arXiv subjects

Shang-Fei Liu

Publications and source records attributed to Shang-Fei Liu.

At least 19 recordsLinked to original sources

arXiv2026

Quasi-periodic Eruptions from Stellar-mass Black Holes Impacting Accretion Disks in Galactic Nuclei

We investigate the origins of quasi-periodic eruptions (QPEs) in galactic nuclei using global three-dimensional meshless finite-mass (MFM) simulations. By modeling stellar and black-hole impactors traversing accretion disks under various inclinations and surface densities, we evaluate their consistency with the observed properties of QPEs. Stellar impacts produce highly asymmetric bipolar ejecta with forward outbursts dominating by over an order of magnitude in energy and luminosity due to the star blocking downstream flow and creating a low-density wake. This shock-compression mechanism often renders backward events unobservable, implying one detectable burst per orbit, and challenging the standard assumption of two bursts. It also fails to explain alternating long--short recurrence patterns and places several sources near or within twice the tidal disruption radius for solar-mass stars, raising severe stability concerns. Whereas a stellar-mass black hole (sBH) gravitationally focuses and heats disk gas over an effective interaction scale that extends beyond its Bondi radius $R_{\rm B}$ and is naturally bounded by its Hill radius $R_{\rm H}$ during an impact, yielding nearly symmetric ejecta with mild contrasts. This gravitational-drag mechanism generates higher energy budgets at low inclinations due to enhanced mass accumulation. We suggest an ad hoc effective interaction radius $ R_{\rm eff} \simeq 0.5\, R_{\rm B}^{1/3} R_{\rm H}^{2/3} $ to quantify this trend. Incorporating this effective radius substantially increases the energy that sBH-disk collisions can produce compared to previous Bondi-only estimates, improving the viability of stellar-mass black holes as the impactors for a wide range of observed QPE energies and properties.

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arXiv2026

Parameter Effects in Circumplanetary Disk Spectra and Prospects for Spectral Fitting

With the commissioning of the James Webb Space Telescope (JWST), near- and mid-infrared observations are rapidly extending into the wavelength regime where emission from small dust grains in circumplanetary disks (CPDs) is expected to dominate. We aim to systematically investigate how individual physical parameters of CPDs shape their infrared spectra and to improve the robustness of spectral fitting and physical interpretation of current and future observations. Building on our previous parametric CPD models, we employ a parameter-grid approach combined with radiative transfer simulations to explore the dependence of observable spectra on disk structure and dust properties. We identify the physical mechanisms responsible for the main spectral features and parameter degeneracies, and present the global trends emerging from the parameter study. We also demonstrate the applicability of the models by fitting representative observational data. Our results provide a structured theoretical framework for interpreting near- and mid-infrared observations of CPDs with JWST and related facilities.

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arXiv2026

The 3D Architecture of a pair of 6:1 Resonant Brown Dwarfs around the Naked-eye star $ν$ Ophiuchi

We present a revisiting study of the brown dwarf pair orbiting the naked-eye ($V=3.3$) K-giant $ν$~Ophiuchi, located only 44\,pc from our Solar system. By jointly analysing archival radial-velocity measurements together with astrometric data from \textit{Hipparcos} and the \textit{Gaia} second and third data releases, we determine the three-dimensional architecture of the system and robustly constrain the masses of both companions. We find brown dwarf masses of $m_{\mathrm{b}} = 24.2^{+6.4}_{-2.8}\,M_{\mathrm{J}}$ and $m_{\mathrm{c}} = 26.8^{+4.3}_{-2.9}\,M_{\mathrm{J}}$. The mathematical constraint, derived from the posterior distribution of the mutual inclination based on MCMC samples, yields a mutual inclination of $ψ_{\mathrm{bc}}=46^{+27}_{-24}\!\,^{\circ}$, while direct calculations based on the maximum a posteriori and posterior median orbital parameters yield values of $\sim$$10^{\circ}$ and $\sim$$20^{\circ}$, respectively. Resonance analysis indicates that the two companions can still be trapped in a 6:1 mean-motion resonance in the maximum a posteriori configuration. To place an upper limit for the mutual inclination, dynamical stability analysis over a 1~Myr timescale further constrains it to be no larger than $\sim$$15^{\circ}$. Systems hosting brown dwarf pairs are rare, yet they provide important constraints on theories of planetary formation and dynamical evolution. Current detections suggest that brown dwarf pairs preferentially reside at large separations from their host stars and are more common in less mature systems. This supports a star-like formation pathway via gravitational instability in disk.

