Apsidal Alignment Research Articles

Counter-rotation and slow precession in aligned eccentric nuclear discs due to gravitational wave recoil kicks

ABSTRACT The M31 nucleus contains a supermassive black hole embedded in a massive stellar disc of apsidally aligned eccentric orbits. It has recently been shown that this disc is slowly precessing at a rate consistent with zero. Here, we demonstrate using N-body methods that apsidally aligned eccentric discs can form with a significant ($\sim$0.5) fraction of orbits counter-rotating as the result of a gravitational wave recoil kick of merging supermassive black holes. Higher amplitude kicks map to a larger retrograde fraction in the surrounding stellar population, which in turn results in slow precession. We furthermore show that discs with significant counter-rotation are more stable (i.e. apsidal alignment is most pronounced and long lasting), more eccentric, and have the highest rates of stars entering the black hole’s tidal disruption radius.

Tidal Disruption of Planetesimals from an Eccentric Debris Disk Following a White Dwarf Natal Kick

The surfaces of many white dwarfs are polluted by metals, implying a recent accretion event. The tidal disruption of planetesimals is a viable source of white dwarf pollution and offers a unique window into the composition of exoplanet systems. The question of how planetary material enters the tidal disruption radius of the white dwarf is currently unresolved. Using a series of N-body simulations, we explore the response of the surrounding planetesimal debris disk as the white dwarf receives a natal kick caused by anisotropic mass loss on the asymptotic giant branch. We find that the kick can form an apse-aligned, eccentric debris disk in the range 30–240 au, which corresponds to the orbits of Neptune, the Kuiper Belt, and the scattered disk in our solar system. In addition, many planetesimals beyond 240 au flip to counterrotating orbits. Assuming an isotropic distribution of kicks, we predict that approximately 80% of white dwarf debris disks should exhibit significant apsidal alignment and a fraction of counterrotating orbits. The eccentric disk is able to efficiently and continuously torque planetesimals onto radial, star-grazing orbits. We show that the kick causes both an initial burst in tidal disruption events as well as an extended period of 100 Myr where tidal disruption rates are consistent with observed mass accretion rates on polluted white dwarfs.

Anisotropic Star Clusters around Recoiling Supermassive Black Holes

Gravitational-wave recoil kicks from merging supermassive black hole binaries can have a profound effect on the surrounding stellar population. In this work, we study the dynamic and kinematic properties of nuclear star clusters following a recoil kick. We show that these postkick structures present unique signatures that can provide key insight to observational searches for recoiling supermassive black holes. In our previous paper, we showed that an in-plane recoil kick turns a circular disk into a lopsided, eccentric disk such as the one we observe in the Andromeda nucleus. Building on this work, here we explore many recoil kick angles as well as initial stellar configurations. For a circular disk of stars, an in-plane kick causes strong apsidal alignment with a significant fraction of the disk becoming retrograde at large radii. If the initial orbits are highly eccentric, an in-plane kick forms a bar-like structure made up of two antialigned lopsided disks. An out-of-plane kick causes clustering in the argument of periapsis, ω, regardless of the initial eccentricity distribution. Initially, isotropic configurations form anisotropies in the form of a torus of eccentric orbits oriented perpendicular to the recoil kick. Postkick surface density and velocity maps are presented in each case to highlight the distinct, observable structures of these systems.

Giant Impacts and Debris Disk Morphology

Certain debris disks have nonaxisymmetric shapes in scattered light which are unexplained. The appearance of a disk depends on how its constituent Keplerian ellipses are arranged. The more the ellipses align apsidally, the more nonaxisymmetric the disk. Apsidal alignment is automatic for fragments released from a catastrophic collision between solid bodies. We synthesize scattered light images, and thermal emission images, of such giant impact debris. Depending on the viewing geometry, and if and how the initial apsidal alignment is perturbed, the remains of a giant impact can appear in scattered light as a one-sided or two-sided “fork,” a lopsided “needle,” or a set of “double wings.” The last shape is difficult to reproduce in other scenarios involving gravitational forcing or gas drag, which do not align orbits as well. We compare our images with observations and offer a scorecard assessing whether the scattered light asymmetries in HD 15115, HD 32297, HD 61005, HD 111520, HD 106906, β Pic, and AU Mic are best explained by giant impacts, gravitational perturbations, or sculpting by the interstellar medium.

