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Thread: Disintegrating Planets - some recent news

  1. #31
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    More on ultrahot gas giants losing their atmospheres.


    https://phys.org/news/2018-12-exopla...here-tail.html

    An exoplanet loses its atmosphere in the form of a tail

    December 6, 2018, Instituto de Astrofísica de Canarias

    A new study led by scientists from the Instituto de Astrofísica de Canarias (IAC) reveals that the giant exoplanet WASP-69b carries a comet-like tail made up of helium particles escaping from its gravitational field and propelled by the ultraviolet radiation of its star. The results of this work are published today in the journal Science.

    The planet was observed during a transit, when it passed in front of its host star. During this event, the planet and its atmosphere eclipse part of the starlight. "We observed a stronger and longer-lasting dimming of the starlight in a region of the spectrum where helium gas absorbs light," says Lisa Nortmann, a researcher at the IAC and lead author of the article published today in the journal Science. "The longer duration of this absorption allows us to infer the presence of a tail," she adds.

    But this is not the only result described in the article. The authors have also analyzed four other planets in a similar way. These are the hot exoplanets HD 189733b and HD 209458b, which have a mass similar to that of Jupiter, the extremely hot giant planet KELT-9b and the warm Neptune-sized exoplanet GJ 436b. The analysis does not show extensive helium exospheres around the last three planets, which defies previous theoretical predictions. The hot Jupiter HD 189733b, on the other hand, does reveal a clear signal of absorbing helium, although here, the helium envelope is more compact and does not form a tail.
    There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.
    — Mark Twain, Life on the Mississippi (1883)

  2. #32
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    Our hot disintegrating friend KELT-9b is back in the news.

    https://arxiv.org/abs/1812.02773

    A ground-based NUV secondary eclipse observation of KELT-9b

    Matthew J. Hooton, Christopher A. Watson, Ernst J. W. de Mooij, Neale P. Gibson, Daniel Kitzmann (Submitted on 6 Dec 2018)

    KELT-9b is a recently discovered exoplanet with a 1.49 d orbit around a B9.5/A0-type star. The unparalleled levels of UV irradiation it receives from its host star put KELT-9b in its own unique class of ultra-hot Jupiters, with an equilibrium temperature > 4000 K. The high quantities of dissociated hydrogen and atomic metals present in the dayside atmosphere of KELT-9b bear more resemblance to a K-type star than a gas giant. We present a single observation of KELT-9b during its secondary eclipse, taken with the Wide Field Camera on the Isaac Newton Telescope (INT). This observation was taken in the U-band, a window particularly sensitive to Rayleigh scattering. We do not detect a secondary eclipse signal, but our 3σ upper limit of 181 ppm on the depth allows us to constrain the dayside temperature of KELT-9b at pressures of ~30 mbar to 4995 K (3σ ). Although we can place an observational constraint of A g < 0.14, our models suggest that the actual value is considerably lower than this due to H − opacity. This places KELT-9b squarely in the albedo regime populated by its cooler cousins, almost all of which reflect very small components of the light incident on their daysides. This work demonstrates the ability of ground-based 2m-class telescopes like the INT to perform secondary eclipse studies in the NUV, which have previously only been conducted from space-based facilities.

    ===============

    I would think deformation comes before disintegration.

    https://arxiv.org/abs/1812.04538

    Detectability of shape deformation in short-period exoplanets

    B. Akinsanmi, S. C. C. Barros, N. C. Santos, A. C. M. Correia, P. F. L. Maxted, G. Boué, J. Laskar (Submitted on 11 Dec 2018)

