arXiv subjects

Melvyn B. Davies

Publications and source records attributed to Melvyn B. Davies.

At least 37 records · Page 2Linked to original sources

arXiv2021

Formation of supermassive black holes in galactic nuclei II: retention and growth of seed intermediate-mass black holes

In many galactic nuclei, a nuclear stellar cluster (NSC) co-exists with a supermassive black hole (SMBH). In this work, we explore the idea that the NSC forms before the SMBH through the merger of several stellar clusters that may contain intermediate-mass black holes (IMBHs). These IMBHs can subsequently grow by mergers and accretion to form an SMBH. To check the observable consequences of this proposed SMBH seeding mechanism, we created an observationally motivated mock population of galaxies, in which NSCs are constructed by aggregating stellar clusters that may or may not contain IMBHs. We model the growth of IMBHs in the NSCs through gravitational wave (GW) mergers with other IMBHs and gas accretion. In the case of GW mergers, the merged BH can either be retained or ejected depending on the GW recoil kick it receives. The likelihood of retaining the merged BH increases if we consider growth of IMBHs in the NSC through gas accretion. We find that nucleated lower-mass galaxies ($\rm M_{\star} \lesssim 10^{9} \ M_{\odot}$; e.g. M33) have an SMBH seed occupation fraction of about 0.3 to 0.5. This occupation fraction increases with galaxy stellar mass and for more massive galaxies ($\rm 10^{9} \ M_{\odot} \lesssim \rm M_{\star} \lesssim 10^{11} \ M_{\odot}$), it is between 0.5 and 0.8, depending on how BH growth is modelled. These occupation fractions are consistent with observational constraints. Furthermore, allowing for BH growth also allows us to reproduce the observed diversity in the mass range of SMBHs in the $\rm M_{\rm NSC} - M_{\rm BH}$ plane.

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arXiv2021

Hot Jupiters, cold kinematics: High phase space densities of host stars reflect an age bias

Context. The birth environments of planetary systems are thought to influence planet formation and orbital evolution, through external photoevaporation and stellar flybys. Recent work has claimed observational support for this, in the form of a correlation between the properties of planetary systems and the local Galactic phase space density of the host star. In particular, Hot Jupiters are found overwhelmingly around stars in regions of high phase space density, which may reflect a formation environment with high stellar density. Aims. We instead investigate whether the high phase space density may have a galactic kinematic origin: Hot Jupiter hosts may be biased towards being young and therefore kinematically cold, because tidal inspiral leads to the destruction of the planets on Gyr timescales, and the velocity dispersion of stars in the Galaxy increases on similar timescales. Methods. We use 6D positions and kinematics from Gaia for the Hot Jupiter hosts and their neighbours, and construct distributions of the phase space density. We investigate correlations between the stars' local phase space density and peculiar velocity. Results. We find a strong anticorrelation between the phase space density and the host star's peculiar velocity with respect to the Local Standard of Rest. Therefore, most stars in "high-density" regions are kinematically cold, which may be caused by the aforementioned bias towards detecting Hot Jupiters around young stars before the planets' tidal destruction. Conclusions. We do not find evidence in the data for Hot Jupiter hosts preferentially being in phase space overdensities compared to other stars of similar kinematics, nor therefore for their originating in birth environments of high stellar density.

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arXiv2021

Dynamical orbital evolution scenarios of the wide-orbit eccentric planet HR 5183b

The recently-discovered giant exoplanet HR5183b exists on a wide, highly-eccentric orbit ($a=18$\,au, $e=0.84$). Its host star possesses a common proper-motion companion which is likely on a bound orbit. In this paper, we explore scenarios for the excitation of the eccentricity of the planet in binary systems such as this, considering planet-planet scattering, Lidov-Kozai cycles from the binary acting on a single-planet system, or Lidov-Kozai cycles acting on a two-planet system that also undergoes scattering. Planet-planet scattering, in the absence of a binary companion, has a $2.8-7.2\%$ probability of pumping eccentricities to the observed values in our simulations, depending on the relative masses of the two planets. Lidov-Kozai cycles from the binary acting on an initially circular orbit can excite eccentricities to the observed value, but require very specific orbital configurations for the binary and overall there is a low probability of catching the orbit at the high observed high eccentricity ($0.6\%$). The best case is provided by planet-planet scattering in the presence of a binary companion: here, the scattering provides the surviving planet with an initial eccentricity boost that is subsequently further increased by Kozai cycles from the binary. We find a success rate of $14.5\%$ for currently observing $e\ge0.84$ in this set-up. The single-planet plus binary and two-planet plus binary cases are potentially distinguishable if the mutual inclination of the binary and the planet can be measured, as the latter permits a broader range of mutual inclinations. The combination of scattering and Lidov-Kozai forcing may also be at work in other wide-orbit eccentric giant planets, which have a high rate of stellar binary companions.

