The STAROX stellar evolution code

This paper describes the STAROX stellar evolution code for the calculation of the evolution of a model of a spherical star. The code calculates a model at time t _k , that is the run of pressure, density, temperature, radius, energy flux and related variables on a mesh in mass M _i , given the distribution of chemical elements X _j (i) at t _k and the model at the previous time step t _k?1. It then advances the chemical composition to the next time step t _k+1 and calculates a new model at time t _k+1. This process is iterated to convergence. The model equations are solved by Newton–Raphson relaxation; the chemical equations are solved by an iterative procedure, each element being advanced in turn, and the process repeated to convergence. Convection is modelled by a mixing length model and convective mixing is treated as a diffusive process; chemical overshooting can be incorporated in parametric form. The equation of state is taken from OPAL tables and the opacity from a blend of OPAL and Alexander tables. Nuclear reaction rates are from NACRE but only cover the p–p chain and CNO cycle. The atmospheric layers are incorporated in the model by applying the surface boundary condition at small optical depth (τ≈0.001). The mesh in mass M _i is usually taken as fixed except that there is a moveable mesh point at the boundary of a convective core. Results are given for models of mass 0.9 and 5.0M _⊙ with initial composition X=0.7,Z=0.02 evolved to a state where the central hydrogen abundance is X _c =0.35, and for a model of mass 2.0M _⊙ with initial X=0.72,Z=0.02, evolved to X _c =0.01 and with core overshooting. In this latter case we compute two models one with and one without a moveable mesh point at the boundary of the convective core to illustrate the importance of having such a moveable mesh point for the determination of the Brunt–Väisälä frequency in the layers outside the core.     

Thermal relic abundance and anisotropy due to modified gravity

Alternative cosmologies, based on extensions of General Relativity, predict modified thermal histories in the early universe during the pre Big Bang Nucleosynthesis (BBN) era. When the expansion rate is enhanced with respect to the standard case, thermal relics typically decouple with larger relic abundances. In this paper, we study the dynamical evolution of an f(R) model of gravity in a homogeneous and anisotropic background which is given by a Bianchi type-I model of the universe filled with dark matter, which is described by a perfect fluid with a barotropic equation of state. As an example of a consistent analysis of modified gravity, we apply the formalism to a simple background solution of R+βR ⁿ gravity. Our analysis shows that f(R) cosmology allows dark matter masses lesser than 100 GeV, in the regime ρ ^c ?ρ ^m . We finally discuss how these limits apply to some specific realizations of standard cosmologies: an f(R) gravity model, Einstein frame model.     

Spatial periodicity of galaxy number counts, CMB anisotropy, and SNIa Hubble diagram based on the universe accompanied by a non-minimally coupled scalar field

We have succeeded in establishing a cosmological model with a non-minimally coupled scalar field φ that can account not only for the spatial periodicity or the picket-fence structure exhibited by the galaxy N-z relation of the 2dF survey but also for the spatial power spectrum of the cosmic microwave background radiation (CMB) temperature anisotropy observed by the WMAP satellite. The Hubble diagram of our model also compares well with the observation of Type Ia supernovae. The scalar field of our model universe starts from an extremely small value at around the nucleosynthesis epoch, remains in that state for sufficiently long periods, allowing sufficient time for the CMB temperature anisotropy to form, and then starts to grow in magnitude at the redshift z of ～1, followed by a damping oscillation which is required to reproduce the observed picket-fence structure of the N-z relation. To realize such behavior of the scalar field, we have found it necessary to introduce a new form of potential V(φ)∝ φ ²exp?(?q φ ²), with q being a constant. Through this parameter q, we can control the epoch at which the scalar field starts growing.     

