WARM JUPITERS NEED CLOSE "FRIENDS" FOR HIGH-ECCENTRICITY MIGRATION—A STRINGENT UPPER LIMIT ON THE PERTURBER'S SEPARATION

The origin of warm Jupiters (gas giants with period 10 days < P < 100 days) presents a similar puzzle to that of hot Jupiters (P ≲ 10 days)—neither population can form in situ according to popular theories of planet formation—yet much less attention has been paid to the former.

Rossiter–Mclaughlin measurements reveal that a considerable fraction of transiting hot Jupiters have orbits misaligned with host star spin axes (e.g., Winn et al. 2010; Triaud et al. 2010), which provide indirect support to high-eccentricity migration mechanisms (Rasio & Ford 1996; Wu & Murray 2003; Fabrycky & Tremaine 2007; Wu & Lithwick 2011; Socrates et al. 2012b). These high-e mechanisms involve the initial excitation of hot Jupiter progenitors at a few AU to very high eccentricity due to gravitational perturbations by additional objects in the system. The excitation is then followed by successive close pericenter passages (r_p ≲ 0.05 AU) that drain the orbital energy via tidal dissipation. The hot Jupiter progenitors eventually become hot Jupiters at a < 0.1 AU.

The majority of known warm Jupiters are sufficiently distant from their hosts (a_f = a(1 − e²) > 0.1 AU) to forbid efficient tidal dissipation, due to the strong distance dependence of tidal effects. However, if the orbital eccentricity of a warm Jupiter is experiencing Kozai–Lidov oscillations due to an external perturber (Holman et al. 1997; Takeda & Rasio 2005), then it may be presently at the low-e stage in the cycle and over a secular timescale, reach an eccentricity high enough for tidal dissipation to cause significant migration (e.g., Wu & Lithwick 2011). A schematic illustration of such a high-e migration scenario is shown in Figure 1 (red solid line). Warm Jupiters detected by radial velocity (RV) are shown in dots in Figure 1 within the black dashed lines. We define Jovian planets to have minimum mass M_psin i > 0.3 M_Jup and set an upper limit in the semimajor axis of 0.5 AU for warm Jupiters. This upper bound is well below the theoretical "snow line" of in situ core-accretion formation at about 2.5–3 AU for solar-type stars (e.g., Kennedy & Kenyon 2008) and the observed "jump" in the a distribution of giant planets at ∼1 AU (e.g., Wright et al. 2009). In planet–planet scattering, a Jupiter can migrate without tidal dissipation by a factor of ∼2 if another Jupiter is ejected (Rasio & Ford 1996), and our upper bound in distance is set to disfavor such a process.

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We discuss a stringent observational constraint on warm Jupiter formation via high-e migration—they must be accompanied by close, easily observable perturbers. These close perturbers are strong enough to overcome the precession caused by general relativity (GR) to reach close enough periapses for effective tidal dissipation within Kozai–Lidov cycles. In contrast, high-e migration for hot Jupiters is not subject to such a stringent constraint on perturbers. Hot Jupiter progenitors can be excited to close periapses at their initial, relatively large semimajor axes with distant perturbers, and throughout the subsequent migration, their periapses may keep close enough for tidal dissipation. Hot Jupiters formed by high-e migration can thus have distant perturbers that are difficult to detect.

We derive below a lower limit on the perturber strength for warm-Jupiter in high-e migration due to tidal dissipation. We adopt a conservative criterion that tidal dissipation may operate when a Jovian planet reaches a_f = a(1 − e²) < 0.1 AU. Observationally, the eccentricities of Jovian planets circularize at a_f ∼ 0.06 AU (e.g., Socrates et al. 2012a). Given that tidal dissipation has a strong dependence on planet–star separation, it is safe to assume that tidal dissipation ceases to be efficient when a_f > a_{f, crit} = 0.1 AU. We stress that the criterion presented below is a necessary but not an adequate condition for high-e migration. Without satisfying the criterion, the migration cannot occur, while fulfilling this requirement does not guarantee migration.