    Short-period planets suffer from extreme tidal forces from their parent stars causing them to deform and attain non-spherical shapes. The planet shapes, modeled here as triaxial ellipsoids, can have an impact on the observed transit light curves and the planetary parameters derived. We investigate the detectability of tidal deformation in short-period planets from their transit light curves and the instrumental precision needed. We show how the detection of deformation from the light curve allows us to obtain an observational estimate of the second fluid Love number which gives valuable insight about the planet's internal structure. We adopted a model to calculate the shape of a planet due to the external potentials acting on it and used this model to modify the ellc transit tool. Our model is parameterized by the Love number, hence for a given light curve we can derive the value of the Love number that best matches the observations. We simulated the known cases of WASP-103b and WASP-121b expected to be highly deformed. Our analyses showed that instrumental precision ≤ 50ppm/min is needed to reliably estimate the Love number and detect tidal deformation. This precision can be achieved for WASP-103b in ~40 transits using HST and in ~300 transits using the forthcoming CHEOPS. However, fewer transits will be required for short-period planets that may be found around bright stars in the TESS and PLATO survey missions. The unprecedented precisions expected from PLATO and JWST can permit the detection of deformation with a single transit. However, the effects of instrumental and astrophysical noise must be well-considered as they can increase the number of transits required to reach the 50 ppm/min detection limit. We also show that improper modeling of limb darkening can act to bury signals related to the planet's shape thereby leading us to infer sphericity for a deformed planet.
    There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.
    — Mark Twain, Life on the Mississippi (1883)

  3. #33
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    A fast-boiling Neptune-type planet that will disappear before long...

    https://phys.org/news/2018-12-hubble...ay-planet.html

    Hubble finds far-away planet vanishing at record speed
    December 13, 2018, Johns Hopkins University

    The speed and distance at which planets orbit their respective blazing stars can determine each planet's fate—whether the planet remains a longstanding part of its solar system or evaporates into the universe's dark graveyard more quickly. In their quest to learn more about far-away planets beyond our own solar system, astronomers discovered that a medium-sized planet roughly the size of Neptune, GJ 3470b, is evaporating at a rate 100 times faster than a previously discovered planet of similar size, GJ 436b.

    The findings, published today in the journal of Astronomy & Astrophysics, advance astronomers' knowledge about how planets evolve. "This is the smoking gun that planets can lose a significant fraction of their entire mass. GJ 3470b is losing more of its mass than any other planet we seen so far; in only a few billion years from now, half of the planet may be gone," said David Sing, Bloomberg Distinguished Professor at Johns Hopkins and an author on the study.
    There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.
    — Mark Twain, Life on the Mississippi (1883)

  4. #34
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    More on same "evaporating" exoplanet.

    Hubble PanCET: An extended upper atmosphere of neutral hydrogen around the warm Neptune GJ 3470 b

    V.Bourrier, et al. (Submitted on 12 Dec 2018)

    GJ 3470b is a warm Neptune transiting an M-dwarf star at the edge of the evaporation desert. It offers the possibility of investigating how low-mass, close-in exoplanets evolve under the irradiation from their host stars. We observed three transits of GJ 3470b in the Lyman-alpha line with the Hubble Space Telescope (HST) as part of the Panchromatic Comparative Exoplanet Treasury (PanCET) program. Absorption signatures are detected with similar properties in all three independent epochs, with absorption depths of 35+-7% in the blue wing of the line, and 23+-5% in the red wing. The repeatability of these signatures, their phasing with the planet transit, and the radial velocity of the absorbing gas allow us to conclude that there is an extended upper atmosphere of neutral hydrogen around GJ 3470 b. We determine from our observations the stellar radiation pressure and XUV irradiation from GJ 3470 and use them to perform numerical simulations of the upper atmosphere of GJ 3470b with the EVaporating Exoplanets (EVE) code. The unusual redshifted signature can be explained by the damping wings of dense layers of neutral hydrogen that extend beyond the Roche lobe and are elongated in the direction of the planet motion. This structure could correspond to a shocked layer of planetary material formed by the collision of the expanding thermosphere with the wind of the star. The blueshifted signature is well explained by neutral hydrogen atoms escaping at rates of about 1e10 g s-1 that are blown away from the star by its strong radiation pressure and are quickly photoionized, resulting in a smaller exosphere than that of the warm Neptune GJ 436b. The stronger escape from GJ 3470b, however, may have led to the loss of about 4-35% of its current mass over its 2 Gyr lifetime.
    There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.
    — Mark Twain, Life on the Mississippi (1883)

  5. #35
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    Two new articles on disintegrating/evaporating planets...