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arXiv2021

Metal pollution of the solar white dwarf by solar system small bodies

White dwarfs (WDs) often show metal lines in their spectra, indicating accretion of asteroidal material. Our Sun is to become a WD in several Gyr. Here, we examine how the solar WD accretes from the three major small body populations: the main belt asteroids (MBAs), Jovian trojan asteroids (JTAs), and trans-Neptunian objects (TNOs). Owing to the solar mass loss during the giant branch, 40\% of the JTAs are lost but the vast majority of MBAs and TNOs survive. During the WD phase, objects from all three populations are sporadically scattered onto the WD, implying ongoing accretion. For young cooling ages $\lesssim 100$ Myr, accretion of MBAs predominates; our predicted accretion rate $\sim10^6$ g/s falls short of observations by two orders of magnitude. On Gyr timescales, thanks to the consumption of the TNOs that kicks in $\gtrsim 100$ Myr, the rate oscillates around $10^6-10^7$ g/s until several Gyr and drops to $\sim10^5$ g/s at 10 Gyr. Our solar WD accretion rate from 1 Gyr and beyond agrees well with those of the extrasolar WDs. We show that for the solar WD, the accretion source region evolves in an inside-out pattern. Moreover, in a realistic small body population with individual sizes covering a wide range as WD pollutants, the accretion is dictated by the largest objects. As a consequence, the accretion rate is lower by an order of magnitude than that from a population of bodies of a uniform size and the same total mass and shows greater scatter.

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arXiv2021

Accretion of tidally disrupted asteroids onto white dwarfs: direct accretion versus disk processing

Atmospheric heavy elements have been observed in more than a quarter of white dwarfs (WDs) at different cooling ages, indicating ongoing accretion of asteroidal material, whilst only a few per cent of the WDs possess a dust disk, and all these WDs are accreting metals. Here, assuming that a rubble-pile asteroid is scattered inside a WD's Roche lobe by a planet, we study its tidal disruption and the long-term evolution of the resulting fragments. We find that after a few pericentric passages, the asteroid is shredded into its constituent particles, forming a flat, thin ring. On a timescale of Myr, tens of per cent of the particles are scattered onto the WD, and are therefore directly accreted without first passing through a circularised close-in disk. Fragment mutual collisions are most effective for coplanar fragments, and are thus only important in $10^3-10^4$ yr before the orbital coplanarity is broken by the planet. We show that for a rubble pile asteroid with a size frequency distribution of the component particles following that of the near earth objects, it has to be roughly at least 10 km in radius such that enough fragments are generated and $\ge10\%$ of its mass is lost to mutual collisions. At relative velocities of tens of km/s, such collisions grind down the tidal fragments into smaller and smaller dust grains. The WD radiation forces may shrink those grains' orbits, forming a dust disk. Tidal disruption of a monolithic asteroid creates large km-size fragments, and only parent bodies $\ge100$ km are able to generate enough fragments for mutual collisions to be significant. Hence, those large asteroids experience a disk phase before being accreted.

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arXiv2021

Transit detection of the long-period volatile-rich super-Earth $ν^2$ Lupi d with $CHEOPS$

