnetic braking therefore corresponds to a smaller equilibrium obliquity. This can be understood because a smaller equilibrium rotation rate requires a smaller spin-orbit misalignment angle in order to maintain enough spin-orbit precession to maintain the equilibrium.

3.3 Numerical solutions with magnetic braking

Magnetic braking has the effect of moving the equilibrium Ω_s to lower values. The position of CS1 and CS2 as a function of r_mag, for different values of η_sync, is shown in Fig. 1. These are calculated by numerically solving for the stationary point of Ω_s using equation (16), in combination with stationarity of ϕ from equation (7):

Here, the ± sign refers to CS1 and CS2, which have ϕ ≃ 0 and |$\pi$| , respectively.

Cassini state equilibria with varying r_mag are shown in Fig. 1. For these calculations, we used I = 20° and η_sync = 0.01, 0.06 and 0.2 for the top, middle, and bottom panels, respectively. Both CS1 and CS2 are moved to smaller spin rotation rates as the strength of magnetic braking increases (decreasing r_mag ≡ t_mag/t_s). However, for CS1, increased magnetic braking increases the spin-orbit obliquity angle, whereas for CS2 this relationship is reversed. We have a lower bound on r_B from equation (23), below which CS1 no longer exists, for example, r_B ≥ 0.011 for the η_sync = 0.06 case.

We also investigate the time-dependent spin-orbit evolution of a few systems, as shown in Fig. 2. To make Fig. 2, we integrated equation (16) along with equations (6) and 7, using |g|t_s = 100, I = 20°, η_sync = 0.06, and various values of r_mag. The different coloured trajectories correspond to different initial conditions. Some trajectories are captured into CS2 and others go to CS1. Both stationary points are at slower spins due to magnetic braking.

4 DYNAMICAL TIDES

The weak tidal friction theory used above and in nearly all prior work assumes a tidal lag time that is the same for each component of the tidal forcing. This must be modified when dissipation occurs via dynamical tides, where the tidal dissipation is strongly dependent on the tidal forcing frequency. The relative synchronization and alignment torques are modified, along with the Cassini equilibria. Here we derive the modified Cassini equilibria for dissipation via internal gravity waves and inertial waves.

4.1 Gravity waves

To investigate possible implications of dynamical tides for tidal Cassini states, we begin with equation (27) of Lai (2012)

where |$T_0 = (3 \pi G/20)(M_2^2 R^5/a_{\rm in}^5)$|⁠, s = sin θ, and c = cos θ. Here, |$\tau _{m m^{\prime }}$| is the effective tidal lag time of each tidal component, whose frequency in the rotating frame of the star is |$\omega _{m m^{\prime }} = m^{\prime } n-m \Omega _s$|⁠. We are interested in CS2 where we expect cos θ ≪ 1 and Ω_s ≪ n. In this case, the tidal forcing frequency is nearly equal for ±m components, i.e. |$\omega _{m m^{\prime }} \simeq \omega _{-m m^{\prime }} \simeq m^{\prime } n$|⁠. We therefore assume τ₂₂ ≃ τ₋₂₂ ≃ τ₁₂ ≃ τ₋₁₂. Expanding to first order in cos θ and Ω_s, equation (29) becomes

We now consider different tidal dissipation mechanisms. For tidally excited gravity waves, the dissipation is much more efficient at high frequencies (Goodman & Dickson 1998), so we expect τ₂₂ ≫ τ₂₀, τ₁₀. In this case, the spin-equilibrium occurs when

Then at CS2 where

we have

The spin rate is thus a factor of |$\sqrt{2}$| larger than the weak tidal friction case.

