The 3D skeleton: tracing the filamentary structure of the Universe

The skeleton formalism, which aims at extracting and quantifying the filamentary structure of our Universe, is generalized to 3D density fields. A numerical method for computing a local approximation of the skeleton is presented and validated here on Gaussian random fields. It involves solving equation     , where  ∇ρ  and     are the gradient and Hessian matrix of the field. This method traces well the filamentary structure in 3D fields such as those produced by numerical simulations of the dark matter distribution on large scales, and is insensitive to monotonic biasing.
Two of its characteristics, namely its length and differential length, are analysed for Gaussian random fields. Its differential length per unit normalized density contrast scales like the probability distribution function of the underlying density contrast times the total length times a quadratic Edgeworth correction involving the square of the spectral parameter. The total length-scales like the inverse square smoothing length, with a scaling factor given by  0.21 (5.28 + n )  where n is the power index of the underlying field. This dependency implies that the total length can be used to constrain the shape of the underlying power spectrum, hence the cosmology.
Possible applications of the skeleton to galaxy formation and cosmology are discussed. As an illustration, the orientation of the spin of dark haloes and the orientation of the flow near the skeleton is computed for cosmological dark matter simulations. The flow is laminar along the filaments, while spins of dark haloes within 500 kpc of the skeleton are preferentially orthogonal to the direction of the flow at a level of 25 per cent.     

Steady magnetic reconnection in three dimensions

The concepts of magnetic reconnection that have been developed in two dimensions need to be generalised to three-dimensional configurations. Reconnection may be defined to occur when there is an electric field (E) parallel to field lines (known as potential singular lines) which are potential reconnection locations and near which the field has an X-type topology in a plane normal to that field line. In general there is a continuum of neighbouring potential singular lines, and which one supports reconnection depends on the imposed flow or electric field. For steady reconnection the nearby flow and electric field are severely constrained in the ideal region by the condition that E = 0 there. Potential singular lines may occur in twisted prominence fields or in the complex magnetic configuration above sources of mixed polarity of an active region or a supergranulation cell. When reconnection occurs there is dynamic MHD behaviour with current concentration and strong plasma jetting along the singular line and the singular surfaces which map onto them.     

Impact of a non-Gaussian density field on Sunyaev–Zeldovich observables

The main statistical properties of the Sunyaev–Zeldovich (S–Z) effect – the power spectrum, cluster number counts and angular correlation function – are calculated and compared within the framework of two density fields which differ in their predictions of the cluster mass function at high redshifts. We do so for the usual Press & Schechter mass function, which is derived on the basis of a Gaussian density fluctuation field, and for a mass function based on a  χ²  distributed density field. These three S–Z observables are found to be very significantly dependent on the choice of the mass function. The different predictions of the Gaussian and non-Gaussian density fields are probed in detail by investigating the behaviour of the three S–Z observables in terms of cluster mass and redshift. The formation time distribution of clusters is also demonstrated to be sensitive to the underlying mass function. A semiquantitative assessment is given of its impact on the concentration parameter and the temperature of intracluster gas.     

Freezing-in condition a magnetic field and current sheets in plasma

In the ideal magnetohydrodynamic approximation it is shown that for physically permissible boundary conditions there may exist some lines on which freezing-in condition is not valid. Such singular lines are closed magnetic lines of force and lines with both ends on the boundary surface. By analogy with the singular lines of a potential magnetic field the conclusion is made that X-type singular lines are the place where current sheets (sheet pinches) appear in plasma, whereas on O-type singular lines quasi-cylindrical pinches of a usual type appear.Paper dedicated to Professor Hannes Alfvén on the occasion of his 70th birthday, 30 May, 1978.     

On string cosmology with loop corrections

Homogeneous cosmological models are investigated within the framework of low- energy string gravitation with loop corrections. Various conformai representations of the effective action are considered. Without specifying the correction functions in the Lagrangian, cosmological solutions are found with an arbitrary curvature and with dilaton fields, moduli fields, and Kalb- Ramond fields corresponding to a source with an extremely stiff equation of state. They generalize previously known solutions of the tree approximation. The behavior of the solutions in different asymptotic domains is investigated. Translated from Astrofizika, Vol. 41, No. 2, pp. 277–295, April-June, 1998.     

On the magnetic field line topology in solar active regions

The field lines of closed magnetic structures above the photosphere define a mapping from the photosphere to itself. This mapping is discontinuous, and the field line connectivity to the boundary can change discontinuously in response to continuous changes of field strength and direction, if field lines either end in a singular point of the field or are tangential to the photosphere at one end. Whereas the general existence of singular points is questionable, the field has typically a cell structure due to the presence of segments of the zero line of the photospheric longitudinal field on which the transversal field is directed from negative (pointing into the Sun) to positive fields. The cell boundaries are made up of field lines which all touch the photosphere on one of these line segments. Within each of the cells the field line mapping is continuous. When during a slow evolution a substantial part of a coronal loop or of an arcade has passed from one cell into another a fast dynamic instability may set in which was previously prevented by the anchoring of field lines in the dense photosphere.     

