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The dynamics of a system composed of inelastic hard spheres or disks that are confined between two parallel vertically vibrating walls is studied (the vertical direction is defined as the direction perpendicular to the walls). The distance between the two walls is supposed to be larger than twice the diameter of the particles so that the particles can pass over each other, but still much smaller than the dimensions of the walls. Hence, the system can be considered to be quasi-two-dimensional (quasi-one-dimensional) in the hard spheres (disks) case. For dilute systems, a closed evolution equation for the one-particle distribution function is formulated that takes into account the effects of the confinement. Assuming the system is spatially homogeneous, the kinetic equation is solved approximating the distribution function by a two-temperatures (horizontal and vertical) gaussian distribution. The obtained evolution equations for the partial temperatures are solved, finding a very good agreement with Molecular Dynamics simulation results for a wide range of the parameters (inelasticity, height and density) for states whose projection over a plane parallel to the walls is homogeneous. In the stationary state, where the energy lost in collisions is compensated by the energy injected by the walls, the pressure tensor in the horizontal direction is analyzed and its relation with an instability of the homogenous state observed in the simulations is discussed., Comment: 36 pages, 14 figures, Accepted in Journal of Statistical Mechanics: Theory and Experiment

We consider four subsets of the complexified spacetime algebra, namely the real even part, the real odd part, the imaginary even part and the imaginary odd part. This naturally leads to the four connected components of the Lorentz group, supplemented each time by an additional symmetry. We then examine how these four parts impact the Dirac equation and show that four types of matter arise with positive and negative masses as well as positive and negative charges., Comment: 21 pages. Reports on Mathematical Physics (2022)

Comet C/2016 R2 PanSTARRS (hereafter C/2016 R2) presents an unusually high N2/CO abundance ratio, as well as a heavy depletion in H2O, making it the only known comet of its kind. Understanding its dynamical history is therefore of essential importance as it would allow us to gain a clearer understanding of the evolution of planetesimal formation in our Solar System. Two studies have independently estimated the possible origin of this comet from building blocks formed in a peculiar region of the protoplanetary disk, near the ice line of CO and N2. We intend to investigate the fates of objects formed from the building blocks in these regions. We hope to find a possible explanation for the lack of C/2016 R2-like comets in our Solar System. Using a numerical simulation of the early stages of Solar System formation, we track the dynamics of these objects in the Jumping Neptune scenario based on five different initial conditions for the protosolar disk. We integrate the positions of 250 000 planetesimals over time in order to analyze the evolution of their orbits and create a statistical profile of their expected permanent orbit. Results. We find that objects formed in the region of the CO- and N2- ice lines are highly likely to be sent towards the Oort Cloud or possibly ejected from the Solar System altogether on a relatively short timescale. In all our simulations, over 90% of clones formed in this region evolved into a hyperbolic trajectory, and between 1% and 10% were potentially captured by the Oort Cloud. The handful of comets that remained were either on long-period, highly eccentric orbits like C/2016 R2, or absorbed into the Edgeworth-Kuiper belt. Comets formed <15 au were predominantly ejected early in the formation timeline. As this is the formation zone likely to produce comets of this composition, this process could explain the lack of similar comets observed in the Solar System, Comment: 7 pages, 5 figures

The Gaia mission of the European Space Agency (ESA) has been routinely observing Solar System objects (SSOs) since the beginning of its operations in August 2014. The Gaia data release three (DR3) includes, for the first time, the mean reflectance spectra of a selected sample of 60 518 SSOs, primarily asteroids, observed between August 5, 2014, and May 28, 2017. Each reflectance spectrum was derived from measurements obtained by means of the Blue and Red photometers (BP/RP), which were binned in 16 discrete wavelength bands. We describe the processing of the Gaia spectral data of SSOs, explaining both the criteria used to select the subset of asteroid spectra published in Gaia DR3, and the different steps of our internal validation procedures. In order to further assess the quality of Gaia SSO reflectance spectra, we carried out external validation against SSO reflectance spectra obtained from ground-based and space-borne telescopes and available in the literature. For each selected SSO, an epoch reflectance was computed by dividing the calibrated spectrum observed by the BP/RP at each transit on the focal plane by the mean spectrum of a solar analogue. The latter was obtained by averaging the Gaia spectral measurements of a selected sample of stars known to have very similar spectra to that of the Sun. Finally, a mean of the epoch reflectance spectra was calculated in 16 spectral bands for each SSO. The agreement between Gaia mean reflectance spectra and those available in the literature is good for bright SSOs, regardless of their taxonomic spectral class. We identify an increase in the spectral slope of S-type SSOs with increasing phase angle. Moreover, we show that the spectral slope increases and the depth of the 1 um absorption band decreases for increasing ages of S-type asteroid families., Comment: 30 pages, 26 figures

The third data release by the Gaia mission of the European Space (DR3) is the first release to provide the community with a large sample of observations for more than 150 thousand Solar System objects, including asteroids and natural planetary satellites. The release contains astrometry (over 23 million epochs) and photometry, along with average reflectance spectra of 60518 asteroids and osculating elements. We present an overview of the procedures that have been implemented over several years of development and tests to process Solar System data at the level of accuracy that Gaia can reach. We illustrate the data properties and potential with some practical examples. In order to allow the users of DR3 to best exploit the data, we explain the assumptions and approaches followed in the implementation of the data processing pipeline for Solar System processing, and their effects in terms of data filtering, optimisation, and performances. We then test the data quality by analysing post-fit residuals to adjusted orbits, the capacity of detecting subtle dynamical effects (wobbling due to satellites or shape and Yarkovsky acceleration), and to reproduce known properties of asteroid photometry (phase curves and rotational light curves). The DR3 astrometric accuracy is a clear improvement over the data published in DR2, which concerned a very limited sample of asteroids. The performance of the data reduction is met, and is illustrated by the capacity of detecting milliarcsecond-level wobbling of the asteroid photocentre that is due to satellite or shape effects and contributes to Yarkovsky effect measurements. The third data release can in terms of data completeness and accuracy be considered the first full-scale realisation of the Solar System survey by Gaia., Comment: Gaia DR3 article

