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The Astrophysical Journal (ApJ)

Resumen/Descripción – provisto por la editorial en inglés
The Astrophysical Journal is an open access journal devoted to recent developments, discoveries, and theories in astronomy and astrophysics. Publications in ApJ constitute significant new research that is directly relevant to astrophysical applications, whether based on observational results or on theoretical insights or modeling.
Palabras clave – provistas por la editorial

astronomy; astrophysics

Disponibilidad
Institución detectada Período Navegá Descargá Solicitá
No detectada desde jul. 1995 / hasta dic. 2023 IOPScience

Información

Tipo de recurso:

revistas

ISSN impreso

0004-637X

ISSN electrónico

1538-4357

Editor responsable

American Astronomical Society (AAS)

Idiomas de la publicación

  • inglés

País de edición

Reino Unido

Información sobre licencias CC

https://creativecommons.org/licenses/by/4.0/

Cobertura temática

Tabla de contenidos

On the Dynamics of Low-viscosity Warped Disks around Black Holes

N. C. Drewes; C. J. NixonORCID

<jats:title>Abstract</jats:title> <jats:p>Accretion disks around black holes can become warped by Lense–Thirring precession. When the disk viscosity is sufficiently small, such that the warp propagates as a wave, then steady-state solutions to the linearized fluid equations exhibit an oscillatory radial profile of the disk tilt angle. Here we show, for the first time, that these solutions are in good agreement with three-dimensional hydrodynamical simulations, in which the viscosity is isotropic and measured to be small compared to the disk angular semi-thickness, and in the case that the disk tilt—and thus the warp amplitude—remains small. We show, using both the linearized fluid equations and hydrodynamical simulations, that the inner disk tilt can be more than several times larger than the original disk tilt, and we provide physical reasoning for this effect. We explore the transition in disk behavior as the misalignment angle is increased, finding increased dissipation associated with regions of strong warping. For large enough misalignments the disk becomes unstable to disk tearing and breaks into discrete planes. For the simulations we present here, we show that the total (physical and numerical) viscosity at the time the disk breaks is small enough that the disk tearing occurs in the wave-like regime, substantiating that disk tearing is possible in this region of parameter space. Our simulations demonstrate that high spatial resolution, and thus low numerical viscosity, is required to accurately model the warp dynamics in this regime. Finally, we discuss the observational implications of our results.</jats:p>

Palabras clave: Space and Planetary Science; Astronomy and Astrophysics.

Pp. 243

Swift/XRT Deep Galactic Plane Survey Discovery of a New Intermediate Polar Cataclysmic Variable, Swift J183920.1-045350

Nicholas M. GorgoneORCID; Patrick A. WoudtORCID; David BuckleyORCID; Koji MukaiORCID; Chryssa KouveliotouORCID; Daniela HuppenkothenORCID; Ersin GöğüşORCID; Eric BellmORCID; Justin D. LinfordORCID; Alexander J. van der HorstORCID; Matthew G. BaringORCID; Dieter HartmannORCID; Paul BarrettORCID; Bradley Cenko; Melissa GrahamORCID; Johnathan Granot; Fiona Harrison; Jamie KenneaORCID; Brendan M. O’ConnorORCID; Stephen Potter; Daniel SternORCID; Patrick SlaneORCID; Ralph WijersORCID