4.2 Inertial waves

Here we follow the procedure of Fuller & Felce (in preparation) to examine the case of inertial wave dissipation. Following Lai (2012), we consider the dissipation via inertial waves due to the m = 1, m′ = 0 component of the tidal forcing, which is the only component that resonates with inertial waves when Ω_s ≪ n. Considering equilibrium tidal dissipation of tidal lag time τ, in addition to extra tidal dissipation via inertial waves via a factor X₁₀ = τ₁₀/τ − 1, the spin evolution equations are modified to

The equilibrium spin rate is modified to

As in the case of magnetic braking, this modification changes the shape of the stationary Ω_s line, as shown in Fig. 3. At high values of X₁₀, the stationary omega line becomes multivalued in θ and may cross CS1 or CS2 lines twice, allowing for multiple Cassini equilibria.

We can analytically solve for Ω_{s, eq} assuming cos θ_eq  ≪ 1 at CS2. Following appendix A of Su & Lai (2020), we set (|g|t_s)⁻¹ = 0, and expand equation (7) in powers of small η = η_sync/Ω, to get the CS2 equilibrium of ϕ = |$\pi$|⁠, θ = |$\pi$|/2 − ηcos I. Then using cos (⁠|$\pi$|/2 − ηcos I) ≃ sin (ηcos I) and rearranging equation (37) gives

Now using η = η_syncn/Ω_s to second order in η_sync we get

Rearranging for Ω_{s, eq} gives

and a corresponding misalignment angle

We see that when X₁₀η_synccos I/4 > 1, the CS2 equilibrium no longer exists, or at least it does not satisfy the assumption that cos θ_eq  ≪ 1. In this case, there can be a different equilibrium as discussed in Fuller & Felce (in preparation). Fig. 3 demonstrates this effect: At moderate values of X₁₀, the CS2 equilibrium remains near θ_eq  ≈ |$\pi$|/2, but the changing shape of the stationary Ω_s line causes θ_eq to decrease, eventually approaching θ_eq  ≈ I or θ_eq  ≈ |$\pi$| − I in the limit of very large X₁₀.

We compute the CS equilibria for inertial wave dissipation for different values of X₁₀. As before, we assume a large t_s and take ϕ = 0, |$\pi$|⁠. Then we numerically solve equations (7) and (35) in equilibrium. Fig. 4 shows results using η_sync = 0.06, and I = 20°. As X₁₀ increases, the location of CS1 and CS2 both move towards lower spin rates. The misalignment angle of CS1 increases, while it decreases for CS2.

Finally, including inertial wave dissipation will alter the minimum value of t_{s, c} below which CS2 is destabilized. As in Su & Lai (2021), this limit is obtained from equation (6), which for equilibrium tides yields

Including inertial waves and using equation (36), this is modified to

When X₁₀η_synccos I/4 ≪ 1, one can show that the new term in equation (43) is smaller than the second and can be ignored. In the limit X₁₀η_synccos I/4 ≫ 1, the equilibrium spin rate from Fuller & Felce (in preparation) is Ω_s ≃ 4n/(X₁₀ sin ²θ cos θ). Using this relation yields

Hence, the critical value of t_{s, c} is only slightly altered for systems in CS2 where sin θ is of the order of unity.

5 COMPARISON WITH OBSERVATIONS

In the following section, we examine several sub-synchronously rotating stars we have discovered in the literature. Additional such systems may also exist, so this list may be very incomplete. Based on the properties of each system we examine, we derive constraints for the mass and inclination of a putative third body, assuming that the systems are trapped in CS2. Since all of the systems we consider have intermediate-mass primary stars, we expect magnetic braking to have a small effect. This justifies the assumption that sub-synchronicity in these systems is better explained by their being in CS2 rather than in a magnetically braked CS1.

5.1 System properties

The system properties described in this section are summarized in Table 1. In what follows, M₁ is the mass of the brighter star in the inner binary, P_rot is its rotation period, P_orb is the orbital period of the inner binary, and P_out is the outer orbital period.

