arXiv subjects

Beibei Liu

Publications and source records attributed to Beibei Liu.

At least 73 records · Page 4Linked to original sources

arXiv2019

Surgery on links of linking number zero and the Heegaard Floer $d$-invariant

We study Heegaard Floer homology and various related invariants (such as the $h$-function) for two-component L-space links with linking number zero. For such links, we explicitly describe the relationship between the $h$-function, the Sato-Levine invariant and the Casson invariant. We give a formula for the Heegaard Floer $d$-invariants of integral surgeries on two-component L-space links of linking number zero in terms of the $h$-function, generalizing a formula of Ni and Wu. As a consequence, for such links with unknotted components, we characterize L-space surgery slopes in terms of the $ν^{+}$-invariants of the knots obtained from blowing down the components. We give a proof of a skein inequality for the $d$-invariants of $+1$ surgeries along linking number zero links that differ by a crossing change. We also describe bounds on the smooth four-genus of links in terms of the $h$-function, expanding on previous work of the second author, and use these bounds to calculate the four-genus in several examples of links.

math.GT↗

arXiv2019

Hausdorff dimension of non-conical limit sets

Geometrically infinite Kleinain groups have nonconical limit sets with the cardinality of the continuum. In this paper, we construct a geometrically infinite Fuchsian group such that the Hausdorff dimension of the nonconical limit set equals zero. For finitely generated, geometrically infinite Kleinian groups, we prove that the Hausdorff dimension of the nonconical limit set is positive.

math.GR↗

arXiv2019

Super-Earth masses sculpted by pebble isolation around stars of different masses

We develop a pebble-driven model to study the formation and evolution of planets around stars in the mass range of 0.08 and 1 solar mass. The growth and migration of a large number of individual protoplanetary embryos are simulated in a population synthesis manner. We test two hypotheses for the birth locations of embryos: at the water ice line or log-uniformly distributed over entire protoplanetary disks. Two types of disks with different turbulent viscous parameters alpha of 1e-3 and 1e-4 are investigated, to shed light on the role of outward migration of protoplanets. The forming planets are compared with the observed exoplanets in terms of masses, semimajor axes, metallicities, and water contents. We find that gas giant planets are likely to form when the characteristic disk sizes are larger, the disk accretion rates are higher, the disks are more metal-rich and/or their stellar hosts are more massive. Our model shows that 1) the characteristic mass of super-Earth is set by the pebble isolation mass. Super-Earth masses increase linearly with the mass of its stellar host, corresponding to one Earth mass around a late M-dwarf star and 20 Earth masses around a solar-mass star. 2) The low-mass planets up to 20 Earth masses can form around stars with a wide range of metallicities, while massive gas giant planets are preferred to grow around metal-rich stars. 3) Super-Earth planets that are mainly composed of silicates, with relatively low water fractions can form from protoplanetary embryos at the water ice line in weakly turbulent disks where outward migration is suppressed. However, if the embryos are formed over a wide range of radial distances, the super-Earths would end up having a distinctive, bimodal composition in water mass. Our model succeeds in quantitatively reproducing several important observed properties of exoplanets and correlations with their stellar hosts.

