Self-similar Model Research Articles

Analytical modelling of AGN outflows

We study AGN jet formation by means of exact solutions of the relativistic mag- netohydrodynamic equations. This is an original extension in Schwarzchild's metric of previous meridionally self-similar models. The outflow is mainly thermally driven to high asymptotic speeds from the central region where the escape speed is also high. The jet solutions are mag- netically collimated on the sub-parsec scale. Almost all AGNs are associated with outflows. These phenomena of high energy affect the evolution of the central object and its surrounding medium. In fact, observations show that jets are launched very close to the black hole where the thermal and mag- netic energies are sufficiently high such that the plasma escapes from the gravitational potential. Many analytical and numerical works have been developed to examine the mechanisms of acceleration and collimation of jets, few of them though in the frame of general relativity. Jet formation seems to be strictly related to the presence of large-scale magnetic fields and the existence of gaseous disks (Konigl & Pudritz 2000). The origin can be the Keplerian disk or the spherical corona surrounding the central part of the AGN or the microquasar as discussed by several authors. All MHD models sustain the idea that the gravitational energy of the central object is transferred to the accreted matter and eventually the jet. This energy is carried by the flow along the magnetic field lines anchored in the corona of the rotating disk. The hot ionized fluid close to the axis where the rotation is weak may escape from the corona thermally in the form of a wind and forced to follow the field lines. Then rotation combined to the magnetic field colli- mates the flow beyond the Alfvsurface. In this contribution we present a model for such thermally driven and magnetically confined outflows from a spherical hot corona. The terminal velocity is, as expected, of the order of the escape speed at the footpoints.

Evaluation and estimation of second-order self-similar network traffic

The second-order character of self-similar network traffic, i.e. its correlation existing at multiple time scales, has an enormous impact on network performance, which has been widely studied. In studying the impact of self-similar traffic on network performance and utilizing its correlation structure to design network control scheme and on-line inspection, a real-time estimate of the autocorrelation of network traffic is often necessary. For an effective and quick estimate, it is very useful to determine the required sampled data length according to the requirement of precision. In this paper, the relationship between Hurst parameter ( H), the precision of estimated autocorrelation and required sampled data length is discussed on the basis of fractional Gaussian noise (FGN) model and a simple calculating formula is proposed. Furthermore, a sharp increase is discovered in the variances of estimated autocorrelation with the same data length, when H>0.75. This is a new phenomenon we call ‘jumping burstiness’, which has not been reported before. This phenomenon shows that Hurst parameter could reflect the self-similar character of network traffic, but it is not enough to capture all the features. Experiments confirm that our results are not only valid to FGN model, but also able to offer reference to estimate the autocorrelation of other self-similar models, such as fractional auto-regression integrated moving average. At the same time, trace driven simulation shows that the existence of jumping burstiness in traffic has a remarkable impact on queueing performance.

The X-ray luminosity-velocity dispersion relation in the REFLEX cluster survey

We present an estimate of the bolometric X-ray luminosity - velocity dispersion (Lxv) relation measured from a new, large and homogeneous sample of 171 low redshift, X-ray selected galaxy clusters. The linear fitting of log(Lx) log(�v) gives Lx = 10 32.72±0.08 � 4.1±0.3 v erg s 1 h 2 50 . Furthermore, a study of 54 clusters, for which the X-ray temperature of the intracluster medium T is available, allows us to explore two other scaling relations, Lx T andv T. From this sample we obtain Lx / T 3.1±0.2 andv / T 1.00±0.16 , which are fully consistent with the above result for the Lx-�v. The slopes of Lx T andv T are incompatible with the values predicted by self-similarity (Lx / T 2 / � 4), thus suggesting the presence of non-gravitational energy sources heating up the intracluster medium, in addition to the gravitational collapse, in the early stages of cluster formation. On the other hand, the result on log(Lx) log(�v) supports the self-similar model.