Consider a warm Jupiter with mass M_p at semimajor axis a and eccentricity e₀ orbiting a star with mass M accompanied by a perturber of mass M_per at a_per and e_per. The criterion is to require the warm Jupiter to reach a(1 − e²) < a_{f, crit} = 0.1 AU during Kozai–Lidov oscillation (see Figure 2 for an example). At a given a, the amplitude of Kozai–Lidov oscillation in eccentricity is limited by sources of precession other than those induced by the perturber and is insensitive to tidal dissipation. At a_f ∼ 0.1 AU, the precessions due to tides and the rotating bulge of the host are negligible compared to GR for typical hosts. Below we consider the Kozai–Lidov oscillation at the warm Jupiter's current a due to the gravitational perturbation and GR precession. We ignore tidal dissipation and precession.

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An analytical constraint is derived under the simplest assumptions: (1) quadrupole approximation in perturbing potential, (2) the warm Jupiter treated as test particle, (3) and the equation of motion is averaged over outer and inner orbits ("double-averaging"). We show below with numerical simulations that these are excellent approximations in deriving this constraint. Under these approximations, the following is a constant (e.g., Fabrycky & Tremaine 2007):

$$\begin{equation} e^2(2-5\sin ^2i\sin ^2\omega)+\frac{\epsilon _{{\rm GR}}}{\sqrt{1-e^2}}={\rm const}, \end{equation} \tag{ 1 }$$

where

$$\begin{eqnarray} \epsilon _{{\rm GR}} & = \frac{8{\it GM}^2b_{{\rm per}}^3}{c^2a^4M_{{\rm per}}} \cr &\approx 1.3 \left(\frac{M}{M_\odot } \right)^2 \left (\frac{a}{0.2 \,\rm AU} \right)^{-4} \left(\frac{M_{{\rm per}}}{M_\odot } \right)^{-1} \left(\frac{b_{{\rm per}}}{30 \,\rm AU} \right)^{3} \cr &\approx 1.4 \left(\frac{M}{M_\odot } \right)^2 \left (\frac{a}{0.2 \,\rm AU} \right)^{-4} \left(\frac{M_{{\rm per}}}{M_{\rm Jup}} \right)^{-1} \left(\frac{b_{{\rm per}}}{3 \,\rm AU} \right)^{3} \end{eqnarray} \tag{ 2 }$$

represents the relative strength of GR compared to the perturber, i is the planet–perturber mutual inclination, ω is the planet's argument of pericenter, and $b_{{\rm per}} = a_{{\rm per}}(1-e_{{\rm per}}^2)^{1/2}$ is the perturber's semiminor axis.

From Equation (1), to reach an eccentricity e from e₀, the following criterion must be satisfied regardless of the values of i and ω:

$$\begin{equation} \epsilon _{{\rm GR}} \left(\frac{1}{\sqrt{1-e^2}}- \frac{1}{\sqrt{1-e_0^2}} \right) < 2e_0^2 + 3e^2. \end{equation} \tag{ 3 }$$

We then put a lower limit on the "strength" of the perturber to reach a(1 − e²) < a_{f, crit} (and an upper limit on the separation ratio between the perturber and the warm Jupiter),

$$\begin{eqnarray} \frac{b_{\rm per}}{a} &<& \left(\frac{8{\it GM}}{c^2a}\right)^{-1/3} \left(\frac{M}{M_{\rm per}}\right)^{-1/3} \nonumber\\ &\times& \left[2e_0^2+3\left(1-\frac{a_{\rm f,crit}}{a}\right)\right]^{1/3} \left(\sqrt{\frac{a}{a_{\rm f,crit}}}-\frac{1}{\sqrt{1-e_0^2}}\right)^{-1/3}. \nonumber\\ \end{eqnarray} \tag{ 4 }$$

Figure 3 shows the constraints on b_per for M_per = M_☉ and M_per = M_Jup in the upper and lower panels, respectively, derived from Equation (4) for a_{f, crit} = 0.1 AU. The blue lines from above to below are for e₀ = 0.5, 0.3, and 0.0, respectively (e₀ = 0.3 in dashed lines while others in solid lines).