Exoplanets transiting bright nearby stars are key objects for advancing our knowledge of planetary formation and evolution. The wealth of photons from the host star gives detailed access to the atmospheric, interior, and orbital properties of the planetary companions. $ν^2$ Lupi (HD 136352) is a naked-eye ($V = 5.78$) Sun-like star that was discovered to host three low-mass planets with orbital periods of 11.6, 27.6, and 107.6 days via radial velocity monitoring (Udry et al. 2019). The two inner planets (b and c) were recently found to transit (Kane et al. 2020), prompting a photometric follow-up by the brand-new $CHaracterising\:ExOPlanets\:Satellite\:(CHEOPS)$. Here, we report that the outer planet d is also transiting, and measure its radius and mass to be $2.56\pm0.09$ $R_{\oplus}$ and $8.82\pm0.94$ $M_{\oplus}$, respectively. With its bright Sun-like star, long period, and mild irradiation ($\sim$5.7 times the irradiation of Earth), $ν^2$ Lupi d unlocks a completely new region in the parameter space of exoplanets amenable to detailed characterization. We refine the properties of all three planets: planet b likely has a rocky mostly dry composition, while planets c and d seem to have retained small hydrogen-helium envelopes and a possibly large water fraction. This diversity of planetary compositions makes the $ν^2$ Lupi system an excellent laboratory for testing formation and evolution models of low-mass planets.

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arXiv2021

Close stellar encounters at the Galactic Centre I: The effect on the observed stellar populations

We model the effects of collisions and close encounters on the stellar populations observed in the Milky Way nuclear stellar cluster (NSC). Our analysis is based on $N$-body simulations in which the NSC forms by accretion of massive stellar clusters around a supermassive black hole. We attach stellar populations to our $N$-body particles and follow the evolution of their stars, and the rate of collisions and close encounters. The most common encounters are collisions between pairs of main-sequence stars, which lead to mergers: destructive collisions between main-sequence stars and compact objects are rare. We find that the effects of collisions on the stellar populations are small for three reasons. First, our models possess a core which limits the maximum stellar density. Secondly, the velocity dispersion in the NSC is similar to the surface escape velocities of the stars, which minimises the collision rate. Finally, whilst collisions between main-sequence stars destroy bright giants by accelerating their evolution, they also create them by accelerating the evolution of lower-mass stars. These two effects approximately cancel out. We also investigate whether the G2 cloud could be a fuzzball: a compact stellar core which has accreted a tenuous envelope in a close encounter with a red giant. We conclude that fuzzballs with cores below $2\,M_\odot$ have thermal times-scales too short to reproduce G2. A fuzzball with a black-hole core could reproduce the surface properties of G2 but the production rate of such objects in our model is low.

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arXiv2021

Formation of super-massive black holes in galactic nuclei I: delivering seed intermediate-mass black holes in massive stellar clusters

Supermassive black holes (SMBHs) are found in most galactic nuclei. A significant fraction of these nuclei also contain a nuclear stellar cluster (NSC) surrounding the SMBH. In this paper, we consider the idea that the NSC forms first, from the merger of several stellar clusters that may contain intermediate-mass black holes (IMBHs). These IMBHs can subsequently grow in the NSC and form an SMBH. We carry out $N$-body simulations of the simultaneous merger of three stellar clusters to form an NSC, and investigate the outcome of simulated runs containing zero, one, two and three IMBHs. We find that IMBHs can efficiently sink to the centre of the merged cluster. If multiple merging clusters contain an IMBH, we find that an IMBH binary is likely to form and subsequently merge by gravitational wave emission. We show that these mergers are catalyzed by dynamical interactions with surrounding stars, which systematically harden the binary and increase its orbital eccentricity. The seed SMBH will be ejected from the NSC by the recoil kick produced when two IMBHs merge, if their mass ratio $q\gtrsim 0.15$. If the seed is ejected then no SMBH will form in the NSC. This is a natural pathway to explain those galactic nuclei that contain an NSC but apparently lack an SMBH, such as M33. However, if an IMBH is retained then it can seed the growth of an SMBH through gas accretion and tidal disruption of stars.

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arXiv2020

Effects of capturing a wide-orbit planet on planetary systems: system stability and Habitable Zone bombardment rates

A large fraction of stars are formed in dense clusters. In the cluster, close encounters between stars at distances less than 100 au are common. It has been shown that during close encounters planets can transfer between stars. Such captured planets will be on different orbits compared to planets formed in the system, often on very wide, eccentric and inclined orbits. We examine how these captured planets affect Kuiper-belt like asteroid belts in their new systems, and how this affects habitable planets in the system. We show that these captured planets can destabilize the asteroid belt, and we show that the fraction of the asteroid that make it past the giant planets into the system to impact the habitable planet is independent of the captured planets orbital plane, whereas the fraction of the asteroids that are removed and the rate at which they are removed depend strongly on the captured planets pericentre and inclination. We then examine all possible outcomes of planet capture and find that when a Jupiter-mass planet is captured it will in 40\% of cases destabilize the planets in the system, in 40\% of cases deplete the asteroid belt in a few Myr, i.e. not posing much risk to life on terrestrial planets which would be expected to develop later. In the final 20\% of cases the result will be a flux of impactors 5-10 times greater than that on Earth that can persist for several Gyr, quite detrimental to the development of life on the planet.