Dynamics of Circumstellar Planets in Binary Star Systems

AbstractCircumstellar planets in binary star systems provide unique constraints on the formation and dynamical evolution of planets. We present an empirical formula for the stability boundary of coplanar retrograde orbits, similar to the classic one for coplanar prograde orbits. We discuss two of the tightest binaries with circumstellar planets: HD 59686 and ν Octantis. For HD 59686, dynamical fitting of the radial velocity data and stability analysis show that the planet must be either on a nearly coplanar retrograde orbit or in one of the narrow regions of prograde orbits stabilized by secular apsidal alignment. For ν Octantis, a nearly coplanar retrograde planetary orbit is the only option for dynamical stability. We also discuss the mysterious case of ε Cygni. It shows short-period radial velocity variations that closely resemble the signal of a Jupiter-mass planet, but the period and amplitude change over time and dynamical stability analysis rules out a planet.

Apsidal alignment and anti-alignment of planets in mean-motion resonance: disc-driven migration and eccentricity driving

ABSTRACT Planets migrating in their natal discs can be captured into mean-motion resonance (MMR), in which the planets’ periods are related by integer ratios. Recent observations indicate that planets in MMR can be either apsidally aligned or anti-aligned. How these different configurations arise is unclear. In this paper, we study the MMR capture process of migrating planets, focusing on the property of the apsidal angles of the captured planets. We show that the standard picture of MMR capture, in which the planets undergo convergent migration and experience eccentricity damping due to planet–disc interactions, always leads to apsidal anti-alignment of the captured planets. However, when the planets experience eccentricity driving from the disc, apsidally aligned configuration in MMR can be produced. In this configuration, both planets’ resonance angles circulate, but a ‘mixed’ resonance angle librates and traps the planets near the nominal resonance location. The MMR capture process in the presence of disc eccentricity driving is generally complex and irregular, and can lead to various outcomes, including apsidal alignment and anti-alignment, as well as the disruption of the resonance. We suggest that the two resonant planets in the K2-19 system, with their moderate eccentricities and aligned apsides, have experienced eccentricity driving from their natal disc in the past.

Stability analysis of apsidal alignment in double-averaged restricted elliptic three-body problem

Stability analysis of apsidal alignment in double-averaged restricted elliptic three-body problem

Planet formation in stellar binaries: global simulations of planetesimal growth

Planet formation around one component of a tight, eccentric binary system such as γ Cephei (with semimajor axis around 20 AU) is theoretically challenging because of destructive high-velocity collisions between planetesimals. Despite this fragmentation barrier, planets are known to exist in such (so-called S-type) orbital configurations. Here we present a novel numerical framework for carrying out multi-annulus coagulation-fragmentation calculations of planetesimal growth, which fully accounts for the specifics of planetesimal dynamics in binaries, details of planetesimal collision outcomes, and the radial transport of solids in the disk due to the gas drag-driven inspiral. Our dynamical inputs properly incorporate the gravitational effects of both the eccentric stellar companion and the massive non-axisymmetric protoplanetary disk in which planetesimals reside, as well as gas drag. We identify a set of disk parameters that lead to successful planetesimal growth in systems such as γ Cephei or α Centauri starting from 1 to 10 km size objects. We identify the apsidal alignment of a protoplanetary disk with the binary orbit as one of the critical conditions for successful planetesimal growth: It naturally leads to the emergence of a dynamically quiet location in the disk (as long as the disk eccentricity is of order several percent), where favorable conditions for planetesimal growth exist. Accounting for the gravitational effect of a protoplanetary disk plays a key role in arriving at this conclusion, in agreement with our previous results. These findings lend support to the streaming instability as the mechanism of planetesimal formation. They provide important insights for theories of planet formation around both binary and single stars, as well as for the hydrodynamic simulations of protoplanetary disks in binaries (for which we identify a set of key diagnostics to verify).