All systems have intermediate-mass (⁠|$1.4{-}3 \, {\rm M}_\odot$|⁠) primaries on the main sequence, with inner orbital periods in the range 2−14 d, most of which are nearly circular. Five of the systems have known tertiary companions with orbits of 0.4−9 yr, but the others may also have nearby tertiaries that have not been detected. Nearly all of these systems have pulsating primaries (γ-Doradus or SPB pulsations), and the very slow rotation period was measured via asteroseismology. Below, we describe key details of each system.

5.1.1 KIC 4480321

KIC 4 480 321 forms part of a known triple system, with the tertiary A-type star more luminous than either star in the inner binary (Lampens et al. 2018), which is composed of nearly equal-mass F-type stars whose temperatures lie in the range |$6300 \, {\rm K} \lesssim T_{\rm eff} \lesssim 6900 \, {\rm K}$|⁠. Unfortunately, this system is not eclipsing, and the outer star would contaminate a Gaia radius estimate, so the inner binary masses and radii are very uncertain. Based on stellar models in Fuller (2017), we estimate masses in the range |$1.3{-}1.8\, {\rm M_\odot}$| and radii in the range |$1.4{-}2.4\,R_\odot$|⁠, which are consistent with the observed temperatures and orbital velocities.

Lampens et al. (2018) also give a constraint on the inclination relative to the line of sight, i, of the system as 36° < i_in < 63°, and 42° < i_out < 57°. One of the inner binary stars is a γ-Doradus pulsator whose spin period was asteroseismically measured to be |$P_{\rm rot}=121^{+4.0}_{-4.0} \, {\rm d}$| by Li et al. (2020).

5.1.2 KIC 8197761

KIC 8 197 761 is a γ Doradus – δ Scuti hybrid pulsator in a binary system. We use the stellar masses, radius, orbital, and spin periods measured by Sowicka et al. (2017), who used RVs in combination with transit photometry. They also asteroseismically measured a rotation period of the primary to be |$P_{\rm rot} = 301.0^{+3.0}_{-3.0} \, {\rm d}$|⁠. There are no published limits on a possible third body in this system.

5.1.3 KIC 4142768

Unlike the other systems studied here, KIC 4 142 768 consists of two evolved A-type stars in an eccentric orbit with e = 0.582 ± 0.002. In addition to γ-Doradus pulsations, KIC 4 142 768 is a heartbeat star exhibiting tidally excited pulsations (Guo et al. 2019). Masses and radii were measured via RVs and light-curve modelling, and the asteroseismically measured spin period (Li et al. 2020) is extremely slow, |$P_{\rm rot} = 2702.7^{+1300.0}_{-662.0} \, {\rm d}$|⁠, though with large uncertainty.

5.1.4 KIC 8429450

KIC 8 429 450 is a γ-Doradus star in an eclipsing binary which is part of a triple system. We adopt the spectroscopically measured primary mass (Zhang et al. 2019), and the orbital period from Windemuth et al. (2019), who also measure a mass ratio q = 0.87. These properties are derived via photometric modelling and the observed spectral energy distribution of the system, combined with stellar evolution models. The rotation period is asteroseismically measured by Li et al. (2020): |$P_{\rm rot}=38^{+128}_{-17} \, {\rm d}$|⁠. The orbital period of the outer tertiary is roughly constrained by Borkovits et al. (2015) to be |$P_{\rm out}=3088^{+1700}_{-1700} \, {\rm d}$| based on eclipse timing measurements, the longest measured period in our sample. The projected motion of the inner binary along the line of sight, a₁₂sin i, is measured to be |$128^{+75}_{-75}\,{R}_{\odot }$|⁠.

5.1.5 HD 201433

HD 201 433 is the primary in a triple system containing a slowly pulsating B9V star, investigated in great detail by Kallinger et al. (2017). The primary mass of |$M_1=3.05^{+0.025}_{-0.025}$| is the highest in our sample and the asteroseismically measured rotation period is |$P_{\rm rot}=292^{+76}_{-76} \, {\rm d}$|⁠. This system also contains the shortest period tertiary in our sample, with |$P_{\rm out}=154.2^{+0.03}_{-0.03} \, {\rm d}$|⁠.