astro-ph.EP↗

arXiv2019

Pebble-driven planet formation for TRAPPIST-1 and other compact systems

Recently, seven Earth-sized planets were discovered around the M-dwarf star TRAPPIST-1. Thanks to transit-timing variations, the masses and therefore the bulk densities of the planets have been constrained, suggesting that all TRAPPIST-1 planets are consistent with water mass fractions on the order of 10%. These water fractions, as well as the similar planet masses within the system, constitute strong constraints on the origins of the TRAPPIST-1 system. In a previous work, we outlined a pebble-driven formation scenario. In this paper we investigate this formation scenario in more detail. We used a Lagrangian smooth-particle method to model the growth and drift of pebbles and the conversion of pebbles to planetesimals through the streaming instability. We used the N-body code \texttt{MERCURY} to follow the composition of planetesimals as they grow into protoplanets by merging and accreting pebbles. This code is adapted to account for pebble accretion, type-I migration, and gas drag. In this way, we modelled the entire planet formation process (pertaining to planet masses and compositions, not dynamical configuration). We find that planetesimals form in a single, early phase of streaming instability. The initially narrow annulus of planetesimals outside the snowline quickly broadens due to scattering. Our simulation results confirm that this formation pathway indeed leads to similarly-sized planets and is highly efficient in turning pebbles into planets ($\sim$50% solids-to-planets conversion efficiency). [...] The water content of planets resulting from our simulations is on the order of 10%, and our results predict a `V-shaped' trend in the planet water fraction with orbital distance: from relatively high (innermost planets) to relatively low (intermediate planets) to relatively high (outermost planets).

astro-ph.EP↗

arXiv2019

L-space surgeries on 2-component L-space links

In this paper, we analyze L-space surgeries on two component L-space links. We show that if one surgery coefficient is negative for the L-space surgery, then the corresponding link component is an unknot. If the link admits very negative (i.e. $d_{1}, d_{2}\ll0$) L-space surgeries, it is the Hopf link. We also give a way to characterize the torus link $T(2, 2l)$ by observing an L-space surgery $S^{3}_{d_{1}, d_{2}}(\mathcal{L})$ with $d_{1}d_{2}<0$ on a 2-component L-space link with unknotted components. For some 2-component L-space links, we give explicit descriptions of the L-space surgery sets.

math.GT↗

arXiv2019

Variational Representation Learning for Vehicle Re-Identification

Vehicle Re-identification is attracting more and more attention in recent years. One of the most challenging problems is to learn an efficient representation for a vehicle from its multi-viewpoint images. Existing methods tend to derive features of dimensions ranging from thousands to tens of thousands. In this work we proposed a deep learning based framework that can lead to an efficient representation of vehicles. While the dimension of the learned features can be as low as 256, experiments on different datasets show that the Top-1 and Top-5 retrieval accuracies exceed multiple state-of-the-art methods. The key to our framework is two-fold. Firstly, variational feature learning is employed to generate variational features which are more discriminating. Secondly, long short-term memory (LSTM) is used to learn the relationship among different viewpoints of a vehicle. The LSTM also plays as an encoder to downsize the features.

cs.CV↗

arXiv2019

Growth after the streaming instability: from planetesimal accretion to pebble accretion

Streaming instability is a key mechanism in planet formation, clustering pebbles into planetesimals. It is triggered at a particular disk location where the local volume density of solids exceeds that of the gas. After their formation, planetesimals can grow by feeding from other planetesimals in the birth ring as well as by accreting inwardly drifting pebbles from the outer disk. To investigate the growth of planetesimals at a single location by the streaming instability, we test the conditions under which super-Earths are able to form within the lifetime of the gaseous disk. We modify the \texttt{Mercury} N-body code to trace the growth and dynamical evolution of a swarm of planetesimals at the ice line for a solar-mass star. Three distributions of planetesimal sizes are investigated: (i) a mono-dispersed population of 400 km radius planetesimals, (ii) a poly-dispersed populations of planetesimals from 200 km up to 1000 km, (iii) a bimodal distribution with a single runaway body and a swarm of smaller, 100 km size planetesimals. The mono-disperse population of 400 km size planetesimals cannot form $\gtrsim$ Earth mass protoplanets. Their velocity dispersions are quickly excited, which suppresses both planetesimal and pebble accretion. Planets can form from the poly-dispersed and bimodal distributions. In these circumstances, the two-component nature damps the random velocity of the large embryo by small planetesimals' dynamical friction, allowing the embryo to accrete pebbles efficiently when it approaches $10^{-2}$ Earth mass. We find that super-Earth planets are preferred to form when the pebble mass flux is higher, the disk turbulence is lower, or the Stokes number of the pebbles is higher.