Gravitational collapse of polytropic, magnetized, filamentary clouds

For the case in which the gas of a magnetized filamentary cloud obeys a polytropic equation of state, gravitational collapse of the cloud is studied using a simplified model. We concentrate on the radial distribution and restrict ourselves to a purely toroidal magnetic field. If the axial motions and poloidal magnetic fields are sufficiently weak, we could reasonably expect our solutions to be a good approximation. We show that while the filament experiences gravitational condensation and the density at the centre increases, the toroidal flux-to-mass ratio remains constant. A series of spatial profiles of density, velocity and magnetic field for several values of the toroidal flux-to-mass ratio and the polytropic index, is obtained numerically and discussed.     

Large-scale galaxy distribution in the Las Campanas Redshift Survey

We make use of three-dimensional clustering analysis, inertia tensor methods, and the minimal spanning tree technique to estimate some physical and statistical characteristics of the large-scale galaxy distribution and, in particular, of the sample of overdense regions seen in the Las Campanas Redshift Survey (LCRS). Our investigation provides additional evidence for a network of structures found in our core sampling analysis of the LCRS : a system of rich sheet-like structures, which in turn surround large underdense regions criss-crossed by a variety of filamentary structures.
We find that the overdense regions contain ∼40–50 per cent of LCRS galaxies and have proper sizes similar to those of nearby superclusters. The formation of such structures can be roughly described as a non-linear compression of protowalls of typical cross-sectional size ∼ 20–25  h ⁻¹ Mpc; this scale is ∼5 times the conventional value for the onset of non-linear clustering – to wit, r ₀, the autocorrelation length for galaxies.
The comparison with available simulations and theoretical estimates shows that the formation of structure elements with parameters similar to those observed is presently possible only in low-density cosmological models, Ω_m h ∼0.2–0.3, with a suitable large-scale bias between galaxies and dark matter.     

Particle Acceleration in Reconnecting Current Sheets in Impulsive Electron-Rich Solar Flares

Electron and proton acceleration in reconnecting current sheets in electron-rich solar flares is considered. A significant three-dimensional magnetic field is assumed in the current sheet where the particles are accelerated by the DC electric field. The tearing instability of a pre-flare current sheet leads to the formation of multiple singular lines of magnetic field where the electric and magnetic fields are coaligned. Magnetized electrons are shown to be accelerated to a few tens of MeV before they leave the vicinity of a singular line. The acceleration time is estimated to be less than 10^–3 s. By contrast, much heavier protons are unmagnetized and their energy gain is more modest. The model explains a high electron-to-proton ratio and the unusually intense gamma-ray continuum above 1 MeV observed in the electron-rich flares.     

Helical fields and filamentary molecular clouds – I

We study the equilibrium of pressure truncated, filamentary molecular clouds that are threaded by rather general helical magnetic fields. We first apply the virial theorem to filamentary molecular clouds, including the effects of non-thermal motions and the turbulent pressure of the surrounding ISM. When compared with the data, we find that many filamentary clouds have a mass per unit length that is significantly reduced by the effects of external pressure, and that toroidal fields play a significant role in squeezing such clouds.
We also develop exact numerical MHD models of filamentary molecular clouds with more general helical field configurations than have previously been considered. We examine the effects of the equation of state by comparing 'isothermal' filaments, with constant total (thermal plus turbulent) velocity dispersion, with equilibria constructed using a logatropic equation of state.
Our theoretical models involve three parameters: two to describe the mass loading of the toroidal and poloidal fields, and a third that describes the radial concentration of the filament. We thoroughly explore our parameter space to determine which choices of parameters result in models that agree with the available observational constraints. We find that both equations of state result in equilibria that agree with the observational results. Moreover, we find that models with helical fields have more realistic density profiles than either unmagnetized models or those with purely poloidal fields; we find that most isothermal models have density distributions that fall off as r ^−1.8 to r ⁻², while logatropes have density profiles that range from r ⁻¹ to r ^−1.8. We find that purely poloidal fields produce filaments with steep radial density gradients that are not allowed by the observations.     