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arXiv2026

Where Do Hot Jupiters Come From? Revisiting Tidal Disruption and Ejection in High-Eccentricity Migration

The origin of hot Jupiters remains a key open question. In the high-eccentricity migration scenario, traditional coreless models predict a strict tidal exclusion zone within $\sim 2.7$ tidal radii $r_\textrm{t}$, in which giant planets are either fully disrupted or ejected. We revisit this limit using three-dimensional hydrodynamic simulations of giant planets with realistic dense cores (10 - 20 $M_\oplus$). We find that even a few-percent-mass core fundamentally changes the outcome: \textbf{no total disruptions} occur within the previously suggested destruction zone ($\lesssim 2.7 \, r_\textrm{t}$). For deep encounters ($\lesssim 1.7 \, r_\textrm{t}$) planets suffer severe envelope stripping and are either progressively downsized to dense remnants or ejected after a few close encounters, possibly contributing to the free-floating planet population. In the intermediate regime ($ \sim 1.7 $--$2.0, r_\mathrm{t}$), planets experience significant partial mass loss over repeated encounters. For wider encounters ($ \gtrsim 2.0\, r_\mathrm{t} $), mass loss is minimal, allowing the planets gradually circularize into hot Jupiters. Furthermore, we show that for highly eccentric orbits ($e\gtrsim 0.9$), the change in specific orbital energy $ ΔE_{\mathrm{orb}} $ depends primarily on periastron distance $ r_\mathrm{p} $ rather than semi-major axis $ a $. This enables us to extrapolate our fixed-$ a $ results across a broad ($a$, $e$) parameter space and identify a well-defined tidal ejection zone whose sharp boundaries converge asymptotically. Our results highlight the crucial role of planetary internal structure in high-eccentricity migration and suggest that the survival and transformation of core-bearing giant planets are far more common than previously thought.

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arXiv2026

Sub-Neptunes Show a Stronger Correlation with Cold Jupiters than Super-Earths Especially in Metal-rich Systems

Correlations between the inner small planets and cold giants encodes the formation and evolution of planetary systems. It remains unclear if the correlation differs on the two sides of the radius valley. In this work, we compute the conditional frequency of cold Jupiters in systems with only inner sub-Neptunes $P(\rm CJ|SN)$ and those with only inner super-Earths $P(\rm CJ|SE)$. We find that, around transiting sample around metal-rich stars, $P(\rm CJ|SN, [Fe/H]>0)$ and $P(\rm CJ|SE, [Fe/H]>0)$ are $42.6^{+10.6}_{-9.9}\%$ and $14.5^{+12.7}_{-6.9}\%$. Comparing with the field giant frequency ($14.3^{+2.0}_{-1.8}\%$), we show that inner sub-Neptunes and cold Jupiters exhibit a significant positive correlation for metal-rich systems with a confidence level of 99.95\%, whereas this correlation is absent for systems with super-Earths. We also consider a homogeneous Kepler-Keck subsample and derive similar results, with $P(\rm CJ|SN, [Fe/H]>0)$ of $45.8^{+18.6}_{-16.3}\%$ and $P(\rm CJ|SE, [Fe/H]>0)$ of $13.3^{+17.0}_{-6.8}\%$. Radial velocity sample shows consistent results, with metal-rich systems hosting massive inner planets exhibiting a strong positive correlation (confidence level of 99.11\%) with outer cold Jupiters ($P(\rm CJ|M_{p}>10M_\oplus, [Fe/H]>0) = 34.6^{+11.0}_{-9.1}\%$). These results can be naturally understood since metal-rich disks are expected to more efficiently produce both outer cold Jupiters and inner planets with larger radii and masses. Our findings highlight the critical role of stellar metallicity in shaping planetary architectures, particularly for large/massive planets.

