Double-peaked Distribution Research Articles

Shifts in the Distribution of Mass Densities Is a Signature of Caloric Restriction in Caenorhabditis elegans

Although the starvation response of the model multicellular organism Caenorhabditis elegans is a subject of much research, there is no convenient phenotypic readout of caloric restriction that can be applicable to large numbers of worms. This paper describes the distribution of mass densities of populations of C. elegans, from larval stages up to day one of adulthood, using isopycnic centrifugation, and finds that density is a convenient, if complex, phenotypic readout in C. elegans. The density of worms in synchronized populations of wildtype N2 C. elegans grown under standard solid-phase culture conditions was normally distributed, with distributions peaked sharply at a mean of 1.091 g/cm3 for L1, L2 and L3 larvae, 1.087 g/cm3 for L4 larvae, 1.081 g/cm3 for newly molted adults, and 1.074 g/cm3 at 24 hours of adulthood. The density of adult worms under starvation stress fell well outside this range, falling to a mean value of 1.054 g/cm3 after eight hours of starvation. This decrease in density correlated with the consumption of stored glycogen in the food-deprived worms. The density of the worms increased when deprived of food for longer durations, corresponding to a shift in the response of the worms: worms sacrifice their bodies by retaining larvae, which consume the adults from within. Density-based screens with the drug Ivermectin on worms cultured on single plates resulted in a clear bimodal (double-peaked) distribution of densities corresponding to drug exposed and non-exposed worms. Thus, measurements of changes in density could be used to conduct screens on the effects of drugs on several populations of worms cultured on single plates.

The Black Hole Masses and Radio Characteristics of Radio Quasars

In this paper, we collect the redshift, bolometric luminosity, the full- width at half maximum of the Hβ emission line, the monochromatic luminosity at 5100Å and the radio loudness for the sample of 117 quasars, including 20 radio-quiet quasars (RQQs) and 97 radio-loud quasars (RLQs). With the reverberation mapping method we calculate the black hole mass and Eddington ratio for this sample, as well as the radio luminosity from the total 5GHz flux density. By analyzing the correlations among them, we obtain the following conclusions: (1) The black hole mass has weak correlations with the bolometric luminosity, radio loudness and radio luminosity for the RQQs, and has strong correlations with the bolometric luminosity, radio loudness and radio luminosity for the RLQs; (2) For the RQQs, the bolometric luminosity has weak correlations with the radio luminosity and 5 100Å monochromatic luminosity, and for the RLQs, the bolometric luminosity has strong correlations with the radio luminosity and 5 100Å monochromatic luminosity; (3) The RQQs and RLQs differ in the distributions of the black hole mass, emission line width and Eddington ratio. Based on these results, we suggest: the difference of emission line width between RQQs and RLQs is probably caused by the difference of black hole mass; the fundamental difference between RQQs and RLQs is caused by the difference of their intrinsic physical nature; the black hole mass, black hole spin, Eddington ratio, and host galaxy morphology are the important parameters to explain the origin of radio loudness and the double-peaked distribution; and the radio jet is closely related with the accretion rate of disk.

