Hot Jupiters in binary star systems

Statistics from radial velocity planet searches (Marcy et al., 2005; Butler et al., 2006) show that the occurrence rate of giant planets within 0.1 AU (“hot-Jupiters”) is $\sim 1\%$; extrapolating to 20 AU the occurrence is $12\%$. There is a clear ”pile-up” of planets with orbital periods near 3 days (Fig. 1). Transit observations yield a similar fraction of hot Juptiers (Gould et al., 2006; Fressin et al., 2007). What migration mechanisms can produce the observed feature in semi-major axis distributions represented by hot Jupiters? In this article we focus on the mechanism known as Kozai migration.

Consider a planet circling a star that is a member of a binary system. The mutual torques between the binary and planetary orbits transfer angular momentum between the two while leaving the orbital energies nearly unchanged. For mutual inclinations $I\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}\hss}\hbox{$>$}}}40^{\circ}$ a resonance between the precession rate of the planet’s nodal and apsidal lines greatly enhances the effectiveness of this exchange of angular momentum, producing large oscillations in the planet’s angular momentum (Kozai cycles, Kozai, 1962). The planet eccentricity ($e_{p}$) and periapse ($r_{p}\equiv a_{p}(1-e_{p})$) oscillate with a characteristic timescale (Holman et al., 1997)

where $m_{*}$ and $m_{c}$ are the masses of the central and companion stars, while $P_{c}$ and $P_{p}$ are the periods of the binary and planetary orbits, respectively. The binary eccentricity is denoted by $e_{c}$. Holman et al. (1997) and Takeda & Rasio (2005), among others, have studied the role of these Kozai cycles in producing the eccentricities observed in known exo-planets.

For sufficiently large $I$, $r_{p}$ can reach very small values, allowing tidal dissipation to erode the orbit of the planet. Eggleton & Kiseleva-Eggleton (2001) were the first to propose that Kozai cycles, in combination with tidal friction, can shrink the orbit of a inner binary in a hierarchical triple system, leading to the formation of contact binaries. Wu & Murray (2003, hereafter WM03) have studied Kozai migration in application to exo-planets and found it to be the only plausible explanation for the migration of the planetary object HD80606b.

In the absence of any other modification of the gravitational potential, the minimum $r_{p}$ may fall below the Roche radius ($r_{R}$) and the planet may be destroyed. However, there are a number of competing torques that can limit the amount of angular momentum that the Kozai torque can extract from the orbit of the planet, including general relativistic (GR) corrections to Newtonian gravity, and torques associated with the extended mass distribution of both the primary star and the planet. The latter includes rotationally induced planetary oblateness, the tidal bulge raised by the star on the planet, the misalignment of this bulge produced by friction, and the stellar counterparts of all these. These torques can halt the Kozai-induced collapse in $r_{p}$ and promote planetary survival.

Which torque becomes competitive with the Kozai torque depends on the system; for systems with very large binary semi-major axis ($a_{c}$) and therefore very weak Kozai torque, the GR precession can halt the reduction of $r_{p}$ before tides become important. However, for tighter or more inclined binaries, tidal friction sets the minimum $r_{p}$. Since the tidal torques depend strongly on $r_{p}$, binary systems with a wide range of $a_{c}/a_{p}$ will be stalled at essentially the same $r_{p}$, leading to a pile-up of hot Jupiters at $a_{p}\sim 2r_{p}$ when the planet orbits are later circularized.

We quantify the effect of Kozai migration by considering an ensemble of binary systems following that in Takeda & Rasio (2005). These binaries are initially comprised of a solar-mass host star, a jupiter-mass planet ($m_{p}=M_{J}$) orbiting at $5$ AU with an eccentricity of $0.05$, and a binary companion of mass $0.23M_{\odot}$ – this is the peak of the observed mass ratio distribution in the solar neighbourhood (Duquennoy & Mayor, 1991). The distribution in binary separation ($P(a_{c})$) is assumed flat in logarithmic $a_{c}$ ($a_{c}$ ranging from $20$ to $20,000$ AU). We set $P(e_{b})=2e_{b}$, a thermal distribution often adopted in binary population synthesis. This latter choice hardly affects the results. The last ansatz, our most sensitive yet most uncertain assumption, takes $I$ to be isotropically distributed. Based on studies of stellar spin and binary orbits (Hale, 1994), this seems reasonable for $a_{c}>40$ AU, but may be less appropriate for tighter binaries; polarimetry studies of protostellar disks suggest that the circumstellar disk and the binary plane are correlated for $a_{c}$ up to a few hundred AU (Jensen et al., 2004; Monin et al., 2006). However, polarimetry estimates only the projected angle between the two planes, and is strongly plagued by interstellar polarization. The results should be taken with caution at present.