    http://cdsads.u-strasbg.fr/abs/2019AAS...23315303D

    The Search for Disintegrating Planets Orbiting White Dwarfs

    Difo Cheri, Gerlinder
    American Astronomical Society, AAS Meeting #233, id.#153.03 (01/2019)

    White dwarfs are the end state for low to medium mass stars like our sun and are essentially the exposed cores of such stars. Once a star exhausts its hydrogen and leaves the main sequence, mass loss occurs which can disturb the orbital path of objects within its gravitational influence. In some cases, the star's planets will fall into a much closer orbit and begin to disintegrate which can occur either though destructive gravitational forces or through the vaporization of rocky surface material. This material is then accreted onto the surface of the white dwarf where it can be analysed to gain a better understanding of the composition of the objects which were destroyed. Recently, transits of disrupted planetary material have been detected around the white dwarf WD 1145+017, giving researchers a new way to study this phenomenon. Here we present a search for new WD 1145+017-like transiting systems through optical observational data conducted by both ground and space-based surveys. Our search utilizes white dwarf databases which have identified thousands of white dwarfs. The NASA Exoplanet Archive was used for the retrieval of light curve data for targets which were observed by the SuperWasp, KELT, and K2 exoplanet surveys. The light curve data was then processed using a Fourier Transform which can reveal periodic dips in stellar flux. So far we have not detected any new WD 1145+017-like systems, but if we are successful in the future, observation of such events provide researchers with a better understanding of the way planets are disrupted around white dwarfs and the planets' exact inner compositions.

    ======

    http://cdsads.u-strasbg.fr/abs/2019AAS...23322606O

    Probing the Escaping Exoplanet Atmospheres with the Helium 1083 nm Line

    Oklopcic, Antonija
    American Astronomical Society, AAS Meeting #233, id.#226.06 (01/2019)

    Atmospheric escape or mass loss is an important process in the evolution of atmospheres of extrasolar planets. However, there are many aspects of atmospheric escape that remain poorly understood, in part due to a small number of direct observations, obtained mostly via transit spectroscopy in the difficult-to-observe hydrogen Lyman-alpha line. In recent theoretical work (Oklopcic & Hirata, 2018), we demonstrated that the absorption line of helium at 1083 nm can be used as a powerful new diagnostic of escaping atmospheres. This line is accessible for ground-based observations using high-resolution spectrographs, which can enable more detailed studies of extended atmospheres for a much larger number of exoplanets than realistically possible with UV spectroscopy. Shortly after the theoretical prediction, excess absorption in the helium line was observed in WASP-107b (Spake et al., 2018) and HAT-P-11b (Allart et al., in press; Mansfield et al., in prep.). I will present new, improved theoretical models of upper planetary atmospheres used to interpret the observed 1083 nm transit absorption signatures and place constraints on the physical properties of extended exoplanet atmospheres. Observations in the helium 1083 nm line opened a new wavelength window into escaping atmospheres; by comparing the data with theoretical models, we can improve our understanding of the physical processes that drive atmospheric mass loss and, consequently, affect planetary evolution and demographics of planetary systems.
    There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.
    — Mark Twain, Life on the Mississippi (1883)

  6. #36
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    A planet that did not disintegrate but did undergo atmospheric erosion, leaving it airless. No doubt the first thing that happens to planets that actually disintegrate.

    https://arxiv.org/abs/1901.04558

    EPIC 247418783b : A rocky super-Earth in a 2.2 day orbit

    Molly R. Kosiarek, et al. (Submitted on 14 Jan 2019)

    EPIC 247418783 is a solar-type star with a radius of R_star = 0.899 ± 0.034 R_sun and mass of M_star=0.934 ± 0.038 M_sun. From K2 C13 data, we found one super-Earth planet (R_p = 1.589+0.095-0.072 R_Earth) transiting this star on a short period orbit (P = 2.225177 +6.6e-5 -6.8e-5 days). We followed this system up with adaptive-optic imaging and spectroscopy to derive stellar parameters, search for stellar companions, and determine a planet mass. From our 75 radial velocity measurements using HIRES on Keck I and HARPS-N on Telescopio Nazionale Galileo, we constrained the mass of EPIC 247418783b to M_p = 6.49 ± 1.16 M_Earth. We found it necessary to model correlated stellar activity radial velocity signals with a Gaussian process in order to more accurately model the effect of stellar noise on our data; the addition of the Gaussian process also improved the precision of this mass measurement. With a bulk density of 8.84+2.50-2.03 g cm-3, the planet is consistent with an Earth-like rock/iron composition and no substantial gaseous envelope. Such an envelope, if it existed in the past, was likely eroded away by photo-evaporation during the first billion years of the star's lifetime.
    There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.
    — Mark Twain, Life on the Mississippi (1883)