astro-ph.EP↗

arXiv2019

Geometric infiniteness in negatively pinched Hadamard manifolds

We generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, $\mathbb{C}$) to geometrically infinite discrete isometry subgroups in the case of rank 1 symmetric spaces, and, under the assumption of bounded torsion, to the case of negatively pinched Hadamard manifolds. Every such geometrically infinite isometry subgroup $Γ$ has a set of nonconical limit points with cardinality of continuum.

math.GR↗

arXiv2018

Geometric finiteness in negatively pinched Hadamard manifolds

In this paper, we generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, $\mathbb{C}$) to geometrically infinite discrete subgroups $Γ$ of isometries of negatively pinched Hadamard manifolds $X$. We then generalize a theorem of Bishop to prove that every discrete geometrically infinite isometry subgroup $Γ$ has a set of nonconical limit points with the cardinality of the continuum.

math.GR↗

arXiv2018

Ping-pong in Hadamard manifolds

In this paper, we prove a quantitative version of the Tits alternative for negatively pinched manifolds $X$. Precisely, we prove that a nonelementary discrete isometry subgroup of $\mathrm{Isom}(X)$ generated by two non-elliptic isometries $g$, $f$ contains a free subgroup of rank $2$ generated by isometries $f^N , h$ of uniformly bounded word length. Furthermore, we show that this free subgroup is convex-cocompact when $f$ is hyperbolic.

math.GR↗

arXiv2018

Four genera of links and Heegaard Floer homology

For links with vanishing pairwise linking numbers, the link components bound pairwise disjoint surfaces in $B^{4}$. In this paper, we describe the set of genera of such surfaces in terms of the $h$-function, which is a link invariant from Heegaard Floer homology. In particular, we use the $h$-function to give lower bounds for the 4-genus of the link. For $L$-space links, the $h$-function is explicitly determined by Alexander polynomials of the link and sublinks. We show some $L$-space links where the lower bounds are sharp, and also describe all possible genera of disjoint surfaces bounded by such links.

math.GT↗

arXiv2018

Catching drifting pebbles I. Enhanced pebble accretion efficiencies for eccentric planets

Coagulation theory predicts that micron-sized dust grains grow into pebbles which drift inward towards the star, when they reach sizes of mm-cm. When they cross the orbit of a planet, a fraction of these drifting pebbles will be accreted. In the pebble accretion mechanism, the combined effects of the planet's gravitational attraction and gas drag greatly increase the accretion rate. We calculate the pebble accretion efficiency $\varepsilon_{2D}$ -- the probability a pebble is accreted by the planet -- in the 2D limit (pebbles reside in the midplane). In particular, we investigate the dependence of $\varepsilon_{2D}$ on the planet eccentricity and its implications for planet formation models. We conduct N-body simulations to calculate the pebble accretion efficiency in both the local frame and the global frame. With the global method we investigate the pebble accretion efficiency when the planet is on an eccentric orbit. We find that the local and the global methods generally give consistent results. The efficiency increases with the planet's eccentricity once the relative velocity between the pebble and the planet is determined by the planet's eccentric velocity. At large eccentricities, however, the relative velocity becomes too large for pebble accretion. The efficiency then drops significantly and the accretion enters the ballistic regime. We present general expressions for $\varepsilon_{2D}$. Applying the obtained formula to the formation of a secondary planet, in resonance with an already-formed giant planet, we find that the embryo grows quickly due to its larger eccentricity. The maximum $\varepsilon_{2D}$ for a planet on an eccentric orbit is several times higher than for a planet on a circular orbit, but this increase gives the planet an important headstart and boosts its following mass growth.