Simulation of f-Mode Propagation Through a Cluster of Small Identical Magnetic Flux Tubes

Motivated by the question of how to distinguish seismically between monolithic and cluster models of sunspots, we have simulated the propagation of an f-mode wave packet through two identical small magnetic flux tubes (R=200 km), embedded in a stratified atmosphere. We want to study the effect of separation d and incidence angle χ on the scattered wave. We have demonstrated that the horizontal compact pair of tubes (d/R=2, χ=0) oscillate as a single tube when the incident wave is propagating, which gives a scattered wave amplitude of about twice that from a single tube. The scattered amplitude decreases with increasing d when d is about λ/2π where λ is the wavelength of the incident wave packet. In this case the individual tubes start to oscillate separately in the manner of near-field scattering. When d is about twice λ/2π, scattering from individual tubes reaches the far-field regime, giving rise to coherent scattering with an amplitude similar to the case of the compact pair of tubes. For perpendicular incidence (χ=π/2), the tubes oscillate simultaneously with the incident wave packet. Moreover, simulations show that a compact cluster oscillates almost as a single individual small tube and acts more like a scattering object, while a loose cluster shows multiple-scattering in the near field and the absorption is largest when d within the cluster is about λ/2π. This is the first step to understand the seismic response of a bundle of magnetic flux tubes in the context of sunspot and plage helioseismology.     

A better way to reconstruct dark energy models?

To reconstruct dark energy models the redshift z _eq , marking the end of radiation era and the beginning of matter-dominated era, can play a role as important as z _t , the redshift at which deceleration parameter experiences a signature flip. To implement the idea we propose a variable equation of state for matter that can bring a smooth transition from radiation to matter-dominated era in a single model. A popular Λ∝ ρ dark energy model is chosen for demonstration but found to be unacceptable. An alternative Λ∝ ρ a ³ model is proposed and found to be more close to observation.     

Exact physical Maxwell-Einstein Tolman-VII solution and its use in stellar models

We derive an exact Maxwell-Einstein metric for a spherically symmetric static perfect fluid with mass and charge (Q), that electrifies the Tolman-VII metric and meets applicable physical boundary conditions. We show that there is more than one way to electrify any Q=0 metric, depending on whether metric component g _tt , g _rr , or neither, is independent of Q. We illustrate an approach not yet in the literature in which g _rr is independent of Q. When applied to a baryonic stellar model, this metric is versatile, capable of producing a range of values for radius, mass, Q, and far-field mass due to the presence of charge.     

Non-linear oscillator models for the X-ray bursting of the microquasar GRS 1915+105

The microquasar GRS 1915+105, exhibits a large variety of characteristic states, according to its luminosity, spectral state, and variability. The most interesting one is the so-called ρ-state, whose light curve shows recurrent bursts. This paper presents a model based on Fitzhugh-Nagumo equations containing two variables: x, linked to the source photon emission rate detected by the MECS, and y related to the mean photon energy. We aim at providing a simple mathematical framework composed by non-linear differential equations useful to predict the observed light curve and the energy lags for the ρ-state and possibly other classes of the source. We studied the equilibrium state and the stability conditions of this system that includes one external parameter, J, that can be considered a function of the disk accretion rate. Our work is based on observations performed with the MECS on board BeppoSAX when the source was in ρ and ν mode, respectively. The evolution of the mean count rate and photon energy were derived from a study of the trajectories in the count rate—photon energy plane. Assuming J constant, we found a solution that reproduces the x profile of the ρ class bursts and, unexpectedly, we found that y exhibited a time modulation similar to that of the mean energy. Moreover, assuming a slowly modulated J the solutions for x quite similar to those observed in the ν class light curves is reproduced. According these results, the outer mass accretion rate is probably responsible for the state transitions, but within the ρ-class it is constant. This finding makes stronger the heuristic meaning of the non-linear model and suggests a simple relation between the variable x and y. However, how a system of dynamical equations can be derived from the complex mathematical apparatus of accretion disks remains to be furtherly explored.     

The circular polarization of sources of synchrotron radiation

The degree of circular polarizationp _c is calculated for two models of a source of synchrotron radiation:

1. A source with an inhomogeneous magnetic field and isotropic angular distribution of the electrons with respect to the magnetic field;
2. A source with a homogeneous magnetic field and anisotropic angular distribution of the electrons in which the anisotropy of angular distribution substantially increases with the electron energy.

