A radial limit on polar circumbinary orbits from general relativity

Many planets have been detected orbiting around binary star systems (e.g. Doyle et al., 2011; Orosz et al., 2012). All of the currently observed systems are in orbits that are close to coplanar to their host binary orbit. This may be representative of nature (Li et al., 2016; Martin et al., 2019) but it may be a result of selection effects since the Kepler binaries are all around very short orbital period binaries. Transits for even slightly misaligned circumbinary planets are less frequent than for circumstellar planets because of their nodal precession, however, the probability of a transit at some point is higher (Martin & Triaud, 2015; Martin, 2017). Eclipse timing variations may provide a way to detect highly misaligned planets in the future (Zhang & Fabrycky, 2019).

We expect that planets may form in misaligned orbits around wider binaries (with semi-major axis $\gtrsim 0.5\,\rm au$) since protoplanetary gas disks around these binaries may be more highly misaligned (Czekala et al., 2019). Observations show that misaligned circumbinary gas disks are common (e.g. Chiang & Murray-Clay, 2004; Capelo et al., 2012; Brinch et al., 2016). HD 98800 was the first circumbinary gas disk to be found in a polar configuration (Kennedy et al., 2019). In a polar alignment, a low mass disk (or planet) is misaligned by close to $90^{\circ}$ to the central binary orbit with its angular momentum aligned to the eccentricity vector of the binary. This is a stable configuration (Aly et al., 2015; Martin & Lubow, 2017; Lubow & Martin, 2018; Zanazzi & Lai, 2018; Cuello & Giuppone, 2019).

The dynamics of misaligned circumbinary planet orbits have been studied in depth both analytically and numerically (Verrier & Evans, 2009; Farago & Laskar, 2010; Doolin & Blundell, 2011; Zanardi et al., 2017; Chen et al., 2019). Around a circular orbit binary, the nodal precession is always around the binary angular momentum vector, we call these circulating orbits. However, around an eccentric binary, a sufficiently misaligned orbit can precess instead about the stationary inclination that is aligned with the binary eccentricity vector, we call these librating orbits. Most previous work has ignored the effects of general relativity that causes an additional apsidal precession of the binary orbit in the same direction as orbital motion (Naoz et al., 2017). Zanardi et al. (2018) examined the effects of GR on the orbit of an asteroid around a star with an inner eccentric orbit planet. Childs & Martin (2021) found that general relativity (GR) does not affect the dynamics of very close in (orbital radius $r<4\,\rm au$) terrestrial planets around a $0.5\,\rm au$ semi-major axis binary since the planet orbits precess with the binary and remain polar aligned.

In this work, for the first time, we consider how the additional precession due to GR affects the dynamics of circumbinary planet orbits in wide orbits around close binaries. Without GR, circumbinary orbital dynamics do not change much with the planet semi-major axis (e.g. Doolin & Blundell, 2011). However, in Section 2 we use $n$-body simulations to show that GR significantly alters the orbits of wider orbit planets relative to the binary orbit. We find that there is a critical radius outside of which the orbits are circulating for all initial inclinations. In Section 3 we show how this critical radius depends upon the the properties of the binary. In Section 4 we discuss the implications of our results for both the dynamics of circumbinary planets and the evolution of circumbinary disks.

The calculations for this paper were made using the REBOUND $n$-body code (Rein & Liu, 2012) and the REBOUNDx package extension to include GR effects using the gr_full package (Tamayo et al., 2020; Newhall et al., 1983). All simulations used the WHFAST integrator, a symplectic Wisdom-Holman integrator (Rein & Tamayo, 2015; Wisdom & Holman, 1991), but the critical points were checked with the IAS15, a 15th order Gauss-Radau integrator (Rein & Spiegel, 2015) and the SimulationArchive format was used to have a fully stored state of the system for analysis (Rein & Tamayo, 2017).

We analyse the particle orbits in the frame of the binary. The inclination is the angle between the direction of the angular momentum vector of the test particles orbit and the direction of the angular momentum vector of the binary given by

where $\hat{\bm{l}}_{\rm b}$ is a unit vector in the direction of the binary’s angular momentum and $\hat{\bm{l}}_{\rm p}$ a unit vector in the direction the test particles angular momentum would be if it were not zero mass. The nodal phase angle measured relative to the eccentricity vector of the binary is given by

(Chen et al., 2019, 2020) where $\hat{\bm{e}}_{\rm b}$ is the eccentricity vector of the binary.

Without GR, the stationary inclination is at $90^{\circ}$ to the binary orbit and is independent of the particle semi-major axis (see Section 2.1). With GR, there is a critical semi-major axis, $r_{\rm c}$, where the stationary inclination reaches $180^{\circ}$ and beyond this there are no librating orbits (see section 2.2). We found this radius numerically in Fig. 5 for varying binary eccentricity, semi-major axis, total binary mass, $M_{\rm b}$ and binary mass fraction $f_{\rm b}=m_{2}/M_{b}$.

We can also solve for the critical radius, $r_{\rm c}$, by setting the stationary inclination to $180^{\circ}$ in Equation 4 and solving for the critical semi-major axis to get

For some typical parameters we can write this as

where $F_{1}(f_{\rm b})$ and $F_{2}(e_{\rm b})$ are functions with a maximum value of one characterizing the effects of the mass fraction and eccentricity on the critical semi-major axis. They are given by

and

where the maximum value for these functions occur at $f_{\rm b}=0.5$ and $e_{\rm b}=\sqrt{3/8}\approx 0.6$, respectively. These analytic results are plotted also in Fig 5 along with the numerical simulations discussed above and they match very well.