A SINFONI view of flies in the Spiderweb: a galaxy cluster in the making

The environment of the high-z radio galaxy PKS 1138-262 at z~2.2 is a prime example of a forming galaxy cluster. We use deep SINFONI data to perform a detailed study of the kinematics of the galaxies within 60 kpc of the radio core and we link this to the kinematics of the protocluster on the megaparsec scale. Identification of optical emission lines shows that 11 galaxies are at the redshift of the protocluster. The density of line emitters is more than an order of magnitude higher in the core of the protocluster than the larger scale environment. This implies a matter overdensity in the core of delta_m~70 which is similar to the outskirts of local galaxy clusters. The velocity distribution of the confirmed satellite galaxies shows a broad, double-peaked velocity structure with sigma=1360+/-206 km/s. A similar broad, double-peaked distribution was found in a previous study targeting the large scale protocluster structure, indicating that a common process is acting on both small and large scales. Including all spectroscopically confirmed protocluster galaxies, a velocity dispersion of 1013+/-87 km/s is found. We show that the protocluster has likely decoupled from the Hubble flow and is a dynamically evolved structure. Comparison to the Millenium simulation indicates that the protocluster velocity distribution is consistent with that of the most massive haloes at z~2, but we rule out that the protocluster is a fully virialized structure based on dynamical arguments and its X-ray luminosity. Comparison to merging haloes in the Millennium simulation shows that the structure as observed in and around the Spiderweb galaxy is best interpreted as being the result of a merger between two massive haloes. We propose that this merger can result in an increase in star formation and AGN activity in the protocluster core and is possibly an important stage in the evolution of massive cD galaxies.

SECULAR RESONANCE SWEEPING OF THE MAIN ASTEROID BELT DURING PLANET MIGRATION

We calculate the eccentricity excitation of asteroids produced by the sweeping $\nu_6$ secular resonance during the epoch of planetesimal-driven giant planet migration in the early history of the solar system. We derive analytical expressions for the magnitude of the eccentricity change and its dependence on the sweep rate and on planetary parameters; the $\nu_6$ sweeping leads to either an increase or a decrease of eccentricity depending on an asteroid's initial orbit. Based on the slowest rate of $\nu_6$ sweeping that allows a remnant asteroid belt to survive, we derive a lower limit on Saturn's migration speed of $\sim0.15\AU\My^{-1}$ during the era that the $\nu_6$ resonance swept through the inner asteroid belt (semimajor axis range 2.1--$2.8\AU$). This rate limit is for Saturn's current eccentricity, and scales with the square of Saturn's eccentricity; the limit on Saturn's migration rate could be lower if Saturn's eccentricity were lower during its migration. Applied to an ensemble of fictitious asteroids, our calculations show that a prior single-peaked distribution of asteroid eccentricities would be transformed into a double-peaked distribution due to the sweeping of the $\nu_6$. Examination of the orbital data of main belt asteroids reveals that the proper eccentricities of the known bright ($H \leq10.8$) asteroids may be consistent with a double-peaked distribution. If so, our theoretical analysis then yields two possible solutions for the migration rate of Saturn and for the dynamical states of the pre-migration asteroid belt: a dynamically cold state (single-peaked eccentricity distribution with mean of $\sim0.05$) linked with Saturn's migration speed $\sim 4\AU\My^{-1}$, or a dynamically hot state (single-peaked eccentricity distribution with mean of $\sim0.3$) linked with Saturn's migration speed $\sim 0.8\AU\My^{-1}$.

Universality in the equilibration of quantum systems after a small quench

A sudden change of the Hamiltonian parameter drives a quantum system out of equilibrium. For a finite-size system, expectations of observables start fluctuating in time without converging to a precise limit. A new equilibrium state emerges only in probabilistic sense, when the probability distribution for the observables expectations over long times concentrate around their mean value. In this paper we study the full statistic of generic observables after a small quench. When the quench is performed around a regular (i.e. non-critical) point of the phase diagram, generic observables are expected to be characterized by Gaussian distribution functions (``good equilibration''). Instead, when quenching around a critical point a new universal double-peaked distribution function emerges for relevant perturbations. Our analytic predictions are numerically checked for a non-integrable extension of the quantum Ising model.

First-order transitions from singly peaked distributions

Suppose that, in the thermodynamic limit, a single-component particle system exhibits a standard first-order transition marked by a jump in the density, ρ , at a chemical potential μ σ ( T ) . In grand canonical simulations of model fluids that realize such a transition when L → ∞ (where L is the linear dimension of the simulation volume) the presence of the transition is typically signaled by the appearance of a double-peaked structure in the distribution function, P N ( T , μ σ ; L ) , of the particle number, N . A simple, explicit counterexample is presented, however, that proves, contrary to popular beliefs, that the converse proposition is false: i.e., a single-peaked distribution, P N ( T , μ σ ; L ) , may, when L → ∞ , give rise to a first-order transition. Alternatively, the existence of a first-order transition does not imply a double-peaked distribution. Systems that may exhibit such single-peaked, first-order behavior are discussed and a possible route to constructing explicit models exhibiting the phenomenon is described. Strategies to use in simulating such systems are briefly considered in the light of related studies.

