Leaky Box Model Research Articles

Acceleration and Spectral Redistribution of Cosmic Rays in Radio-jet Shear Flows

A steady-state, semi-analytical model of energetic particle acceleration in radio-jet shear flows due to cosmic-ray viscosity obtained by Webb et al. is generalized to take into account more general cosmic-ray boundary spectra. This involves solving a mixed Dirichlet–Von Neumann boundary value problem at the edge of the jet. The energetic particle distribution function f 0(r, p) at cylindrical radius r from the jet axis (assumed to lie along the z-axis) is given by convolving the particle momentum spectrum with the Green’s function , which describes the monoenergetic spectrum solution in which as r → ∞ . Previous work by Webb et al. studied only the Green’s function solution for . In this paper, we explore for the first time, solutions for more general and realistic forms for . The flow velocity u = u(r) e z is along the axis of the jet (the z-axis). u is independent of z, and u(r) is a monotonic decreasing function of r. The scattering time in the shear flow region 0 < r < r 2, and , where s > 0 in the region r > r 2 is outside the jet. Other original aspects of the analysis are (i) the use of cosmic ray flow lines in (r, p) space to clarify the particle spatial transport and momentum changes and (ii) the determination of the probability distribution that particles observed at (r, p) originated from r → ∞ with momentum . The acceleration of ultrahigh-energy cosmic rays in active galactic nuclei jet sources is discussed. Leaky box models for electron acceleration are described.

Cosmic-Ray Boron Flux Measured from 8.4 GeV/n to 3.8 TeV/n with the Calorimetric Electron Telescope on the International Space Station.

We present the measurement of the energy dependence of the boron flux in cosmic rays and its ratio to the carbon flux in an energy interval from 8.4 GeV/n to 3.8 TeV/n based on the data collected by the Calorimetric Electron Telescope (CALET) during ∼6.4 yr of operation on the International Space Station. An update of the energy spectrum of carbon is also presented with an increase in statistics over our previous measurement. The observed boron flux shows a spectral hardening at the same transition energy E_{0}∼200 GeV/n of the C spectrum, though B and C fluxes have different energy dependences. The spectral index of the B spectrum is found to be γ=-3.047±0.024 in the interval 25<E<200 GeV/n. The B spectrum hardens by Δγ_{B}=0.25±0.12, while the best fit value for the spectral variation of C is Δγ_{C}=0.19±0.03. The B/C flux ratio is compatible with a hardening of 0.09±0.05, though a single power-law energy dependence cannot be ruled out given the current statistical uncertainties. A break in the B/C ratio energy dependence would support the recent AMS-02 observations that secondary cosmic rays exhibit a stronger hardening than primary ones. We also perform a fit to the B/C ratio with a leaky-box model of the cosmic-ray propagation in the Galaxy in order to probe a possible residual value λ_{0} of the mean escape path length λ at high energy. We find that our B/C data are compatible with a nonzero value of λ_{0}, which can be interpreted as the column density of matter that cosmic rays cross within the acceleration region.

Power-law spectrum of energetic particles in classical thermal equilibrium by pitch-angle scattering process

The Boltzmann–Gibbs thermodynamic equilibrium state of charged particles pitch-angle scattered by weak plasma waves is discussed. Degrees of freedom of these waves play a fundamental role in constructing the grand canonical ensemble. Via the gyro-resonance condition, fast particles have an inverse break power-law spectrum for ɛ−μ≪T, where ɛ is the particle energy, μ is the chemical potential, T is the temperature. The break energies are the rest energy and −μ. For ɛ≪−μ≪T, the energy spectral index α is δ/2+1 and δ+1 for non- and ultra-relativistic particles, respectively, with δ an effective fractal dimension of background magnetic field lines. The spectral index for −μ≪ɛ≪T is α+1. This thermal equilibrium scenario, combined with the leaky-box model and cosmic-ray observations, seems to suggest that the Galactic magnetic field is super-diffusive with δ≈1.4.