  7. #37
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    New item elaborating briefly on K2-22b's situation.

    https://phys.org/news/2019-01-disint...et-k2-22b.html

    The disintegrating exoplanet K2-22b
    January 21, 2019, Harvard-Smithsonian Center for Astrophysics

    Exoplanet surveys have yielded many surprises over the years, and the discovery of "disintegrating" exoplanets was one of them. These are planets that produce asymmetric shapes in the dips of the light curves seen as they transit across the faces of their stars. The asymmetry is hypothesized to be due to tails of dusty material from the planets' disintegration. At present, only three such planets known around main sequence stars, one being K2-22b. There are currently over 3800 confirmed exoplanets, suggesting either that such objects are intrinsically rare or that they have very short lifetimes, in which case it is lucky to catch any in the act of disintegration. These systems have been under intense study to better understand their formation and evolution and to constrain the properties of the grains in the dust tails.

    CfA astronomers George Zhou, Karen Collins, Allyson Bieryla, and Dave Latham were members of a team that obtained forty-five ground-based observations of the K2-22 system in their study of the evolution of its transit. K2-22b is a Neptune-sized exoplanet that orbits its star in only about nine hours; it is unusual in that it appears to have not only a trailing dust tail but a leading trail as well. The team's observations of the dust tails included observing the transits at multiple wavelengths to try to use color to characterize the dust grain size or composition, but except in one transit event no differences were seen. The color information is, however, consistent with the previous model of dust grains as being small—comparable to or smaller than optical light wavelengths.

    The astronomers also confirmed the variability of the transits, thought to be evidence of the continuing rapid evolution of the dust tails. The scientists point out that this variability appears in all three disintegrating planets, and the shape variability occurs on all the timescales observed, from transit to transit and over several years. They conclude that a continuous observing campaign would be a valuable tool in unraveling the mystery of these dusty trails.
    There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.
    — Mark Twain, Life on the Mississippi (1883)

  8. #38
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    Recent paper investigating the disintegration/ evaporation of superhot giant planets, also exploring what they look like as they boil away. Ionosphere idea as prelude to boiling is nifty, too.

    https://arxiv.org/abs/1901.08640

    Sparkling nights and very hot days on WASP-18b: the formation of clouds and the emergence of an ionosphere

    Ch. Helling, P. Gourbin, P. Woitke, V. Parmentier (Submitted on 24 Jan 2019)

    WASP-18b is an utra-hot Jupiter with a temperature difference of upto 2500K between day and night. Such giant planets begin to emerge as planetary laboratory for understanding cloud formation and gas chemistry in well-tested parameter regimes in order to better understand planetary mass loss and for linking observed element ratios to planet formation and evolution. We aim to understand where clouds form, their interaction with the gas phase chemistry through depletion and enrichment, the ionisation of the atmospheric gas and the possible emergence of an ionosphere on ultra-hot Jupiters. We utilize 1D profiles from a 3D atmosphere simulations for WASP-18b as input for kinetic cloud formation and gas-phase chemical equilibrium calculations. We solve our kinetic cloud formation model for these 1D profiles that sample the atmosphere of WASP-18b at 16 different locations along the equator and in the mid-latitudes and derive consistently the gas-phase composition. The dayside of WASP-18b emerges as completely cloud-free due to the very high atmospheric temperatures. In contrast, the nightside is covered in geometrically extended and chemically heterogeneous clouds with disperse particle size distributions. The atmospheric C/O increases to >0.7 and the enrichment of the atmospheric gas with cloud particles is ρ d /ρ gas >10 −3 . The clouds that form at the limbs appear located farther inside the atmosphere and they are the least extended. Not all day-night terminator regions form clouds. The gas-phase is dominated by H 2 , CO, SiO, H 2 O, H 2 S, CH 4 , SiS. In addition, the dayside has a substantial degree of ionisation due to ions like Na + , K + , Ca + , Fe + . Al + and Ti + are the most abundant of their element classes. We find that WASP-18b, as one example for ultra-hot Jupiters, develops an ionosphere on the dayside.
    There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.
    — Mark Twain, Life on the Mississippi (1883)