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arXiv2020

Encounters involving planetary systems in birth environments: the significant role of binaries

Most stars form in a clustered environment. Both single and binary stars will sometimes encounter planetary systems in such crowded environments. Encounter rates for binaries may be larger than for single stars, even for binary fractions as low as 10-20 per cent. In this work, we investigate scatterings between a Sun-Jupiter pair and both binary and single stars as in young clusters. We first perform a set of simulations of encounters involving wide ranges of binaries and single stars, finding that wider binaries have larger cross sections for the planet's ejection. Secondly, we consider such scatterings in a realistic population, drawing parameters for the binaries and single stars from the observed population. The scattering outcomes are diverse, including ejection, capture/exchange and collision. The binaries are more effective than single stars by a factor of several or more in causing the planet's ejection and collision. Hence, in a cluster, as long as the binary fraction is larger than about 10 per cent, the binaries will dominate the scatterings in terms of these two outcomes. For an open cluster of a stellar density 50 pc$^{-3}$, a lifetime 100 Myr and a binary fraction 0.5, we estimate that of the order of 1 per cent of the Jupiters are ejected, 0.1 per cent collide with a star, 0.1 per cent change ownership and 10 per cent of the Sun-Jupiter pairs acquire a stellar companion during scatterings. These companions are typically 1000s of au distant and in half of the cases (so 5 per cent of all Sun-Jupiter pairs), they can excite the planet's orbit through Kozai--Lidov mechanism before stripped by later encounters. Our result suggests that the Solar System may have once had a companion in its birth cluster.

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arXiv2020

Oumuamuas passing through molecular clouds

The detections of 1I/Oumuamua and 2I/Borisov within just two years demonstrate impressively that interstellar objects (ISOs) must be common in the Milky Way. Once released from their parent system, these ISOs travel for Gyr through interstellar space. While often imagined as empty, interstellar space contains gas and dust most prominent in the form of molecular clouds. Performing numerical simulations, we test how often ISOs cross such molecular clouds. We find that the ISOs pass amazingly often through molecular clouds. In the solar neighbourhood, ISOs typically spend 0.1-0.2% of their journey inside molecular clouds, for relative slow ISOs ($<$ 5 km/s) this can increase to 1-2%, equivalent to 10 - 20 Myr per Gyr. Thus the dynamically youngest ISOs spend the longest time in molecular clouds. In other words, molecular clouds must mainly contain relatively young ISOs ($<$ 1-2 Gyr). Thus the half-life of the seeding process by ISOs is substantially shorter than a stellar lifetime. The actual amount of time spent in MCs decreases with distance to the Galactic Centre. We find that ISOs pass so often through MCs that backtracing their path to find their parent star beyond 250 Myr seems beyond the point. Besides, we give a first estimate of the ISO density depending on the galactic centre distance based on the stellar distribution.

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arXiv2020

Global torques and stochasticity as the drivers of massive black hole pairing in the young Universe

The forthcoming Laser Interferometer Space Antenna (LISA) will probe the population of coalescing massive black hole (MBH) binaries up to the onset of structure formation. Here we simulate the galactic-scale pairing of $\sim10^6 M_\odot$ MBHs in a typical, non-clumpy main-sequence galaxy embedded in a cosmological environment at $z = 7-6$. In order to increase our statistical sample, we adopt a strategy that allows us to follow the evolution of six secondary MBHs concomitantly. We find that the magnitude of the dynamical-friction induced torques is significantly smaller than that of the large-scale, stochastic gravitational torques arising from the perturbed and morphologically evolving galactic disc, suggesting that the standard dynamical friction treatment is inadequate for realistic galaxies at high redshift. The dynamical evolution of MBHs is very stochastic, and a variation in the initial orbital phase can lead to a drastically different time-scale for the inspiral. Most remarkably, the development of a galactic bar in the host system either significantly accelerates the inspiral by dragging a secondary MBH into the centre, or ultimately hinders the orbital decay by scattering the MBH in the galaxy outskirts. The latter occurs more rarely, suggesting that galactic bars overall promote MBH inspiral and binary coalescence. The orbital decay time can be an order of magnitude shorter than what would be predicted relying on dynamical friction alone. The stochasticity, and the important role of global torques, have crucial implications for the rates of MBH coalescences in the early Universe: both have to be accounted for when making predictions for the upcoming LISA observatory.