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arXiv2026

Misaligned circumbinary discs around unequal-mass eccentric binaries: alignment, morphology, and binary accretion variability

Binary systems are ubiquitous in the Universe and often host circumbinary discs that are misaligned with the binary orbital plane. Such misalignments can affect disc evolution and binary accretion variability. We here present 3D hydrodynamical simulations of circumbinary discs with initial tilts $i_0$ from $0^\circ$ to $180^\circ$, around eccentric binaries with secondary-to-primary mass ratios of $0.11-0.67$. We find that both the initial tilt and mass ratio can affect the long-term accretion variability in our simulations. Discs evolving towards polar and coplanar retrograde generally favour accretion onto the primary star, while discs evolving towards coplanar prograde generally favour accretion onto the secondary. We find preferential accretion ratio $η=\langle\dot{M_2}\rangle/\langle\dot{M_\mathrm{b}}\rangle$ to be a non-monotonic function of the mass ratio. For discs close to coplanar prograde alignment, $η$ increases with decreasing mass ratio, whereas for discs with $30^\circ \le i_0 \le 135^\circ$, $η$ decreases for smaller mass ratios. Polar discs show the lowest mass loss rates, slightly lower than those of coplanar prograde discs, while retrograde discs lose mass faster than their prograde counterparts. Discs that undergo strong warping or breaking experience rapid mass loss. Our findings provide insights into observed circumbinary discs and have implications for circumbinary planet formation.

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arXiv2026

Nii-body: Bayesian Inference of Multiplanet Dynamics via N-body Simulations

Many exoplanetary systems are multiplanet configurations whose long-term dynamics are governed by N-body gravitational interactions. Consequently, their detection signatures cannot be adequately described by Keplerian orbits. Accurately interpreting the observational data of these systems -- including radial velocity (RV), astrometry, and transit timing variations (TTVs) -- requires N-body integration. Motivated by this need, we developed a Bayesian fitting framework that couples N-body integration with Markov chain Monte Carlo (MCMC) to retrieve the system parameters of multiplanet systems. The code, named \texttt{Nii-body}, integrates an adaptive Runge--Kutta--Fehlberg 7(8) (RKF78) solver with an automated parallel tempering MCMC algorithm. Using simplified synthetic astrometric observations, we evaluated the efficiency and robustness of \texttt{Nii-body}'s N-body orbit retrieval on an idealized two-planet model, demonstrating its potential for future application to real observational data. The N-body fitting workflow can be readily extended to RV, TTVs, or combined datasets, providing a versatile engine for high-precision orbital inference in multiplanet systems.

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arXiv2026

Evidence for a Nonzero Eccentricity Superpuff Exoplanet WASP-107 b Using JWST Occultation Observation

WASP-107~b is an extremely low-density super-puff exoplanet whose inflated radius and evidence of strong internal heating make it a key target for understanding planetary structure and evolution. Its orbital eccentricity is a critical parameter for testing mechanisms such as tidal heating and high-eccentricity migration, yet previous measurements have remained inconclusive. Due to the large radial velocity jitter caused by stellar activity, and the presence of at least one additional planet in the system, previous radial velocity measurements could not robustly determine the eccentricity of WASP-107~b. Here we combine the new JWST secondary eclipse data with transit timing data from HST, TESS, and JWST to measure the eccentricity of WASP-107~b. Our joint analysis shows that WASP-107~b has an eccentricity of $0.09\pm0.02$, a mass of $0.096\pm0.005 \, M_J$, and an orbital period of $5.721487\pm0.000001$~days. We find the $99.7\%$ lower limit of the eccentricity is about 0.04. These new measurements are consistent with the scenario in which WASP-107~b is in the final stage of high-eccentricity migration. Preliminary estimate shows that eccentricity-driven tidal dissipation can provide a significant contribution to the energy required to sustain the observed radius inflation of WASP-107~b. Our results establish the dynamical status of one of the most intriguing low-density exoplanets known, and offer new insights into its formation and evolution history.

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arXiv2025

Breaking the Mass Inclination Degeneracy of Radial Velocity Measurements via Monitoring von Zeipel-Lidov-Kozai Cycles: Implications in the HD 41004 System

We investigate the dynamical stability of the S-type planet in the compact binary HD 41004. Using $N$-body simulations, we find that the planet could be dynamically stable at a mutual angle inclination up to $\sim75^\circ$. The von Zeipel-Lidov-Kozai (vZLK) mechanism becomes active when the mutual inclination is greater than 39.2$^\circ$. High-inclination orbits exhibit coupled oscillations in eccentricity and inclination, along with apsidal precession. Synthetic radial velocity (RV) modeling shows that these secular variations produce measurable signatures across a broad range of timescales, from full vZLK cycles to observationally accessible decades. For instance, a high mutual inclination at 75$^\circ$ can induce RV drifts exceeding 5 m s$^{-1}$ per planetary orbit ($\sim 1.9 \,\text{m s}^{-1}\, \text{yr}^{-1}$) in circular binary configurations. The presence of eccentric vZLK further accelerates these drifts, enhancing the detectability. Long-term RV observations of this system offer a unique pathway to dynamically constrain the orbital inclination, and thus determine the true mass of HD 41004 Ab. The degeneracy of mass-inclination is well known when using RV measurements alone. Our results highlight that HD 41004Ab and potentially other S-type planets in compact binaries are promising targets for breaking such a degeneracy by studying the dynamics induced by the vZLK mechanism through long-term high-precision RV monitoring.

