The Perturber

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 e 0 orbiting a star with mass M accompanied by a perturber of mass M per at a per and e per. 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.

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. At higher e lower a f , the tidal dissipation may be efficient. The eccentricity as a function of time from direct three-body integration is shown by a red line and that from double-averaging calculations to octupole order in the perturbing potential is shown by a black dashed line. The two integrations show excellent agreement, validating the double-averaging approximation. As can be seen in the inset, the three-body integration shows slight variations from the double-averaging calculations within each orbital period of the outer perturber.

However, their impact on the long term evolution "averages out" to essentially zero, meaning that they play no role in the current study. An analytical constraint is derived under the simplest assumptions: We show below with numerical simulations that these are excellent approximations in deriving this constraint. Under these approximations, the following is a constant e. Perturber constraint of warm Jupiters for high- e migration. Upper and lower panels are for solar-mass and Jupiter-mass perturbers, respectively.

The blue lines show the analytical upper limit Equation 4 in the ratio between semiminor axis of the perturber and the warm Jupiters' semimajor axis as a function of a.

This is verified by 10, numerical simulations with random initial orbital orientations that include the double-averaging octupole-order approximation and without neglecting the effect of the mass of the warm Jupiter. Each simulation is shown as a dot and the initial eccentricity of the planet is fixed at 0.

Recently, it was realized that corrections due to various approximations above may lead to significant effects in several scenarios e. 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.

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These simulations employ the double averaged approximation but include the octupole term and are not restricted to the test particle approximation. The eccentricities of the outer perturbers are uniformly distributed within 0—0. The orbital orientations of the outer and inner orbits are randomly distributed isotropically. The results are shown in Figure 3. 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.

For the scenarios considered, the Kozai—Lidov timescale is much longer than the outer and inner orbital timescales, justifying the double averaging assumption.

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To illustrate this, the results of a direct three-body integration are compared to those of a double-averaging integration in Figure 2. The results of two integrations are practically indistinguishable black dashed: 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 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. 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: 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. 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. This is indicative that the majority of low- e warm Jupiters are unlikely due to high- e migration induced by planet perturbers.

Eccentricity distribution of warm Jupiters. The red solid histogram is for warm Jupiters with an external Jovian perturber. All satisfy the constraint in Equation 4. The fraction of warm Jupiters with detected Jovian perturbers is a growing function of eccentricities. A large fraction of low- e warm Jupiters lack such perturbers. 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. 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.

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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.

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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. Extreme mass-ratio inspirals, in which a stellar-mass object orbits a supermassive black hole, are prime sources for space-based gravitational wave detectors because they will facilitate tests of strong gravity and probe the spacetime around rotating compact objects.

In the last few years of such inspirals, the total phase is in the millions of radians and details of the waveforms are sensitive to small perturbations. We show that one potentially detectable perturbation is the presence of a second supermassive black hole within a few tenths of a parsec. The acceleration produced by the perturber on the extreme mass-ratio system produces a steady drift that causes the waveform to deviate systematically from that of an isolated system.

If the perturber is a few tenths of a parsec from the extreme mass-ratio system plausible in as many as a few percent of cases higher derivatives of motion might also be detectable.

In that case, the mass and distance of the perturber can be derived independently, which would allow a new probe of merger dynamics. Library subscriptions will be modified accordingly. This arrangement will initially last for two years, up to the end of The 60th anniversary edition of the Review of Particle Physics , published in Physical Review D , provides a comprehensive review of the field of particle physics and of related areas in cosmology.

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