We produce an ensemble of $100,000$ systems. Out of these we select systems that can potentially perturb the planet to $r_{p}\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}0.1$ AU. To reach this distance, a planet starting at semi-major axis $a_{p}$ (with a small eccentricity) will have to attain $e_{\rm max}\geq 1-0.1/a_{p}$. Ignoring tidal dissipation,¹¹1Tidal dissipation increases the Kozai integral and slightly raises the minimum requirement on $I$ (WM03). the Kozai integral (the planet’s orbital angular momentum in the normal of the binary plane) $H_{K}=\sqrt{(1-e_{p}^{2})}\cos I={\rm constant}$. Taking a minimum $I\approx 40\deg$ during the Kozai cycles (see, e.g., Holman et al., 1997), this yields a minimum initial inclination required for producing hot Jupiters: $I\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}\hss}\hbox{$>$}}}81\deg$. This value is independent of the binary separation or mass. The fraction of isotropically inclined systems that have such a misalignment is $\sim 15\%$.

We then weed out planets that are likely dynamically unstable according to the following fitting formula

This expression is obtained by integrating the orbits of our initial system for $10^{4}$ binary orbits, taking $I=90\deg$. This non-coplanar stability limit is $15\%$ to $30\%$ more restrictive than the coplanar stability limit found by Holman & Wiegert (1999). It is used here as a rough proxy for systems that either eject their planets quickly after formation, or are unable to form planets due to the strong tidal influence of the companion. This proceedure eliminates many systems with $a_{c}<100$ AU; we are left with $\sim 10\%$ of the original ensemble that could potentially reach $<0.1$ AU, if they are not stalled by other torques at larger distances.

These remaining systems are integrated using secular equations obtained by averaging over the orbital motions of both the planet and the binary companion (Eggleton & Kiseleva-Eggleton, 2001). These equations include the effects of Kozai perturbation, tidal dissipation, GR precession, and tidal and rotational bulge precessions.²²2In this study, we rely exclusively on these secular equations. The actual dynamics may deviate due to short term noises and mean-motion perturbations and should be studied with N-body integration codes. We use a Runge-Kutta integrator with an adaptive step size set to keep the integration error below a preset limit. We follow the procedure described in WM03, which also lists values for the various parameters involved. In particular, we choose the initial stellar spin direction to be aligned with the initial orbit normal for the planet.

The integration is stopped after $5$ Gyrs have passed, or when $5$ million timesteps are exhausted, or when $r_{p}<2R_{\odot}$. The last condition roughly corresponds to the planet overflowing its Roche lobe; however, none of the planet in our simulation reached this state.³³3This is due to the strong dependence of the tidal timescale on $r_{p}$; tidal distortions act as a barrier, maintaining $r_{p}\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}\hss}\hbox{$>$}}}r_{R}$. The limit on the number of integration timesteps is usually reached if Kozai oscillations have been effectively halted by rapid tidal or other precessions; in that case the subsequent dynamical evolution of the planet simply reduces $e_{p}$. We then use a simplified code, including only the effects of tidal dissipation on the planet orbit and planet/stellar spins, to finish integrating to $5$ Gyrs.

We find that about $2.5\%$ of our ensemble eventually migrate inward of $0.1$ AU. The distribution of final semi-major axes is concentrated between $0.02$ AU and $0.05$ AU with a peak at $0.03$ AU. Our hot Jupiters exhibit a pile-up at $\sim 3$ day periods similar to the observed population (Fig. 1).

Given the same initial $I$, tighter binaries produce a closer-in hot Jupiter in a shorter amount of time. Many of the hot Jupiters are tidally ensnared on their first close approach to the host star (Fig. 3), with the Kozai period between $10^{4}$ to $10^{8}$ yrs. Tidal circularization of these orbits then takes upward of $10^{7}$ years.

The 3-day feature in the computed $a_{p}$ distribution appears wider than the observed distribution. However, as just noted, closer-in planets are migrated in earlier, so they still have larger radii and larger stalling peraipses. Experimenting with the following time evolution of planet radius,

with $\tau_{\rm shrink}$ taken to be $3\times 10^{7}\,\rm yrs$, we find that the 3-day bump narrows significantly (Fig. 2).