astro-ph.EP↗

arXiv2018

Catching drifting pebbles II. A stochastic equation of motions for pebbles

Turbulence plays a key role in the transport of pebble-sized particles. It also affects the ability of pebbles to be accreted by protoplanets, because it stirs pebbles out of the disk midplane. In addition, turbulence can suppress pebble accretion once the relative velocities become too large for the settling mechanism to be viable. Following Paper I, we aim to quantify these effects by calculating the pebble accretion efficiency $\varepsilon$ using three-body simulations. To model the effect of turbulence on the pebbles, we derive a stochastic equation of motion (SEOM) applicable to stratified disk configurations. In the strong coupling limit (ignoring particle inertia) the limiting form of this equation agrees with previous works. We conduct a parameter study and calculate $\varepsilon$ in 3D, varying pebble and gas (turbulence) properties and accounting for the planet inclination. We find that strong turbulence suppresses pebble accretion through turbulent diffusion, agreeing within factors of order unity with previous works. Another reduction of $\varepsilon$ occurs when the turbulent rms motions are large and the settling mechanism fails. Efficiency-wise, the outer disk regions are more affected by turbulence than the inner regions. At the location of the H$_2$O iceline, planets around low-mass stars achieve much higher efficiencies. Including the results from Paper I, we present a framework to obtain $\varepsilon$ under general circumstances.

astro-ph.EP↗

arXiv2017

Dynamical rearrangement of super-Earths during disk dispersal II. Assessment of the magnetospheric rebound model for planet formation scenarios

Context.The Kepler mission has provided a large sample to statistically analyze the orbital properties of the super-Earth planets. We hypothesize that these planets formed early and consider the problem of matching planet formation theory to the current observations. Two scenarios, disk migration and in-situ formation, have been proposed to explain their origin. In the migration scenario planets migrate inward due to planet-disk interaction, whereas in the in-situ scenario planets assemble locally. Therefore, planets formed by migration are expected to end up in resonances, whereas those formed in-situ are expected to stay in short period ratios and in non-resonant orbits. Both predictions are at odds with observations. Aims. We investigate whether a preferred formation scenario can be identified through a comparison between the magnetospheric rebound model and the Kepler data. Methods. We conduct N-body simulations of two-planet systems during the disk dispersal phase, and make a statistical comparison between the simulations and the Kepler observations. Results. Comparing the two scenarios, we find that magnetospheric rebound tends to erase the difference in the orbital configuration that was initially presented. After disk dispersal, not all planets are in resonance in the migration scenario, whereas planets do not remain in compact configurations in the in-situ scenario. In both scenarios, the orbits of planets increase with the cavity expansion, and their period ratios have a wider distribution. Conclusions. From a statistical perspective, the magnetospheric rebound model reproduces several observed properties of Kepler planets, such as the significant number of planets are not in resonances and planet pairs can end up at large period ratios. The disparity in orbital configuration between the two formation scenarios is substantially reduced after disk dispersal.

astro-ph.EP↗

arXiv2017

Formation of TRAPPIST-1 and other compact systems

TRAPPIST-1 is a nearby 0.08 M M-star, which was recently found to harbor a planetary system of at least seven Earth-mass planets, all within 0.1 au. The configuration confounds theorists as the planets are not easily explained by either in situ or migration models. In this Paper we present a scenario for the formation and orbital architecture of the TRAPPIST-1 system. In our model, planet formation starts at the H2O iceline, where pebble-size particles -- whose origin is the outer disk -- concentrate to trigger streaming instabilities. After their formation, planetary embryos quickly mature by pebble accretion. Planet growth stalls at Earth masses, where the planet's gravitational feedback on the disk keeps pebbles at bay. Planets are transported by Type I migration to the inner disk, where they stall at the magnetospheric cavity and end up in mean motion resonances. During disk dispersal, the cavity radius expands and the inner-most planets escape resonance. We argue that the model outlined here can also be applied to other compact systems and that the many close-in super-Earth systems are a scaled-up version of TRAPPIST-1. We also hypothesize that few close-in compact systems harbor giant planets at large distances, since they would have stopped the pebble flux from the outer disk.