5.1.6 KIC 9850387

KIC 9 850 387 is a γ Doradus-δ Scuti star in an eclipsing binary (Zhang et al. 2020). Based on the gravity mode period spacing, the primary could be on the pre-main-sequence with an estimated age of only ≈7 Myr. However, this conclusion is not confirmed by the detailed binary and asteroseismic modelling of Sekaran et al. (2020, 2021), who accurately measure the masses and radii of the stars in the inner binary. The rotation period is estimated by Li et al. (2020) to be |$P_{\rm rot}=188.68^{+74.5}_{-41.6} \, {\rm d}$|⁠. The outer orbital period of the triple system is derived by Borkovits et al. (2015) using eclipse timing variations to be |$P_{\rm out}=671.0^{+2.0}_{-2.0} \, {\rm d}$|⁠, with |$a_{12} \sin {i} = 98^{+1}_{-1} \, R_\odot$|⁠.

5.1.7 HD 126516

Unlike all the stars above, HD 126 516 (Fekel et al. 2019) does not contain a pulsating primary star. It contains an F-type star in a circular eclipsing binary with an M-dwarf. A third body that is likely a K- or M-dwarf is detected via radial-velocity measurements of the inner F-star. The primary’s spin rate is |$v \sin i \simeq 4 \pm 1 \, {\rm km\,s^{-1}}$|⁠, which is about 10 times lower than the expected value for an aligned and synchronous rotation. We compute the corresponding spin period in Table 1 using the measured vsin i, radius, and an inclination distribution ranging from 30^o < i < 90^o.

5.2 Derived constraints

Using the measurements for each system, we now assume that the stars’ sub-synchronicity is due to the systems being captured in CS2, and predict the properties of the outer star. We first calculate the expected value of η_sync if the system is in CS2 (assuming η ≪ 1), for equilibrium tidal theory:

We then calculate the semimajor axis of the outer orbit via a re-arrangement of equation (11):

The projected radial velocity of the inner binary about the tertiary system’s centre of mass is

For the mutual inclination, we consider the most probable values, |$I = 60.0^{+27.1}_{-41.8}$|⁠, using the mean and a 90  per cent confidence interval for a uniform distribution. For k₂ and k, we interpolate in mass and radius between values given in Claret (2019). For the ratio 3k/k₂, we find KIC 4142768: 39.6, KIC 9850387: 31.4, KIC 4480321: 38.8, KIC 8429450: 40.3, KIC 8197761: 36.1, HD 201433: 31.0.

The results of these calculations are shown in Figs 5–11. We plot the expected outer orbital period (P_out), radial velocity of the inner binary about the system’s centre of mass (K₁₂), and angular separation of the outer tertiary, as a function of the mass of the tertiary star. For each case, shaded blue regions span the range of possible values using the uncertainty range of the measured quantities.

For the known triple systems, we can check consistency of our results with observational constraints. In Figs 5–11, the red bars/lines indicate the measured properties of the outer orbit, when available. We use the values shown in Table 1 to calculate these values. The outer tertiaries have all been measured via RVs or eclipse timing variations, and the latter translates directly RVs given the outer orbital period via

where a₁₂sin i is measured via eclipse timing.

The angular separation between the inner binary and outer tertiary can potentially be measured via direct imaging of the third body. The outer semimajor axis is related to the measured properties as

The corresponding angular separation is then computed given the distance to each system. In Figs 5–11, the red-shaded area indicates the possible range in angular separation computed via equation (49).

In addition, there is a minimum outer companion mass that can be computed from the mass function,

We assume a maximum tertiary mass of M₁ for all systems apart from KIC 4480321, where the bright tertiary dominates the spectrum and appears to be less than |$3.5 \, {\rm M}_\odot$| (Lampens et al. 2018).