There have been 77 TNOs discovered to be librating in the distant transneptunian resonances (beyond the 2:1 resonance, at semimajor axes greater than 47.7~AU) in four well-characterized surveys: the Outer Solar System Origins Survey (OSSOS) and three similar prior surveys. Here we use the OSSOS Survey Simulator to measure their intrinsic orbital distributions using an empirical parameterized model. Because many of the resonances had only one or very few detections, $j$:$k$ resonant objects were grouped by $k$ in order to have a better basis for comparison between models and reality. We also use the Survey Simulator to constrain their absolute populations, finding that they are much larger than predicted by any published Neptune migration model to date; we also find population ratios that are inconsistent with published models, presenting a challenge for future Kuiper Belt emplacement models. The estimated population ratios between these resonances are largely consistent with scattering-sticking predictions, though further discoveries of resonant TNOs with high-precision orbits will be needed to determine whether scattering-sticking can explain the entire distant resonant population or not., Comment: Accepted for publication in Planetary Sciences Journal (PSJ)

We reconsider the classical Schwarzschild solution in the context of a Janus cosmological model. We show that the central singularity can be eliminated through a simple coordinate change and that the subsequent transit from one fold to the other is accompanied by mass inversion. In such scenario matter swallowed by black holes could be ejected as invisible negative mass and dispersed in space., Comment: 11 pages , 12 figures

We present the early installment of the third Gaia data release, Gaia EDR3, consisting of astrometry and photometry for 1.8 billion sources brighter than magnitude 21, complemented with the list of radial velocities from Gaia DR2. Gaia EDR3 contains celestial positions and the apparent brightness in G for approximately 1.8 billion sources. For 1.5 billion of those sources, parallaxes, proper motions, and the (G_BP-G_RP) colour are also available. The passbands for G, G_BP, and G_RP are provided as part of the release. For ease of use, the 7 million radial velocities from Gaia DR2 are included in this release, after the removal of a small number of spurious values. New radial velocities will appear as part of Gaia DR3. Finally, Gaia EDR3 represents an updated materialisation of the celestial reference frame (CRF) in the optical, the Gaia-CRF3, which is based solely on extragalactic sources. The creation of the source list for Gaia EDR3 includes enhancements that make it more robust with respect to high proper motion stars, and the disturbing effects of spurious and partially resolved sources. The source list is largely the same as that for Gaia DR2, but it does feature new sources and there are some notable changes. The source list will not change for Gaia DR3. Gaia EDR3 represents a significant advance over Gaia DR2, with parallax precisions increased by 30 percent, proper motion precisions increased by a factor of 2, and the systematic errors in the astrometry suppressed by 30--40 percent for the parallaxes and by a factor ~2.5 for the proper motions. The photometry also features increased precision, but above all much better homogeneity across colour, magnitude, and celestial position. A single passband for G, G_BP, and G_RP is valid over the entire magnitude and colour range, with no systematics above the 1 percent level., Comment: Accepted for A&A Special Issue on Gaia EDR3, 21 pages, 2 figures. This version includes the updates in the erratum (https://doi.org/10.1051/0004-6361/202039657e)

Since July 2014, the ESA Gaia mission has been surveying the entire sky down to magnitude 20.7 in the visible. In addition to the millions of stars, thousands of Solar System Objects (SSOs) are observed daily. By comparing their positions to those of known objects, a daily processing pipeline filters known objects from potential discoveries. However, owing to Gaia's specific scanning law designed for stars, potential newly discovered moving objects are characterized by very few observations, acquired over a limited time. This aspect was recognized early in the design of the Gaia data processing. A daily processing pipeline dedicated to these candidate discoveries was set up to release calls for observations to a network of ground-based telescopes. Their aim is to acquire follow-up astrometry and to characterize these objects. From the astrometry measured by Gaia, preliminary orbital solutions are determined, allowing to predict the position of these potentially new discovered objects in the sky accounting for the large parallax between Gaia and the Earth (separated by 0.01 au). A specific task within the Gaia Consortium has been responsible for the distribution of requests for follow-up observations of potential Gaia SSO discoveries. Since late 2016, these calls for observations (called alerts) are published daily via a Web interface, freely available to anyone world-wide. Between November 2016 and July 2020, over 1700 alerts have been published, leading to the successful recovery of more than 200 objects. Among those, six have provisional designation assigned with the Gaia observations, the others being previously known objects with poorly characterized orbits, precluding identification at the time of Gaia observations. There is a clear trend for objects with a high inclination to be unidentified, revealing a clear bias in the current census of SSOs against high inclination populations., Comment: Submitted to Astronomy & Astrophysics

The Cold Classical Kuiper Belt, a class of small bodies in undisturbed orbits beyond Neptune, are primitive objects preserving information about Solar System formation. The New Horizons spacecraft flew past one of these objects, the 36 km long contact binary (486958) Arrokoth (2014 MU69), in January 2019. Images from the flyby show that Arrokoth has no detectable rings, and no satellites (larger than 180 meters diameter) within a radius of 8000 km, and has a lightly-cratered smooth surface with complex geological features, unlike those on previously visited Solar System bodies. The density of impact craters indicates the surface dates from the formation of the Solar System. The two lobes of the contact binary have closely aligned poles and equators, constraining their accretion mechanism.

The out-of-equilibrium dynamics of quantum systems is one of the most fascinating problems in physics, with outstanding open questions on issues such as relaxation to equilibrium. An area of particular interest concerns few-body systems, where quantum and thermal fluctuations are expected to be especially relevant. In this contribution, we present numerical results demonstrating the impact of conserved quantities (or 'charges') in the outcomes of out-of-equilibrium measurements starting from realistic equilibrium states on a few-body system implementing the Dicke model., Comment: 12 pages, 1 fig. Contribution to Proceedings of the 24th European Conference on Few-Body Problems in Physics (EFB24). Matches journal version published under CC BY 4.0

The ESA Euclid mission is a space telescope that will survey ~15,000 square degrees of the sky, primarily to study the distant universe (constraining cosmological parameters through the lensing of galaxies). It is also expected to observe ~150,000 Solar System Objects (SSOs), primarily in poorly understood high inclination populations, as it will mostly avoid +/-15 degrees from the ecliptic plane. With a launch date of 2022 and a 6 year survey, Euclid and LSST will operate at the same time, and have complementary capabilities. We propose a LSST mini-survey to coordinate quasi-simultaneous observations between these two powerful observatories, when possible, with the primary aim of greatly improving the orbits of SSOs discovered by these facilities. As Euclid will operate from a halo orbit around the Sun-Earth L2 Lagrangian point, there will be significant parallax between observations from Earth and Euclid (0.01 AU). This means that simultaneous observations will give an independent distance measurement to SSOs, giving additional constraints on orbits compared to single Euclid visits., Comment: white paper submitted as part of the LSST survey strategy call