Stationary Accretion Disks Launching Super-fast-magnetosonic Magnetohydrodynamic Jets

We present self-similar models of resistive viscous Keplerian disks driving nonrelativistic MHD jets and becoming super-fast-magnetosonic. We show that in order to obtain such solutions, the thermal pressure must be a sizeable fraction of the poloidal magnetic pressure at the Alfvén surface. These steady solutions that undergo a recollimation shock causally disconnected from the driving engine account for structures with a high-temperature plasma in the sub-Alfvénic region. We suggest that only unsteady outflows with typical timescales of several disk dynamical timescales can be produced if the suitable pressure conditions are not fulfilled.

Effects of Magnetic Field and Opacity on Self-Similar YSO Flow Models

During star formation, both infall and outflows are present around protostellar cores. Here we show solutions of a self-similar model that study the two flows with only one set of equations. We focus here on the effects of magnetic field and dust on solutions. Unmagnetized solutions have also been found. This shows that magnetic field is not the main driving mechanism of the circulation process. We have found that a reduction of magnetic field produces denser, slower and narrower outflows. When the opacity is less dominated by dust, density increases in the equatorial region, allowing larger accretion rates to occur. The comprehension of massive star formation could be related to this latter effect.

A self-similar view of massive star formation

Despite their important luminosity, massive stars keep unveiled most of their formation mainly because they are deeply embedded within their native cloud. This difficulty is accentuated by their small number and by their larger distances compared to low mass stars. This explains why models for the various stages of their formation are still in their infancy. We briefly present here the initial conditions and the most common models for the formation of massive stars. Also, we focus onto a family of self-similar models that allow accretion and outflows around massive protostars. Finally, we try to sketch the temporal evolution of massive star formation.

Ideal Magnetohydrodynamic Solution to the σ Problem in Crab‐like Pulsar Winds and General Asymptotic Analysis of Magnetized Outflows

Using relativistic, steady, axisymmetric, ideal magnetohydrodynamics (MHD), we analyze the super-Alfvénic regime of a pulsar wind by solving the momentum equation along the flow, as well as in the transfield direction. Employing a self-similar model, we demonstrate that ideal MHD can account for the full acceleration from high (≫1) to low (≪1) values of σ, the Poynting-to-matter energy flux ratio. The solutions also show a transition from a current-carrying to a return-current regime, partly satisfying the current-closure condition. We discuss the kind of boundary conditions near the base of the ideal MHD regime that are necessary in order to have the required transition from high to low σ in realistic distances and argue that this is a likely case for an equatorial wind. Examining the MHD asymptotics in general, we extend the analysis of Heyvaerts & Norman and Chiueh, Li, & Begelman by including two new elements: classes of quasi-conical and parabolic field line shapes that do not preclude an efficient and much faster than logarithmic acceleration, and the transition σ = σc after which the centrifugal forces (poloidal and azimuthal) are the dominant terms in the transfield force-balance equation.

The Environment of YSO Jets

It is commonly accepted that stars form in molecular clouds by the gravitational collapse of dense gas. However, it is precisely not the infalling but the outflowing material that is primarily observed. Outflow motions prevail around both low and high mass young stellar objects. We present here results from a family of self-similar models that could possibly help to understand this paradox. The models take into account the heating of the central protostar for the deflection and acceleration of the gas. The models make room for all the ingredients observed around the central objects, i.e. molecular outflows, fast jets, accretion disks and infalling envelopes. We suggest that radiative heating and magnetic field may ultimately be the main energy sources driving outflows for both low and high mass stars. The models show that the ambient medium surrounding the jet is unhomogeneous in density, velocity, magnetic field. Consequently, we suggest that jets and outflows have a prehistory that is inprinted in their environment, and that this should have direct consequences on the setting of jet numerical simulations.

Scaling invariant distributions of firms’ exit in OECD countries

Self-similar models are largely used to describe the extinction rate of biological species. In this paper we analyse the extinction rate of firms in eight OECD countries. Firms are classified by industrial sectors and sizes. We find that while a power-law distribution with exponent close to 2 fits the extinction rate very well by sector, a Weibull distribution is more appropriate if one analyses the firms’ size.