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arXiv2024

Fate of the remnant in tidal stripping event: repeating and non-repeating

Tidal disruption events (TDE) occur when a star ventures too close to a massive black hole. In a partial TDE (pTDE), the star only grazes the tidal radius, causing the outer envelope of the star to be stripped away while the stellar core survives. Previous research has shown that a star, once tidally stripped in a parabolic orbit, can acquire enough orbital energy for its remnant to become a high-velocity star potentially capable of escaping the galaxy. Conversely, some studies have reported that the remnant may lose orbital energy and undergo re-disruption, leading to a recurring pTDE. This study aims to uncover the physical mechanisms and determine the conditions that lead to these divergent outcomes. We find that the orbital energy change only depends on the impact factor and the stellar structure, and barely depends on the mass of the black hole or the exact mass or orbital eccentricity of the star. For a $γ=5/3$ (or $γ=4/3$) polytropic star, after a pTDE its remnant gains orbital energy when the impact factor $β\gtrsim 0.62$ (or $\gtrsim 1.1$) or loses energy vice versa. Additionally, we verify an analytical equation for orbital energy change that is applicable across various systems. Through hydrodynamic simulations, we also explore the structure of the stellar remnant post-tidal stripping. Our findings provide critical insights for interpreting observed pTDEs and advancing knowledge on the orbital evolution and event rate of these events.

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arXiv2024

Surviving in the Hot Neptune Desert: The Discovery of the Ultra-Hot Neptune TOI-3261b

The recent discoveries of Neptune-sized ultra-short period planets (USPs) challenge existing planet formation theories. It is unclear whether these residents of the Hot Neptune Desert have similar origins to smaller, rocky USPs, or if this discrete population is evidence of a different formation pathway altogether. We report the discovery of TOI-3261b, an ultra-hot Neptune with an orbital period $P$ = 0.88 days. The host star is a $V = 13.2$ magnitude, slightly super-solar metallicity ([Fe/H] $\simeq$ 0.15), inactive K1.5 main sequence star at $d = 300$ pc. Using data from the Transiting Exoplanet Survey Satellite and the Las Cumbres Observatory Global Telescope, we find that TOI-3261b has a radius of $3.82_{-0.35}^{+0.42}$ $R_{\oplus}$. Moreover, radial velocities from ESPRESSO and HARPS reveal a mass of $30.3_{-2.4}^{+2.2}$ $M_{\oplus}$, more than twice the median mass of Neptune-sized planets on longer orbits. We investigate multiple mechanisms of mass loss that can reproduce the current-day properties of TOI-3261b, simulating the evolution of the planet via tidal stripping and photoevaporation. Thermal evolution models suggest that TOI-3261b should retain an envelope potentially enriched with volatiles constituting $\sim$5% of its total mass. This is the second highest envelope mass fraction among ultra-hot Neptunes discovered to date, making TOI-3261b an ideal candidate for atmospheric follow-up observations.

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arXiv2024

Observational characteristics of circum-planetary-mass-object disks in the era of James Webb Space Telescope

Recent observations have confirmed circumplanetary disks (CPDs) embedded in parental protoplanetary disks (PPDs). On the other hand, planetary-mass companions (PMCs) and planetary-mass objects (PMOs) are likely to harbor their own accretion disks. Unlike PPDs, CPDs and other disks around planet analogues are generally too compact to be spatially resolved by current instrumentation. In this study, we generate over 4,000 spectral energy distributions (SEDs) of circum-PMO-disks (CPMODs) with various host temperature and disk properties, which can be categorized into four prototypes, i.e., full, pre-transitional, transitional and evolved CPMODs. We propose a classification scheme based on their near-to-mid-infrared colors. Using those CPMOD models, we synthesize JWST (NIRCam and MIRI) photometry for F444W, F1000W and F2550W wide filters. We show F444W - F1000W and F444 - F2550W colors can be applied to distinguish different types of CPMODs, especially for those around hot hosts. Our results indicate that the ongoing and future JWST observations are promising to unveil structures and properties of CPMODs.

