arXiv subjects

Rosemary A. Mardling

Publications and source records attributed to Rosemary A. Mardling.

At least 19 recordsLinked to original sources

arXiv2023

Analysis of Kozai Cycles in Equal-Mass Hierarchical Triple Supermassive Black Hole Mergers in the Presence of a Stellar Cluster

Supermassive black holes (SMBHs) play an important role in galaxy evolution. Binary and triple SMBHs can form after galaxy mergers. A third SMBH may accelerate the SMBH merging process, possibly through the Kozai mechanism. We use N -body simulations to analyze oscillations in the orbital elements of hierarchical triple SMBHs with surrounding star clusters in galaxy centers. We find that SMBH triples spend only a small fraction of time in the hierarchical merger phase (i.e., a binary SMBH with a distant third SMBH perturber). Most of the time, the enclosed stellar mass within the orbits of the innermost or the outermost SMBH is comparable to the SMBH masses, indicating that the influence of the surrounding stellar population cannot be ignored. We search for Eccentric Kozai-Lidov (EKL) oscillations for which (i) the eccentricity of the inner binary and inclination are both oscillate and are anti-phase or in-phase and (ii) the oscillation period is consistent with EKL timescale. We find that EKL oscillations are short-lived and rare: the triple SMBH spends around 3% of its time in this phase over the ensemble of simulations, reaching around 8% in the best-case scenario. This suggests that the role of the EKL mechanism in accelerating the SMBH merger process may have been overestimated in previous studies. We follow-up with three-body simulations, using initial conditions extracted from the simulation, and the result can to some extent repeat the observed EKL-like oscillations. This comparison provides clues about why those EKL oscillations with perturbing stars are short-lived.

astro-ph.GA↗

arXiv2023

Quadruple-star systems are not always nested triples: a machine learning approach to dynamical stability

The dynamical stability of quadruple-star systems has traditionally been treated as a problem involving two `nested' triples which constitute a quadruple. In this novel study, we employed a machine learning algorithm, the multi-layer perceptron (MLP), to directly classify 2+2 and 3+1 quadruples based on their stability (or long-term boundedness). The training data sets for the classification, comprised of $5\times10^5$ quadruples each, were integrated using the highly accurate direct $N$-body code MSTAR. We also carried out a limited parameter space study of zero-inclination systems to directly compare quadruples to triples. We found that both our quadruple MLP models perform better than a `nested' triple MLP approach, which is especially significant for 3+1 quadruples. The classification accuracies for the 2+2 MLP and 3+1 MLP models are 94% and 93% respectively, while the scores for the `nested' triple approach are 88% and 66% respectively. This is a crucial implication for quadruple population synthesis studies. Our MLP models, which are very simple and almost instantaneous to implement, are available on GitHub, along with Python3 scripts to access them.

cs.LG↗

arXiv2022

Algebraic and machine learning approach to hierarchical triple-star stability

We present two approaches to determine the dynamical stability of a hierarchical triple-star system. The first is an improvement on the Mardling-Aarseth stability formula from 2001, where we introduce a dependence on inner orbital eccentricity and improve the dependence on mutual orbital inclination. The second involves a machine learning approach, where we use a multilayer perceptron (MLP) to classify triple-star systems as `stable' and `unstable'. To achieve this, we generate a large training data set of 10^6 hierarchical triples using the N-body code MSTAR. Both our approaches perform better than previous stability criteria, with the MLP model performing the best. The improved stability formula and the machine learning model have overall classification accuracies of 93 % and 95 % respectively. Our MLP model, which accurately predicts the stability of any hierarchical triple-star system within the parameter ranges studied with almost no computation required, is publicly available on Github in the form of an easy-to-use Python script.