The first model can be used to describe extended radio-sources; and the second, to describe compact radio-sources. For those sources, whose observed polarization properties correspond to the first model, we obtain an integral equation which connects the observed distribution of the sources with the extent of their linear and circular polarization (p _l andp _c ) and the unknown distribution of the sources over the strengthB and the degree of homogeneity ?=(B ₀/B)² of the magnetic field;B ₀ is a homogenous field,B ₀?B. A solution of the integral equation obtained is found for a particular case. This solution makes it possible to determine the distribution of different types of sources over ? if the distribution of these sources in the extent of linear polarization is known. The formulae obtained make it possible to indicate which sources with a known degree of linear polarization should be expected to exhibit highest circular polarization. In the discussion of the first model the question is raised as to the information one can get about the magnetic field by using observations of both linear and circular polarization for a separate source, and for a number of sources. It is shown that the determination of the most probable values ofB and ? in a separate source based on the known values ofp _l andp _c for the source, is possible only if one knows the distribution overB and ? of the sources of the type to which the source in question belongs. The observational data now available make it possible to find the distribution of the sources only over ?. Since the distribution overB and ? is at present unknown, even a very strong upper limit forp _c in the case of a separate source does not enable us to give an exact upper limit for the strength of the magnetic field in this source. In the first model the upper limit for the magnetic field can be obtained only if the upper limit ofp _c is known for a certain number of sourcesN, withN?1. This limit allows for much stronger fields than are usually admitted. This last fact should be taken into consideration when one deals with the results of observations of circular polarization in sources with strong magnetic fields. The first model presents some difficulties when we compare it with observations of some compact sources. The second model can explain why one observes in these sources a violation of the lawp _c ～v ^?1/2 and a change of sign inp _c when the frequency of the observationsv changes.     

Homologous gravitational collapse in Lagrangian representation

The classical problem of spherical homologous gravitational collapse with a polytropic equation of state with γ=4/3 is examined in Lagrangian fluid coordinate. The fluid velocity v(t)=dr(t)/dt=ηdy/dt is derived from the evolution function y(t) where η=r(0) is the radial fluid label in Lagrangian formulation. The evolution function y(t), which describes the collapse time history of a finite pressure cloud, is solved which happens to be identical to the well established parametric form of Mestel (Mon. Not. R. Astron. Soc., 6:161–198, 1965) for cold cloud collapse. The spatial structure is described by a nonlinear equation of the density profile function q(z) with z=aη. Due to the nonlinearity, the collapse profile is highly non-uniform in space. For moderate values of q(0), the solutions are homologous. For large q(0), homology is broken leading to the formation of a central core and a central cavity. The stellar envelope bounding the central cavity collapses under the additional external gravity of the central core, generating eventually a sequence of cavity-shell structure in the envelope, until the entire mass of the original cloud is accounted for.     

Local Lorentz transformation and exact solution in f(T) gravity theories

A general tetrad fields, with an arbitrary function of radial coordinate, preserving spherical symmetry, is provided. Such tetrad is split into two matrices: The first matrix represents a Local Lorentz Transformation (LLT), which contains an arbitrary function. The second matrix represents a proper tetrad fields which satisfy the field equations of f(T) gravitational theory. This general tetrad is applied to the field equations of f(T). We derive a solution with one constant of integration to the resulting field equations of f(T). This solution gives a vanishing value of the scalar torsion. We calculate the energy associated with this solution to investigate what is the nature of the constant of integration.     

Symetries du Probleme desN Corps,Regularisation des Solutions Centrales Singulieres

The newtonian problem ofn mass points bodies is invariant by several changes of spatio-temporal variables. These symmetries correspond to arbitrary choices of the referential and they are related via Noether's theorem or by its generalization to conservative quantities of the motion. Forn=2 the author has defined two families of symmetriesS ₁ andS ₂ changing the eccentricity of a solution. The family of symmetries,S ₁, is associated to the arbitrary choice of thezero level of the potential and may related unbounded and bounded solutions. The family of symmetries,S ₂, is related to a possibleaffinity of the configurations space. Via a symmetry of theS ₂ family a zero angular momentum solution is equivalent to a non-zero angular momentum solution. Via a product of two symmetries of each family, denoted byS ₁.S ₂, any solution of the two-body problem is equivalent to a circular solution. In this paper it is shown that some of these transformations may be generalized to symmetries changing the quantityC ² H in then-body problem, whereC is the angular momentum andH is the energy. The extension is easily made to central solutions of then-body problem because involving several synchroneous two-body problems. We consider for exposition then=3 case. The principal results may be resumed by the following propositions:

1. The two families of symmetriesS ₁ andS ₂ are described by a spatial transformation product of an instantaneous homothethy and an instantaneous rotation completed by a change of temporal variable.
2. TheS ₁ family of symmetries may relate unbounded and bounded central solutions of the same type, i.e. unaligned or aligned.
3. TheS ₂ family of symmetries may regularize multiple collisions among central solutions of the same type.