A slotted mac protocol for efficient bandwidth utilization in wdm metropolitan access ring networks

The paper proposes an original wavelength-division multiplexing slotted-ring network architecture with nodes that use one fixed transmitter and fixed receivers. It is shown, through theoretical predictions and simulation results, how a simple slotted medium access control protocol can be implemented to achieve fairness between access nodes and efficient bandwidth utilization when used in a metropolitan access network. Performance for both source and destination stripping schemes has been analytically derived and simulated with a realistic self-similar traffic model. A modified node architecture using a tunable receiver has also been considered and the impact of receiver collisions has been evaluated and analyzed. Finally, the effect of using unbalanced traffic sources has also been analyzed and simulated.

Self‐similar Models for the Mass Profiles of Early‐Type Lens Galaxies

We introduce a self-similar mass model for early-type galaxies and constrain it using the aperture mass-radius relations determined from the geometries of 22 gravitational lenses. The model consists of two components: a concentrated component, which traces the light distribution, and a more extended power-law component (ρ ∝ r-n), which represents the dark matter. We find that lens galaxies have total mass profiles that are nearly isothermal, or slightly steeper, on the several-kiloparsec radial scale spanned by the lensed images. In the limit of a single-component, power-law radial profile, the model implies n = 2.07 ± 0.13, consistent with isothermal (n = 2). Models in which mass traces light are excluded at higher than 99% confidence. An n = 1 cusp (such as the Navarro-Frenk-White profile) requires a projected dark matter mass fraction of fcdm = 0.22 ± 0.10 inside two effective radii. These are the best statistical constraints yet obtained on the mass profiles of lenses and provide clear evidence for a small but nonzero dark matter mass fraction in the inner regions of early-type galaxies. In addition, we derive the first strong-lensing constraint on the relation between the stellar mass-to-light ratio ϒ and galaxy luminosity L, ϒ ∝ L, which is consistent with the relation suggested by the fundamental plane. Finally, we apply our self-similar mass models to current problems regarding the interpretation of time delays and flux ratio anomalies in gravitational lens systems.

Entropy scaling in galaxy clusters: Insights from anXMM-Newtonobservation of the poor cluster A1983

An XMM-Newton observation of the cool (kT=2.1 keV) cluster A1983, at z=0.044, is presented. Gas density and temperature profiles are calculated for the inner 500 h_{50}^{-1} kpc (~0.35 r_200). The outer regions of the surface brightness profile are well described with a beta model with beta=0.74, but the central regions require the introduction of a second component. The temperature profile is flat at the exterior with a slight dip towards the centre. The total mass profile, calculated assuming hydrostatic equilibrium, is consistent with an NFW profile, but with a low concentration parameter c=3.75 +/- 0.74. The M/L_B ratio profile shows that, at large scale, light traces mass to a reasonable extent, and the M/L_B ratio at 0.35 r_200 is consistent with the trends with mass observed in the optical. The M_Fe/L_B ratio is about two times less than that observed for a cluster at 5 keV. The gas mass fraction rises rapidly to level off at ~200 kpc; the value at 0.35 r_200 is ~8%. The scaling properties of the emission measure profile are consistent with the empirical relation \mgas \propto \Tx^{1.94}, and not with the self-similar relation \mgas \propto \Tx^{1.5}. Comparison of the entropy profile of A1983 with that of the hot cluster A1413 shows that the profiles are well scaled using the empirically determined relation S \propto \Tx^{0.65}, suggesting that the slope of the S-T relation is shallower than in the self-similar model. The form of the entropy profiles is remarkably similar, and there is no sign of a larger isentropic core in the cooler cluster. These data provide powerful agruments against preheating models. In turn, there is now increasing observational support for a trend of f_gas with system mass, which may go some way towards explaining the observed scaling behaviour. (Abridged.)