astro-ph.SR↗

arXiv2021

The Stability Boundary of the Distant Scattered Disk

The scattered disk is a vast population of trans-Neptunian minor bodies that orbit the sun on highly elongated, long-period orbits. The stability of scattered disk objects is primarily controlled by a single parameter - their perihelion distance. While the existence of a perihelion boundary that separates chaotic and regular motion of long-period orbits is well established through numerical experiments, its theoretical basis as well as its semi-major axis dependence remain poorly understood. In this work, we outline an analytical model for the dynamics of distant trans-Neptunian objects and show that the orbital architecture of the scattered disk is shaped by an infinite chain of $2:j$ resonances with Neptune. The widths of these resonances increase as the perihelion distance approaches Neptune's semi-major axis, and their overlap drives chaotic motion. Within the context of this picture, we derive an analytic criterion for instability of long-period orbits, and demonstrate that rapid dynamical chaos ensues when the perihelion drops below a critical value, given by $q_{\rm{crit}}=a_{\rm{N}}\,\big(\ln((24^2/5)\,(m_{\rm{N}}/M_{\odot})\,(a/a_{\rm{N}})^{5/2})\big)^{1/2}$. This expression constitutes a boundary between the "detached" and actively "scattering" sub-populations of distant trans-Neptunian minor bodies. Additionally, we find that within the stochastic layer, the Lyapunov time of scattered disk objects approaches the orbital period, and show that the semi-major axis diffusion coefficient is approximated by $\mathcal{D}_a\sim(8/(5\,π))\,(m_{\rm{N}}/M_{\odot})\,\sqrt{\mathcal{G}\,M_{\odot}\,a_{\rm{N}}}\,\exp\big[-(q/a_{\rm{N}})^2/2\big]$. We confirm our results with numerical simulations and highlight the connections between scattered disk dynamics and the Chirikov Standard Map. Implications of our results for the long-term evolution of the distant solar system are discussed.

astro-ph.EP↗

arXiv2019

Gravitational waves from dynamical tides in white dwarf binaries

We study the effect of tidal forcing on gravitational wave signals from tidally relaxed white dwarf pairs in the LISA, DECIGO and BBO frequency band ($0.1-100\,{\rm mHz}$). We show that for stars not in hydrostatic equilibrium (in their own rotating frames), tidal forcing will result in energy and angular momentum exchange between the orbit and the stars, thereby deforming the orbit and producing gravitational wave power in harmonics not excited in perfectly circular synchronous binaries. This effect is not present in the usual orbit-averaged treatment of the equilibrium tide, and is analogous to transit timing variations in multiplanet systems. It should be present for all LISA white dwarf pairs since gravitational waves carry away angular momentum faster than tidal torques can act to synchronize the spins, and when mass transfer occurs as it does for at least eight LISA verification binaries. With the strain amplitudes of the excited harmonics depending directly on the density profiles of the stars, gravitational wave astronomy offers the possibility of studying the internal structure of white dwarfs, complimenting information obtained from asteroseismology of pulsating white dwarfs. Since the vast majority of white-dwarf pairs in this frequency band are expected to be in the quasi-circular state, we focus here on these binaries, providing general analytic expressions for the dependence of the induced eccentricity and strain amplitudes on the stellar apsidal motion constants and their radius and mass ratios. Tidal dissipation and gravitation wave damping will affect the results presented here and will be considered elsewhere.

astro-ph.HE↗

arXiv2015

Dynamical Interactions Make Hot Jupiters in Open Star Clusters

Explaining the origin and evolution of exoplanetary "hot Jupiters" remains a significant challenge. One possible mechanism for their production is planet-planet interactions, which produces hot Jupiters from planets born far from their host stars but near their dynamical stability limits. In the much more likely case of planets born far from their dynamical stability limits, can hot Jupiters can be formed in star clusters? Our N-body simulations of planetary systems inside star clusters answer this question in the affirmative, and show that hot Jupiter formation is not a rare event. We detail three case studies of the dynamics-induced births of hot Jupiters on highly eccentric orbits that can only occur inside star clusters. The hot Jupiters' orbits bear remarkable similarities to those of some of the most extreme exoplanets known: HAT-P-32 b, HAT-P-2 b, HD 80606 b and GJ 876 d. If stellar perturbations formed these hot Jupiters then our simulations predict that these very hot, inner planets are often accompanied by much more distant gas giants in highly eccentric orbits.

astro-ph.EP↗

arXiv2013

New developments for modern celestial mechanics. I. General coplanar three-body systems. Application to exoplanets