<jats:title>Abstract</jats:title> <jats:p>We report on the Swift/XRT Deep Galactic Plane Survey discovery and multiwavelength follow-up observations of a new intermediate polar (IP) cataclysmic variable, Swift J183920.1-045350. A 449.7 s spin period is found in XMM-Newton and NuSTAR data, accompanied by a 459.9 s optical period that is most likely the synodic, or beat period, produced from a 5.6 hr orbital period. The orbital period is seen with moderate significance in independent long-baseline optical photometry observations taken with the ZTF and SAAO telescopes. We find that the X-ray pulse fraction of the source decreases with increasing energy. The X-ray spectra are consistent with the presence of an Fe emission line complex with both local and interstellar absorption. In the optical spectra, strong H<jats:italic>α</jats:italic>, H <jats:sc>i</jats:sc>, He <jats:sc>i,</jats:sc> and He <jats:sc>ii</jats:sc> emission lines are observed, all common features in magnetic CVs. The source properties are thus typical of known IPs, with the exception of its estimated distance of <jats:inline-formula> <jats:tex-math> <?CDATA ${2.26}_{-0.83}^{+1.93}$?> </jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mn>2.26</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>0.83</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>1.93</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjac2738ieqn1.gif" xlink:type="simple" /> </jats:inline-formula> kpc, which is larger than typical, extending the reach of the CV population in our Galaxy.</jats:p>

Palabras clave: Space and Planetary Science; Astronomy and Astrophysics.

Pp. 243

Effects of Haze Radiation and Eddy Heat Transport on the Thermal Structure of Pluto’s Lower Atmosphere

Linfeng Wan; Xi Zhang; Tanguy Bertrand

<jats:title>Abstract</jats:title> <jats:p>The temperature profile of Pluto’s atmosphere has generally been assumed in a radiative–conductive equilibrium. Recent studies further highlighted the importance of radiative heating and cooling effects by haze particles. In this study, we update results from Zhang et al. by taking into account the icy haze composition proposed by Lavvas et al., and find that radiation of such an icy haze could still dominate the energy balance in the middle and upper atmosphere and explain the cold temperature observed by New Horizons. However, additional considerations are needed to explain the rapid decrease in temperature toward the icy surface at altitudes &lt;25 km. We propose that vertical eddy heat transport might help maintain radiative–diffusive equilibrium in the lower atmosphere. In this scenario, our radiative–conductive–diffusive model (including both gas and haze) would match observations if the eddy diffusivity is on the order of 10<jats:sup>3</jats:sup> cm<jats:sup>2</jats:sup> s<jats:sup>−1</jats:sup>. Alternatively, if eddy heat transport is not effective on Pluto, in order to match observations, haze albedo must increase rapidly with decreasing altitude and approach unity near the surface. This is a plausible result of additional ice condensation and/or cloud formation. In this scenario, haze radiation might still dominate over gas radiation and heat conduction to maintain radiative equilibrium. Better constraints on haze albedo at ultraviolet and visible wavelengths would be a key to distinguish these two scenarios. Future mid-infrared observations from the James Webb Space Telescope could also constrain the thermal emission and haze properties in Pluto’s lower atmosphere.</jats:p>

Palabras clave: Space and Planetary Science; Astronomy and Astrophysics.

Pp. 244

Nonstandard Modeling of a Possible Blue Straggler Star, KIC 11145123

Yoshiki HattaORCID; Takashi SekiiORCID; Masao TakataORCID; Othman BenomarORCID

<jats:title>Abstract</jats:title> <jats:p>Nonstandard modeling of KIC 11145123, a possible blue straggler star, has been asteroseismically carried out based on a scheme to compute stellar models with the chemical compositions in their envelopes arbitrarily modified, mimicking the effects of some interactions with other stars through which blue straggler stars are thought to be born. We have constructed a nonstandard model of the star with the following parameters: <jats:italic>M</jats:italic> = 1.36 <jats:italic>M</jats:italic> <jats:sub>⊙</jats:sub>, <jats:italic>Y</jats:italic> <jats:sub>init</jats:sub> = 0.26, <jats:italic>Z</jats:italic> <jats:sub>init</jats:sub> = 0.002, and <jats:italic>f</jats:italic> <jats:sub>ovs</jats:sub> = 0.027, where <jats:italic>f</jats:italic> <jats:sub>ovs</jats:sub> is the extent of overshooting described as an exponentially decaying diffusive process. The modification is down to the depth of <jats:italic>r</jats:italic>/<jats:italic>R</jats:italic> ∼ 0.6 and the extent Δ<jats:italic>X</jats:italic>, which is a difference in surface hydrogen abundance between the envelope-modified and unmodified models, is 0.06. The residuals between the model and the observed frequencies are comparable with those for the previous model computed assuming standard single-star evolution, suggesting that it is possible that the star was born with a relatively ordinary initial helium abundance of ∼0.26 compared with that of the previous models (∼0.30–0.40), then experienced some modification of the chemical compositions and gained helium in the envelope. Detailed analyses of the nonstandard model have implied that the elemental diffusion in the deep radiative region of the star might be much weaker than that assumed in current stellar evolutionary calculations; we need some extra mechanisms inside the star, rendering the star a much more intriguing target to be further investigated.</jats:p>