Apsidal asymmetric-alignment of Jupiter Trojans

ABSTRACT The most distant Kuiper belt objects exhibit the clustering in their orbits, and this anomalous architecture could be caused by Planet 9 with large eccentricity and high inclination. We then suppose that the orbital clustering of minor planets may be observed somewhere else in the Solar system. In this paper, we consider the over 7000 Jupiter Trojans from the Minor Planet Center, and find that they are clustered in the longitude of perihelion ϖ, around the locations ϖJ + 60○ and ϖJ − 60○ (ϖJ is the longitude of perihelion of Jupiter) for the L4 and L5 swarms, respectively. Then we build a Hamiltonian system to describe the associated dynamical aspects for the co-orbital motion. The phase space displays the existence of the apsidally aligned islands of libration centred on Δϖ = ϖ − ϖJ ≈ ±60○, for the Trojan-like orbits with eccentricities e < 0.1. Through a detailed analysis, we have shown that the observed Jupiter Trojans with proper eccentricities ep < 0.1 spend most of their time in the range of |Δϖ| = 0°–120○, while the more eccentric ones with ep > 0.1 are too few to affect the orbital clustering within this Δϖ range for the entire Trojan population. Our numerical results further prove that, even starting from a uniform Δϖ distribution, the apsidal alignment of simulated Trojans similar to the observation can appear on the order of the age of the Solar system. We conclude that the apsidal asymmetric-alignment of Jupiter Trojans is robust, and this new finding can be helpful to design the survey strategy in the future.

The stationary points of the hierarchical three-body problem

ABSTRACT We study the stationary points of the hierarchical three body problem in the planetary limit (m1, m2 ≪ m0) at both the quadrupole and octupole orders. We demonstrate that the extension to octupole order preserves the principal stationary points of the quadrupole solution in the limit of small outer eccentricity e2 but that new families of stable fixed points occur in both prograde and retrograde cases. The most important new equilibria are those that branch off from the quadrupolar solutions and extend to large e2. The apsidal alignment of these families is a function of mass and inner planet eccentricity, and is determined by the relative directions of precession of ω1 and ω2 at the quadrupole level. These new equilibria are also the most resilient to the destabilizing effects of relativistic precession. We find additional equilibria that enable libration of the inner planet argument of pericentre in the limit of radial orbits and recover the non-linear analogue of the Laplace–Lagrange solutions in the coplanar limit. Finally, we show that the chaotic diffusion and orbital flips identified with the eccentric Kozai–Lidov mechanism and its variants can be understood in terms of the stationary points discussed here.

The CARMENES search for exoplanets around M dwarfs

Context. GJ 1148 is an M-dwarf star hosting a planetary system composed of two Saturn-mass planets in eccentric orbits with periods of 41.38 and 532.02 days. Aims. We reanalyze the orbital configuration and dynamics of the GJ 1148 multi-planetary system based on new precise radial velocity measurements taken with CARMENES. Methods. We combined new and archival precise Doppler measurements from CARMENES with those available from HIRES for GJ 1148 and modeled these data with a self-consistent dynamical model. We studied the orbital dynamics of the system using the secular theory and direct N-body integrations. The prospects of potentially habitable moons around GJ 1148 b were examined. Results. The refined dynamical analyses show that the GJ 1148 system is long-term stable in a large phase-space of orbital parameters with an orbital configuration suggesting apsidal alignment, but not in any particular high-order mean-motion resonant commensurability. GJ 1148 b orbits inside the optimistic habitable zone (HZ). We find only a narrow stability region around the planet where exomoons can exist. However, in this stable region exomoons exhibit quick orbital decay due to tidal interaction with the planet. Conclusions. The GJ 1148 planetary system is a very rare M-dwarf planetary system consisting of a pair of gas giants, the inner of which resides in the HZ. We conclude that habitable exomoons around GJ 1148 b are very unlikely to exist.