Modern applications of celestial mechanics include the study of closely packed systems of exoplanets, circumbinary planetary systems, binary-binary interactions in star clusters, and the dynamics of stars near the galactic centre. While developments have historically been guided by the architecture of the Solar System, the need for more general formulations with as few restrictions on the parameters as possible is obvious. Here we present clear and concise generalisations of two classic expansions of the three-body disturbing function, simplifying considerably their original form and making them accessible to the non-specialist. Governing the interaction between the inner and outer orbits of a hierarchical triple, the disturbing function in its general form is the conduit for energy and angular momentum exchange and as such, governs the secular and resonant evolution of the system and its stability characteristics. Focusing here on coplanar systems, the first expansion is one in the ratio of inner to outer semimajor axes and is valid for all eccentricities, while the second is an expansion in eccentricity and is valid for all semimajor axis ratios [...]. Our generalizations make both formulations valid for arbitrary mass ratios. [...]. We demonstrate the equivalence of the new expansions, identifying the role of the spherical harmonic order m in both and its physical significance in the three-body problem, and introducing the concept of principal resonances. Several examples of the accessibility of both expansions are given including resonance widths and the secular rates of change of the elements. Results in their final form are gathered together at the end of the paper for the reader mainly interested in their application, including a guide for the choice of expansion.

astro-ph.EP↗

arXiv2010

Confirmation of a Retrograde Orbit for Exoplanet WASP-17b

We present high-precision radial velocity observations of WASP-17 throughout the transit of its close-in giant planet, using the MIKE spectrograph on the 6.5m Magellan Telescope at Las Campanas Observatory. By modeling the Rossiter-McLaughlin effect, we find the sky-projected spin-orbit angle to be lambda = 167.4 \pm 11.2 deg. This independently confirms the previous finding that WASP-17b is on a retrograde orbit, suggesting it underwent migration via a mechanism other than just the gravitational interaction between the planet and the disk. Interestingly, our result for lambda differs by 45 \pm 13 deg from the previously announced value, and we also find that the spectroscopic transit occurs 15 \pm 5 min earlier than expected, based on the published ephemeris. The discrepancy in the ephemeris highlights the need for contemporaneous spectroscopic and photometric transit observations whenever possible.

astro-ph.EP↗

arXiv2010

On the Possibility of Tidal Formation of Binary Planets Around Ordinary Stars

The planet formation process and subsequent planet migration may lead to configurations resulting in strong dynamical interactions among the various planets. Well-studied possible outcomes include collisions between planets, scattering events that eject one or more of the planets, and a collision of one or more of the planets with the parent star. In this work we consider one other possibility that has seemingly been overlooked in the various scattering calculations presented in the literature: the tidal capture of two planets which leads to the formation of a binary planet (or binary brown dwarf) in orbit about the parent star. We carry out extensive numerical simulations of such dynamical and tidal interactions to explore the parameter space for the formation of such binary planets. We show that tidal formation of binary planets is possible for typical planet masses and distances from the host star. The detection (or lack thereof) of planet-planet binaries can thus be used to constrain the properties of planetary systems, including their mutual spacing during formation, and the fraction of close planets in very eccentric orbits which are believed to form by a closely related process.

astro-ph.EP↗

arXiv2010

The determination of planetary structure in tidally relaxed inclined systems

[Abridged] The recent discovery of a transiting planet on a non-circular orbit with a massive highly eccentric companion orbiting HAT-P-13 offers the possibility of probing the structure of the short-period planet. The ability to do this relies on the system being in a quasi-equilibrium state in the sense that the eccentricities are constant on the usual secular timescale, and decay on a timescale which is much longer than the age of the system. Since the equilibrium eccentricity is effectively a function only of observable system parameters and the unknown Love number of the short-period planet, the latter can be determined with accurate measurements of the planet's eccentricity and radius. However, this analysis relies on the unlikely assumption that the system is coplanar. Here we generalize our recent analysis of this fixed-point phenomenon to mutually inclined systems and show that the fixed point of coplanar systems is replaced by a limit cycle, with the average value of the eccentricity decreasing and its amplitude of variation increasing with increasing mutual inclination. This behaviour significantly reduces the ability to unambiguously determine the Love number of the short-period planet if the mutual inclination is higher than around 10^o. We show that for Q-values less than 10^6, the HAT-P-13 system cannot have a mutual inclination between 54 and 126^o because Kozai oscillations coupled with tidal dissipation would act to quickly move the inclination outside this range, and that the behaviour of retrograde systems is the mirror image of that for prograde systems. We derive a relationship between the equilibrium radius of the short-period planet, its Q-value and its core mass, and show that given current estimates of e_b and the planet radius, the HAT-P-13 system is likely to be close to coplanar [...]