Non-radial motion and the NFW profile

The self-similar infall model (SSIM) is normally discussed in the context of radial orbits in spherical symmetry. However it is possible to retain the spherical symmetry while permitting the particles to move in Keplerian ellipses, each having the squared angular momentum peculiar to their shell. The spherical shell, defined for example by the particles turning at a given radius, then moves according to the radial equation of motion of a particle. The itself has no physical existence except as an ensemble of particles, but it is convenient to sometimes refer to the shells since it is they that are followed by a shell code. In this note we find the distribution of squared angular momentum as a function of radius that yields the NFW density profile for the final dark matter halo. It transpires that this distribution is amply motivated dimensionally. An effective lambda spin parameter is roughly constant over the shells. We also study the effects of angular momentum on the relaxation of a dark matter system using a three dimensional representation of the relaxed phase space.

A self-similarity model for effective thermal conductivity of porous media

A new model, the self-similarity model, for effective thermal conductivity of porous media is proposed based on the thermal–electrical analogy technique and on statistical self-similarity of porous media. The proposed thermal conductivity model is expressed as a function of porosity (related to stage n of Sierpinski carpet), ratio of areas, ratio of component thermal conductivities, and contact resistance. A recursive algorithm for the thermal conductivity is obtained using the proposed model and is found to be quite simple. The predictions of the model are compared with those of other models and with existing measurements and good agreement was found. This proves that the proposed model is a valid one.

The evolution of slab jets and self-similar models of extragalactic radio sources

Simple general arguments suggest that, like round jets, initially divergent hypersonic slab jets propagating through a uniform atmosphere are eventually recollimated by the high pressure of their cocoons and the subsequent evolution is approximately self-similar. Although slab jets are not directly relevant to astrophysical jets, they allow us to study the effects of enforced symmetry without resorting to expensive three-dimensional numerical simulations. In this paper we show that, although both jets with enforced mirror symmetry and those without eventually become self-similar, the flow depends significantly on the symmetry. In particular, the aspect ratio of the cocoon created by asymmetric jets seems to be independent of the initial jet opening angle for small opening angles, contrary to what is found for symmetric jets. This suggests that the hydrodynamics of the jet and its cocoon is much more complicated than has been assumed in current theoretical models of extragalactic radio sources and that results of two-dimensional axisymmetric numerical simulations can be rather misleading.

Envelope Emission in Young Stellar Systems: A Subarcsecond Survey of Circumstellar Structure

We present modeling results for 6 of the 11 deeply embedded systems from our subarcsecond ? = 2.7 mm continuum interferometric survey. The modeling, performed in the u-v plane, assumes dust properties, allows for a power-law density profile, uses a self-consistent, luminosity-conserving temperature profile, and has an embedded point source to represent a circumstellar disk. Even though we have the highest spatial resolution to date at these wavelengths, only the highest signal-to-noise ratio systems can adequately constrain the simple self-similar collapse models. Of the six sources modeled, all were fitted with a density power-law index of 2.0; however, in half of the systems, those with the highest signal-to-noise ratio, a density power-law index of 1.5 can be rejected at the 95% confidence level. Further, we modeled the systems using the pure Larson-Penston (LP) and Shu solutions, with only age and sound speed as parameters. Overall, the LP solution provides a better fit to the data, in both likelihood and providing the observed luminosity, but the ages of the systems required by the fits are surprisingly low (1000-2000 yr). We suggest that either there is some overall time scaling of the self-similar solutions that invalidates the age estimates or, more likely, we are at the limit of the usefulness of these models. With our observations we have begun to reach the stage at which models need to incorporate more of the fundamental physics of the collapse process, probably including magnetic fields and/or turbulence. In addition to constraining collapse solutions, our modeling allows the separation of large-scale emission from compact emission, enabling the probing of the circumstellar disk component embedded within the protostellar envelope. Typically, 85% or more of the total emission is from the extended circumstellar envelope component. Using HL Tauri as a standard candle, the range of circumstellar disk masses allowed in our models is 0.0-0.12 M?; our Class 0 systems do not have disks that are significantly more massive than those in Class I/II systems. This implies that the disk in Class 0 systems must quickly and efficiently process ~1 M? of material from the envelope onto the protostar.