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arXiv2020

Making massive stars in the Galactic Centre via accretion onto low-mass stars within an accretion disc

The origin of the population of very massive stars observed within $\sim 0.4$ pc of the supermassive black hole in the Galactic Centre is a mystery. Tidal forces from the black hole would likely inhibit {\it in situ} star formation whilst the youth of the massive stars would seem to exclude formation elsewhere followed by transportation (somehow) into the Galactic centre. Here we consider a third way to produce these massive stars from the lower-mass stars contained in the nuclear stellar cluster which surrounds the supermassive black hole. A passing gas cloud can be tidally shredded by the supermassive black hole forming an accretion disc around the black hole. Stars embedded within this accretion disc will accrete gas from the disc via Bondi-Hoyle accretion, where the accretion rate onto a star, $\dot{M}_\star \propto M_\star^2$. This super-exponential growth of accretion can lead to a steep increase in stellar masses, reaching the required 40-50 M$_\odot$ in some cases. The mass growth rate depends sensitively on the stellar orbital eccentricities and their inclinations. The evolution of the orbital inclinations and/or their eccentricities as stars are trapped by the disc, and their orbits are circularised, will increase the number of massive stars produced. Thus accretion onto low-mass stars can lead to a top heavy stellar mass function in the Galactic Centre and other galactic nuclei. The massive stars produced will pollute the environment via supernova explosions and potentially produce compact binaries whose mergers may be detectable by the LIGO-VIRGO gravitational waves observatories.

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arXiv2020

The path to instability in compact multi-planetary systems

The dynamical stability of tightly packed exoplanetary systems remains poorly understood. While for a two-planet system a sharp stability boundary exists, numerical simulations of three and more planet systems show that they can experience instability on timescales up to billions of years. Moreover, an exponential trend between the planet orbital separation measured in units of Hill radii and the survival time has been reported. While these findings have been observed in numerous numerical simulations, little is known of the actual mechanism leading to instability. Contrary to a constant diffusion process, planetary systems seem to remain dynamically quiescent for most of their lifetime before a very short unstable phase. In this work, we show how the slow chaotic diffusion due to the overlap of three-body resonances dominates the timescale leading to the instability for initially coplanar and circular orbits. While the last instability phase is related to scattering due to two-planet mean motion resonances (MMR), for circular orbits the two-planets MMR are too far separated to destabilize systems initially away from them. We develop an analytical model to generalize the empirical trend obtained for equal mass and equally-spaced planets to general systems. We obtain an analytical estimate of the survival time consistent with simulations over four orders of magnitude for the planet to star mass ratio $ε$, and 6 to 8 orders of magnitude for the instability time. We also confirm that measuring the orbital spacing in terms of Hill radii is not adapted and that the right spacing unit scales as $ε^{1/4}$. We predict that beyond a certain spacing, the three-planet resonances are not overlapped, which results in an increase of the survival time. We finally discuss the extension of our result to more general systems, containing more planets on initially non circular orbits.

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arXiv2020

Resonance in the K2-19 system is at odds with its high reported eccentricities

K2-19 hosts a planetary system composed of two outer planets, b and c, with size of $7.0\pm 0.2~R_\oplus$ and $4.1\pm0.2~R_\oplus$ , and an inner planet, d, with a radius of $1.11\pm 0.05 R_\oplus$. A recent analysis of Transit-Timing Variations (TTVs) suggested b and c are close to but not in 3:2 mean motion resonance (MMR) because the classical resonant angles circulate. Such an architecture challenges our understanding of planet formation. Indeed, planet migration through the protoplanetary disk should lead to a capture into the MMR. Here, we show that the planets are in fact, locked into the 3:2 resonance despite circulation of the conventional resonant angles and aligned periapses. However, we show that such an orbital configuration cannot be maintained for more than a few hundred million years due to the tidal dissipation experienced by planet d. The tidal dissipation remains efficient because of a secular forcing of the innermost planet eccentricity by planets b and c. While the observations strongly rule out an orbital solution where the three planets are on close to circular orbits, it remains possible that a fourth planet is affecting the TTVs such that the four planet system is consistent with the tidal constraints.