Apsidal alignment in migrating dust - Crescent features caused by eccentric planets

AbstractCircumstellar discs are known to exist in great variety, from gas-rich discs around the youngest stars to evolved debris discs such as the solar system’s zodiacal cloud. Through gravitational interaction, exoplanets embedded in these discs can generate density variations, imposing potentially observable structural features on the disc such as rings or gaps. Here we report on a mirrored double crescent pattern arising in simulations of discs harbouring a small, moderately eccentric planet - such as Mars. We show that the structure is a result of a directed apsidal precession occurring in particles that migrate the planet’s orbital region under Poynting-Robertson drag. We further analyze the strength of this effect with respect to planet and particle parameters.

Sculpting Eccentric Debris Disks with Eccentric Gas Rings

Abstract Many debris disks seen in scattered light have shapes that imply their dust grains trace highly eccentric, apsidally aligned orbits. Apsidal alignment is surprising, especially for dust. Even when born from an apse-aligned ring of parent bodies, dust grains have their periastra dispersed in all directions by stellar radiation pressure. The periastra cannot be reoriented by planets within the short dust lifetimes at the bottom of the collisional cascade. We propose that what realigns dust orbits is drag exerted by second-generation gas. Gas is largely immune to radiation pressure, and when released by photodesorption or collisions within an eccentric ring of parent bodies should occupy a similarly eccentric, apse-aligned ring. Dust grains launched onto misaligned orbits cross the eccentric gas ring supersonically and can become dragged into alignment within collisional lifetimes. The resultant dust configurations, viewed nearly but not exactly edge-on, with periastra pointing away from the observer, appear moth-like, with kinked wings and even doubled pairs of wings, explaining otherwise mysterious features in HD 61005 (“The Moth”) and HD 32297, including their central bulbs when we account for strong forward scattering from irregularly shaped particles. Around these systems we predict gas at Kuiper-belt-like distances to move on highly elliptical streamlines that owe their elongation, ultimately, to highly eccentric planets. Unresolved issues and an alternative explanation for apsidal alignment are outlined.

Dynamical Analysis of the Circumprimary Planet in the Eccentric Binary System HD 59686

Abstract We present a detailed orbital and stability analysis of the HD 59686 binary-star planet system. HD 59686 is a single-lined, moderately close (a B = 13.6 au) eccentric (e B = 0.73) binary, where the primary is an evolved K giant with mass M = 1.9 M ⊙ and the secondary is a star with a minimum mass of m B = 0.53 M ⊙. Additionally, on the basis of precise radial velocity (RV) data, a Jovian planet with a minimum mass of m p = 7 M Jup, orbiting the primary on a nearly circular S-type orbit with e p = 0.05 and a p = 1.09 au, has recently been announced. We investigate large sets of orbital fits consistent with HD 59686's RV data by applying bootstrap and systematic grid search techniques coupled with self-consistent dynamical fitting. We perform long-term dynamical integrations of these fits to constrain the permitted orbital configurations. We find that if the binary and the planet in this system have prograde and aligned coplanar orbits, there are narrow regions of stable orbital solutions locked in a secular apsidal alignment with the angle between the periapses, Δω, librating about 0°. We also test a large number of mutually inclined dynamical models in an attempt to constrain the three-dimensional orbital architecture. We find that for nearly coplanar and retrograde orbits with mutual inclination 145° ≲ Δi ≤ 180°, the system is fully stable for a large range of orbital solutions.