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Recently, it was realized that corrections due to various approximations above may lead to significant effects in several scenarios (e.g., Ford et al. 2000; Naoz et al. 2011; Katz et al. 2011b; Lithwick & Naoz 2011; Katz & Dong 2012).

We show that the analytic constraint given by Equation (4) are not affected by the inaccuracies of the adopted approximations using numerical integrations without these approximations. The effects of the quadrupole and test particle approximations are studied by performing 20,000 simulations (10,000 for M_per = M_☉ and 10,000 for M_per = M_Jup). These simulations employ the double averaged approximation but include the octupole term and are not restricted to the test particle approximation. The warm Jupiters have e₀ = 0.3 and a uniformly distributed (randomly) between 0.15 and 0.5 AU. The eccentricities of the outer perturbers are uniformly distributed within 0–0.5. The ratio b_per/a are uniformly distributed within 100–300 (10–30) for solar-mass (Jupiter-mass) perturbers. The orbital orientations of the outer and inner orbits are randomly distributed isotropically. All runs were integrated to 5 Gyr. The results are shown in Figure 3. The integrations in which the warm Jupiter reaches a(1 − e²) < a_{f, crit} = 0.1 AU are plotted as red dots and others in black. The analytical constraint given by Equation (4) for the appropriate eccentricity, e₀ = 0.3 (dashed blue lines), accurately traces the border of required perturbers for achieving the required eccentricity. Note that at the limit given in Equation (4), the strength of the octupole, ${\sim} {\epsilon} _{\rm oct} = a/a_{\rm per}[e_{{\rm per}}/(1-e_{{\rm per}}^2)] \lesssim 1/10 (M_{\rm per}/M_{\rm Jup})^{-1/3}$, is negligible compared to GR, _GR ∼ 1. While a small octupole can change the orbital inclination and lead to Kozai cycles with growing eccentricities, the eccentricity cannot surpass the limit Equation (4), which is the maximal value for all mutual orientations.⁵

For the scenarios considered, the Kozai–Lidov timescale is much longer than the outer (and inner) orbital timescales, justifying the double averaging assumption. To illustrate this, the results of a direct three-body integration are compared to those of a double-averaging integration in Figure 2. The considered warm Jupiter is at a = 0.3 AU and e₀ = 0.3 and has a solar-mass perturber corresponding to the limit derived from Equation (4) with e_per = 0.5 and b_per = 68.8 AU. The initial inclination is at 90°. The results of two integrations are practically indistinguishable (black dashed: double-averaging; red solid: direct three-body), validating the double-averaging approximation. The approximation is even better for a Jupiter-mass perturber satisfying the same constraint. This is because it has a shorter period and a similar Kozai–Lidov timescale.

In the limit of a ≫ a_{f, crit} and e₀ → 0, the following useful approximation can be obtained using Equation (4):

$$\begin{eqnarray} \frac{b_{\rm per}}{a} &<& \left(\frac{8{\it GM}}{3c^2 \sqrt{a a_{\rm f,crit}}}\right)^{-1/3} \left(\frac{M}{M_{\rm per}}\right)^{-1/3} \nonumber\\ &\approx& 175 \left(\frac{M_{\rm per}}{M_\odot }\right)^{1/3} \left(\frac{M}{M_\odot }\right)^{-2/3} \left(\frac{a}{0.2\,\rm AU}\right)^{1/6} \left(\frac{a_{\rm f,crit}}{0.1\,\rm AU}\right)^{1/6} \nonumber\\ &\approx & 17 \left(\frac{M_{\rm per}}{M_{\rm Jup}}\right)^{1/3} \left(\frac{M}{M_\odot }\right)^{-2/3} \left(\frac{a}{0.2\,\rm AU}\right)^{1/6} \left(\frac{a_{\rm f,crit}}{0.1\,\rm AU}\right)^{1/6}. \nonumber\\ \end{eqnarray} \tag{ 5 }$$

We stress that this approximation should be used for order-of-magnitude estimates since it is only accurate in the limit e ∼ 0 and a ≫ a_{f, crit} = 0.1 AU.