Palabras clave: Space and Planetary Science; Astronomy and Astrophysics.

Pp. 244

Phases of Mass Transfer from Hot Subdwarfs to White Dwarf Companions and Their Photometric Properties

Evan B. BauerORCID; Thomas KupferORCID

<jats:title>Abstract</jats:title> <jats:p>Binary systems of a hot subdwarf B (sdB) star + a white dwarf (WD) with orbital periods less than 2–3 hr can come into contact due to gravitational waves and transfer mass from the sdB star to the WD before the sdB star ceases nuclear burning and contracts to become a WD. Motivated by the growing class of observed systems in this category, we study the phases of mass transfer in these systems. We find that because the residual outer hydrogen envelope accounts for a large fraction of an sdB star’s radius, sdB stars can spend a significant amount of time (∼tens of megayears) transferring this small amount of material at low rates (∼10<jats:sup>−10</jats:sup>–10<jats:sup>−9</jats:sup> <jats:italic>M</jats:italic> <jats:sub>⊙</jats:sub> yr<jats:sup>−1</jats:sup>) before transitioning to a phase where the bulk of their He transfers at much faster rates ( ≳10<jats:sup>−8</jats:sup> <jats:italic>M</jats:italic> <jats:sub>⊙</jats:sub> yr<jats:sup>−1</jats:sup>). These systems therefore spend a surprising amount of time with Roche-filling sdB donors at orbital periods longer than the range associated with He star models without an envelope. We predict that the envelope transfer phase should be detectable by searching for ellipsoidal modulation of Roche-filling objects with <jats:italic>P</jats:italic> <jats:sub>orb</jats:sub> = 30–100 minutes and <jats:italic>T</jats:italic> <jats:sub>eff</jats:sub> = 20,000–30,000 K, and that many (≥10) such systems may be found in the Galactic plane after accounting for reddening. We also argue that many of these systems may go through a phase of He transfer that matches the signatures of AM CVn systems, and that some AM CVn systems associated with young stellar populations likely descend from this channel.</jats:p>

Palabras clave: Space and Planetary Science; Astronomy and Astrophysics.

Pp. 245

Whistler-regulated Magnetohydrodynamics: Transport Equations for Electron Thermal Conduction in the High-β Intracluster Medium of Galaxy Clusters

J. F. DrakeORCID; C. PfrommerORCID; C. S. ReynoldsORCID; M. Ruszkowski; M. SwisdakORCID; A. Einarsson; T. Thomas; A. B. Hassam; G. T. Roberg-ClarkORCID