Large and small-scale structures of the local Galactic disc

An analytical model based on the maximum entropy approach is proposed to describe the eventual asymmetries of the velocity distribution, which are collected through its sample moments. If an extended set of moments is available, the current method provides a linear algorithm, associated with a Gramian system of equations, that leads to a fast and suitable estimation of the velocity distribution. In particular, it could be used to model multimodal distributions that cannot be described through Gaussian mixtures. The method is used with several samples from the HIPPARCOS and Geneva-Copenhagen survey catalogues. For the large-scale distribution, the phase density function may be obtained by fitting moments up to sixth order as a product of two exponential functions, one giving a background ellipsoidal shape of the distribution and the other accounting for the skewness and for the slight shift in the ellipsoidal isocontours in terms of the rotation velocity. The small-scale distribution can be deduced from truncated distributions, such as velocity-bounded samples with km s<sup>-1<sup/>, which contain a complex mixture of early-type and young disc stars. By fitting up to ten-order moments, the maximum entropy approach gives a realistic portrait of actual asymmetries, showing a clear bimodal pattern: (i) around the Hyades-Pleiades stream, with negative radial mean velocity and (ii) around the Sirius-UMa stream, with slightly positive radial mean velocity. Among metallicity, colour, and other star properties, the eccentricity of the star's orbit behaves as a very good sampling parameter to find a more detailed structure for the disc velocity distribution, allowing distinctions between different eccentricity layers. For subsamples with eccentricities <i>e<0.15<i/>, star velocities are approximately symmetrically distributed around the LSR in the radial direction, with a dearth of stars at the LSR. For <i>e=0.15<i/>, the core distribution of the thin disc is supported by two major stellar groups with opposite radial velocities. Several simulations confirm that such a double-peaked distribution comes from the lognormal distribution of the velocity amplitudes. For maximum eccentricity 0.3 and maximum distance to the Galactic plane 0.5 kpc a representative thin disc sample is obtained. The “U-anomaly” along the radial direction is estimated straightforwardly km s<sup>-1<sup/> from the contour plots. An explanation of the apparent vertex deviation of the disc from the swinging of those major kinematic groups around the LSR is then possible, which predicts a continuously changing orientation of the disc's pseudo ellipsoid.

The flaring and quiescent components of the solar corona

Context. The solar corona is a template to understand stellar activity. The Sun is a moderately active star, and its corona differs from that of active stars: for instance, active stellar coronae have a double-peaked emission measure distribution $EM(T)$ with a hot peak at $8{-}20$ MK, while the non-flaring solar corona has one peak at $1{-}2$ MK and, typically, much cooler plasma. Aims. We study the average contribution of flares to the solar emission measure distribution to investigate indirectly the hypothesis that the hot peak in the $EM(T)$ of active stellar coronae is due to a large number of unresolved solar-like flares, and to infer properties about the flare distribution from nano- to macro-flares. Methods. We measure the disk-integrated time-averaged emission measure, $EM_{\rm F}(T)$, of an unbiased sample of solar flares, analyzing uninterrupted GOES/XRS light curves over time intervals of one month. We obtain the $EM_{\rm Q}(T)$ of quiescent corona for the same time intervals from Yohkoh/SXT data. To investigate how $EM_{\rm F}(T)$ and $EM_{\rm Q}(T)$ vary during the solar cycle, we evaluate them at different phases of the cycle between December 1991 and April 1998. Results. Irrespective of the solar cycle phase, $EM_{\rm F}(T)$ appears as a peak in the distribution, and it is significantly larger than the values of $EM_{\rm Q}(T)$ for $T\sim5{-}10$ MK. As a result, the time-averaged $EM(T)$ of the entire solar corona is double-peaked, with the hot peak, due to time-averaged flares, being located at temperatures similar to those of active stars, but less enhanced. The $EM_{\rm F}(T)$ shape supports the hypothesis that the hot $EM(T)$ peak of active coronae is due to unresolved solar-like flares. If this is the case, quiescent and flare components should follow different scaling laws for increasing stellar activity. In the assumption that the heating of the corona is entirely due to flares, from nano- to macro-flares, then either the flare distribution or the confined plasma response to flares, or both, are bimodal.