Therefore any central solution, via a symmetryS ₁ orS ₂ orS ₁.S ₂, is equivalent to a central circular solution of the same type. That is a form of regularization.     

Dynamical portrait of the Lixiaohua asteroid family

In this paper we present a comprehensive analysis of the dynamics in the region of the (3556) Lixiaohua asteroid family. The family lies in a particularly interesting region of the phase space, crossed by several two-body and three-body mean motion resonances. Also, members of this family can have close encounters with large asteroids, such as Ceres. We have identified the mean motion resonances which contribute to the long-term dynamical evolution of the family and our results confirm that the members of this family can be classified into a number of groups, exhibiting different dynamical behavior. We show for the first time that in the Lixiaohua region, apart from the chaotic diffusion in proper eccentricity and inclination (e _p and I _p ), there is at least one extended chaotic zone where several resonances overlap, thus giving rise to chaotic diffusion in proper semi-major axis (a _p ) as well. Using a code of Monte Carlo type, we simulate the evolution of the family, according to the model which combines the chaotic diffusion (in a _p , e _p and I _p ), Yarkovsky/YORP thermal effect and random walk in a _p due to the close encounters with massive asteroids. These simulations show that all these effects should be taken into account in order to accurately explain the observed distribution of family members in the space of proper elements, although a “minimal” model that accounts for chaotic diffusion in (e _p , I _p ), Yarkovsky-induced drift in a _p and random walk in a _p due to the close encounters with the most massive asteroids is enough to grossly characterize the shape of the family.     

On the ring current energy injection rate

Assuming that the formation of the ring current belt is a direct consequence of an enhanced crosstail electric field and hence of an enhanced convection, we calculate the total ring current kinetic energy (K_R) and the ring current energy injection rate (U_R) as a function of the cross-tail electric field (E_CT); the cross-tail electric field is assumed to have a step function-like increase. The loss of ring current particles due to recombination and charge-exchange is assumed to be distributed over the whole ring current region. It is found that: (1) the steady-state ring current energy K_R is approximately linearly proportional to E_CT; (2) the characteristic time t_c for K_R to reach the saturation level is 3–4 h; (3) the injection rate U_R is proportional to E_CT^β where β ? 1.33?1.52; and (4) the characteristic time t_p for U_R to reach the peak value is 1–2 h and the peak U_R value is 50% higher than the steady-state value. Since β is now determined specifically for an enhanced convection, an observational determination of the relationship between E_CT(or φ_CT) and U_R is essential to a better understanding of ring current formation processes. If the observed β is greater than 1.5, additional processes (e.g. an injection of heavy ions from the ionosphere to the plasma sheet and subsequently to the ring current region) may be required.     

Combined color indices and photometric structures of the galaxies NGC 834 and NGC 1134

Surface BVRI photometry is presented for two spiral galaxies with a complex photometric structure: NGC 834 and NGC 1134. We propose to introduce the combined color indices Q _BVI and Q _VRI to investigate the photometric structure of the galaxies. These color indices depend only slightly on selective absorption, which allows them to be used to study the photometric structure of “dusty” galaxies. Evolutionary stellar-population models show that Q _BVI is most sensitive to the presence of blue stars, while Q _VRI depends on local Hα equivalent width. A ring with active star formation manifests itself on the Q _BVI map for NGC 834 at a distance of ～15 from its center, and a spiral structure shows up on the Q _VRI map for NGC 1134 in its inner region. The Q _BVI –Q _VRI diagram can provide information about the current stage of a star's formation in various galactic regions. A comparison of the color indices for the galaxies with their model values allows us to estimate the color excesses and extinction in various galactic regions.     