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arXiv2024

White dwarf magnetospheres: Shielding volatile content of icy objects and implications for volatile pollution scarcity

Context. About 25% -- 50% of white dwarfs are found to be contaminated by heavy elements, which are believed to originate from external sources such as planetary materials. Elemental abundances suggest that most of the pollutants are rocky objects and only a small fraction of white dwarfs bear traces of volatile accretion. Aims. In order to account for the scarcity of volatile pollution, we investigate the role of the white dwarfs' magnetospheres in shielding the volatile content of icy objects. Methods. We estimated the volatile sublimation of inward-drifting exocomets. We assume the orbits of the exocomets are circularized by the Alfven wing drag that is effective for long-period comets. Results. Volatile material can sublimate outside the corotation radius and be shielded by the magnetic field. {The two conditions for this volatile-shielded mechanism are that the magnetosphere radius must be larger than the corotation radius and that the volatiles are depleted outside the corotation radius, which requires a sufficiently slow orbital circularization process.} We applied our model to nine white dwarfs with known rotational periods, magnetic fields, and atmosphere compositions. Our volatile-shielded model may explain the excess of volatile elements such as C and S in the disk relative to the white dwarf atmosphere in WD2326+049 (G29-38). Nevertheless, given the sensitivity of our model to the circularization process and material properties of icy objects, there remains considerable uncertainty in our results. Conclusions. Our work suggests a possible explanation for the scarcity of volatile-accretion signatures among white dwarfs. We also identify a correlation between the magnetic field strength, the spin period, and the composition of pollutants in white dwarf atmospheres.

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arXiv2024

The Impact-driven Atmospheric Loss of Super-Earths around Different Spectral Type Host Stars

The planet's mass loss is important for the planet's formation and evolution. The radius valley (RV) is believed to be triggered by evaporation-induced mass loss. As an alternative mechanism for the RV, the mass loss of post-impact planets is thoroughly investigated in this work. The impact energy is converted to the planet's internal energy, enhancing its core energy and accelerating mass loss and orbital migration. As the host star changes from K-type to F-type, the planet's mass loss and orbital migration increase. When the initial gas-to-core mass ratio (GCR) is small, the migration efficiency for planets around K-type stars will increase, which helps to suppress mass loss and retain the planet's mass and radius within a specific range. On the contrary, planets around more massive F-type stars experience more substantial mass loss, potentially leading to complete mass loss, and migrate to orbits with longer periods. Our calculation shows that planets around different spectral types of host stars give rise to an RV ranging from 1.3-2.0 $R_{\oplus}$, consistent with the observed range of 1.3-2.6 $R_{\oplus}$. Despite the presence of uncertain parameters, the planetesimal impact can promote the RV establishment for planets around host stars of different spectral types.

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arXiv2024

Observability of substructures in planet-forming disk in (sub)cm wavelength with SKA and ngVLA

Current imaging observations of protoplanetary disks using ALMA primarily focus on the sub-millimeter wavelength, leaving a gap in effective observational approaches for centimeter-sized dust, which is crucial to the issue of planet formation. The forthcoming SKA and ngVLA may rectify this deficiency. In this paper, we employ multi-fluid hydrodynamic numerical simulations and radiative transfer calculations to investigate the potential of SKA1-Mid, ngVLA, and SKA2 for imaging protoplanetary disks at sub-cm/cm wavelengths. We create mock images with ALMA/SKA/ngVLA at multi-wavelengths based on the hydrodynamical simulation output, and test different sensitivity and spatial resolutions. We discover that both SKA and ngVLA will serve as excellent supplements to the existing observational range of ALMA, and their high resolution enables them to image substructures in the disk's inner region ($\sim$ 5 au from the stellar). Our results indicate that SKA and ngVLA can be utilized for more extended monitoring programs in the centimeter waveband. While in the sub-centimeter range, ngVLA possesses the capability to produce high-fidelity images within shorter observation times ($\sim$ 1 hour on source time) than previous research, holding potential for future survey observations. We also discuss for the first time the potential of SKA2 for observing protoplanetary disks at a 0.7 cm wavelength.