Structure characterization and oxidation mechanism study of porous biomorphic carbon template derived from basswood

A porous biomorphic carbon template (BCT), retained its biological feature, was prepared by carbonized basswood. Microstructure and component of BCT were performed by scanning electron microscope (SEM), energy dispersive spectrometry (EDS) and Raman spectroscopy. Non-isothermal oxidation mechanism of BCT was studied by thermogravimetry analysis (TGA). Experimental results show that BCT has a honeycomb structure and double-peaked porous diameter distribution. It is a typical turbostratic network structure, with increasing carbonization temperature, the ratio of the integrated intensities I D/ I G increase, the lateral extension L a of the graphene units decrease. The non-isothermal oxidation mechanism of BCT exhibits a self-catalytic characteristic, which is explained by molecular structure schematic model.

Towards a spectral technique for determining material geometry around evolved stars: application to HD 179821

HD 179821 is an evolved star of unknown progenitor mass range (either post-asymptotic giant branch or post-red supergiant) exhibiting a double-peaked spectral energy distribution (SED) with as harp rise from∼ 8t o 20μm. Such features have been associated with ejected dust shells or inwardly truncated circumstellar discs. In order to compare SEDs from both systems, we employ a spherically symmetric radiative transfer code and compare it to a radiative, inwardly truncated disc code. As a case study, we model the broad-band SED of HD 179821 using both codes. Shortward of 40 μm, we find that both models produce equivalent fits to the data. However, longward of 40 μm, the radial density distribution and corresponding broad range of disc temperatures produce excess emission above our spherically symmetric solutions and observations. For HD 179821, our best fit consists of a T eff = 7000 K central source characterized by τV ∼ 1.95 and surrounded by a radiatively driven, spherically symmetric dust shell. The extinction of the central source reddens the broad-band colours so that they resemble aT eff = 5750 K photosphere. We believe that HD 179821 contains a hotter central star than previously thought. Our results provide an initial step towards a technique to distinguish geometric differences from spectral modelling.

The Human Development Trap in Mexico

Summary A dynamic poverty trap model describing long-term human development is defined in the context of endogenous technological change. Increasing returns are not required: market failures and indivisibilities imply a human capital undersupply and hence above-equilibrium returns. Evidence for this trap is provided for Mexico. High returns to education and early child development, untapped by about 75% of the population, imply an undersupply of human capital. A double-peaked schooling distribution for male and female spouses attests to multiple equilibria. One peak lies beyond complete higher secondary, the other below complete lower secondary schooling. Supporting early child development can help eliminate the trap.