Cosmic-Ray Acceleration in Radio-jet Shear Flows: Scattering Inside and Outside the Jet

Abstract A steady-state, analytical model of energetic particle acceleration in radio-jet shear flows due to cosmic-ray viscosity is explored, including particle scattering both into and out of the shear flow acceleration region. This involves solving a mixed Dirichlet–Von Neumann boundary value problem at the edge of the jet. The spectrum of the accelerated particles is harder than the free-escape case from the edge of the jet. The flow velocity u = u(r) e z is along the axis of jet (the z-axis). u is independent of distance z along the jet axis, and u(r) is a monotonically decreasing function of cylindrical radius r from the jet axis. The scattering time where p is the particle momentum in the fluid frame in the shear flow region 0 &lt; r &lt; r 2, and outside the jet (r &gt; r 2). Green’s functions are obtained for monoenergetic injection of particles with momentum p = p 0 at radius r = r 1 (0 &lt; r 1 &lt; r 2). The Green’s function and Green’s formula are used to determine solutions for a general spectrum of particles at . Solutions are obtained corresponding to a monoenergetic spectrum at infinity. We discuss the implications of these results for the acceleration of ultra-high-energy cosmic-rays in active galactic nucleus jet sources. Leaky box models of particle acceleration in shear flows, including synchrotron losses and particle escape, are used to describe the momentum spectrum of accelerated particles. The use of the relativistic telegrapher transport equation model is discussed.

Both starvation and outflows drive galaxy quenching

ABSTRACT Star-forming galaxies can in principle be transformed into passive systems by a multitude of processes that quench star formation, such as the halting of gas accretion (starvation) or the rapid removal of gas in AGN-driven outflows. However, it remains unclear which processes are the most significant, primary drivers of the SF-passive bimodality. We address this key issue in galaxy evolution by studying the chemical properties of 80 000 local galaxies in Sloan Digital Sky Survey Data Release 7. In order to distinguish between different quenching mechanisms, we analyse the stellar metallicities of star-forming, green valley, and passive galaxies. We find that the significant difference in stellar metallicity between passive galaxies and their star-forming progenitors implies that for galaxies at all masses, quenching must have involved an extended phase of starvation. However, some form of gas ejection also has to be introduced into our models to best match the observed properties of local passive galaxies, indicating that, while starvation is likely to be the prerequisite for quenching, it is the combination of starvation and outflows that is responsible for quenching the majority of galaxies. Closed-box models indicate that the duration of the quenching phase is 2–3 Gyr, with an e-folding time of 2–4 Gyr, after which further star formation is prevented by an ejective/heating mode. Alternatively, leaky-box models find a longer duration for the quenching phase of 5–7 Gyr and an e-folding time of ∼1 Gyr, with outflows becoming increasingly important with decreasing stellar mass. Finally, our analysis of local green valley galaxies indicates that quenching is slower in the local Universe than at high redshift.

Usine: Semi-analytical models for Galactic cosmic-ray propagation

I present the first public releases (v3.4 and v3.5) of the usine code for cosmic-ray propagation in the Galaxy (https://lpsc.in2p3.fr/usine). It contains several semi-analytical propagation models previously used in the literature (leaky-box model, 2-zone 1D and 2D diffusion models) for the calculation of nuclei (Z=1−30), anti-protons, and anti-deuterons. For minimisations, the geometry, transport, and source parameters of all models can be enabled as free parameters, whereas nuisance parameters are enabled on solar modulation levels, cross sections (inelastic and production), and systematics of the CR data. With a single ASCII initialisation file to configure runs, its many displays, and the speed associated to semi-analytical approaches, usine should be a useful tool for beginners, but also for experts to perform statistical analyses of high-precision cosmic-ray data. Program summaryProgram Title:usine.Program Files doi:http://dx.doi.org/10.17632/66dvnt642g.1Licensing provisions: GPLv3Programming language: C++External routines/libraries:root cern.Nature of problem: Semi-analytical models for GCR transport.Solution method: Partial derivatives on spatial coordinates solved analytically, partial derivatives on energy coordinates solved numerically (finite difference).Restrictions: Electronic capture decay not fully implemented.