Context. Centaurs are icy objects in transition between the transneptunian region and the inner solar system, orbiting the Sun in the giant planet region. Some Centaurs display cometary activity, which cannot be sustained by the sublimation of water ice in this part of the solar system, and has been hypothesized to be due to the crystallization of amorphous water ice. Aims. In this work, we look at Centaurs discovered by the Outer Solar System Origins Survey (OSSOS) and search for cometary activity. Tentative detections would improve understanding of the origins of activity among these objects. Methods. We search for comae and structures by fitting and subtracting both Point Spread Functions (PSF) and Trailed point-Spread Functions (TSF) from the OSSOS images of each Centaur. When available, Col-OSSOS images were used to search for comae too. Results. No cometary activity is detected in the OSSOS sample. We track the recent orbital evolution of each new Centaur to confirm that none would actually be predicted to be active, and we provide size estimates for the objects. Conclusions. The addition of 20 OSSOS objects to the population of 250 known Centaurs is consistent with the currently understood scenario, in which drastic drops in perihelion distance induce changes in the thermal balance prone to trigger cometary activity in the giant planet region., Comment: 7 pages, 6 figures, Accepted for publication in Astronomy and Astrophysics

The migration of Neptune's resonances through the proto-Kuiper belt has been imprinted in the distribution of small bodies in the outer Solar System. Here we analyze five published Neptune migration models in detail, focusing on the high pericenter distance (high-q) trans-Neptunian Objects (TNOs) near Neptune's 5:2 and 3:1 mean-motion resonances, because they have large resonant populations, are outside the main classical belt, and are relatively isolated from other strong resonances. We compare the observationally biased output from these dynamical models with the detected TNOs from the Outer Solar System Origins Survey, via its Survey Simulator. All of the four new OSSOS detections of high-q non-resonant TNOs are on the Sunward side of the 5:2 and 3:1 resonances. We show that even after accounting for observation biases, this asymmetric distribution cannot be drawn from a uniform distribution of TNOs at 2sigma confidence. As shown by previous work, our analysis here tentatively confirms that the dynamical model that uses grainy slow Neptune migration provides the best match to the real high-q TNO orbital data. However, due to extreme observational biases, we have very few high-q TNO discoveries with which to statistically constrain the models. Thus, this analysis provides a framework for future comparison between the output from detailed, dynamically classified Neptune migration simulations and the TNO discoveries from future well-characterized surveys. We show that a deeper survey (to a limiting r-magnitude of 26.0) with a similar survey area to OSSOS could statistically distinguish between these five Neptune migration models., Comment: Accepted for publication in the Astronomical Journal

We highlight the power of the Gaia DR2 in studying many fine structures of the Hertzsprung-Russell diagram (HRD). Gaia allows us to present many different HRDs, depending in particular on stellar population selections. We do not aim here for completeness in terms of types of stars or stellar evolutionary aspects. Instead, we have chosen several illustrative examples. We describe some of the selections that can be made in Gaia DR2 to highlight the main structures of the Gaia HRDs. We select both field and cluster (open and globular) stars, compare the observations with previous classifications and with stellar evolutionary tracks, and we present variations of the Gaia HRD with age, metallicity, and kinematics. Late stages of stellar evolution such as hot subdwarfs, post-AGB stars, planetary nebulae, and white dwarfs are also analysed, as well as low-mass brown dwarf objects. The Gaia HRDs are unprecedented in both precision and coverage of the various Milky Way stellar populations and stellar evolutionary phases. Many fine structures of the HRDs are presented. The clear split of the white dwarf sequence into hydrogen and helium white dwarfs is presented for the first time in an HRD. The relation between kinematics and the HRD is nicely illustrated. Two different populations in a classical kinematic selection of the halo are unambiguously identified in the HRD. Membership and mean parameters for a selected list of open clusters are provided. They allow drawing very detailed cluster sequences, highlighting fine structures, and providing extremely precise empirical isochrones that will lead to more insight in stellar physics. Gaia DR2 demonstrates the potential of combining precise astrometry and photometry for large samples for studies in stellar evolution and stellar population and opens an entire new area for HRD-based studies., Comment: Published in the A&A Gaia Data Release 2 special issue. Tables 2 and A.4 corrected. Tables available at http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/616/A10

The scattering trans-Neptunian Objects (TNOs) can be measured to smaller sizes than any other distant small-body population. We use the largest sample yet obtained, 68 discoveries, primarily by the Outer Solar System Origins Survey (OSSOS), to constrain the slope of its luminosity distribution, with sensitivity to much fainter absolute $H$ magnitudes than previous work. Using the analysis technique in Shankman et al. (2016), we confirm that a single slope for the $H$-distribution is not an accurate representation of the scattering TNOs and Centaurs, and that a break in the distribution is required, in support of previous conclusions. A bright-end slope of $\alpha_b=0.9$ transitioning to a faint-end slope $\alpha_f$ of 0.4-0.5 with a differential number contrast $c$ from 1 (a knee) to 10 (a divot) provides an acceptable match to our data. We find that break magnitudes $H_b$ of 7.7 and 8.3, values both previously suggested for dynamically hot Kuiper belt populations, are equally non-rejectable for a range of $\alpha_f$ and $c$ in our statistical analysis. Our preferred divot $H$-distribution transitions to $\alpha_f=0.5$ with a divot of contrast $c=3$ at $H_b=8.3$, while our preferred knee $H$-distribution transitions to $\alpha_f=0.4$ at $H_b=7.7$. The intrinsic population of scattering TNOs required to match the OSSOS detections is $3\times10^6$ for $H_r<12$, and $9\times10^4$ for $H_r<8.66$ ($D\gtrsim100$~km), with Centaurs having an intrinsic population two orders of magnitude smaller., Comment: accepted for publication in AJ