  9. #39
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    More on hot exo-Jupiters that undergo severe mass loss from stellar heating.


    https://arxiv.org/abs/1901.10223

    Swift UVOT near-UV transit observations of WASP-121 b

    M. Salz, P. C. Schneider, L. Fossati, S. Czesla, K. France, J. H. M. M. Schmitt (Submitted on 29 Jan 2019)

    Close-in gas planets are subject to continuous photoevaporation that can erode their volatile envelopes. Today, ongoing mass loss has been confirmed in a few individual systems via transit observations in the ultraviolet spectral range. We demonstrate that the Ultraviolet/Optical Telescope (UVOT) onboard the Neil Gehrels Swift Observatory enables photometry to a relative accuracy of about 0.5% and present the first near-UV (200-270 nm, NUV) transit observations of WASP-121 b, a hot Jupiter with one of the highest predicted mass-loss rates. The data cover the orbital phases 0.85 to 1.15 with three visits. We measure a broad-band NUV transit depth of 2.10±0.29 %. While still consistent with the optical value of 1.55%, the NUV data indicate excess absorption of 0.55% at a 1.9σ level. Such excess absorption is known from the WASP-12 system, and both of these hot Jupiters are expected to undergo mass loss at extremely high rates. With a CLOUDY simulation, we show that absorption lines of Fe II in a dense extended atmosphere can cause broad-band NUV absorption at the 0.5% level. Given the numerous lines of low-ionization metals, the NUV range is a promising tracer of photoevaporation in the hottest gas planets.
    There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.
    — Mark Twain, Life on the Mississippi (1883)

  10. #40
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    This news fits better here than anywhere else: Asteroid disruptions near the Sun, and the reason why we don't seem to have Vulcanoids (near-Sun asteroids where the hypothetical planet Vulcan was thought to be).


    https://arxiv.org/abs/1902.01758

    Debris of asteroid disruptions close to the Sun

    Quanzhi Ye, Mikael Granvik (Submitted on 5 Feb 2019)

    The under-abundance of asteroids on orbits with small perihelion distances suggests that thermally-driven disruption may be an important process in the removal of rocky bodies in the Solar System. Here we report our study of how the debris streams arise from possible thermally-driven disruptions in the near-Sun region. We calculate that a small body with a diameter \gtrsim0.5 km can produce a sufficient amount of material to allow the detection of the debris at the Earth as meteor showers, and that bodies at such sizes thermally disrupt every \sim2 kyrs. We also find that objects from the inner parts of the asteroid belt are more likely to become Sun-approacher than those from the outer parts. We simulate the formation and evolution of the debris streams produced from a set of synthetic disrupting asteroids drawn from Granvik et al. (2016)'s near-Earth object population model, and find that they evolve 10--70 times faster than streams produced at ordinary solar distances. We compare the simulation results to a catalog of known meteor showers on Sun-approaching orbits. We show that there is a clear overabundance of Sun-approaching meteor showers, which is best explained by a combining effect of comet contamination and an extended disintegration phase that lasts up to a few kyrs. We suggest that a few asteroid-like Sun-approaching objects that brighten significantly at their perihelion passages could, in fact, be disrupting asteroids. An extended period of thermal disruption may also explain the widespread detection of transiting debris in exoplanetary systems.
    There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.
    — Mark Twain, Life on the Mississippi (1883)

  11. #41
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    Relevant to the issue of disintegrating planets around white dwarfs--what happens to the planets earlier in their existence while the WDs are in red giant stage.

    https://arxiv.org/abs/1902.02795

    Speeding past planets? Asteroids radiatively propelled by giant branch Yarkovsky effects

    Dimitri Veras, Arika Higuchi, Shigeru Ida (Submitted on 7 Feb 2019)