<jats:title>Abstract</jats:title> <jats:p>Transport equations for electron thermal energy in the high-<jats:italic>β</jats:italic> <jats:sub> <jats:italic>e</jats:italic> </jats:sub> intracluster medium (ICM) are developed that include scattering from both classical collisions and self-generated whistler waves. The calculation employs an expansion of the kinetic electron equation along the ambient magnetic field in the limit of strong scattering and assumes whistler waves with low phase speeds <jats:italic>V</jats:italic> <jats:sub> <jats:italic>w</jats:italic> </jats:sub> ∼ <jats:italic>v</jats:italic> <jats:sub> <jats:italic>te</jats:italic> </jats:sub>/<jats:italic>β</jats:italic> <jats:sub> <jats:italic>e</jats:italic> </jats:sub> ≪ <jats:italic>v</jats:italic> <jats:sub> <jats:italic>te</jats:italic> </jats:sub> dominate the turbulent spectrum, with <jats:italic>v</jats:italic> <jats:sub> <jats:italic>te</jats:italic> </jats:sub> the electron thermal speed and <jats:italic>β</jats:italic> <jats:sub> <jats:italic>e</jats:italic> </jats:sub> ≫ 1 the ratio of electron thermal to magnetic pressure. We find: (1) temperature-gradient-driven whistlers dominate classical scattering when <jats:italic>L</jats:italic> <jats:sub> <jats:italic>c</jats:italic> </jats:sub> &gt; <jats:italic>L</jats:italic>/<jats:italic>β</jats:italic> <jats:sub> <jats:italic>e</jats:italic> </jats:sub>, with <jats:italic>L</jats:italic> <jats:sub> <jats:italic>c</jats:italic> </jats:sub> the classical electron mean free path and <jats:italic>L</jats:italic> the electron temperature scale length, and (2) in the whistler-dominated regime the electron thermal flux is controlled by both advection at <jats:italic>V</jats:italic> <jats:sub> <jats:italic>w</jats:italic> </jats:sub> and a comparable diffusive term. The findings suggest whistlers limit electron heat flux over large regions of the ICM, including locations unstable to isobaric condensation. Consequences include: (1) the Field length decreases, extending the domain of thermal instability to smaller length scales, (2) the heat flux temperature dependence changes from <jats:inline-formula> <jats:tex-math> <?CDATA ${T}_{e}^{7/2}/L$?> </jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>7</mml:mn> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:mi>L</mml:mi> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjac1ff1ieqn1.gif" xlink:type="simple" /> </jats:inline-formula> to <jats:inline-formula> <jats:tex-math> <?CDATA ${V}_{w}{{nT}}_{e}\sim {T}_{e}^{1/2}$?> </jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>w</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mrow> <mml:mi mathvariant="italic">nT</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> </mml:msub> <mml:mo>∼</mml:mo> <mml:msubsup> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjac1ff1ieqn2.gif" xlink:type="simple" /> </jats:inline-formula>, (3) the magneto-thermal- and heat-flux-driven buoyancy instabilities are impaired or completely inhibited, and (4) sound waves in the ICM propagate greater distances, as inferred from observations. This description of thermal transport can be used in macroscale ICM models.</jats:p>

Palabras clave: Space and Planetary Science; Astronomy and Astrophysics.

Pp. 245

Energy-dependent Boundaries of Earth's Radiation Belt Electron Slot Region

Yang MeiORCID; Yasong GeORCID; Aimin Du; Xudong GuORCID; Danny SummersORCID; Xinlin LiORCID; Song FuORCID; Zheng XiangORCID

<jats:title>Abstract</jats:title> <jats:p>The variations in radiation belt boundaries reflect competition between acceleration and loss physical processes of energetic electrons, which is an important issue for radiation belts of planets with an internal magnetic field (e.g., Earth, Jupiter, and Saturn). Based on high-quality measurements from Van Allen Probes spanning the years 2014–2018, we develop an empirical model of the energy-dependent boundaries of Earth's electron radiation belt slot region, showing that the lower boundary follows a logarithmic function of the electron energy while the upper boundary is controlled by two competing energy-dependent processes, namely compression and recovery. The compression process relates linearly to a 15 hr averaged <jats:italic>K</jats:italic>p index, while the recovery process is found to be approximately proportional to time. Detailed data-model comparisons demonstrate that our model, using only the <jats:italic>K</jats:italic>p index and time epoch as inputs, reconstructs the slot region boundaries in real time for 200 keV to 2 MeV electrons under varying geomagnetic conditions. Such a data-driven empirical model is prerequisite to understanding the dynamic changes of the slot region in response to both solar and geomagnetic activities. The model can be readily incorporated into future global simulations of radiation belt electron dynamics in Earth's inner magnetosphere and provide new insights into the study of Saturn's and Jupiter's radiation belt variability.</jats:p>

Palabras clave: Space and Planetary Science; Astronomy and Astrophysics.