All surveys include observational biases, which makes it impossible to directly compare properties of discovered trans-Neptunian Objects (TNOs) with dynamical models. However, by carefully keeping track of survey pointings on the sky, detection limits, tracking fractions, and rate cuts, the biases from a survey can be modelled in Survey Simulator software. A Survey Simulator takes an intrinsic orbital model (from, for example, the output of a dynamical Kuiper belt emplacement simulation) and applies the survey biases, so that the biased simulated objects can be directly compared with real discoveries. This methodology has been used with great success in the Outer Solar System Origins Survey (OSSOS) and its predecessor surveys. In this chapter, we give four examples of ways to use the OSSOS Survey Simulator to gain knowledge about the true structure of the Kuiper Belt. We demonstrate how to statistically compare different dynamical model outputs with real TNO discoveries, how to quantify detection biases within a TNO population, how to measure intrinsic population sizes, and how to use upper limits from non-detections. We hope this will provide a framework for dynamical modellers to statistically test the validity of their models., Comment: accepted as a chapter in the "Frontiers in Astronomy and Space Sciences" research topic "From Comets to Pluto and Beyond"

The non-equilibrium dynamics of quantum many-body systems is one of the most fascinating problems in physics. Open questions range from how they relax to equilibrium to how to extract useful work from them. A critical point lies in assessing whether a system has conserved quantities (or 'charges'), as these can drastically influence its dynamics. Here we propose a general protocol to reveal the existence of charges based on a set of exact relations between out-of-equilibrium fluctuations and equilibrium properties of a quantum system. We apply these generalised quantum fluctuation relations to a driven quantum simulator, demonstrating their relevance to obtain unbiased temperature estimates from non-equilibrium measurements. Our findings will help guide research on the interplay of quantum and thermal fluctuations in quantum simulation, in studying the transition from integrability to chaos and in the design of new quantum devices., Comment: v3: substantially improved presentation and discussion, matches published version; accompanied by numerical data in figures

Parallaxes for 331 classical Cepheids, 31 Type II Cepheids and 364 RR Lyrae stars in common between Gaia and the Hipparcos and Tycho-2 catalogues are published in Gaia Data Release 1 (DR1) as part of the Tycho-Gaia Astrometric Solution (TGAS). In order to test these first parallax measurements of the primary standard candles of the cosmological distance ladder, that involve astrometry collected by Gaia during the initial 14 months of science operation, we compared them with literature estimates and derived new period-luminosity ($PL$), period-Wesenheit ($PW$) relations for classical and Type II Cepheids and infrared $PL$, $PL$-metallicity ($PLZ$) and optical luminosity-metallicity ($M_V$-[Fe/H]) relations for the RR Lyrae stars, with zero points based on TGAS. The new relations were computed using multi-band ($V,I,J,K_{\mathrm{s}},W_{1}$) photometry and spectroscopic metal abundances available in the literature, and applying three alternative approaches: (i) by linear least squares fitting the absolute magnitudes inferred from direct transformation of the TGAS parallaxes, (ii) by adopting astrometric-based luminosities, and (iii) using a Bayesian fitting approach. TGAS parallaxes bring a significant added value to the previous Hipparcos estimates. The relations presented in this paper represent first Gaia-calibrated relations and form a "work-in-progress" milestone report in the wait for Gaia-only parallaxes of which a first solution will become available with Gaia's Data Release 2 (DR2) in 2018., Comment: 29 pages, 25 figures. Accepted for publication by A&A

Context. The first Gaia Data Release contains the Tycho-Gaia Astrometric Solution (TGAS). This is a subset of about 2 million stars for which, besides the position and photometry, the proper motion and parallax are calculated using Hipparcos and Tycho-2 positions in 1991.25 as prior information. Aims. We investigate the scientific potential and limitations of the TGAS component by means of the astrometric data for open clusters. Methods. Mean cluster parallax and proper motion values are derived taking into account the error correlations within the astrometric solutions for individual stars, an estimate of the internal velocity dispersion in the cluster, and, where relevant, the effects of the depth of the cluster along the line of sight. Internal consistency of the TGAS data is assessed. Results. Values given for standard uncertainties are still inaccurate and may lead to unrealistic unit-weight standard deviations of least squares solutions for cluster parameters. Reconstructed mean cluster parallax and proper motion values are generally in very good agreement with earlier Hipparcos-based determination, although the Gaia mean parallax for the Pleiades is a significant exception. We have no current explanation for that discrepancy. Most clusters are observed to extend to nearly 15 pc from the cluster centre, and it will be up to future Gaia releases to establish whether those potential cluster-member stars are still dynamically bound to the clusters. Conclusions. The Gaia DR1 provides the means to examine open clusters far beyond their more easily visible cores, and can provide membership assessments based on proper motions and parallaxes. A combined HR diagram shows the same features as observed before using the Hipparcos data, with clearly increased luminosities for older A and F dwarfs., Comment: Accepted for publication by A&A. 21 pages main text plus 46 pages appendices. 34 figures main text, 38 figures appendices. 8 table in main text, 19 tables in appendices

Methods to manipulate the individual constituents of an ultracold quantum gas mixture are essential tools for a number of applications, for example the direct quantum simulation of impurity physics. We investigate a scheme in which species-selective control is achieved using magnetic potentials dressed with multiple radiofrequencies, exploiting the different Land\'e g-factors of the constituent atomic species. We describe a mixture dressed with two frequencies, where atoms are confined in harmonic potentials with a controllable degree of overlap between the two atomic species. This is then extended to a four radiofrequency scheme in which a double well potential for one species is overlaid with a single well for the other. The discussion is framed with parameters that are suitable for a 85Rb and 87Rb mixture, but is readily generalised to other combinations., Comment: 9 pages, 3 figures

We report the orbital distribution of the Trans-Neptunian objects (TNOs) discovered during the High Ecliptic Latitude (HiLat) extension of the Canada-France Ecliptic Plane Survey (CFEPS), conducted from June 2006 to July 2009. The HiLat component was designed to address one of the shortcomings of ecliptic surveys (like CFEPS), their lack of sensitivity to high-inclination objects. We searched 701~deg$^2$ of sky ranging from 12$^\circ$ to 85$^\circ$ ecliptic latitude and discovered \lKBO TNOs, with inclinations between 15$^\circ$ to 104$^\circ$. This survey places a very strong constraint on the inclination distribution of the hot component of the classical Kuiper Belt, ruling out any possibility of a large intrinsic fraction of highly inclined orbits. Using the parameterization of \citet{2001AJ....121.2804B}, the HiLat sample combined with CFEPS imposes a width $14^\circ \le \sigma \le 15.5^\circ$, with a best match for $\sigma = 14.5^\circ$. HiLat discovered the first retrograde TNO, 2008~KV$_{42}$, with an almost polar orbit with inclination 104$^\circ$, and (418993), a scattering object with perihelion in the region of Saturn's influence, with $a \sim 400$~AU and $i = 68^\circ$., Comment: 27 pages, 6 figures, 5 tables