For two of the systems (KIC 4480321 and KIC 9850387), the observed outer orbital period is shorter by a factor of an ∼10 compared to the most probable values. Similarly, the RVs of the outer orbits are larger than expected by a factor of a few, and the angular separations are smaller by a factor of several. This problem also exists and is most severe for HD 201433, whose observed orbital period is smaller by a factor of nearly 100 compared to what is expected.

This issue was also pointed out in Fuller & Felce (in preparation), though they did not perform detailed analyses of each system. The problem can be equivalently stated that the observed stellar rotation periods are slower than expected given the known locations of tertiaries in those systems. A long stellar rotation period entails a small value of η_sync (equation 45), and hence a large outer orbital semimajor axis (equation 46) and orbital period.

For HD 126516, the observed tertiary orbital period and inner binary RV are consistent with the predicted values. For KIC 8429450, the observed tertiary orbital period is roughly consistent (albeit slightly longer) with what is expected. For the systems without observational constraints on tertiaries, we can predict the range of expected outer orbital periods by assuming a companion mass of |$0.1{-}1 \rm {M}_{\rm \odot }$|⁠. This yields estimates of |$P_{\rm out} \sim 5 \times 10^3 {-} 5 \times 10^5$| d for KIC 4 142 768 and |$P_{\rm out} \sim 10^3{-}10^5$| d for KIC 8197761. However, we caution that the eccentric orbit of KIC 4 142 768 will change the spin and orbital precession rates, which should be factored into more accurate predictions.

5.3 Orbital trajectories and capture probability

Since KIC 4 480 321 is the triple with one of the best agreements between predicted and measured tertiary properties, we calculate examples of its prior orbital evolution. We use the properties in Table 1 and calculate η_sync using equation (45). We use equilibrium tidal theory with gt_s = 100. We then integrate the equations of motion: equations (6)–(8).

The results are shown in Fig. 12, for several different choices of initial conditions. For small values of η_sync, capture into CS2 requires an initial obliquity of around or above 90°. Using the analytic expression given in Su & Lai (2021), the probability capture into CS2 is

for an initial spin rate Ω_s,i. Taking an initial spin rate of Ω_s,i = 3n, corresponding to an initial spin period of |$P_{\rm spin} \sim 3 \, {\rm d}$|⁠, we arrive at probabilites between |$\sim 3{\,\rm and\,}10~{{\ \rm per\ cent}}$|⁠, depending on the mutual inclination of the orbits.

The other known triples in this sample also have small values of η_sync ranging from ∼10⁻³ − 1. hence the capture probability into CS2 is typically quite small. This may explain the rarity of such systems relative to those in CS1 which are nearly synchronized and aligned. Evolution into CS2 can occur through tidal dissipation alone, as in Fig. 12. However, other mechanisms such as changing stellar properties during early evolution (e.g. Anderson & Lai 2018), might also contribute to capture into tCS2.

6 INVESTIGATION OF SUB-SYNCHRONOUS SOLAR-TYPE BINARIES

While the systems discussed above are intermediate-mass stars, we also expect lower mass stars to be able to enter CS2. Lurie et al. (2017) investigate 816 EBs composed primarily of FGK-type main sequence star primaries, using star-spot modulations to deduce the rotation period of the primary star in each system. Among short-period systems (⁠|$P \lesssim 10 \, {\rm d}$|⁠), most stars are nearly tidally synchronized. They also find a population of binaries with slightly sub-synchronous rotation rates (on average, 13 per cent slower than their orbital period). In addition, their sample contains a population of potentially very sub-synchronous rotators (25 out of 816 systems) in short-period orbits (P <10 d) with a rotation rate less than half of the orbital frequency. We investigate these systems to determine whether this sub-synchronicity can be explained via the CS2 equilibrium.

