A number of protoplanetary disks observed with ALMA potentially provide direct examples of initial conditions for planetary systems. In particular, the HL Tau disk has been intensively studied, and its rings/gaps are conventionally interpreted to be a result of unseen massive planets embedded in the gaps. Based on this interpretation, we carried out N-body simulations to investigate orbital evolution of planets within the protoplanetary disk and after the disk dispersal. Before the disk dispersal, our N-body simulations include both migration and mass-growth of the planet coupled with evolution of the disk. By varying the disk parameters, we produce a variety of widely-separated planetary systems consisting of three super-Jupiters at the end of disk dispersal. We found the outer planet is more massive than the inner one, and the migration of the innermost planet is inefficient due to the accretion of outer planet(s). We also showed how the final configuration and the final planetary mass depend on disk parameters. The migration is found to be convergent and no planet-pair has a period ratio less than 2. After the disk dispersal, we switch to pure gravitational N-body simulations and integrate the orbits up to 10 Gyr. Most simulated systems remain stable for at least 10 Gyr. We discuss implications of our result in terms of the observed widely-separated planetary systems HR 8799 and PDS 70.

We examine the stability of hierarchical triple systems using direct N-body simulations without adopting a secular perturbation assumption. We estimate their disruption timescales in addition to the mere stable/unstable criterion, with particular attention to the mutual inclination between the inner and outer orbits. First, we improve the fit to the dynamical stability criterion by Mardling & Aarseth widely adopted in the previous literature. Especially, we find that that the stability boundary is very sensitive to the mutual inclination; coplanar retrograde triples and orthogonal triples are much more stable and unstable, respectively, than coplanar prograde triples. Next, we estimate the disruption timescales of triples satisfying the stability condition up to 109 times the inner orbital period. The timescales follow the scaling predicted by Mushkin & Katz, especially at high e out where their random walk model is most valid. We obtain an improved empirical fit to the disruption timescales, which indicates that the coplanar retrograde triples are significantly more stable than the previous prediction. We furthermore find that the dependence on the mutual inclination can be explained by the energy transfer model based on a parabolic encounter approximation. We also show that the disruption timescales of triples are highly sensitive to tiny changes of the initial parameters, reflecting the genuine chaotic nature of the dynamics of those systems.

There are several ongoing projects to search for stars orbiting around an invisible companion. A fraction of such candidates may be a triple, instead of a binary, consisting of an inner binary black hole (BBH) and an outer orbiting star. In this paper, we propose a methodology to search for a signature of such an inner BBH, possibly a progenitor of gravitational-wave sources discovered by LIGO, from the precise radial velocity (RV) follow-up of the outer star. We first describe a methodology using an existing approximate RV formula for coplanar circular triples. We apply this method and constrain the parameters of a possible inner binary object in 2M05215658+4359220, which consists of a red giant and an unseen companion. Next we consider coplanar but non-circular triples. We compute numerically the RV variation of a tertiary star orbiting around an inner BBH, generate mock RV curves, and examine the feasibility of detection of the BBH for our fiducial models. We conclude that short-cadence RV monitoring of a star–BH binary provides an interesting and realistic method to constrain and/or search for possible inner BBHs. Indeed a recent discovery of the star–BH binary system LB-1 may imply that there are a large number of such unknown objects in our Galaxy, which are ideal targets for the methodology proposed here.

A number of ongoing surveys are likely to discover star–black hole binaries in our Galaxy in the near future. A fraction of them may be triple systems comprising an inner binary, instead of a single black hole, which might be progenitors of binary black holes (BBHs) routinely discovered now from the gravitational wave. We extend our previous proposal to locate inner BBHs from the short-term radial-velocity (RV) variation of a tertiary star in coplanar triples, and we consider noncoplanar triples and their long-term RV variations as well. Specifically, we assume coplanar and noncoplanar triples with an inner BBH of total mass 20 M ⊙, whose outer and inner orbital periods are 80 days and 10 days, respectively. We perform a series of N-body simulations and compare the results with analytic approximate solutions based on quadrupole perturbation theory. For coplanar triples, the pericenter shift of the outer star can be used to detect the hidden inner BBH. For noncoplanar triples, the total RV semiamplitude of the outer star is modulated periodically on the order of 100 km s−1, due to its precession over roughly the Kozai–Lidov oscillation timescale. Such long-term modulations would be detectable within a decade, independent of the short-term RV variations on the order of 100 m s−1 at roughly twice the orbital frequency of the inner binary. Thus the RV monitoring of future star–black hole binary candidates offers a promising method for searching for their inner hidden BBHs in optical bands.

While there have been many studies examining the stability of hierarchical triple systems, the meaning of “stability” is somewhat vague and has been interpreted differently in previous literatures. The present paper focuses on “Lagrange stability,” which roughly refers to the stability against the escape of a body from the system, or “disruption” of the triple system, in contrast to “Lyapunov-like stability,” which is related to the chaotic nature of the system dynamics. We compute the evolution of triple systems using direct N-body simulations up to 107 P out, which is significantly longer than previous studies (with P out being the initial orbital period of the outer body). We obtain the resulting disruption timescale T d as a function of the triple orbital parameters with particular attention to the dependence on the mutual inclination between the inner and outer orbits, i mut. By doing so, we have clarified explicitly the difference between Lagrange and Lyapunov stabilities in astronomical triples. Furthermore, we find that the von Zeipel–Kozai–Lidov oscillations significantly destabilize inclined triples (roughly with 60° < i mut < 150°) relative to those with i mut = 0°. On the other hand, retrograde triples with i mut > 160° become strongly stabilized with much longer disruption timescales. We show the sensitivity of the normalized disruption timescale T d/P out to the orbital parameters of triple system. The resulting T d/P out distribution is practically more useful in a broad range of astronomical applications than the stability criterion based on the Lyapunov divergence.