Pp. 246

Central X-Ray Point Sources Found to Be Abundant in Low-mass, Late-type Galaxies Predicted to Contain an Intermediate-mass Black Hole

Alister W. GrahamORCID; Roberto SoriaORCID; Benjamin L. DavisORCID; Mari Kolehmainen; Thomas Maccarone; James Miller-JonesORCID; Christian MotchORCID; Douglas A. Swartz

<jats:title>Abstract</jats:title> <jats:p>Building upon three late-type galaxies in the Virgo cluster with both a predicted black hole mass of less than ∼10<jats:sup>5</jats:sup> <jats:italic>M</jats:italic> <jats:sub>⊙</jats:sub> and a centrally located X-ray point source, we reveal 11 more such galaxies, more than tripling the number of active intermediate-mass black hole candidates among this population. Moreover, this amounts to a ∼36 ± 8% X-ray detection rate (despite the sometimes high, X-ray-absorbing, H <jats:sc>i</jats:sc> column densities), compared to just 10 ± 5% for (the largely H <jats:sc>i</jats:sc>-free) dwarf early-type galaxies in the Virgo cluster. The expected contribution of X-ray binaries from the galaxies’ inner field stars is negligible. Moreover, given that both the spiral and dwarf galaxies contain nuclear star clusters, the above inequality appears to disfavor X-ray binaries in nuclear star clusters. The higher occupation, or rather detection, fraction among the spiral galaxies may instead reflect an enhanced cool gas/fuel supply and Eddington ratio. Indeed, four of the 11 new X-ray detections are associated with known LINERs or LINER/H <jats:sc>ii</jats:sc> composites. For all (four) of the new detections for which the X-ray flux was strong enough to establish the spectral energy distribution in the Chandra band, it is consistent with power-law spectra. Furthermore, the X-ray emission from the source with the highest flux (NGC 4197: <jats:italic>L</jats:italic> <jats:sub> <jats:italic>X</jats:italic> </jats:sub> ≈ 10<jats:sup>40</jats:sup> erg s<jats:sup>−1</jats:sup>) suggests a non-stellar-mass black hole if the X-ray spectrum corresponds to the “low/hard state”. Follow-up observations to further probe the black hole masses, and prospects for spatially resolving the gravitational spheres of influence around intermediate-mass black holes, are reviewed in some detail.</jats:p>

Palabras clave: Space and Planetary Science; Astronomy and Astrophysics.

Pp. 246

AT 2018lqh and the Nature of the Emerging Population of Day-scale Duration Optical Transients

E. O. OfekORCID; S. M. Adams; E. WaxmanORCID; A. Sharon; D. Kushnir; A. Horesh; A. Ho; M. M. KasliwalORCID; O. YaronORCID; A. Gal-YamORCID; S. R. KulkarniORCID; E. Bellm; F. MasciORCID; D. ShupeORCID; R. DekanyORCID; M. GrahamORCID; R. RiddleORCID; D. DuevORCID; I. AndreoniORCID; A. MahabalORCID; A. Drake