For warm Jupiters with a ∼ 0.1–0.5 AU with Jovian-planet perturbers, this constraint leads to an upper limit in orbital separation of ∼1.5–10 AU (period 2–30 yr). The RV semiamplitude is ≳ 10 m s⁻¹, accessible to available high-precision RV instruments. The perturbers at the high end in period range (∼20–30 yr) are more challenging since they may not have completed the full orbits yet during the monitoring projects. While for more massive perturbers, the upper limit in orbital separation implies much longer periods (P_per∝M_per^1/2), they can generally be identified from the easily detectable RV linear trends,

$$\begin{eqnarray} |\dot{v_\perp }| &=& \left|\frac{{\it GM}_{\rm per}}{r_{\rm per}^2} \sin i_{\rm per}\sin \theta _{\rm per} \right| \nonumber\\ &\sim & 200\; {\rm m\, s^{-1}\, yr^{-1}} \left(\frac{M_{{\rm per}}}{1\; M_\odot } \right) \left(\frac{r_{{\rm per}}}{30 \,\rm AU} \right)^{-2} \nonumber\\ &\sim & 20\; {\rm m\, s^{-1} \,yr^{-1}} \left(\frac{M_{{\rm per}}}{1\; M_J} \right) \left(\frac{r_{{\rm per}}}{3 \,\rm AU} \right)^{-2}, \end{eqnarray} \tag{ 6 }$$

where i_per, θ_per, and r_per are the inclination, position angle with respect to the line of the node, and orbital separation of the perturber, respectively.

In RV surveys, close binaries are commonly excluded to avoid contamination of the spectra, making the bias for estimating stellar perturbers challenging. We focus on planetary perturbers. There are 34 warm Jupiters discovered with RV at a_f > 0.1 AU and a < 0.5 AU listed in the exoplanets.org database (Wright et al. 2011). Ten of them have additional Jovian planets at longer orbits (see Figure 1, in which planets with outer Jovian companions are plotted in cyan). All perturbers satisfy the constraint in Equation (4). Figure 4 shows the eccentricity distribution for all warm Jupiters (blue dashed) and for those with external Jovian perturbers (red solid). The fraction of warm Jupiters with detected Jovian perturbers appears to be a growing function of their eccentricities. This trend appears to be significant in spite of the uncertainties due to the small number statistics. If true, there are two interesting implications: (1) there seems to exist a connection between eccentricity and the existence of a planet perturber capable of exciting such eccentricity. This implies that the eccentricities for the eccentric warm Jupiters are likely excited by their perturbers. High-e migration is therefore an attractive scenario for their formation. (2) The majority of low-e (e < 0.2) warm Jupiters lack strong perturbers necessary for high-e migration (Equation (4)). Given that for eccentric warm Jupiters, similar perturbers are indeed detected around a considerable fraction of the systems, this deficiency seems to be unlikely due to detection sensitivity. Moreover, out of the five warm Jupiter systems with e < 0.4 with Jovian perturbers, three are in compact multiple planet systems with three or more planets (55 Cnc b, GJ 876 c, and HIP 57274 c), which are challenging to explain with high-e migration. In contrast, for the five eccentric warm Jupiters at e > 0.4, there are no known additional planets in the system other than their Jovian perturbers, all of which are located further than 2 AU yet close enough to satisfy the constraint from Equation (4). This is indicative that the majority of low-e warm Jupiters are unlikely due to high-e migration induced by planet perturbers. It is worth noting that M_psin i rather than M_p is constrained from RV, so the above results are statistical. Note that the perturbers for all five eccentric warm Jupiters have larger M_psin i than the inner planets, consistent with simple expectations from the planet–planet scattering scenario that less massive planets are easier to get excited into high-e orbits.