Secondary Eclipses of HAT-P-13b

Abstract We present Spitzer secondary-eclipse observations of the hot Jupiter HAT-P-13 b in the 3.6 and 4.5 μm bands. HAT-P-13 b inhabits a two-planet system with a configuration that enables constraints on the planet’s second Love number, , from precise eccentricity measurements, which in turn constrains models of the planet’s interior structure. We exploit the direct measurements of from our secondary-eclipse data and combine them with previously published radial velocity data to generate a refined model of the planet’s orbit and thus an improved estimate on the possible interval for . We report eclipse phases of and and corresponding estimates of and . Under the assumptions of previous work, our estimate of of 0.81 ± 0.10 is consistent with the lower extremes of possible core masses found by previous models, including models with no solid core. This anomalous result challenges both interior models and the dynamical assumptions that enable them, including the essential assumption of apsidal alignment. We also report eclipse depths of 0.081% ± 0.008% in the 3.6 μm channel and 0.088% ± 0.028% in the 4.5 μm channel. These photometric results are non-uniquely consistent with solar-abundance composition without any thermal inversion.

Constraints on Planet Nine’s Orbit and Sky Position within a Framework of Mean-motion Resonances

Abstract A number of authors have proposed that the statistically significant orbital alignment of the most distant Kuiper Belt Objects (KBOs) is evidence of an as-yet undetected planet in the outer solar system, now referred to colloquially as “Planet Nine.” Dynamical simulations by Batygin & Brown have provided constraints on the range of the planet’s possible orbits and sky locations. We extend these investigations by exploring the suggestion of Malhotra et al. that Planet Nine is in small integer ratio mean-motion resonances (MMRs) with several of the most distant KBOs. We show that the observed KBO semimajor axes present a set of commensurabilities with an unseen planet at ∼654 au (P ∼ 16,725 years) that has a greater than 98% chance of stemming from a sequence of MMRs rather than from a random distribution. We describe and implement a Monte-Carlo optimization scheme that drives billion-year dynamical integrations of the outer solar system to pinpoint the orbital properties of perturbers that are capable of maintaining the KBOs’ apsidal alignment. This optimization exercise suggests that the unseen planet is most consistently represented with mass, m ∼ 6–12 M ⊕, semimajor axis, a ∼ 654 au, eccentricity, e ∼ 0.45, inclination, i ∼ 30°, argument of periastron, ω ∼ 150°, longitude of ascending node, Ω ∼ 50°, and mean anomaly, M ∼ 180°. A range of sky locations relative to this fiducial ephemeris are possible. We find that the region 30° ≲ R.A. ≲ 50°, −20° ≲ decl. ≲ 20° is promising.

SECURE MASS MEASUREMENTS FROM TRANSIT TIMING: 10 KEPLER EXOPLANETS BETWEEN 3 AND 8 M ⊕ WITH DIVERSE DENSITIES AND INCIDENT FLUXES

ABSTRACT We infer dynamical masses in eight multiplanet systems using transit times measured from Kepler's complete data set, including short-cadence data where available. Of the 18 dynamical masses that we infer, 10 pass multiple tests for robustness. These are in systems Kepler-26 (KOI-250), Kepler-29 (KOI-738), Kepler-60 (KOI-2086), Kepler-105 (KOI-115), and Kepler-307 (KOI-1576). Kepler-105 c has a radius of 1.3 R ⊕ and a density consistent with an Earth-like composition. Strong transit timing variation (TTV) signals were detected from additional planets, but their inferred masses were sensitive to outliers or consistent solutions could not be found with independently measured transit times, including planets orbiting Kepler-49 (KOI-248), Kepler-57 (KOI-1270), Kepler-105 (KOI-115), and Kepler-177 (KOI-523). Nonetheless, strong upper limits on the mass of Kepler-177 c imply an extremely low density of ∼0.1 g cm−3. In most cases, individual orbital eccentricities were poorly constrained owing to degeneracies in TTV inversion. For five planet pairs in our sample, strong secular interactions imply a moderate to high likelihood of apsidal alignment over a wide range of possible eccentricities. We also find solutions for the three planets known to orbit Kepler-60 in a Laplace-like resonance chain. However, nonlibrating solutions also match the transit timing data. For six systems, we calculate more precise stellar parameters than previously known, enabling useful constraints on planetary densities where we have secure mass measurements. Placing these exoplanets on the mass–radius diagram, we find that a wide range of densities is observed among sub-Neptune-mass planets and that the range in observed densities is anticorrelated with incident flux.