When examining the 25 very sub-synchronous systems, we noticed that nearly all of them lie near the bottom end of the distribution of primary eclipse depths, i.e. they have shallow transits. This is unlikely to occur by chance, and is shown in Fig. 13. Since shallow transits could indicate an eclipsing planet rather than a binary star system, we cross-referenced each system with the NASA exoplanet catalogue (KOI). Of the 25 systems, 24 are listed in the NASA exoplanet FP list. Of these, four are listed as not examined, and four are listed as possible planets. These systems could be genuine planetary transits, in which case the tidal spin-up of the host star occurs on long time-scales due to the planet’s low relative mass. Rather than being in CS2, the star’s slow rotation is most likely due to a lack of tidal spin-up.

Out of the 16 certified FPs, 11 are categorized as likely to be eclipsing binaries by the Kepler False Positive Working Group (Bryson et al. 2017). In these cases, the shallow observed eclipse depths can be explained if these systems are blended eclipsing binaries, where a background eclipsing binary is on the same pixel as a bright foreground star, or a bright tertiary companion. The measured spin rate would most likely be that of the foreground star, rather than either star in the binary.

Using the FP probabilities listed (see Montet et al. 2015), we compare the probabilities of different binary scenarios, listed in Table 2. Un-blended binaries (UEB) describe the systems we are looking for, where the observed eclipse is that of the primary star, with shallow transits produced by grazing eclipses. Blended binaries (BEB) are systems where the primary star (which produces the observed spot modulation) is not in a binary system, but is bleneded with light from a foreground/background eclipsing binary. Hierarchical-triple eclipsing binaries are denoted by HEB. Another possible scenario is a double period system (dbl), where similar primary and secondary eclipses are mistaken for two primary eclipses, resulting in the misclassification of the system as a low-mass ratio system with half the actual period. FP denotes total probability that the system represents an FP, i.e. some type of stellar binary rather than a planetary transit. We look for systems where the total probability to be either a UEB or a hierarchical triple is higher than the total probability to be a BEB. This search yields only six systems, which are shown in bold in Table 2.

KIC 8 938 628 has a high chance of being either a UEB or a hierarchical triple system, while KIC 3 848 972 and KIC 3 443 790 have high chances of being UEB. These three systems are good candidates for being in CS2. Two systems are most likely double period eclipsing binaries (KIC 4 456 622 and KIC 10613718). KIC 4 456 622 would not be a candidate example of a tidal Cassini equilibrium, since its corrected orbital period would not satisfy our criterion for very sub-synchronous rotation. However, KIC 10 613 718 is highly sub-synchronous and would still qualify. KIC 11 560 037 could be in CS2, but it has a substantial chance of being a planetary transit (low FP).

7 DISCUSSION

7.1 Slow observed rotation rates

We have identified several short-period stellar binaries (Table 1) with a very slowly rotating primary star, which are good candidates to be trapped in the sub-synchronous CS2 state. Previously, Fuller (2020) postulated that an ‘inverse tidal’ process could produce either very slowly rotating or very rapidly rotating stars in close binaries containing pulsating primaries. We believe this hypothesis is less likely than the CS2 hypothesis because (a) one of the observed systems and the new candidates from Section 6 do not contain pulsating stars (b) five out of seven of the systems in Table 2 are already known to have tertiaries at about the right separation, (c) we are not aware of a population of very rapidly rotating stars in close binaries as predicted by the inverse tides theory, and (d) the inverse tidal process can only act over a narrow range of binary orbital period, whereas the observed systems span a range of 2–14 d.

However, there are problems with the CS2 hypothesis. Three of the observed systems have primaries that rotate slower than predicted by the weak tides assumption, suggesting additional physical mechanisms may be at work. One possibility we have explored is magnetic braking, which slows the CS2 equilibrium rotation period by a factor of (1 + t_s/t_mag)^−1/2 (equation 26). However, this is unlikely to be the explanation because all but one of the systems in Table 1 are hot stars (⁠|$T_{\rm eff} \gtrsim 6500 \, {\rm K}$|⁠) that are γ-Doradus or SPB pulsators, which typically rotate rapidly and are not magnetically braked.