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We caution that the conclusions may be affected if the chance for detecting an outer perturber strongly depends on the eccentricity of the inner planet. The observing strategies in RV surveys can be complicated, especially for those involving multiple planets. For example, Wright et al. (2009) pointed out that a system was observed more frequently after a planet was found, so the detection of a massive planet would likely facilitate the discovery of smaller planets. A similar selection effect may make the detection of perturbers of eccentric warm Jupiters easier if their eccentricities attract particular attentions. A comprehensive sensitivity study would be helpful. Additionally, there are a number of possible modeling degeneracies that may masquerade a double low-e planet system as an eccentric warm Jupiter (Rodigas & Hinz 2009; Anglada-Escudé et al. 2010). Systematic modeling efforts are possibly needed to evaluate such degeneracies. There might be other mechanisms that produce eccentric warm Jupiters associated with a perturber, including scattering followed by disk migration similar to Guillochon et al. (2011) and scattering of three planets with the third planet being ejected (C. Petrovich et al., in preparation).

High-precision RV surveys (≲ 5 m s⁻¹) on thousands of stars have lasted for ∼15 yr (e.g., Mayor et al. 2011; Wright et al. 2009; Wittenmyer et al. 2011). For a considerable fraction of their targets, they can detect Jupiters at ≲ 5 AU with full orbits (though note that some discoveries from these surveys remain unpublished). Given our constraint on the axis ratio of ∼20 for Jovian perturbers, this implies the present observational constraint on planet perturbers are likely relatively incomplete for warm Jupiters at ≳ 0.3 AU. For these systems, a thorough analysis of incomplete orbits and trends in RV is required. Unlike close solar-type companions, low-mass stellar and brown dwarf companions are unlikely to be excluded from the RV samples to search for planets. The combined efforts of RV linear trends and high-contrast imaging will yield excellent constraints for such perturbers (e.g., Crepp et al. 2012).

Rossiter–Mclaughlin effects for transiting planets are an important diagnostic for high-e migration, to which the spin–orbit misalignments have been commonly attributed. No ground-based surveys have so far detected transiting warm Jupiters (a_f = a(1 − e²) > a_{f, crit} = 0.1 AU).⁶ Yet it is interesting to note that, among the ground-based transiting planets with the longest period, possibly requiring eccentricity oscillations for tidal migration, several have known additional planet companions or have large RV linear trends (e.g., HAT-P-17b, Howard et al. 2012; WASP-8b, Queloz et al. 2010; KELT-6b, Collins et al. 2013). Future ground-based surveys or space-based surveys targeting bright stars are likely to discover warm Jupiters suitable for spin–orbit alignment measurements (note a possible transiting warm Jupiter candidate with a strong perturber identified by Dawson et al. 2012). They will be particularly interesting candidates subject to our proposed observational test on perturbers. If the spin–orbit misalignments are solely due to high-e migration, and given that the majority of low-e warm Jupiters do not seem to have strong enough perturbers for high-e migration, we expect that the majority of warm Jupiters with low-e (e ≲ 0.2) will be found to be aligned with the spin axes of their hosts.

Finally, if the warm Jupiters are indeed migrating due to tidal dissipation at the high-e stage during Kozai–Lidov oscillations, they should be tidally powered and luminous enough to be detected by the future high-contrast imaging facilities such as those to be installed at the Thirty Meter Telescope, the Giant Magellan Telescope, and the European Extremely Large Telescope (Dong et al. 2013a). Similar high-e migration mechanisms have also been raised for the formation of close binary stars at P ≲ 10 days (Fabrycky & Tremaine 2007; Dong et al. 2013b), and the constraint we derive in this work can also be applied to test the formation of binaries at 10 days ≲ P ≲ 100 days due to high-e mechanisms.

We thank Andy Gould, Scott Tremaine, and Cristobal Petrovich for discussions. We are grateful to the referee for a helpful report. S.D. was partly supported through a Ralph E. and Doris M. Hansmann Membership at the IAS and by NSF grant AST-0807444. B.K. is supported by NASA through the Einstein Postdoctoral Fellowship awarded by Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060. B.K. and A.S. acknowledge support from a John N. Bahcall Fellowship at the Institute for Advanced Study, Princeton. This research has made use of the Exoplanet Orbit Database and the Exoplanet Data Explorer at exoplanets.org.