We measure the absolute magnitude, $H$, distribution, $dN(H) \propto 10^{\alpha H}$ of the scattering Trans-Neptunian Objects (TNOs) as a proxy for their size-frequency distribution. We show that the H-distribution of the scattering TNOs is not consistent with a single-slope distribution, but must transition around $H_g \sim 9$ to either a knee with a shallow slope or to a divot, which is a differential drop followed by second exponential distribution. Our analysis is based on a sample of 22 scattering TNOs drawn from three different TNO surveys, the Canada-France Ecliptic Plane Survey (CFEPS, Petit et al. 2011), Alexandersen et al. (2014), and the Outer Solar System Origins Survey (OSSOS, Bannister et al. 2016), all of which provide well characterized detection thresholds, combined with a cosmogonic model for the formation of the scattering TNO population. Our measured absolute magnitude distribution result is independent of the choice of cosmogonic model. Based on our analysis, we estimate that number of scattering TNOs is (2.4-8.3)$\times 10^5$ for $H_r < 12$. A divot $H$-distribution is seen in a variety of formation scenarios and may explain several puzzles in Kuiper Belt science. We find that a divot $H$-distribution simultaneously explains the observed scattering TNO, Neptune Trojan, Plutino, and Centaur $H$-distributions while simultaneously predicting a large enough scattering TNO population to act as the sole supply of the Jupiter-Family Comets.

We perform a comprehensive analysis of the spectral statistics of the molecular resonances in $^{166}$Er and $^{168}$Er observed in recent ultracold collision experiments [Frisch et al., Nature {\bf 507}, 475 (2014)] with the aim of determining the chaoticity of this system. We calculate different independent statistical properties to check their degree of agreement with random matrix theory (RMT), and analyze if they are consistent with the possibility of having missing resonances. The analysis of the short-range fluctuations as a function of the magnetic field points to a steady increase of chaoticity until $B \sim 30$ G. The repulsion parameter decreases for higher magnetic fields, an effect that can be interpreted as due to missing resonances. The analysis of long-range fluctuations allows us to be more quantitative and estimate a $20-25\%$ fraction of missing levels. Finally, a study of the distribution of resonance widths provides additional evidence supporting missing resonances of small width compared with the experimental magnetic field resolution. We conclude that further measurements with increased resolution will be necessary to give a final answer to the problem of missing resonances and the agreement with RMT., Comment: 9 pages, 6 figures

The Canada-France Ecliptic Plane Survey discovered four trans-Neptunian objects with semi-major axes near the 5:1 resonance, revealing a large and previously undetected intrinsic population. Three of these objects are currently resonant with Neptune, and the fourth is consistent with being an object that escaped the resonance at some point in the past. The non-resonant object may be representative of a detached population that is stable at slightly lower semi-major axes than the 5:1 resonance. We generated clones of these objects by resampling the astrometric uncertainty and examined their behavior over a 4.5 Gyr numerical simulation. The majority of the clones of the three resonant objects (>90%) spend a total of 10^7 years in resonance during their 4.5 Gyr integrations; most clones experience multiple periods of resonance capture. Our dynamical integrations reveal an exchange between the 5:1 resonance, the scattering objects, and other large semi-major axis resonances, especially the 4:1, 6:1, and 7:1. The multiple capture events and relatively short resonance lifetimes after capture suggest that these objects are captured scattering objects that stick in the 5:1 resonance. These 5:1 resonators may be representative of a temporary population, requiring regular contributions from a source population. We examined the dynamical characteristics (inclination, eccentricity, resonant island, libration amplitude) of the detected objects and their clones in order to provide an empirical model of the orbit structure of the 5:1 resonance. This resonance is dynamically hot and includes primarily symmetric librators. Given our orbit model, the intrinsic population necessary for the detection of these three objects in the 5:1 resonance is 1900(+3300 -1400, 95% confidence) with H_g< 8 and e > 0.5.

The formation mechanisms of the ice giants Uranus and Neptune, and the origin of their elemental and isotopic compositions, have long been debated. The density of solids in the outer protosolar nebula is too low to explain their formation, and spectroscopic observations show that both planets are highly enriched in carbon, very poor in nitrogen, and the ices from which they originally formed might had deuterium-to-hydrogen ratios lower than the predicted cometary value, unexplained properties observed in no other planets. Here we show that all these properties can be explained naturally if Uranus and Neptune both formed at the carbon monoxide iceline. Due to the diffusive redistribution of vapors, this outer region of the protosolar nebula intrinsically has enough surface density to form both planets from carbon-rich solids but nitrogen-depleted gas, in abundances consistent with their observed values. Water rich interiors originating mostly from transformed CO ices reconcile the D/H value of Uranus and Neptune's building blocks with the cometary value. Finally, Our scenario generalizes a well known hypothesis that Jupiter formed on an iceline (water snowline) for the two ice giants, and might be a first step towards generalizing this mechanism for other giant planets., Comment: The Astrophysical Journal (in press), 8 pages, 5 figures

Remote sensing observations meet some limitations when used to study the bulk atmospheric composition of the giant planets of our solar system. A remarkable example of the superiority of in situ probe measurements is illustrated by the exploration of Jupiter, where key measurements such as the determination of the noble gases abundances and the precise measurement of the helium mixing ratio have only been made available through in situ measurements by the Galileo probe. This paper describes the main scientific goals to be addressed by the future in situ exploration of Saturn placing the Galileo probe exploration of Jupiter in a broader context and before the future probe exploration of the more remote ice giants. In situ exploration of Saturn's atmosphere addresses two broad themes that are discussed throughout this paper: first, the formation history of our solar system and second, the processes at play in planetary atmospheres. In this context, we detail the reasons why measurements of Saturn's bulk elemental and isotopic composition would place important constraints on the volatile reservoirs in the protosolar nebula. We also show that the in situ measurement of CO (or any other disequilibrium species that is depleted by reaction with water) in Saturn's upper troposphere would constrain its bulk O/H ratio. We highlight the key measurements required to distinguish competing theories to shed light on giant planet formation as a common process in planetary systems with potential applications to most extrasolar systems. In situ measurements of Saturn's stratospheric and tropospheric dynamics, chemistry and cloud-forming processes will provide access to phenomena unreachable to remote sensing studies. Different mission architectures are envisaged, which would benefit from strong international collaborations., Comment: Submitted to Planetary and Space Science