Some aspects of mid-latitude spread-Es, and its relationship with spread-F

After discussing some examples of mid-latitude spread-E_s this paper presents distributions showing the diurnal, annual and sunspot-cycle variations of this phenomenon. The variations are very similar to those found for mid-latitude spread-F. Also, a superposed-epoch analysis involving an investigation of spread-F occurrence relative to controls consisting of nights of high spread-E_s occurrence showed a relationship of high significance. A similar analysis using spread-F occurrence relative to daytime spread-E_s controls also suggested a relationship between these parameters. The results presented in this paper together with results from previous work suggest strongly that the two phenomena (spread-E_s and spread-F) are produced by the same or similar disturbances. Also it seems likely that the disturbances responsible for spread-F during night hours are also present during daylight hours.     

The formation of ring galaxies

The conditions under which a head-on collision between a disk galaxy and a spherical galaxy can lead to ring formation are investigated, using the impulsive approximation. The spherical galaxy is modeled as a polytrope of indexn=4 and radiusR _S and the disk galaxy as an exponential disk whose surface density is given by \(\sigma (r) = \sigma _c e^{ - 4r/R_D } \) , where σ _c is the central density andR _D is the radius of the disk. The formation and properties of the rings are closely related to the fractional change in binding energy of the disk galaxy, given by ΔU/?U?=γ _D β _D , where (GM _S ² R _D )/(V ² M _D R _S ² ),M _S andM _D being the masses of the spherical and disk galaxies, respectively, and β _D ≡β _D (n, σ, ?,i) is a function of the models of the two galaxies, the ratio of the radii of the two galaxies ?=R _S /R _D , and the angle of inclinationi, of the disk to the direction of relative motion of the two galaxies. Calculations are made for the caseR _S =R _D . Since practically the entire mass of the spherical galaxy, for the chosen model, lies within 1/3 of its radius, the radius of the spherical galaxy is effectively \(\tfrac{1}{3}\) that of the disk galaxy. It is found that as a result of the collision, the innermost and the outer parts of the disk galaxy are not much affected, but the intermediate region expands and gets evacuated, leading to the crowding of stars in a preferential region forming a ring structure. The rings are best formed for a normal, on-axis collision. For this case, rings form when ΔU/|U| lies between \(\tfrac{1}{2}\) and 2, while they are very sharp and bright when ΔU/|U| lies between \(\tfrac{1}{2}\) and 1. Within this range, as ΔU/|U| increases, the rings become sharper and their positions shift outwards with respect to the centre of the disk galaxy. The relationship $$\gamma _D = 0.0016 + 0.045s_{{\text{max}}}^2 ,$$ wheres _max is the radial distance of the density maximum of the ring from the centre of the disk galaxy (measured in terms of the radius of the disk galaxy as unit) enables us to finds _max from γ _D and vice versa, and interpret some prominent ring galaxies. The effect of introducing a bulge to the disk is to distribute the tidal disruptive effects more evenly and, hence, reduce the sharpness of the ring.     

Multiplicity functions of dark matter haloes from fractional Brownian motion

We studied the formation of dark matter haloes assuming a Fractional Brownian Motion (FBM) of trajectories in the plane (S,δ). S is the variance of the smoothed overdensity δ. The values of δ are calculated using a filter of radius R that defines a mass scale M, and thus S is considered as a function of mass M. Focusing on a specific point of the initial Universe and calculating δ as a function of the radius R of the filter a random walk on the plane (S,δ) is established. If the above evolution is a FBM motion then there exist correlations between various mass scales. These correlations depend on a Hurst exponent H. Various mass scales are not correlated for H=1 and the evolution is reduced to the classical Brownian motion. Following Zhang and Hui (Astrophys. J. 641:641, 2006), we constructed a Volterra integral equation for the distribution of trajectories after a first crossing of an ellipsoidal barrier. The integral equation is solved numerically and multiplicity functions of dark matter haloes were derived and compared to the results of N-body simulations. Our results show that for H=1.05 the resulting multiplicity functions are in excellent agreement, much better than the agreement predicted by any other model, with the predictions of N-body simulations. This shows that FBM model is a very promising tool for the study of the formation of structures in the Universe.     