    Understanding the fate of planetary systems through white dwarfs which accrete debris crucially relies on tracing the orbital and physical properties of exo-asteroids during the giant branch phase of stellar evolution. Giant branch luminosities exceed the Sun's by over three orders of magnitude, leading to significantly enhanced Yarkovsky and YORP effects on minor planets. Here, we place bounds on Yarkovsky-induced differential migration between asteroids and planets during giant branch mass loss by modelling one exo-Neptune with inner and outer exo-Kuiper belts. In our bounding models, the asteroids move too quickly past the planet to be diverted from their eventual fate, which can range from: (i) populating the outer regions of systems out to 10^4-10^5 au, (ii) being engulfed within the host star, or (iii) experiencing Yarkovsky-induced orbital inclination flipping without any Yarkovsky-induced semi-major axis drift. In these violent limiting cases, temporary resonant trapping of asteroids with radii of under about 10 km by the planet is insignificant, and capture within the planet's Hill sphere requires fine-tuned dissipation. The wide variety of outcomes presented here demonstrates the need to employ sophisticated structure and radiative exo-asteroid models in future studies. Determining where metal-polluting asteroids reside around a white dwarf depends on understanding extreme Yarkovsky physics.

    ==

    A white dwarf is eating something and we aren't sure what it is.

    https://arxiv.org/abs/1902.03219

    A New Generation of Cool White Dwarf Atmosphere Models. III. WD J2356−209: Accretion of a Planetesimal with an Unusual Composition

    S. Blouin, P. Dufour, N.F. Allard, S. Salim, R.M. Rich, L.V.E. Koopmans (Submitted on 8 Feb 2019)

    WD J2356−209 is a cool metal-polluted white dwarf whose visible spectrum is dominated by a strong and broad sodium feature. Although discovered nearly two decades ago, no detailed and realistic analysis of this star had yet been realized. In the absence of atmosphere models taking into account the nonideal high-density effects arising at the photosphere of WD J2356−209, the origin of its unique spectrum had remained nebulous. We use the cool white dwarf atmosphere code presented in the first paper of this series to finally reveal the secrets of this peculiar object and details about the planetesimal that polluted its atmosphere. Thanks to the improved input physics of our models, we find a solution that is in excellent agreement with the photometric observations and the visible spectrum. Our solution reveals that the photosphere of WD J2356−209 has a number density ratio of log Na/Ca = 1.0 ± 0.2, which is the highest ever found in a white dwarf. Since we do not know how long ago the accretion episode stopped (if it has), we cannot precisely determine the composition nor the mass of the accreted planetesimal. Nevertheless, all scenarios considered indicate that its composition is incompatible with that of chondrite-like material and that its mass was at least 10^21 g.
    There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.
    — Mark Twain, Life on the Mississippi (1883)

  12. #42
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    Neutron stars could eat planets, too.


    https://arxiv.org/abs/1902.05203

    Investigation of the asteroid-neutron star collision model for the repeating fast radio bursts

    Jeremy L. Smallwood, Rebecca G. Martin, Bing Zhang (Submitted on 14 Feb 2019)

    The origin of fast radio bursts (FRBs) is still a mystery. One model proposed to interpret the only known repeating object, FRB 121102, is that the radio emission is generated from asteroids colliding with a highly magnetized neutron star (NS). With N-body simulations, we model a debris disc around a central star with an eccentric orbit intruding NS. As the NS approaches the first periastron passage, most of the comets are scattered away rather than being accreted by the NS. To match the observed FRB rate, the debris belt would have to be at least three orders of magnitude more dense than the Kuiper belt. We also consider the rate of collisions on to the central object but find that the density of the debris belt must be at least four orders of magnitude more dense than the Kuiper belt. These discrepancies in the density arise even if (1) one introduces a Kuiper-belt like comet belt rather than an asteroid belt and assume that comet impacts can also make FRBs; (2) the NS moves 2 orders of magnitude slower than their normal proper-motion velocity due to supernova kicks; and (3) the NS orbit is coplanar to the debris belt, which provides the highest rate of collisions.
    There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.
    — Mark Twain, Life on the Mississippi (1883)

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