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arXiv2020

Merger rates in primordial black hole clusters without initial binaries

Primordial black holes formed through the collapse of cosmological density fluctuations have been hypothesised as contributors to the dark matter content of the Universe. At the same time, their mergers could contribute to the recently observed population of gravitational-wave sources. We investigate the scenario in which primordial black holes form binaries at late times in the Universe. Specifically, we re-examine the mergers of primordial black holes in small clusters of ~30 objects in the absence of initial binaries. Binaries form dynamically through Newtonian gravitational interactions. These binaries act as heat sources for the cluster, increasing the cluster's velocity dispersion, which inhibits direct mergers through gravitational-wave two-body captures. Meanwhile, three-body encounters of tight binaries are too rare to tighten binaries sufficiently to allow them to merge through gravitational-wave emission. We conclude that in the absence of initial binaries, merger rates of primordial black holes in the Bird et al. (2016) initial cluster configuration are at least an order of magnitude lower than previously suggested, which makes gravitational-wave detections of such sources improbable.

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arXiv2020

Capture of satellites during planetary encounters A case study of the Neptunian moons Triton and Nereid

Single-binary scattering may lead to an exchange where the single object captures a component of the binary, forming a new binary. This has been well studied in encounters between a star--planet pair and a single star. Here we explore the application of the exchange mechanism to a planet--satellite pair and another planet in the gravitational potential of a central star. As a case study, we focus on encounters between a satellite-bearing object and Neptune. We investigate whether Neptune can capture satellites from that object and if the captured satellites have orbits analogous to the Neptunian moons Triton and Nereid. Using $N$-body simulations, we study the capture probability at different encounter distances. Post-capture, we use a simple analytical argument to estimate how the captured orbits evolve under collisional and tidal effects. We find that the average capture probability reaches $\sim$$10\%$ if Neptune penetrates the donor planet's satellite system. Most moons grabbed by Neptune acquire highly eccentric orbits. Post-capture, around half of those captured, especially those on tight orbits, can be circularised, either by tides only or by collisions+tides, turning into Triton-like objects. Captures further out, on the other hand, stay on wide and eccentric orbits like that of Nereid. Both moon types can be captured in the same encounter and they have wide distributions in orbital inclination. Therefore, Triton naturally has a $\sim$50\% chance of being retrograde. A similar process potentially applies to an exoplanetary system, and our model predicts that exomoons can jump from one planet to another during planetary scattering. Specifically, there should be two distinct populations of captured moons: one on close-in circular orbits and the other on far-out eccentric orbits. The two populations may have highly inclined prograde or retrograde orbits.

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arXiv2020

Fly-by encounters between two planetary systems II: Exploring the interactions of diverse planetary system architectures

Planetary systems formed in clusters may be subject to stellar encounter flybys. Here we create a diverse range of representative planetary systems with different orbital scales and planets' masses and examine encounters between them in a typical open cluster. We first explore the close-in multi-super earth systems $\lesssim0.1$ au. They are resistant to flybys in that only ones inside a few au can destabilise a planet or break the resonance between such planets. But these systems may capture giant planets onto wide orbits from the intruding star during distant flybys. If so, the original close-in small planets' orbits may be tilted together through Kozai--Lidov mechanism, forming a ``cold'' system that is significantly inclined against the equator of the central host. Moving to the intermediately-placed planets around solar-like stars, we find that the planets' mass gradient governs the systems' long-term evolution post-encounter: more massive planets have better chances to survive. Also, a system's angular momentum deficit, a quantity describing how eccentric/inclined the orbits are, measured immediately after the encounter, closely relates to the longevity of the systems -- whether or not and when the systems turn unstable in the ensuing evolution millions of years post-encounter. We compare the orbits of the surviving planets in the unstable systems through (1) the immediate consequence of the stellar fly or (2) internal interplanetary scattering long post-encounter and find that those for the former are systematically colder. Finally, we show that massive wide-orbit multi-planet systems like that of HR 8799 can be easily disrupted and encounters at a few hundreds of au suffice.

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