A local leaky-box model for the local stellar surface density–gas surface density–gas phase metallicity relation

We revisit the relation between the stellar surface density, the gas surface density, and the gas-phase metallicity of typical disk galaxies in the local Universe with the SDSS-IV/MaNGA survey, using the star formation rate surface density as an indicator for the gas surface density. We show that these three local parameters form a tight relationship, confirming previous works (e.g., by the PINGS and CALIFA surveys), but with a larger sample. We present a new local leaky-box model, assuming star formation history and chemical evolution is localized except for outflowing materials. We derive closed-form solutions for the evolution of stellar surface density, gas surface density and gas-phase metallicity, and show that these parameters form a tight relation independent of initial gas density and time. We show that, with canonical values of model parameters, this predicted relation match the observed one well. In addition, we briefly describe a pathway to improving the current semi-analytic models of galaxy formation by incorporating the local leaky-box model in the cosmological context, which can potentially explain simultaneously multiple properties of Milky Way-type disk galaxies, such as the size growth and the global stellar mass-gas metallicity relation.

CHEMISTRY AND KINEMATICS OF THE LATE-FORMING DWARF IRREGULAR GALAXIES LEO A, AQUARIUS, AND SAGITTARIUS DIG*

ABSTRACT We present Keck/DEIMOS spectroscopy of individual stars in the relatively isolated Local Group dwarf galaxies Leo A, Aquarius, and the Sagittarius dwarf irregular galaxy. The three galaxies—but especially Leo A and Aquarius—share in common delayed star formation histories (SFHs) relative to many other isolated dwarf galaxies. The stars in all three galaxies are supported by dispersion. We found no evidence of stellar velocity structure, even for Aquarius, which has rotating H i gas. The velocity dispersions indicate that all three galaxies are dark-matter-dominated, with dark-to-baryonic mass ratios ranging from (SagDIG) to (Aquarius). Leo A and SagDIG have lower stellar metallicities than Aquarius, and they also have higher gas fractions, both of which would be expected if Aquarius were further along in its chemical evolution. The metallicity distribution of Leo A is inconsistent with a closed or leaky box model of chemical evolution, suggesting that the galaxy was pre-enriched or acquired external gas during star formation. The metallicities of stars increased steadily for all three galaxies, but possibly at different rates. The [α/Fe] ratios at a given [Fe/H] are lower than that of the Sculptor dwarf spheroidal galaxy, which indicates more extended SFHs than Sculptor, consistent with photometrically derived SFHs. Overall, the bulk kinematic and chemical properties for the late-forming dwarf galaxies do not diverge significantly from those of less delayed dwarf galaxies, including dwarf spheroidal galaxies.

Positrons and Antiprotons in Galactic Cosmic Rays

I consider the impact of recent measurements of positron and antiproton spectra in cosmic rays on our understanding of the origins and propagation of cosmic rays, as well as on the annihilation and decay characteristics of particles of Galactic dark matter, from the perspective of current models postulating energy-dependent leakage of cosmic rays from the Galaxy and of the nested leaky-box model, in which the leakage from the Galaxy is independent of energy. The nested leaky-box model provides a straightforward and consistent explanation of the observed spectral intensities, and finds no compelling need for a contribution from the annihilation or decay of Galactic dark matter. Improved observations and modeling efforts are needed to probe the properties of dark matter deeply enough to be significant to particle physics and cosmology.

SPECTRAL INTENSITIES OF ANTIPROTONS AND THE NESTED LEAKY-BOX MODEL FOR COSMIC RAYS IN THE GALAXY

ABSTRACT In this paper we note that the spectral intensities of antiprotons observed in Galactic cosmic rays in the energy range ∼1–300 GeV by BESS, PAMELA, and AMS instruments display nearly the same spectral shape as that generated by primary cosmic rays through their interaction with matter in the interstellar medium, without any significant modifications. More importantly, a constant residence time of ∼2.3 ± 0.7 million years in the Galactic volume, independent of the energy of cosmic rays, matches the observed intensities. A small additional component of secondary antiprotons in the energy range below 10 GeV, generated in cocoon-like regions surrounding the cosmic-ray sources, seems to be present. We discuss this result in the context of observations of other secondary components such as positrons and boron, and the bounds on anisotropy of cosmic rays. In the nested leaky-box model the spectral intensities of antiprotons and positrons can be interpreted as secondary products of cosmic-ray interactions.