A study of the relationship between geo-magnetic storms and ionospheric disturbances at mid-latitudes

The problem of the ionospheric disturbances associated with geomagnetic storms is examined with the goal of searching for a relationship between the time-developments of the two phenomena. Faraday rotation measurements of total electron content (N_T) are used to monitor the ionospheric F-region at a mid-latitude site, while a variety of geomagnetic parameters are examined as possible ways of following the geomagnetic variations. The ionospheric and geomagnetic data taken during 28 individual storms from 1967 to 1969 are used to search for a predictive scheme which can be tested using data from 17 storms in 1970. The specific aim is to find the geomagnetic parameter whose time-development can best forecast whether or not the ionospheric response will include an initial positive phase prior to the normally extended period of F-region depletions. Correlations between N_T and the geomagnetic indices K_p, and equatorial Dst(H) prove to be wholly inadequate. The local times of main-phase-onset (MPO) determined from the equatorial Dst(H) indices as well as from local horizontal component data, also prove to be unsatisfactory. The best correlations are obtained using local measurements of the total geomagnetic field (F). These results show that a storm commencement (SC) will produce an enhancement in n_t during the afternoon period following the SC unless there is an intervening post-midnight period with a strong depression of the geomagnetic field. Operationally this is taken to be a depression in F of at least 100γ near 03:00 LT     

NLTE formation of the solar spectrum of silicon: Abundance of silicon in a three-dimensional model of the solar atmosphere

We investigated the NLTE formation of the solar spectrum of neutral silicon using 3D hydrodynamic model of the solar atmosphere and realistic atomic model. We show that, within the intergranular region, combined action of the deficit in the source function and excess in opacity due to the overpopulation of the lower Si I levels leads to a considerably higher increase in the central depth D and equivalent width W of these lines as compared to the granules. We have fitted silicon abundances A _W and A _D from the equivalent widths W and central depths D for 65 Si I lines using a 3D model. We show that a total error in the calculated silicon abundance due to neglecting NLTE and 3D effects, as well as the uncertainty in the van der Waals broadening constant γ₆, turns out to be ?0.1 dex. Using a semiclassical theory by Anstee, Barklem, and O’Mara in calculating γ₆ yields a fair coincidence between the values of A _W and A _D , because the average difference A _W — A _D does not exceed 0.01 dex for both NLTE and LTE. When applying the Unsold’s approximation in calculating γ₆ with an enhancement factor E = 1.5, the abundances A _W and A _D proved to be in disagreement with one another. We analyzed the “solar” oscillator strength scale by Gurtovenko and Kostik, as well as the experimental one by Garz and Becker et al. We show that using “solar” oscillator strengths log gfw leads to a minimum trend with the equivalent widths for NLTE abundances A _W , A _D , their difference A _W — A _D , and standard deviations. The NLTE abundance of silicon obtained using solar oscillator strength scale by Gurtovenko and Kostik is A _W ^NTLE = 7.549 ± 0.016. This value is in good agreement with the value of silicon abundance recommended by Grevesse and Sauval for the CI chondrite meteorites.     

On the asteroid belt's orbital and size distribution

For absolute magnitudes greater than the current completeness limit of H-magnitude ∼15 the main asteroid belt's size distribution is imperfectly known. We have acquired good-quality orbital and absolute H-magnitude determinations for a sample of small main-belt asteroids in order to study the orbital and size distribution beyond H=15, down to sub-kilometer sizes (H>18). Based on six observing nights over a 11-night baseline we have detected, measured photometry for, and linked observations of 1087 asteroids which have one-week time baselines or more. The linkages allow the computation of full heliocentric orbits (as opposed to statistical distances determined by some past surveys). Judged by known asteroids in the field the typical uncertainty in the (a/e/i) orbital elements is less than 0.03 AU/0.03/0.5°. The distances to the objects are sufficiently well known that photometric uncertainties (of 0.3 magnitudes or better) dominate the error budget of their derived H-magnitudes. The detected asteroids range from HR=12-22 and provide a set of objects down to sizes below 1 km in diameter. We find an on-sky surface density of 210 asteroids per square degree in the ecliptic with opposition magnitudes brighter than mR=23, with the cumulative number of asteroids increasing by a factor of 10^0.27/mag from mR=18 down to the mR?23.5 limit of our survey. In terms of absolute H magnitudes, we find that beyond H=15 the belt exhibits a constant power-law slope with the number increasing proportional to ¹⁰0.30H from H?15 to 18, after which incompleteness begins in the survey. Examining only the subset of detections inside 2.5 AU, we find weak evidence for a mildly shallower slope for H=15-19.5. We provide the information necessary such that anyone wishing to model the main asteroid belt can compare a detailed model to our detected sample.