<jats:title>Abstract</jats:title> <jats:p>We report on the discovery of AT 2018lqh (ZTF 18abfzgpl)—a rapidly evolving extragalactic transient in a star-forming host at 242 Mpc. The transient <jats:italic>g</jats:italic>-band light curve’s duration above a half-maximum light is about 2.1 days, where 0.4/1.7 days are spent on the rise/decay, respectively. The estimated bolometric light curve of this object peaked at about 7 × 10<jats:sup>42</jats:sup>erg s<jats:sup>−1</jats:sup>—roughly 7 times brighter than the neutron star (NS)–NS merger event AT 2017gfo. We show that this event can be explained by an explosion with a fast (<jats:italic>v</jats:italic> ∼ 0.08 c) low-mass (≈0.07 <jats:italic>M</jats:italic> <jats:sub>⊙</jats:sub>) ejecta, composed mostly of radioactive elements. For example, ejecta dominated by <jats:sup>56</jats:sup>Ni with a timescale of <jats:italic>t</jats:italic> <jats:sub>0</jats:sub> ≅ 1.6 days for the ejecta to become optically thin for <jats:italic>γ</jats:italic>-rays fits the data well. Such a scenario requires burning at densities that are typically found in the envelopes of neutron stars or the cores of white dwarfs. A combination of circumstellar material (CSM) interaction power at early times and shock cooling at late times is consistent with the photometric observations, but the observed spectrum of the event may pose some challenges for this scenario. We argue that the observations are not consistent with a shock breakout from a stellar envelope, while a model involving a low-mass ejecta ramming into low-mass CSM cannot explain both the early- and late-time observations.</jats:p>

Palabras clave: Space and Planetary Science; Astronomy and Astrophysics.

Pp. 247

New Perspectives on the Exoplanet Radius Gap from a Mathematica Tool and Visualized Water Equation of State

Li ZengORCID; Stein B. JacobsenORCID; Eugenia HyungORCID; Amit LeviORCID; Chantanelle NavaORCID; James KirkORCID; Caroline PiauletORCID; Gaia LacedelliORCID; Dimitar D. SasselovORCID; Michail I. Petaev; Sarah T. StewartORCID; Munazza K. AlamORCID; Mercedes López-MoralesORCID; Mario DamassoORCID; David W. LathamORCID

<jats:title>Abstract</jats:title> <jats:p>Recent astronomical observations obtained with the Kepler and TESS missions and their related ground-based follow-ups revealed an abundance of exoplanets with a size intermediate between Earth and Neptune (1<jats:italic> R</jats:italic> <jats:sub>⊕</jats:sub> ≤ <jats:italic>R</jats:italic> ≤ 4 <jats:italic>R</jats:italic> <jats:sub>⊕</jats:sub>). A low occurrence rate of planets has been identified at around twice the size of Earth (2 × <jats:italic>R</jats:italic> <jats:sub>⊕</jats:sub>), known as the exoplanet radius gap or radius valley. We explore the geometry of this gap in the mass–radius diagram, with the help of a <jats:italic>Mathematica</jats:italic> plotting tool developed with the capability of manipulating exoplanet data in multidimensional parameter space, and with the help of visualized water equations of state in the temperature–density (<jats:italic>T</jats:italic>–<jats:italic>ρ</jats:italic>) graph and the entropy–pressure (<jats:italic>s</jats:italic>–<jats:italic>P</jats:italic>) graph. We show that the radius valley can be explained by a compositional difference between smaller, predominantly rocky planets (&lt;2 × <jats:italic>R</jats:italic> <jats:sub>⊕</jats:sub>) and larger planets (&gt;2 × <jats:italic>R</jats:italic> <jats:sub>⊕</jats:sub>) that exhibit greater compositional diversity including cosmic ices (water, ammonia, methane, etc.) and gaseous envelopes. In particular, among the larger planets (&gt;2 × <jats:italic>R</jats:italic> <jats:sub>⊕</jats:sub>), when viewed from the perspective of planet equilibrium temperature (<jats:italic>T</jats:italic> <jats:sub>eq</jats:sub>), the hot ones (<jats:italic>T</jats:italic> <jats:sub>eq</jats:sub> ≳ 900 K) are consistent with ice-dominated composition without significant gaseous envelopes, while the cold ones (<jats:italic>T</jats:italic> <jats:sub>eq</jats:sub> ≲ 900 K) have more diverse compositions, including various amounts of gaseous envelopes.</jats:p>

Palabras clave: Space and Planetary Science; Astronomy and Astrophysics.

Pp. 247