astro-ph.EP↗

arXiv2009

Signatures of resonant terrestrial planets in long-period systems

The majority of extrasolar planets discovered to date have significantly eccentric orbits, some if not all of which may have been produced through planetary migration. During this process, any planets interior to such an orbit would therefore have been susceptible to resonance capture, and hence may exhibit measurable orbital period variations. Here we summarize the results of our investigation into the possibility of detecting low-mass planets which have been captured into the strong 2:1 resonance. Using analytical expressions together with simulated data we showed that it is possible to identify the existence of a low-mass companion in the internal 2:1 resonance by estimating the time-dependant orbital period for piecewise sections of radial velocity data. This works as long as the amplitude of modulation of the orbital period is greater than its uncertainty, which in practice means that the system should not be too close to exact resonance. Here we provide simple expressions for the libration period and the change in the observed orbital period, these being valid for arbitrary eccentricities and planet masses. They in turn allow one to constrain the mass and eccentricity of a companion planet if the orbital period is sufficiently modulated.

astro-ph.EP↗

arXiv2008

Possibility of Detecting Moons of Pulsar Planets Through Time-of-Arrival Analysis

The perturbation caused by planet-moon binarity on the time-of-arrival signal of a pulsar with an orbiting planet is derived for the case in which the orbits of the moon and the planet-moon barycenter are both circular and coplanar. The signal consists of two sinusoids with frequency (2n_p - 3n_b) and (2n_p - n_b ), where n_p and n_b are the mean motions of the planet and moon around their barycenter, and the planet-moon system around the host, respectively. The amplitude of the signal is equal to the fraction sin I[9(M_p M_m)/16(M_p + M_m)^2] [r/R]^5 of the system crossing time R/c, where M_p and M_m are the the masses of the planet and moon, r is their orbital separation, R is the distance between the host pulsar and planet-moon barycenter, I is the inclination of the orbital plane of the planet, and c is the speed of light. The analysis is applied to the case of PSR B1620-26 b, a pulsar planet, to constrain the orbital separation and mass of any possible moons. We find that a stable moon orbiting this pulsar planet could be detected, if the moon had a separation of about one fiftieth of that of the orbit of the planet around the pulsar, and a mass ratio to the planet of ~5% or larger.

astro-ph↗

arXiv2008

On the long-term tidal evolution of GJ 436b in the presence of a resonant companion

In order to explain the significant orbital eccentricity of the short-period transiting Neptune-mass planet GJ 436b and at the same time satisfy various observational constraints and anomalies, Ribas, Font-Ribera and Beaulieu have proposed the existence of an eccentric low-mass companion planet at the position of the outer 2:1 resonance. The authors demonstrate the viability of their proposal using point-mass three-body integrations, arguing that as long as the system appears to be dynamically stable, the short-term secular variations ought to dominate the long-term dissipative evolution. Here we demonstrate that if one includes tidal dissipation, both orbits circularize after a few times the circularization timescale of the inner planet. We conclude that with or without a nearby companion planet, in or out of the 2:1 resonance, the Q-value of GJ 436b must be near the upper bound estimate for Neptune if the system is as young as 1 Gyr, and an order of magnitude higher if the system is as old as 10 Gyr. We show detail of passage through resonance and conclude that even out of resonance, a companion planet should still be detectable through transit timing variations.

astro-ph↗

arXiv2007

Inflated planets and their low-mass companions

Various mechanisms have been proposed to explain the inflated size of HD 209458b after it became clear that it has no companions capable of producing a stellar reflex velocity greater than around 5 m/s. Had there been such a companion, the hypothesis that it forces the eccentricity of the inflated planet thereby tidally heating it may have been readily accepted. Here we summarize a paper by the author which shows that companion planets with masses as low as a fraction of an Earth mass are capable of sustaining a non-zero eccentricity in the observed planet for at least the age of the system. While such companions produce stellar reflex velocities which are fractions of a meter per second and hence are below the stellar jitter limit, they are consistent with recent theoretical work which suggests that the planet migration process often produces low-mass companions to short-period giants.