METALLICITY EVOLUTION OF THE SIX MOST LUMINOUS M31 DWARF SATELLITES

We present global metallicity properties, metallicity distribution functions (MDFs) and radial metallicity profiles for the six most luminous M31 dwarf galaxy satellites: M32, NGC 205, NGC 185, NGC 147, Andromeda VII, and Andromeda II. The results presented are the first spectroscopic MDFs for dwarf systems surrounding a host galaxy other than the Milky Way. Our sample consists of individual metallicity measurements for 1243 red giant branch (RGB) member stars spread across these six systems. We determine metallicities based on the strength of the Ca II triplet lines using the empirical calibration of Carrera et al.(2013) which is calibrated over the metallicity range -4 < [Fe/H] <+0.5. We find that these M31 satellites lie on the same luminosity-metallicity relationship as the Milky Way dwarf satellites. We do not find a trend between the internal metallicity spread and galaxy luminosity, contrary to previous studies. The MDF widths of And II and And VII are similar to the MW dwarfs of comparable luminosity, however, our four brightest M31 dwarf are more luminous than any of the MW dwarf spheroidals and have broader MDFs. The MDFs of our six M31 dwarfs are consistent with the leaky box model of chemical evolution, although our metallicity errors allow a wide range of evolution models. We find a significant radial gradient in metallicity in only two of our six systems, NGC 185 and Andromeda II, and flat radial metallicity gradients in the rest of our sample with no observed correlation between rotational support and radial metallicity gradients. While the average properties and radial trends of the M31 dwarf galaxies agree with MW counterparts at similar luminosity, the detailed MDFs are different, particularly at the metal-rich end.

Rigidity dependence of cosmic-ray escape length obtained from a comparison of proton and iron spectra in the energy range 3–3000 GeV

A simple leaky-box model of cosmic-ray propagation in the Galaxy is quite suitable for analysis of cosmic-ray nuclei energy spectra and composition at energies under 1 GeV [1, 2]. Within this model the full information about the cosmic-ray propagation in the Galaxy is compressed into a single parameter, viz., escape length, Xe, which characterizes the mean grams of matter that are passed by cosmic-rays from sources to the Earth. In this paper we analyze the worldwide data on proton and iron cosmic-ray spectra that have been collected in experiments and obtain the rigidity dependence of the escape length from the measured rigidity dependence of the proton/iron ratio. This agrees well with the one estimated by a standard manner from the secondary/primary nuclei ratio. However, at R > 300 GV the behavior of Xe(R) distinctly changes, which can point to a change of the proton/iron ratio in cosmic-ray sources.

THE [Fe/H], [C/Fe], AND [α/Fe] DISTRIBUTIONS OF THE BOÖTES I DWARF SPHEROIDAL GALAXY

We present the results of a low-resolution spectral abundance study of 25 stars in the Bootes I dwarf spheroidal (dSph) galaxy. The data were obtained with the LRIS instrument at Keck Observatory, and allow us to measure [Fe/H], [C/Fe], and [alpha/Fe] for each star. We find both a large spread in metallicity (2.1 dex in [Fe/H]) as well as the low average metallicity in this system, <[Fe/H]>=-2.59, matching previous estimates. This sample includes a newly discovered extremely metal-poor star, with [Fe/H]=-3.8, that is one of the most metal-poor stars yet found in a dSph. We compare the metallicity distribution function of Bootes I to analytic chemical evolution models. While the metallicity distribution function of Bootes I is best fit by an Extra Gas chemical evolution model, leaky-box models also provide reasonable fits. We also find that the [alpha/Fe] distribution and the carbon-enhanced metal-poor fraction of our sample (12%) are reasonable matches to Galactic halo star samples in the same metallicity range, indicating that at these low metallicities, systems like the Bootes I ultra-faint dSph could have been contributors to the Galactic halo.

MULTI-ELEMENT ABUNDANCE MEASUREMENTS FROM MEDIUM-RESOLUTION SPECTRA. III. METALLICITY DISTRIBUTIONS OF MILKY WAY DWARF SATELLITE GALAXIES