Even if magnetic braking does act, from equation (15) we have a minimum magnetic braking time-scale of t_mag ∼ 150 Myr for Sun-like stars at spin periods of 5 d. Evaluating equation (9), the tidal spin-up time-scale is

for Sun-like stars with equal mass companions. Hence, we expect t_mag < t_s only for relatively long-period systems (P ≳ 5 d). Using a non-saturated braking prescription with |$t_{\rm mag} \propto P_{\rm s}^2$| would further reduce the importance of magnetic braking at long spin periods, so this effect is likely unimportant for the stars in our sample.

We showed that dissipation via gravity waves slightly increases the expected rotation rate in CS2, so this cannot explain the very slow rotation rates. In contrast, dissipation via inertial waves can substantially reduce the expected rotation rate (see also Fuller & Felce, in preparation). The problem with this explanation is that the stars in our sample are hot stars with extremely thin surface convective zones, decreasing the importance of inertial wave dissipation in the envelope. These stars do have convective cores, which may help contribute to inertial wave dissipation, which should be studied in future work.

7.2 Lack of low-mass stars in Cassini states

As shown in Section 6, there are only ∼4–6 good candidates for stars in CS2 in the sample of Lurie et al. (2017). It is remarkable to find only a few candidates among the several hundred short-period binaries of that sample. For comparison, roughly 4 out of the 35 γ-Doradus systems in close binaries in Li et al. (2020) have very sub-synchronous rotation indicative of being in CS2. Here we explore possible explanations.

One possibility is that Sun-like stars experience more efficient tidal dissipation that accelerates orbital decay. As discussed in Fuller & Felce (in preparation), systems typically only remain in CS2 for a fraction of their main sequence lifetime, because ongoing tidal dissipation shrinks the orbit until CS2 destabilizes and the system transitions to the standard CS1 equilibrium with nearly synchronous rotation. Indeed, many recent works have investigated the eccentricity or obliquity of stellar binaries and exoplanets (Winn et al. 2010; Van Eylen, Winn & Albrecht 2016; Bashi, Mazeh & Faigler 2023), finding that tidal dissipation is more efficient in cool stars (⁠|$T \lesssim 6200 \, {\rm K}$|⁠) relative to hot stars. In this case, cool stars would be caught in CS2 for a smaller fraction of their lifetimes, making it less likely to observe them in this state.

Another possibility is that there are selection effects in these samples that prevent us from finding cooler stars in CS2. Since rotation periods in Lurie et al. (2017) are measured via spot modulation, and spot modulation becomes weaker for slowly rotating stars (Cao & Pinsonneault 2022), it may be difficult to measure the rotation periods of very slowly rotating stars in CS2. It would be worthwhile to further investigate those systems that do not show clear spot modulation, for example, with spectroscopic measurements of vsin i to constrain the stellar rotation rate. Indeed, this is how HD 126 516 (Fekel et al. 2019) was discovered, and so future spectroscopic measurements of short-period binaries could yield many more CS2 candidates.

It could also be possible that cooler stars are less likely to have tertiary companions capable of locking the inner system in CS2. We disfavor this possibility because it is known that the tertiary fraction of short-period binaries is high (Tokovinin et al. 2006), even for cool stars. Moreover, it seems unlikely this effect could create a large difference between the |$\sim \! 1 \, {\rm M}_\odot$| stars of Lurie et al. (2017) and the |$\sim \! 1.5 \, {\rm M}_\odot$| stars of Li et al. (2020), since multiplicity fractions only vary slightly over that mass range (Moe & Di Stefano 2017).

7.3 Secondary star properties

While our analysis has centred around the rotation rate of primary stars, it is worth commenting on some properties of the secondary star.

First, although the spin rotation rate of the secondary star has usually not been measured, it is possible for the secondary star to have also been captured into CS2. Since, to first approximation, the stars within the binary have uncoupled spin-orbit dynamics, their spin rates should be independent. This would allow either or both stars to be caught in CS2.