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arXiv2023

Tidal Stripping of a White Dwarf by an Intermediate-Mass Black Hole

During the inspiralling of a white dwarf (WD) into an intermediate-mass black hole (~ 10^{2-5} M_sun), both gravitational waves (GWs) and electromagnetic (EM) radiation are emitted. Once the eccentric orbit's pericenter radius approaches the tidal radius, the WD would be tidally stripped upon each pericenter passage. The accretion of these stripped mass would produce EM radiation. It is suspected that the recently discovered new types of transients, namely the quasi-periodic eruptions and the fast ultraluminous X-ray bursts, might originate from such systems. Modeling these flares requires a prediction of the amount of stripped mass from the WD and the details of the mass supply to the accretion disk. We run hydrodynamical simulations to study the orbital parameter dependence of the stripped mass. We find that our results match the analytical estimate that the stripped mass is proportional to z^{5/2}, where z is the excess depth by which the WD overfills its instantaneous Roche lobe at the pericenter. The corresponding fallback rate of the stripped mass is calculated, which may be useful in interpreting the individual flaring light curve in candidate EM sources. We further calculate the long-term mass-loss evolution of a WD during its inspiral and the detectability of the GW and EM signals. The EM signal from the mass-loss stage can be easily detected: the limiting distance is ~ 320(M_h/10^4 M_sun)^{1/2} Mpc for Einstein Probe. The GW signal, for the space-borne detectors such as Laser Interferometer Space Antenna or TianQin, can be detected only within the Local Supercluster (~ 33 Mpc).

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arXiv2022

Observable tests for the light-sail scenario of interstellar objects

We scrutinize the light sail scenario of the first interstellar object (ISO) 1I/2017 U1 (`Oumuamua) by making comparisons between physical models and observational constraints. These analyses can be generalized for future surveys of `Oumuamua-like objects. The light sail goes through a drift in the interstellar space due to the magnetic field and gas atoms, which poses challenges to the navigation system. When the light sail enters the inner solar system, the sideways radiation pressure leads to a considerable non-radial displacement. The immensely high dimensional ratio and the tumbling motion could cause a light curve with an extremely large amplitude and could even make the light sail invisible from time to time. These observational features allow us to examine the light sail scenario of interstellar objects. Our results show that the drift of the freely rotating light sail in the interstellar medium is $\sim 100\,$au even if the travel distance is only 1 pc. The probability is < 1.5\% for the expected brightness modulation of the light sail to match with `Oumuamua's observed variation amplitude ($\sim$ 2.5 -- 3). In addition, the probability is 0.4% for the tumbling light-sail to be visible (brighter than V=27) in all 55 observations spread over two months after discovery. Radiation pressure could cause a larger displacement that is normal to the orbital plane for a lightsail than that for `Oumuamua. Also, the ratio of anti-solar to sideways acceleration of `Oumuamua deviates from that of the light sail by ~ 1.5σ. We suggest that `Oumuamua is unlikely a light sail. The dynamics of an intruding light sail, if exist, has distinct observational signatures, which can be quantitatively identified and analyzed with our methods in future surveys.

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arXiv2021

Orbital migration and circularization of tidal debris by Alfvén-wave drag: circumstellar debris and pollution around white dwarfs

A significant fraction of white dwarfs (WDs) exhibit signs of ongoing accretion of refractory elements at rates $\sim10^3$--$10^7$ kg s$^{-1}$, among which, 37 WDs were detected to harbor dusty debris disks. Such a concurrence requires not only fertile reservoirs of planetary material, but also a high duty cycle of metal delivery. It has been commonly suggested that this material could be supplied by Solar System analogs of Main Belt asteroids or Kuiper Belt objects. Here we consider the primary progenitors of WD pollutants as a population of residual high-eccentricity planetesimals, de-volatilized during the stellar giant phases. Equivalent to the Solar System's long-period comets, they are scattered to the proximity of WDs by perturbations from remaining planets, Galactic tides, passing molecular clouds, and nearby stars. These objects undergo downsizing when they venture within the tidal disruption limit. We show quantitatively how the breakup condition and fragment sizes are determined by material strength and gravity. Thereafter, the fragments' semi-major axes need to decay by at least $\sim$6 orders of magnitude before their constituents are eventually accreted onto the surface of WDs. We investigate the orbital evolution of these fragments around WDs and show that WDs' magnetic fields induce an Alfvén-wave drag during their periastron passages and rapidly circularize their orbits. This process could be responsible for the observed accretion rates of heavy-elements and the generation of circum-WD debris disks. A speculative implication is that giant planets may be common around WDs' progenitors and they may still be bound to some WDs today.

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