Halo Star Streams in the Solar Neighborhood

We have assembled a sample of halo stars in the solar neighborhood to look for halo substructure in velocity and angular momentum space. Our sample includes red giants, RR Lyrae, and red horizontal branch stars within 2.5 kpc of the Sun with [Fe/H] less than -1.0. It was chosen to include stars with accurate distances, space velocities, and metallicities as well as well-quantified errors. We confirm the existence of the streams found by Helmi and coworkers, which we refer to as the H99 streams. These streams have a double-peaked velocity distribution in the z direction. We use the results of modeling of the H99 streams by Helmi and collaborators to test how one might use v_z velocity information and radial velocity information to detect kinematic substructure in the halo. We find that detecting the H99 streams with radial velocities alone would require a large sample. We use the velocity distribution of the H99 streams to estimate their age. From our model of the progenitor of the H99 streams, we determine that it was accreted between 6 and 9 Gyr ago. The H99 streams have [alpha/Fe] abundances similar to other halo stars in the solar neighborhood, suggesting that the gas that formed these stars were enriched mostly by Type II SNe. We have also discovered in angular momentum space two other possible substructures, which we refer to as the retrograde and prograde outliers. The retrograde outliers are likely to be halo substructure, but the prograde outliers are most likely part of the smooth halo. The retrograde outliers have significant structure in the v_phi direction and show a range of [alpha/Fe]. The methods presented in this paper can be used to exploit the kinematic information present in future large databases like RAVE, SDSSII/SEGUE, and Gaia.

Confirmation of double-peaked time distribution of mortality among Asian breast cancer patients in a population-based study

IntroductionDouble-peaked time distributions of the mortality hazard function have been reported for breast cancer patients from Western populations treated with mastectomy alone. These are thought to reflect accelerated tumour growth at micrometastatic sites mediated by angiogenesis after primary tumour removal as well as tumor dormancy. Similar data are not available for Asian populations. We sought to investigate whether differences exist in the pattern of mortality hazard function between Western breast cancer patients and their Asian counterparts in Singapore, which may suggest underlying differences in tumor biology between the two populations.MethodsWe performed a retrospective cohort study of female unilateral breast cancer patients diagnosed in Singapore between October 1994 and June 1999. Data regarding patient demographics, tumour characteristics and death were available. Overall survival curves were calculated using the Kaplan-Meier method. The hazard rate was calculated as the conditional probability of dying in a time interval, given that the patient was alive at the beginning of the interval. The life table method was used to calculate the yearly hazard rates.ResultsIn the 2,105 women identified, 956 patients (45.4%) had mastectomy alone. Demographic characteristics were as follows: 86.5% were Chinese, 45.2% were postmenopausal, 38.9% were hormone receptor positive, 54.6% were node negative and 44.1% had high histological grade. We observed a double-peaked mortality hazard pattern, with a first peak in mortality achieving its maximum between years 2 and 4 after mastectomy, and a second large peak in mortality during year 9. Analyses by subgroups revealed a similar pattern regardless of T stage, or node or menopausal status. This pattern was also noted in high-grade tumors but not in those that were well to moderately differentiated. The double-peaked pattern observed in Singaporean women was quantitatively and qualitatively similar to those reported in Western series.ConclusionOur study confirms the existence of a double-peaked process in Asian patients, and it gives further support to the tumour dormancy hypothesis after mastectomy.

Confirmation of double-peaked time distribution of mortality for breast cancer in a population-based study

593 Background: Double-peaked time distributions of mortality hazard function have been reported for breast cancer patients from Western populations treated with mastectomy alone and is thought to reflect accelerated tumor growth at metastatic sites mediated by angiogenesis after primary tumor removal. Similar data is not available for Asian populations. We sought to investigate if differences exist in the pattern of mortality hazard function between Western breast cancer patients and Asian patients in Singapore treated with mastectomy alone which may suggest underlying differences in tumor biology between the two populations. Methods: We performed a retrospective cohort study of 3809 female unilateral breast cancer patients registered in the Singapore Breast Cancer Registry between October 1994 to June 1999. Data regarding patient demographics, tumor characteristics and death data were available. Overall survival curves were calculated using the Kaplan-Meier method and compared by the log-rank test. Death-specific hazard rate was calculated as the conditional probability of dying in a time interval, given that the patient was alive at the beginning of the interval. The life table method was used to calculate the yearly hazard rates. All tests were two-sided and significant at the 5% level. Results: In this cohort, 1169 patients (45.9%) had mastectomy alone without further adjuvant therapy. Demographics were: median age 55; 87.4% Chinese; 62.9% postmenopausal; 51.6% hormone receptor positive; 70.3% node negative; 22.8% high histological grade; 76.9% stage I or II. A double-peaked mortality hazard pattern, with a first major mortality rise reaching the maximum at the 2nd - 4th year after diagnosis and a second small mortality rise near the 9th year was observed. Analyses by subgroups showed a similar pattern in node positive or high grade tumors but not in node negative or well to moderately differentiated tumors. A double peaked pattern was seen regardless of menopausal status. The double-peaked pattern observed in our patients was quantitatively and qualitatively similar to that reported by Western series. Conclusions: Our study confirms a double-peaked process in Asian patients and gives further support to the tumor dormancy hypothesis. No significant financial relationships to disclose.