Self-similar Bianchi models: I. Class A models

We present a study of Bianchi class A tilted cosmological models admitting a proper homothetic vector field together with the restrictions, both at the geometrical and dynamical level, imposed by the existence of the simply transitive similarity group. The general solution of the symmetry equations and the form of the homothetic vector field are given in terms of a set of arbitrary integration constants. We apply the geometrical results for tilted perfect fluid sources and give the general Bianchi II self-similar solution and the form of the similarity vector field. In addition we show that self-similar perfect fluid Bianchi VII0 models and irrotational Bianchi VI0 models do not exist.

Evidence for ring caustics in the Milky Way

The late infall of cold dark matter onto our galaxy produces discrete flows and caustics in its halo. The recently discovered ring of stars near galactocentric distance 20 kpc and a series of sharp rises in the Milky Way rotation curve are interpreted as due to the presence of caustic rings of dark matter in the galactic plane. Their locations are consistent at the 3% level with the predictions of the self-similar infall model for the caustic ring radii. Also, a triangular feature in the IRAS map of the galactic plane is consistent with the imprint of a caustic ring of dark matter upon the baryonic matter. These observations imply that the dark matter in our neighborhood is dominated by a single flow whose density and velocity vector are estimated.

Dark matter caustics

The late infall of cold dark matter onto an isolated galaxy, such as our own, produces streams and caustics in its halo. The outer caustics are topological spheres, whereas the inner caustics axe rings. The self-similar model of galactic halo formation predicts that the caustic ring radii follow the approximate law a n ∝ 1/ n. In a study of 32 extended and well-measured external galactic rotation curves evidence was found for this law. In the case of the Milky Way, the locations of eight rises in the rotation curve fit the prediction of the self-similar model at the 3% level. Moreover, a triangular feature in the IRAs map of the galactic plane is consistent with the imprint of a ring caustic upon the baxyonic matter. These observations imply that the dark matter in our neighborhood is dominated by a single flow. Estimates of the flow's density and velocity vector are given.

On the source of dense outflows from T Tauri stars -- I. Photoionization of cool MHD disc winds

Cool magnetohydrodynamics (MHD) disc wind physics is reviewed by means of a self-similar analytical model, putting special emphasis on the mathematical aspects of the solution. It is found that the key parameter of the theory (μ) measures the relation between magnetic and tidal forces. The generation of MHD winds from accretion discs requires a subtle tuning between both stresses because only a narrow range of μ values is allowed; this range is, indeed, close to the cut-off of the magnetic turbulence induced by the development of the Balbus-Hawley instability. The space of solutions can be separated into two quite distinct classes: low-μsolutions generate magnetically dominated outflows and display a characteristic density change from horizontal to vertical stratification, while in high-μsolutions the density decreases without any intermediate enhancement as the rotation axis is approached. These theoretical (dynamical) results have been used to study the properties of the base of the wind. Density and velocity laws have been derived directly from the dynamics. The effect of the propagation of the stellar X-ray radiation through the wind has been analysed to determine the temperature law at the base of the wind (polar angles θ > 45°). It is shown that a cocoon of photoionized gas is generated around the star. The extent of the photoionized region is small (tenths of au) in dense outflows and close to the disc plane; however, it may cover the whole wind extent in diffuse winds, e.g. disc winds generated by small accretion rates (≤10−9 M⊙ yr−1). Photoionization also modifies the electron density in the plasma. As a consequence, the ambipolar diffusion heating decreases in the inner part of the wind by roughly one order of magnitude with respect to that derived by other authors. In fact, radiative heating controls the thermal properties of the inner 0.3 and 1 au of the disc wind for accretion rates of 10−7 and 10−8 M⊙ yr−1, respectively. The temperature of the densest region (base) of the wind is, at most, ≃10 000 K. Therefore, although densities as high as ∼109 cm−3 can be achieved by disc winds, the temperature is significantly smaller than the ∼5 × 105–8 × 105 K derived from the ultraviolet (UV) observations of the base of the optical jets. Also, it is shown that densities as high as ∼109 cm−3 cannot be achieved at the jet recollimation point for the accretion rates observed in the T Tauri stars. In summary, we conclude that the flow traced by the UV semiforbidden lines is not associated with cold disc winds but, most likely, it is tracing the hot inner jet, postulated in cold disc wind theory, which prevents the radial collapse of the wind.