Secondly, we consider the effect of a very low-mass secondary on the system dynamics. Since the tidal synchronization time is inversely proportional to the mass of the secondary (equation 9), systems with low-mass secondary stars are less likely to reach CS2 within their lifetime. In addition, all of our analysis relies on the assumption that the inner binary orbital AM is larger than that of either star’s spin. This requires

Taking the limit M₁ ≫ M₂, and using Ω_s = Ω_orb, we find

For typical values |$P_{\rm orb} = 3 \, {\rm d}$|⁠, R₁ = 2R_⊙, M₁ = 1 M_⊙, and k = 0.1, we find M₂ ≥ 0.005 M_⊙. This suggests that the mechanism discussed above would not be applicable to systems with planetary-mass secondaries, but it could still be applicable to those with brown dwarf secondaries.

8 CONCLUSION

In this work, we investigated the possibility that slowly rotating stars in close binaries result from capture into CS2, an equilibrium state caused by orbital precession from a tertiary star. In CS2, the primary star can have a very slow rotation and highly misaligned rotation axis relative to the inner binary orbit. We examined the behaviour of Cassini states incorporating the effects of magnetic braking and dynamical tides. We build on the results from Su & Lai (2021) and find that magnetic braking reduces the spin rate for both tidal Cassini states 1 and 2, and moves the obliquities of these two equilibria closer to each other. At a critical value of the magnetic braking strength, tidal Cassini State 1 is destabilized, but this is unlikely to be realized based on estimates of tidal spin-up and magnetic braking time-scales.

We also considered the effect of gravity and inertial waves on the CS2 equilibria, finding slightly increased rotation rates for gravity wave dissipation, and potentially greatly decreased rotation rates when accounting for inertial wave dissipation. Large amounts of inertial wave dissipation can shift the CS2 spin-orbit misalignment away from θ ≃ |$\pi$|/2 and towards θ = I or θ = |$\pi$| − I, where I is the inclination of the outer tertiary relative to the inner orbit.

Comparing to observed systems shown in Table 1, we find that many have longer outer orbital periods than expected from equilibrium tidal theory (or alternatively, they have longer stellar rotation periods than expected given the observed outer orbital period). Since magnetic braking is likely unimportant for those systems which contain hot primaries (⁠|$T_{\rm eff} \gtrsim 6500 \, {\rm K}$|⁠), dissipation via inertial waves is our favored explanation. However, these stars have convective cores and only very thin convective envelopes, so it is not clear that inertial wave dissipation can be effective. We recommend further investigation into the effects of inertial waves in these kinds of systems.

We also search the eclipsing binary catalogue of Lurie et al. (2017) for systems with highly sub-synchronous rotation, finding 25 candidates. However, closer inspection of these objects revealed that most have very shallow eclipse depths, entailing they likely contain blended systems or planetary companions. We found only four good candidates to be in CS2, a remarkably small number relative to the sub-synchronous rotators in the asteroseismic catalogue of Li et al. (2020). It is possible that more efficient tidal dissipation in Sun-like stars causes their orbits to quickly decay to short periods where CS2 is destabilized, making us less likely to observe them in CS2.

The CS2 hypothesis can be tested by searching for tertiary companions, which are expected to have orbital periods of ∼1–100 yr. Figs 5–11 show the predicted tertiary orbital properties as a function of tertiary mass for systems suspected to be in CS2. These types of tertiaries can be detected through either long-term RV measurements, or high-resolution direct imaging. We also recommend future spectroscopic vsin i measurements of stars in short-period binaries to search for slow rotators that could be in CS2.

ACKNOWLEDGEMENTS

We thank Yubo Su for useful discussions and feedback on this work.

This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

9 DATA AVAILABILITY

The orbital integration code, and plotting scripts to make Figs 1–13, are available upon request.

References