The critical radius goes to infinity, matching the non-GR case as $a_{\rm b}$ goes to infinity and as $M_{\rm b}$ goes to zero. This can be explained as in both these limits, the binary precession rate goes to zero which matches the lack of precession in the non-GR case. As the $a_{\rm b}$ goes to zero or the $M_{\rm b}$ goes to infinity the precession rate goes to infinity and the critical semi-major axis goes to zero. The critical semi-major axis is not a strong function of either binary eccentricity or mass fraction. However, the critical semi-major axis goes to zero as the binary eccentricity goes to 1, where the precession rate also goes toward infinity, In the case of the mass fraction, the critical radius goes to zero as the mass fraction goes to zero, but this is because the binary is no longer a binary in this limit.

Finally, we examine the radius at which the stationary inclination reaches a particular angle. The stationary inclination varies from the non-GR case of $90^{\circ}$ to the limit of $180^{\circ}$ which it reaches at $r_{\rm c}$. The radius or semi-major axis where it passes through a particular angle is a fixed fraction of the critical semi-major axis. This fraction is given by

So for example the stationary inclination will be at $i_{s}=95^{\circ}$ at a radius of $F_{3}\approx 0.50$ of the critical radius and the stationary inclination will be at $i_{s}=110^{\circ}$ when the radius is at $F_{3}\approx 0.74$ of $r_{\rm c}$.

Circumbinary orbits around an eccentric binary undergo nodal precession. The precession may be centered on the binary angular momentum vector (circulating orbits) or the stationary inclination (librating orbits). In the absence of general relativity, the stationary inclination is always at $90^{\circ}$ for a test particle and aligned with the binary eccentricity vector. We have shown that the effects of general relativity increase the stationary inclination. For particles that are close to the binary, the stationary inclination remains close to $90^{\circ}$, however it increases with the particle semi-major axis. There is a critical radius outside of which there are no polar librating orbits, only circulating orbits. The critical radius is not very sensitive to the binary eccentricity but increases with the binary semi-major axis. The critical radius is in planet forming regions around close binaries with separation $a_{\rm b}\lesssim 1\,\rm au$.

We have not included the effect of the mass of the particle in our calculations. A massive third body causes a retrograde apsidal precession of the binary. If the particle is massive enough, it may effectively cancel out the prograde precession caused by general relativity. However, the work presented here does not change significantly if the outer test particle is replaced by a Jupiter mass planet (see also Chen et al., 2019; Martin & Lubow, 2019) and does not affect the orbit of the binary (Martin & Triaud, 2016). However, the effect of the mass may become important when considering a massive or radially extended circumbinary disk (see also Migaszewski & Goździewski, 2011).

The particle simulations are run with point masses and are valid for that limit. At small separations, but before the orbits are circularised, tidal forces and the spin of the stars cause precessions which may compete with the GR precession. The precession due to these other processes can be quite complicated and depends on the exact composition and structure of the two stars (Sterne, 1939; Sirotkin & Kim, 2009) or compact objects (Hamilton & Rafikov, 2021). A faster prograde precession would lead to a smaller critical radius for polar orbits to exist.

The dynamics of a circumbinary gas disk are qualitatively similar to those of test particles. Each ring of the disk wants to behave in the same way as a particle, however, communication between the rings leads to differing behaviour. For protoplanetary discs, the communication typically occurs in the bending waves regime (Papaloizou & Pringle, 1983). The disk remains in a coherent structure if the communication timescale (that is about half the sound speed) is shorter than the precession timescale (Papaloizou & Terquem, 1995; Larwood et al., 1996; Papaloizou et al., 1998). In this case, a given radius of the disc may display dynamically different behaviour to a particle at the same radius. However, circumbinary disks can be sufficiently radially extended that they are not in good communication and can undergo disk breaking (Nixon et al., 2013).

Another difference with the disk compared to the particles is that there is viscous dissipation. A misaligned circumbinary gas disc moves towards either coplanar or polar alignment (Papaloizou & Terquem, 1995; Nixon et al., 2013; Martin & Lubow, 2017, 2018). For a binary with a critical radius, $r_{\rm c}$, that is within a circumbinary disk, the evolution may depend on the radial extent of the disk and disk properties such as the sound speed and the viscosity. If the disc extends beyond $r_{\rm c}$, disc breaking may lead to an inner polar disc and an outer disc that is circulating and moving towards coplanar. It is interesting to note that the observed polar gas disc HD 98800 has a exterior companion that could truncate the circumbinary disc and prevent it from spreading to the critical radius that we have found (Martin et al., 2022). The semi-major axis of HD98800 is about $1\,\rm au$ (Kennedy et al., 2019) and this corresponds to a critical radius of about $r_{\rm c}=80\,\rm au$, much larger than the size of the observed disc.

These results have important implications for observations of misaligned circumbinary planets. Stationary polar planets and polar librating planets, can only be inside of the critical radius. Similarly, debris discs, that may typically extend to tens or hundreds of au, must be close to the binary to be polar-aligned. Note that the polar debris disc around 99 Herculis would not be affected by GR since the separation of the binary is about $16.5\,\rm au$ (Kennedy et al., 2012; Smallwood et al., 2020), much larger than those considered in this work. This suggests that we would not expect a solar system analogue with a polar close central binary. The inner planets in the solar system could be polar, but the outer planets and the Kuiper belt or debris disc are more likely to be coplanar if they are the remnants of a circumbinary gas disc.