Color functions of stellar systems

Model calculations of the photometric evolution of rather dense stellar systems, such as globular clusters, are presented. On “luminosity-effective temperature” diagrams of these systems, low-mass stars are concentrated near the minimum and maximum temperatures for a given luminosity and are deficient in the intermediate region. This sort of double-peaked distribution of the stars can be avoided in open models with ejection of excess metals into the surrounding medium. The distributions of the stars with respect to effective temperature on a “ luminosity-effective temperature” diagram are sensitive to the history of star formation in the system and to possible time variations in the initial mass function. In open systems with a single-peak distribution function, the asymmetry in the distribution varies over wide limits with the lower bound for the initial mass function and this can be used to establish whether the first generations of stars might have been more massive than in the present epoch.

The Unique Type Ib Supernova 2005bf: A WN Star Explosion Model for Peculiar Light Curves and Spectra

Observations and modeling for the light curve (LC) and spectra of supernova (SN) 2005bf are reported. This SN showed unique features: the LC had two maxima, and declined rapidly after the second maximum, while the spectra showed strengthening He lines whose velocity increased with time. The double-peaked LC can be reproduced by a double-peaked 56Ni distribution, with most 56Ni at low velocity and a small amount at high velocity. The rapid postmaximum decline requires a large fraction of the γ-rays to escape from the 56Ni-dominated region, possibly because of low-density "holes." The presence of Balmer lines in the spectrum suggests that the He layer of the progenitor was substantially intact. Increasing γ-ray deposition in the He layer due to enhanced γ-ray escape from the 56Ni-dominated region may explain both the delayed strengthening and the increasing velocity of the He lines. The SN has massive ejecta (~6-7 M☉), normal kinetic energy [~(1.0-1.5) × 1051 ergs], a high peak bolometric luminosity (~5 × 1042 ergs s-1) for an epoch as late as ~ 40 days, and a large 56Ni mass (~0.32 M☉). These properties and the presence of a small amount of H suggest that the progenitor was initially massive (M ~ 25-30 M☉) and had lost most of its H envelope, and was possibly a WN star. The double-peaked 56Ni distribution suggests that the explosion may have formed jets that did not reach the He layer. The properties of SN 2005bf resemble those of the explosion of Cassiopeia A.