Context. The observation of carbon-rich disks have motivated several studies questioning the influence of the C/O ratio on their gas phase composition in order to establish the connection between the metallicity of hot-Jupiters and that of their parent stars. Aims. We to propose a method that allows the characterization of the adopted C/O ratio in protoplanetary disks independently from the determination of the host star composition. Titanium and vanadium chemistries are investigated because they are strong optical absorbers and also because their oxides are known to be sensitive to the C/O ratio in some exoplanet atmospheres. Methods. We use a commercial package based on the Gibbs energy minimization technique to compute the titanium and vanadium equilibrium chemistries in protoplanetary disks for C/O ratios ranging from 0.05 to 10. Our calculations are performed for pressures ranging from 1e-6 to 1e-2 bar, and for temperatures ranging from 50 to 2000 K. Results. We find that the vanadium nitride/vanadium oxide and titanium hydride/titanium oxide gas phase ratios strongly depend on the C/O ratio in the hot parts of disks (T > 1000 K). Our calculations suggest that, in these regions, these ratios can be used as tracers of the C/O value in protoplanetary disks., Comment: Accepted for publication in A&A

Amateur contributions to professional publications have increased exponentially over the last decades in the field of Planetary Astronomy. Here we review the different domains of the field in which collaborations between professional and amateur astronomers are effective and regularly lead to scientific publications. We discuss the instruments, detectors, softwares and methodologies typically used by amateur astronomers to collect the scientific data in the different domains of interest. Amateur contributions to the monitoring of planets and interplanetary matter, characterization of asteroids and comets, as well as the determination of the physical properties of Kuiper Belt Objects and exoplanets are discussed., Comment: 123 pages, Accepted for publication in Experimental Astronomy

Via joint analysis of a calibrated telescopic survey, which found scattering Kuiper Belt objects, and models of their expected orbital distribution, we measure the form of the scattering object's size distribution. Ruling out a single power-law at greater than 99% confidence, we constrain the form of the size distribution and find that, surprisingly, our analysis favours a very sudden decrease (a divot) in the number distribution as diameters decrease below 100 km, with the number of smaller objects then rising again as expected via collisional equilibrium. Extrapolating at this collisional equilibrium slope produced enough kilometer-scale scattering objects to supply the nearby Jupiter-Family comets. Our interpretation is that this divot feature is a preserved relic of the size distribution made by planetesimal formation, now "frozen in" to portions of the Kuiper Belt sharing a "hot" orbital inclination distribution, explaining several puzzles in Kuiper Belt science. Additionally, we show that to match today's scattering-object inclination distribution, the supply source that was scattered outward must have already been vertically heated to of order 10 degrees., Comment: accepted 2013 January 8; published 2013 January 22 21 pages, 4 figures

We study the mean-field dynamics of p-wave Feshbach molecule production in an ultra cold gas of Fermi atoms in the same internal state. We derive a separable potential to describe the low-energy scattering properties of such atoms, and use this potential to solve the mean-field dynamics during a magnetic field sweep. Initially, on the negative scattering length side of a Feshbach resonance the gas is described by the BCS theory. We adapt the method by Szyma\'{n}ska et al. [Phys. Rev. Lett. 94, 170402 (2005)] to p-wave interacting Fermi gases and model the conversion dynamics of the gas into a Bose-Einstein condensate of molecules on the other side of the resonance under the influence of a linearly varying magnetic field. We have analyzed the dependence of the molecule production efficiency on the density of the gas, temperature, initial value of the magnetic field, and magnetic field ramp speed. Our results show that in this approximation molecule production by a linear magnetic field sweep is highly dependent on the initial state., Comment: 11 pages, 11 figures. Accepted to appear in Phys. Rev. A

The transneptunian objects (TNOs) trapped in mean-motion resonances with Neptune were likely emplaced there during planet migration late in the giant-planet formation process. We perform detailed modelling of the resonant objects detected in the Canada-France Ecliptic Plane Survey (CFEPS) in order to provide population estimates and, for some resonances, constrain the complex internal orbital element distribution. Detection biases play a critical role because phase relationships with Neptune make object discovery more likely at certain longitudes. This paper discusses the 3:2, 5:2, 2:1, 3:1, 5:1, 4:3, 5:3, 7:3, 5:4, and 7:4 mean-motion resonances, all of which had CFEPS detections, along with our upper limit on 1:1 Neptune Trojans (which is consistent with their small population estimated elsewhere). For the plutinos (TNOs in the 3:2 resonance) we refine the orbital element distribution given in Kavelaars et al. (2009) and show that steep H-magnitude distributions (N(H) proportional to 10aH, with a=0.8-0.9) are favoured in the range Hg=8-9, and confirm that this resonance does not share the inclination distribution of the classical Kuiper Belt. We give the first population estimate for the 5:2 resonance and find that, to within the uncertainties, the population is equal to that of the 3:2 (13,000 TNOs with Hg < 9.16), whereas the 2:1 population is smaller by a factor of 3-4 compared to the other two resonances. We also measure significant populations inhabiting the 4:3, 5:3, 7:3, 5:4, 7:4, 3:1, and 5:1 resonances, with Hg < 9.16 (D >100 km) populations in the thousands. We compare our intrinsic population and orbital-element distributions with several published models of resonant-TNO production; the most striking discrepancy is that resonances beyond the 2:1 are in reality more heavily populated than in published models., Comment: Accepted to AJ. Full dataset, survey simulator, and synthetic models available at http://www.cfeps.net

We propose a fast and non-destructive spectroscopic method for single molecular ions that implements quantum logic schemes between an atomic ion and the molecular ion of interest. Our proposal relies on a hybrid coherent manipulation of the two-ion system, using optical or magnetic forces depending on the types of molecular levels to be addressed (Zeeman, rotational, vibrational or electronic degrees of freedom). The method is especially suited for the non-destructive precision spectroscopy of single molecular ions, and sets a starting point for new hybrid quantum computation schemes that combine molecular and atomic ions, covering the measurement and entangling steps., Comment: v3. Substantially enlarged manuscript with details of derivations and calculations in two appendices. To appear in PRA