astro-ph.EP↗

arXiv2017

Heegaard Floer homology of L-space links with two components

We compute different versions of link Floer homology $HFL^{-}$ and $\widehat{HFL}$ for any $L$-space link with two components. The main approach is to compute the $h$-function of the filtered chain complex which is determined by the Alexander polynomials of every sublink of the $L$-space link. As an application, Thurston polytope and Thurston norm of any 2-component $L$-space link are explicitly determined by Alexander polynomials of the link and the link components.

math.GT↗

arXiv2017

Dynamical rearrangement of super-Earths during disk dispersal I. Outline of the magnetospheric rebound model

The Kepler mission has discovered that multiple close-in super-Earth planets are common around solar-type stars, but their period ratios do not show strong pile-ups near mean motion resonances (MMRs). One scenario is that super-Earths form in a gas-rich disk, and they interact gravitationally with the surrounding gas, inducing their orbital migration. Disk migration theory predicts, however, that planets would end up at resonant orbits due to their differential migration speed. Motivated by the discrepancy between observation and theory, we seek for a mechanism that moves planets out of resonances. We examine the orbital evolution of planet pairs near the magnetospheric cavity during the gas disk dispersal phase. Our study determines the conditions under which planets can escape resonances. We perform two-planet N-body simulations, varying the planet masses, stellar magnetic field strengths, disk accretion rates and gas disk depletion timescales. As planets migrate outward with the expanding magnetospheric cavity, their dynamical configurations can be rearranged. Migration of planets is substantial (minor) in a massive (light) disk. When the outer planet is more massive than the inner planet, the period ratio of two planets increases through outward migration. On the other hand, when the inner planet is more massive, the final period ratio tends to remain similar to the initial one. Larger stellar magnetic field strengths result in planets stopping their migration at longer periods. We highlight \textit{magnetospheric rebound} as an important ingredient able to reconcile disk migration theory with observations. Even when planets are trapped into MMR during the early gas-rich stage, subsequent cavity expansion would induce substantial changes to their orbits, moving them out of resonance.

astro-ph.EP↗

arXiv2016

Migration and Growth of Protoplanetary Embryos III: Mass and Metallicity Dependence for FGKM main-sequence stars

Radial velocity and transit surveys have found that the fraction of FGKM stars with close-in super-Earth(s) ($η_\oplus$) is around $30 \%- 50\%$, independent of the stellar mass $M_\ast$ and metallicity $Z_\ast$. In contrast, the fraction of solar-type stars harboring one or more gas giants ($η_J $) with masses $M_{\rm p} > 100 \ M_\oplus $ is nearly $ 10\%-15\%$, and it appears to increase with both $M_\ast$ and $Z_\ast$. Regardless of the properties of their host stars, the total mass of some multiple super-Earth systems exceeds the core mass of Jupiter and Saturn. We suggest that both super-Earths and supercritical cores of gas giants were assembled from a population of embryos that underwent convergent type I migration from their birthplaces to a transition location between viscously heated and irradiation heated disk regions. We attribute the cause for the $η_\oplus$-$η_{\rm J}$ dichotomy to conditions required for embryos to merge and to acquire supercritical core mass ($M_c \sim 10 \ M_\oplus$) for the onset of efficient gaseous envelope accretion. We translate this condition into a critical disk accretion rate, and our analysis and simulation results show that it weakly depends on $M_\ast$ and decreases with metallicity of disk gas $Z_{\rm d}$. We find that embryos are more likely to merge into supercritical cores around relatively massive and metal-rich stars. This dependence accounts for the observed $η_{\rm J}$-$M_\ast$. We also consider the $Z_{\rm d}$-$Z_\ast$ dispersed relationship and reproduce the observed $η_J$-$Z_\ast$ correlation.

astro-ph.EP↗