We present metallicity distribution functions (MDFs) for the central regions of eight dwarf satellite galaxies of the Milky Way: Fornax, Leo I and II, Sculptor, Sextans, Draco, Canes Venatici I, and Ursa Minor. We use the published catalog of abundance measurements from the previous paper in this series. The measurements are based on spectral synthesis of iron absorption lines. For each MDF, we determine maximum likelihood fits for Leaky Box, Pre-Enriched, and Extra Gas (wherein the gas supply available for star formation increases before it decreases to zero) analytic models of chemical evolution. Although the models are too simplistic to describe any MDF in detail, a Leaky Box starting from zero metallicity gas fits none of the galaxies except Canes Venatici I well. The MDFs of some galaxies, particularly the more luminous ones, strongly prefer the Extra Gas Model to the other models. Only for Canes Venatici I does the Pre-Enriched Model fit significantly better than the Extra Gas Model. The best-fit effective yields of the less luminous half of our galaxy sample do not exceed 0.02 Z_sun, indicating that gas outflow is important in the chemical evolution of the less luminous galaxies. We surmise that the ratio of the importance of gas infall to gas outflow increases with galaxy luminosity. Strong correlations of average [Fe/H] and metallicity spread with luminosity support this hypothesis.

On leaky-box approximation to GALPROP

The Galactic Propagation (GALPROP) numerical code is now accepted as an advanced tool for simulations of cosmic-ray diffusion and interaction in the Galaxy. The code is used for the interpretation of a large body of cosmic-ray data. In some cases, including in particular the case of stable primary and secondary nuclei, one can use a simple leaky-box model for handling of data on cosmic-ray energy spectra and composition. We find an adequate leaky-box approximation to the basic GALPROP model and estimate its accuracy.

A Markov Chain Monte Carlo technique to sample transport and source parameters of Galactic cosmic rays

Context. Propagation of charged cosmic-rays in the Galaxy depends on the transport parameters, whose number can be large depending on the propagation model under scrutiny. A standard approach for determining these parameters is a manual scan, leading to an inefficient and incomplete coverage of the parameter space. Aims. In analyzing the data from forthcoming experiments, a more sophisticated strategy is required. An automated statistical tool is used, which enables a full coverage of the parameter space and provides a sound determination of the transport and source parameters. The uncertainties in these parameters are also derived. Methods. We implement a Markov Chain Monte Carlo (MCMC), which is well suited to multi-parameter determination. Its specificities (burn-in length, acceptance, and correlation length) are discussed in the context of cosmic-ray physics. Its capabilities and performances are explored in the phenomenologically well-understood Leaky-Box Model. Results. From a technical point of view, a trial function based on binary-space partitioning is found to be extremely efficient, allowing a simultaneous determination of up to nine parameters, including transport and source parameters, such as slope and abundances. Our best-fit model includes both a low energy cut-off and reacceleration, whose values are consistent with those found in diffusion models. A Kolmogorov spectrum for the diffusion slope (δ = 1/3) is excluded. The marginalised probability-density function for δ and α (the slope of the source spectra) are δ ≈ 0.55−0.60 and α ≈ 2.14−2.17, depending on the dataset used and the number of free parameters in the fit. All source-spectrum parameters (slope and abundances) are positively correlated among themselves and with the reacceleration strength, but are negatively correlated with the other propagation parameters. Conclusions. The MCMC is a practical and powerful tool for cosmic-ray physic analyses. It can be used to confirm hypotheses concerning source spectra (e.g., whether αi αj )a nd/or determine whether different datasets are compatible. A forthcoming study will extend our analysis to more physical diffusion models.

Galactic propagation of cosmic ray nuclei in a model with an increasing diffusion coefficient at low rigidities: A comparison of the new interstellar spectra with Voyager data in the outer heliosphere

Using both a Monte Carlo Diffusion Model and a Leaky Box Model for propagation in the galaxy, we have determined the interstellar spectra of H, He, and heavier nuclei with emphasis on energies below ∼1–2 GeV nucleon−1. This calculation uses diffusion coefficients that are increasing at rigidities below ∼2–3 GV and which are based on those used to fit the interstellar electron spectrum to the galactic nonthermal radio synchrotron spectrum by Webber and Higbie (2008). The resulting interstellar spectra of the cosmic ray nuclei are thus reduced at low energies in a charge‐dependent way from those derived using a diffusion coefficient = constant as was the case in some earlier calculations. The resulting interstellar intensities at ∼200 MeV nucleon−1 near the peak in the differential spectrum have been reduced by factors of 2.00 and 1.46 for H and He nuclei, respectively. These new interstellar intensities are then compared with recent Voyager measurements beyond the heliospheric termination shock. Solar modulation effects, corresponding to values of ϕ between 60 and 80 MV in simple force field models, are still evident in the 2008.5 Voyager 1 H and He spectra at 107 AU. In 2009, when the 11‐year solar modulation effects are expected to be a minimum in the outer heliosphere, a comparison of future Voyager measurements at ∼110 AU and beyond and these new interstellar spectra along with the corresponding electron spectrum measured by Voyager 1 will provide an important test of both modulation and interstellar propagation models.