Hot temperature minerals have been detected in a large number of comets and were also identified in the samples of Comet Wild 2 that were returned by the Stardust mission. Meanwhile, observations of the distribution of hot minerals in young stellar systems suggest that these materials were produced in the inner part of the primordial nebula and have been transported outward in the formation zone of comets. We investigate the possibility that photophoresis provides a viable mechanism to transport high-temperature materials from the inner solar system to the regions in which the comets were forming. We use a grid of time-dependent disk models of the solar nebula to quantify the distance range at which hot minerals can be transported from the inner part of the disk toward its outer regions as a function of their size and density. The particles considered here are in the form of aggregates that presumably were assembled from hot mineral individual grains ranging down to submicron sizes and formed by condensation within the hottest portion of the solar nebula. Our particle-transport model includes the photophoresis, radiation pressure, and gas drag. Depending on the postulated disk parameters and the density of particles, 10-2 to 10-1 m aggregates can reach heliocentric distances up to 35 AU in the primordial nebula over very short timescales (no more than a few hundred thousand years). 10-3 m particles follow the same trajectory as the larger ones but their maximum migration distance does not exceed 26 AU and is reached at later epochs in the disks. On the other hand, 10-5 to 10-4 m aggregates are continuously pushed outward during the evolution of the solar nebula. Our simulations suggest that irrespective of the employed solar nebula model, photophoresis is a mechanism that can explain the presence of hot temperature minerals in the formation region of comets., Comment: Accepted for publication in Astronomy & Astrophysics

In this work we provide a general methodology to directly measure topological order in cold atom systems. As an application we propose the realisation of a characteristic topological model, introduced by Haldane, using optical lattices loaded with fermionic atoms in two internal states. We demonstrate that time-of-flight measurements directly reveal the topological order of the system in the form of momentum space skyrmions.

Favored theories of giant planet formation center around two main paradigms, namely the core accretion model and the gravitational instability model. These two formation scenarios support the hypothesis that the giant planet metallicities should be higher or equal to that of the parent star. Meanwhile, spectra of the transiting hot Jupiter HD189733b suggest that carbon and oxygen abundances range from depleted to enriched with respect to the star. Here, using a model describing the formation sequence and composition of planetesimals in the protoplanetary disk, we determine the range of volatile abundances in the envelope of HD189733b that is consistent with the 20--80 Earth-masses of heavy elements estimated to be present in the planet's envelope. We then compare the inferred carbon and oxygen abundances to those retrieved from spectroscopy and we find a range of supersolar values that directly fit both spectra and internal structure models. In some cases, we find that the apparent contradiction between the subsolar elemental abundances and the {mass of heavy elements predicted in HD189733b by internal structure models} can be explained by the presence of large amounts of carbon molecules in the form of polycyclic aromatic hydrocarbons and soots in the upper layers of the envelope, as suggested by recent photochemical models. A diagnostic test that would confirm the presence of these compounds in the envelope is the detection of acetylene., Comment: Accepted for publication in The Astrophysical Journal

We report here the discovery of an new near-equal mass Trans-Neptunian Binaries (TNBs) L5c02 and the and the putative detection of a second TNB (L4k12) among the year two and three detections of the Canada-France-Eclipic Plane Survey (CFEPS). These new binaries (internal designation L4k12 and L5c02) have moderate separations of 0.4" and 0.6" respectively. The follow-up observation confirmed the binarity of L5c02, but L4k12 are still lack of more followup observations. L4k12 has a heliocentric orbital inclination of ~ 35?degree, marking this system as having the highest heliocentric orbital inclination among known near-equal mass binaries. Both systems are members of the classical main Kuiper belt population. Based on the sample of objects searched we determine that the fraction of near-equal mass wide binaries with separations > 0.4" is 1.5% to 20% in the cold main classical Kuiper belt and, if our detection the binarity L4k12 holds, 3% to 43% in the hot main classical objects are binary. In this manuscript we describe our detection process, the sample of objects surveyed, our confirmation observations., Comment: 12 pages, 4 figures, accepted for publication in PASP

The dynamics of F=1 spinor condensates initially prepared in a double-well potential is studied in the mean field approach. It is shown that a small seed of $m=0$ atoms on a system with initially well separated m=1 and m=-1 condensates has a dramatic effect on their mixing dynamics, acting as an effective barrier for a remarkably long time. We show that this effect is due to the spinor character of the system, and provides an observable example of the interplay between the internal spin dynamics and the macroscopic evolution of the magnetization in a spinor Bose-Einstein condensate., Comment: Accepted for publication at the Europhysics Letters

We have performed an ecliptic imaging survey of the Kuiper belt with our deepest and widest field achieving a limiting flux of m(g') = 26.4, with a sky coverage of 3.0 square-degrees. This is the largest coverage of any other Kuiper belt survey to this depth. We detect 72 objects, two of which have been previously observed. We have improved the Bayesian maximum likelihood fitting technique presented in Gladman et al. (1998) to account for calibration and sky density variations and have used this to determine the luminosity function of the Kuiper belt. Combining our detections with previous surveys, we find the luminosity function is well represented by a single power-law with slope alpha = 0.65 +/- 0.05 and an on ecliptic sky density of 1 object per square-degree brighter than m(R)=23.42 +/- 0.13. Assuming constant albedos, this slope suggests a differential size-distribution slope of 4.25 +/- 0.25, which is steeper than the Dohnanyi slope of 3.5 expected if the belt is in a state of collisional equilibrium. We find no evidence for a roll-over or knee in the luminosity function and reject such models brightward of m(R) ~ 24.6., Comment: 50 Pages, 8 Figures

Protoplanetary disks start their lives with a dust free inner region where the temperatures are higher than the sublimation temperature of solids. As the star illuminates the innermost particles, which are immersed in gas at the sublimation edge, these particles are subject to a photophoretic force. We examine the motion of dust particles at the inner edge of protoplanetary disks due to photophoretic drag. We find that, in a laminar disk, photophoretic drag increases the size of the inner hole after accretion onto the central body has become subdued. This region within the hole becomes an optically transparent zone containing gas and large dusty particles (>>10 cm), but devoid of, or strongly depleted in, smaller dust aggregates. Photophoresis can clear the inner disk of dust out to 10 AU in less than 1 Myr. At late times, the edge reaches a stable equilibrium between inward drift and photophoretic outward drift, at a distance of some tens of AU. Eventually, the edge will move inwards again as the disk disperses, shifting the equilibrium position back from about 40 AU to below 30 AU in 1-2 Myr in the disk model. In a turbulent disk, diffusion can delay the clearing of a disk by photophoresis. Smaller and/or age-independent holes of radii of a few AU are also possible outcomes of turbulent diffusion counteracting photophoresis. This outward and then inward moving edge marks a region of high dust concentration. This density enhancement, and the efficient transport of particles from close to the star to large distances away, can explain features of comets such as high measured ratios of crystalline to amorphous silicates, and has a large number of other applications., Comment: 11 pages, 6 figures