X‐Ray Emission from Orion Nebula Cluster Stars with Circumstellar Disks and Jets

We investigate the X-ray and near-infrared emission properties of a sample of pre-main-sequence (PMS) stellar systems in the Orion Nebula Cluster (ONC) that display evidence for circumstellar disks (proplyds) and optical jets in Hubble Space Telescope (HST) imaging. Our study uses X-ray data acquired during Chandra Orion Ultradeep Program (COUP) observations, as well as complementary optical and near-infrared data recently acquired with HST and the Very Large Telescope (VLT), respectively. Approximately 70% of ~140 proplyds were detected as X-ray sources in the 838 ks COUP observation of the ONC, including ~25% of proplyds that do not display central stars in HST imaging. In near-infrared imaging, the detection rate of proplyd central stars is >90%. Many proplyds display near-infrared excesses, suggesting disk accretion is ongoing onto the central, PMS stars. About 50% of circumstellar disks that are detected in absorption in HST imaging contain X-ray sources. For these sources, we find that X-ray absorbing column and apparent disk inclination are well correlated, providing insight into the disk scale heights and metal abundances of UV- and X-ray-irradiated protoplanetary disks. Approximately 2/3 of the ~30 proplyds and PMS stars exhibiting jets in Hubble images have COUP X-ray counterparts. These jet sources display some of the largest near-infrared excesses among the proplyds, suggesting that the origin of the jets is closely related to ongoing, PMS stellar accretion. One morphologically complex jet source, d181-825, displays a double-peaked X-ray spectral energy distribution with a prominent soft component that is indicative of strong shocks in the jet collimation region. A handful of similar objects also display X-ray spectra that are suggestive of shocks near the jet source. These results support models in which circumstellar disks collimate and/or launch jets from young stellar objects and, furthermore, demonstrate that star-disk-jet interactions may contribute to PMS X-ray emission.

Near-field spectroscopy of bimodal size distribution of InAs∕AlGaAs single quantum dots

We report on high-resolution photoluminescence (PL) spectroscopy of size distribution of InAs∕AlGaAs quantum dots (QDs) by using a near-field scanning optical microscope (NSOM). The double-peaked distribution of PL spectra is clearly observed, which is associated with the bimodal size distribution of single QDs. In particular, the size difference of single QDs, represented by the doublet spectral distribution, is directly observed by the NSOM images of PL.

Dynamic features of Io's extended sodium distributions

We report on two-dimensional imaging observations of D-line emissions from the extended distribution of iogenic sodium atoms with two fields of view (±20 R J (narrow FOV) and ±400 R J (wide FOV)) simultaneously by using a portable small telescope or camera lens. We derived dynamic feature of the band-shaped and spray-shaped distributions near Io's orbit by means of continuous observation. The observations confirm the phenomenological behavior of the sodium cloud on two spatial scales, as previously observed by Pilcher et al. [Pilcher, C.B., Smyth, W.H., Combi, M.R., Fertel, J.H., 1984. Astrophys. J. 287, 427–444], Schneider et al. [Schneider, N.M., Trauger, J.T., Wilson, J.K., Brown, D.I., Evans, R.W., Shemansky, D.E., 1991. Science 253, 1394–1397], and Mendillo et al. [Mendillo, M., Baumgartner, J., Flynn, B., Hughes, W.S., 1990. Nature 348, 312–314]. We also confirm an elongated oval emission distribution of the sodium nebula and derivation of its detailed east–west asymmetry depending on Io's phase angle, which was first noted by Flynn et al. [Flynn, B., Mendillo, M., Baumgartner, J., 1994. J. Geophys. Res. 99, 8403–8409]. We then did model analyses to investigate the source process for sodium atoms and the dynamics behind their distribution. We conclude that the essential of molecular ion mechanisms to the band-shaped distribution is in agreement with Wilson and Schneider [Wilson, J.K., Schneider, N.M., 1999. J. Geophys. Res. 104, 16567–16583]. We differ from Wilson et al. [Wilson, J.K., Mendillo, M., Baumgartner, J., Schneider, N.M., Trauger, J.T., Flynn, B., 2002. Icarus 157, 476–489] in finding that charge exchange process contributes more to the spray-shaped distribution and sodium nebula than sputtering does. These results derived the double-peaked velocity distribution of released sodium atoms, and re-confirmed the source rates in agreement with past studies.

Velocity distribution in a viscous granular gas

We investigate the velocity relaxation of a viscous one-dimensional granular gas in which neither energy nor momentum is conserved in a collision. Of interest is the distribution of velocities in the gas as it cools, and the time dependence of the relaxation behavior. A Boltzmann equation of instantaneous binary collisions leads to a two-peaked distribution, as do numerical simulations of grains on a line. Of particular note is that in the presence of friction there is no inelastic collapse, so there is no need to invoke additional assumptions such as the quasielastic limit.