Life in the last lane: star formation and chemical evolution in an extremely gas rich dwarf

We present an analysis of HI, H-alpha, and oxygen abundance data for NGC 3741. This galaxy has a very extended gas disk (~8.8 times the Holmberg radius), and a dark to luminous (i.e. stellar) mass ratio of ~149, which makes it one of the ``darkest'' dwarf irregular galaxies known. However its ratio of baryons (i.e. gas + stellar) mass to dark mass is typical of that in galaxies. Our new high-resolution HI images of the galaxy show evidence for a large scale (purely gaseous) spiral arm and central bar. Despite the gaseous spiral feature and the on-going star formation, we find that the global gas density in NGC 3741 is significantly lower than the Toomre instability criterion. We also find that the star formation rate is consistent with that expected from the observed correlations between HI mass and SFR and the global Kennicutt-Schmidt law respectively. We measure the oxygen abundance to be 12+ log(O/H)=7.66$\pm$0.10, which is consistent with that expected from the metallicity-luminosity relation, despite its extreme gas mass ratio. We also examine the issue of chemical evolution of NGC 3741 in the context of closed-box model of chemical evolution. The effective oxygen yield of NGC 3741 is consistent with recent model estimates of closed-box yields, provided one assumes that the gas has been efficiently mixed all the way to edge of the HI disk (i.e. >8 times the optical radius). This seems a priori unlikely. On the other hand, using a sample of galaxies with both interferometric HI maps and chemical abundance measurements, we find that the effective yield is anti-correlated with the total dynamical mass, as expected in leaky box models (slightly abridged).

Radio synchrotron spectra for a leaky box approximation

The synchrotron emission observed from many astrophysical objects is commonly modelled to arise from relativistic electrons in an emission region occupied by a spatially homogeneous, but tangled magnetic field. However, we know that magnetic fields embedded in ionised gases tend to form flux ropes interspersed with regions of much lower magnetic field strengths. Here we develop a full description of the evolution of the energy distribution of relativistic electrons in a plasma divided into two distinct regions with different strengths of the magnetic field. Electrons are able to continuously leak from the low-field region into the high-field region. The model becomes fully analytic for physically reasonable assumptions. We show that such a leaky box model produces two distinct breaks in the electron energy distributions which give rise to three breaks in the resulting synchrotron spectrum. The spectral slopes in between the breaks are in general not constant and thus allow for significant curvature of the spectrum. These spectra are consistent with spatially resolved observations of the radio spectra of the lobes of radio galaxies. The exact form of the spectra depends on the adopted diffusion rate. The leaky box model significantly extends the time over which synchrotron emission can be detected at a given frequency compared to the usually assumed case of homogeneous magnetic fields. The spectral ages inferred for the electron population from standard techniques for the leaky box model are considerably younger than their real age.

Propagation model for cosmic ray species in the Galaxy

In a recent paper [Astrophys. J. 565 (2002) 280], it has been shown that the flux of secondary cosmic ray (CR) antiprotons appears to be contradictory to measurements of secondary to primary nuclei ratios in cosmic rays when calculated in the same Galactic propagation model. The contradiction appears as a value of the diffusion coefficient necessary to match the secondary ratios p ¯ / p and B/C. In particular, it was shown that the reacceleration models designed to match secondary to primary nuclei ratios produce too few antiprotons. It is, however, clear that some reacceleration is unavoidable in the turbulent interstellar medium. Here we discuss an idea of how to improve reacceleration model by allowing for the damping of interstellar turbulences on the small scale by cosmic rays, mostly protons. This would lead to increase in the mean free path lengths at low energies, the well-known phenomena empirically discovered in the Leaky-Box models, thus producing less secondary nuclei. The effect on secondary antiprotons will be discussed elsewhere.