We study the spin dynamics of quasi-one-dimensional F=1 condensates both at zero and finite temperatures for arbitrary initial spin configurations. The rich dynamical evolution exhibited by these non-linear systems is explained by surprisingly simple principles: minimization of energy at zero temperature, and maximization of entropy at high temperature. Our analytical results for the homogeneous case are corroborated by numerical simulations for confined condensates in a wide variety of initial conditions. These predictions compare qualitatively well with recent experimental observations and can, therefore, serve as a guidance for on-going experiments., Comment: 4 pages, 2 figures. v3: matches version appeared in PRL

Binding energies and density profiles of two-dimensional systems of liquid He-4 with different geometries are studied by means of a zero-range density functional adjusted to reproduce the line tension obtained in a previous diffusion Monte Carlo calculation (lambda_{DMC}=0.121 K/A). It is shown that this density functional provides accurate results for the binding energy of large clusters with a reasonable computational effort., Comment: RevTeX4, 11 pages + 2 tables + 6 figures

We present the data acquisition strategy and characterization procedures for the Canada-France Ecliptic Plane Survey (CFEPS), a sub-component of the Canada-France-Hawaii Telescope Legacy Survey. The survey began in early 2003 and as of summer 2005 has covered 430 square degrees of sky within a few degrees of the ecliptic. Moving objects beyond the orbit of Uranus are detected to a magnitude limit of $m_R$=23 -- 24 (depending on the image quality). To track as large a sample as possible and avoid introducing followup bias, we have developed a multi-epoch observing strategy that is spread over several years. We present the evolution of the uncertainties in ephemeris position and orbital elements as the objects progress through the epochs. We then present a small 10-object sample that was tracked in this manner as part of a preliminary survey starting a year before the main CFEPS project. We describe the CFEPS survey simulator, to be released in 2006, which allows theoretical models of the Kuiper Belt to be compared with the survey discoveries since CFEPS has a well-documented pointing history with characterized detection efficiencies as a function of magnitude and rate of motion on the sky. Using the pre-survey objects we illustrate the usage of the simulator in modeling the classical Kuiper Belt., Comment: to be submitted to Icarus

In this work we compare the results of the Gross-Pitaevskii and modified Gross-Pitaevskii equations with ab initio variational Monte Carlo calculations for Bose-Einstein condensates of atoms in axially symmetric traps. We examine both the ground state and excited states having a vortex line along the z-axis at high values of the gas parameter and demonstrate an excellent agreement between the modified Gross-Pitaevskii and ab initio Monte Carlo methods, both for the ground and vortex states., Comment: To appear in Phys Rev A 71

The possibilities of pairing in two-dimensional boson-fermion mixtures are carefully analyzed. It is shown that the boson-induced attraction between two identical fermions dominates the p-wave pairing at low density. For a given fermion density, the pairing gap becomes maximal at a certain optimal boson concentration. The conditions for observing pairing in current experiments are discussed, Comment: 10 pages, 5 figs, revtex4

The binding energies of two-dimensional clusters (puddles) of $^4$He are calculated in the framework of the diffusion Monte Carlo method. The results are very well fitted by a mass formula in powers of $x=N^{-1/2}$, where $N$ is the number of particles. The analysis of the mass formula allows for the extraction of the line tension, which turns out to be 0.121 K/A. Sizes and density profiles of the puddles are also reported., Comment: 19 pages including 2 figs; revtex4. Matches published version: major changes in section V: profiles are now calculated with an unbiased estimator. Conclusions remain unchanged

We analyze the possibilities of pairing between two different fermion species in asymmetric matter at low density. While the direct interaction allows pairing only for very small asymmetries, the pairing mediated by polarization effects is always possible, with a pronounced maximum at finite asymmetry. We present analytical results up to second order in the low-density parameter $k_F a$., Comment: LaTeX 2.09, 12 pages, 4 figures. To appear in Phys. Lett. A

By telescopic tracking, we have established that the orbit of the trans-neptunian object (2000 CR$_{105}$) has a perihelion of $\simeq$44 AU, and is thus outside the domain controlled by strong gravitational close encounters with Neptune. Because this object is on a very large, eccentric orbit (with semimajor axis $a\simeq$216 AU and eccentricity $e\simeq$0.8) this object must have been placed on this orbit by a gravitational perturbation which is {\it not} direct gravitational scattering off of any of the giant planets (on their current orbits). The existence of this object may thus have profound cosmogonic implications for our understanding of the formation of the outer Solar System. We discuss some viable scenarios which could have produced it, including long-term diffusive chaos and scattering off of other massive bodies in the outer Solar System. This discovery implies that there must be a large population of trans-neptunian objects in an `extended scattered disk' with perihelia above the previously-discussed 38 AU boundary., Comment: 22 test pages, 4 figures, 2 tables. Submitted to Icarus, 26 March 2001

Numerical integrations in celestial mechanics often involve the repeated computation of a rotation with a constant angle. A direct evaluation of these rotations yields a linear drift of the distance to the origin. This is due to roundoff in the representation of the sine s and cosine c of the angle theta. In a computer, one generally gets c^2 + s^2 <> 1, resulting in a mapping that is slightly contracting or expanding. In the present paper we present a method to find pairs of representable real numbers s and c such that c^2 + s^2 is as close to 1 as possible. We show that this results in a drastic decrease of the systematic error, making it negligible compared to the random error of other operations. We also verify that this approach gives good results in a realistic celestial mechanics integration., Comment: 24 pages, 3 figures

Grazing incidence x-ray surface scattering has been used to investigate liquid surfaces down to the molecular scale. The free surface of water is well described by the capillary wave model ( ~ q-2 spectrum) up to wavevectors > 10^8 m^-1. At larger wavevectors near-surface acoustic waves must be taken into account. When the interface is bounded by a surfactant monolayer, it exhibits a bending stiffness and the bending rigidity modulus can be measured. However, bending effects generally cannot be described using the Helfrich Hamiltonian and the characteristic exponent in the roughness power spectrum can smaller than 4. Finally, upon compression, tethered monolayers formed on a subphase containing divalent ions are shown to buckle in the third dimension with a characteristic wavelength on the order of 10^8 m^-1.