astro-ph↗

arXiv2007

Long-term tidal evolution of short-period planets with companions

Of the fourteen transiting extrasolar planetary systems for which radii have been measured, at least three appear to be considerably larger than theoretical estimates suggest. It has been proposed by Bodenheimer, Lin & Mardling that undetected companions acting to excite the orbital eccentricity are responsible for these oversized planets, as they find new equilibrium radii in response to being tidally heated. In the case of HD 209458, this hypothesis has been rejected by some authors because there is no sign of such a companion at the 5 m/s level, and because it is difficult to say conclusively that the eccentricity is non-zero. Transit timing analysis [...]. Whether or not a companion is responsible for the large radius of HD 209458b, almost certainly some short-period systems have companions which force their eccentricities to nonzero values. This paper is dedicated to quantifying this effect. The eccentricity of a short-period planet will only be excited as long as its (non-resonant) companion's eccentricity is non-zero. Here we show that the latter decays on a timescale which depends on the structure of the interior planet, a timescale which is often shorter than the lifetime of the system. This includes Earth-mass planets in the habitable zones of some stars. We determine which configurations are capable of sustaining significant eccentricity for at least the age of the system, and show that these include systems with companion masses as low as a fraction of an Earth mass. The orbital parameters of such companions are consistent with recent calculations which show that the migration process can induce the formation of low mass planets external to the orbits of hot Jupiters. Systems with inflated planets are therefore good targets in the search for terrestrial planets.

astro-ph↗

arXiv2004

Spin-Orbit Evolution Of Short-Period Planets

The negligible eccentricity of all extra solar planets with periods less than six days can be accounted for by dissipation of tidal disturbances within their envelopes which are induced by their host stars. In the period range of 7-21 days, planets with circular orbits coexist with planets with eccentric orbits. These will be referred to as the borderline planets. We propose that this discrepancy can be attributed to the variation in spin-down rates of young stars. In particular, prior to spin-down, dissipation of a planet's tidal disturbance within the envelope of a sufficiently rapidly spinning star can excite eccentricity growth, and for a more slowly spinning star, at least reduce the eccentricity damping rate. In contrast, tidal dissipation within the envelope of a slowly spinning low-mass mature star can enhance the eccentricity damping process. Based on these results, we suggest that short-period planets around relatively young stars may have a much larger dispersion in eccentricity than those around mature stars. We also suggest that because the rate of angular momentum loss from G and K dwarfs via stellar winds is much faster than the tidal transfer of angular momentum between themselves and their very-short (3-4 days) period planets, they cannot establish a dynamical configuration in which the stellar and planetary spins are approximately parallel and synchronous with the orbital frequency. In principle, however, such configurations may be established for planets (around G and K dwarfs) with orbital periods of up to several weeks. In contrast to G and K dwarfs, the angular momentum loss due to stellar winds is much weaker in F dwarfs. It is therefore possible for synchronized short-period planets to exist around such stars. The planet around Tau Boo is one such example.

astro-ph↗

arXiv2004

On the Survival of Short-Period Terrestrial Planets

The currently feasible method of detection of Earth-mass planets is transit photometry, with detection probability decreasing with a planet's distance from the star. The existence or otherwise of short-period terrestrial planets will tell us much about the planet formation process, and such planets are likely to be detected first if they exist. Tidal forces are intense for short-period planets, and result in decay of the orbit on a timescale which depends on properties of the star as long as the orbit is circular. However, if an eccentric companion planet exists, orbital eccentricity ($e_i$) is induced and the decay timescale depends on properties of the short-period planet, reducing by a factor of order $10^5 e_i^2$ if it is terrestrial. Here we examine the influence companion planets have on the tidal and dynamical evolution of short-period planets with terrestrial structure, and show that the relativistic potential of the star is fundamental to their survival.

astro-ph↗

arXiv2002

A Critical Examination of Li Pollution and Giant Planet Consumption by a Host Star

We examine the overall likelihood that the consumption of a giant planet could pollute its host star. First, we show that the detection of 6Li in SWPs could be used to distinguish between giant planet formation theories, and can be used to detect the consumption of giant planets independent of planet mass. Second, we examine the probability that giant planets between 1 and 3 M_J could be destroyed in the outer convection zone of stars slightly more massive than the Sun, finding that heated giant planets would be efficiently destroyed near the surface of the star, while the cores of cold giant planets may be able to survive to the base of the star's convection zone. Third, we examine the tidal interaction between a planet and its host star, showing that this interaction can lead to tidal disruption of the planet and accretion of its material, or orbital decay followed by hydrodynamical interaction. Throughout, we consider the case of HD 82943, a star known to have two planets and having a preliminary detection of 6Li. Using stellar models including diffusion, we estimate the mass of the HD 82943 to be about 1.2 M_sun and its age to be about 0.5-1.5 Gyr. The observed 7Li abundance for HD 82943 is consistent with stars of similar T_eff and age in the open cluster NGC 752. We describe a possible dynamical history for a hypothetical planet in the presence of the two resonant planets currently known. (abridged)

astro-ph↗