How to reach the orbital configuration of the inner three planets in HD 40307 planet system?

The formation of the present configuration of three hot super-Earths in the planet system HD 40307 is a challenge to dynamical astronomers. With the two successive period ratios both near and slightly larger than 2, the system may have evolved from pairwise 2:1 mean motion resonances (MMRs). In this paper, we investigate the evolutions of the period ratios of the three planets after the primordial gas disk was depleted. Three routines are found to probably result in the current configuration under tidal dissipation with the center star, they are: (i) through apsidal alignment only; (ii) out of pairwise 2:1MMRs, then through apsidal alignment; (iii) out of the 4:2:1 Laplace Resonance (LR), then through apsidal alignment. All the three scenarios require the initial eccentricities of planets ∼ 0.15, which implies a planetary scattering history during and after the gas disk was depleted. All the three routines will go through the apsidal alignment phase, and enter a state with near-zero eccentricities finally. We also find some special characteristics for each routine. If the system went through pairwise 2:1 MMRs at the beginning, the MMR of the outer two planets would be broken first to reach the current state. As for routine (iii), the planets would be out of the LR at the place where some high-order resonances are located. At the high-order resonances 17:8 or 32:15 of the planets c and d, the system will possibly enter the current state as the final equilibrium.

TWO SUPER-EARTHS ORBITING THE SOLAR ANALOG HD 41248 ON THE EDGE OF A 7:5 MEAN MOTION RESONANCE

The number of multi-planet systems known to be orbiting their host stars with orbital periods that place them in mean motion resonances is growing. For the most part, these systems are in first-order resonances and dynamical studies have focused their efforts towards understanding the origin and evolution of such dynamically resonant commensurabilities. We report here the discovery of two super-Earths that are close to a second-order dynamical resonance, orbiting the metal-poor ([Fe/H]=-0.43 dex) and inactive G2V star HD41248. We analysed 62 HARPS archival radial velocities for this star, that until now, had exhibited no evidence for planetary companions. Using our new Bayesian Doppler signal detection algorithm, we find two significant signals in the data, with periods of 18.357 days and 25.648 days, indicating they could be part of a 7:5 second-order mean motion resonance. Both semi-amplitudes are below 3m/s and the minimum masses of the pair are 12.3 and 8.6Mearth, respectively. Our simulations found that apsidal alignment stabilizes the system, and even though libration of the resonant angles was not seen, the system is affected by the presence of the resonance and could yet occupy the 7:5 commensurability, which would be the first planetary configuration in such a dynamical resonance. Given the multitude of low-mass multiplanet systems that will be discovered in the coming years, we expect more of these second-order resonant configurations will emerge from the data, highlighting the need for a better understanding of the dynamical interactions between forming planetesimals.

RESOLVING THE sin(I) DEGENERACY IN LOW-MASS MULTI-PLANET SYSTEMS

Long-term orbital evolution of multi-planet systems under tidal dissipation often converges to a stationary state, known as the tidal fixed point. The fixed point is characterized by a lack of oscillations in the eccentricities and apsidal alignment among the orbits. Quantitatively, the nature of the fixed point is dictated by mutual interactions among the planets as well as non-Keplerian effects. We show that if a roughly coplanar system hosts a hot, sub-Saturn mass planet, and is tidally relaxed, separation of planet-planet interactions and non-Keplerian effects in the equations of motion leads to a direct determination of the true masses of the planets. Consequently, a "snap-shot" observational determination of the orbital state resolves the sin(I) degeneracy, and opens up a direct avenue towards identification of the true lowest-mass exo-planets detected. We present an approximate, as well as a general, mathematical framework for computation of the line of sight inclination of secular systems, and apply our models illustratively to the 61 Vir system. We conclude by discussing the observability of planetary systems to which our method is applicable and we set our analysis into a broader context by presenting a current summary of the various possibilities for determining the physical properties of planets from observations of their orbital states.