Reconstruction of the early Universe as a convex optimization problem

We show that the deterministic past history of the Universe can be uniquely reconstructed from knowledge of the present mass density field, the latter being inferred from the three-dimensional distribution of luminous matter, assumed to be tracing the distribution of dark matter up to a known bias. Reconstruction ceases to be unique below those scales – a few Mpc – where multistreaming becomes significant. Above 6 h ⁻¹ Mpc we propose and implement an effective Monge–Ampère–Kantorovich method of unique reconstruction. At such scales the Zel'dovich approximation is well satisfied and reconstruction becomes an instance of optimal mass transportation, a problem which goes back to Monge. After discretization into N point masses one obtains an assignment problem that can be handled by effective algorithms with not more than O ( N ³) time complexity and reasonable CPU time requirements. Testing against N -body cosmological simulations gives over 60 per cent of exactly reconstructed points.
We apply several interrelated tools from optimization theory that were not used in cosmological reconstruction before, such as the Monge–Ampère equation, its relation to the mass transportation problem, the Kantorovich duality and the auction algorithm for optimal assignment. A self-contained discussion of relevant notions and techniques is provided.     

DISTRIBUTION OF 2-D MAGNETIC SADDLE POINTS AND MORPHOLOGY OF FLARE KERNELS IN SOLAR ACTIVE REGIONS

We extrapolated the 3-D fields above the photosphere, taking the observed photospheric magnetic fields in the active regions NOAA 6659 and 7321 as the boundary conditions of a linear force-free field model, and detected the singular points of the 2-D fields in a plane at the chromospheric level. These singular points can be described with the Poincaré index. Singular points with the index of +1 correspond to concentrations of magnetic flux, and those with the index of -1 to the saddle points in the plane. All of these singular points are connected by the lanes demarcating the 2-D magnetic cells in the plane. It has been confirmed that these saddle points are the intersections between separators and planes intersecting the 3-D fields. From comparisons between kernels of flares occurring in both regions and the saddle points, we found that there is a close morphological relationship between distributions of the saddle points and flare kernels. The main results are as follows: (a) The flare kernels tend to appear in areas with concentrating 2-D saddle points. (b) The morphology of the kernels is exactly confined by the lanes in the plane at chromospheric level. These facts seem favourable for the viewpoint that solar flares are closely related to magnetic separatrices and separators.     

Filamentary structure in an objectively-derived deep galaxy sample?

The deep galaxy sample of MacGillivray & Dodd (1980), obtained from purely objective means, is investigated using the technique of Kuhn & Uson (1982) for the presence of structure of a filamentary nature. A variety of synthetic fields of galaxies (including both ‘filament’ and ‘non-filament’ models) generated by means of a computer simulation technique have also been studied for comparison purposes. No strong evidence for filamentary structure in the galaxy distribution is found for this deep sample.     

One-dimensional electric field structure of an outer gap accelerator – III. Location of the gap and the gamma-ray spectrum

We investigate a stationary particle acceleration zone in the outer magnetosphere of an obliquely rotating neutron star. The charge depletion as a result of global current causes a large electric field along the magnetic field lines. Migratory electrons and/or positrons are accelerated by this field to radiate curvature gamma-rays, some of which collide with the X-rays to materialize as pairs in the gap. As a result of this pair-production cascade, the replenished charges partially screen the electric field, which is self-consistently solved together with the distribution of particles and gamma-rays. If no current is injected at either of the boundaries of the accelerator, the gap is located around the so-called null surface, where the local Goldreich–Julian charge density vanishes. However, we find that the gap position shifts outwards (or inwards) when particles are injected at the inner (or outer) boundary. We apply the theory to the seven pulsars whose X-ray fields are known from observations. We show that the gap should be located near to or outside of the null surface for the Vela pulsar and PSR B1951+32, so that their expected GeV spectrum may be consistent with observations. We then demonstrate that the intrinsically large TeV flux from the outer gap of PSR B0540–69 is absorbed by the magnetospheric infrared photons, causing it to be undetectable. We also point out that the electrodynamic structure and the resultant GeV emission properties of millisecond pulsars are similar to young pulsars.     

Topology of Magnetic Field and Coronal Heating in Solar Active Regions

Force-free magnetic fields can be computed by making use of a new numerical technique, in which the fields are represented by a boundary integral equation based on a specific Green's function. Vector magnetic fields observed on the photospheric surface can be taken as the boundary conditions of this equation. In this numerical computation, the following two points are emphasized: (1) A new method for data reduction is proposed, for removing uncertainties in boundary data and determining the parameter in this Green's function, which is important for solving the boundary integral equation. In this method, the transverse components of the observed boundary field are calibrated with a linear force-free field model without changing their azimuth. (2) The computed 3-D fields satisfy the divergence-free and force-free conditions with high precision. The alignment of these field lines is mostly in agreement with structures in Hα and Yohkoh soft X-ray images. Since the boundary data are calibrated with a linear force-free field model, the computed 3-D magnetic field can be regarded as a quasi-linear force-free field approximation. The reconstruction of 3-D magnetic field in active region NOAA 7321 was taken as an example to quantitatively exhibit the capability of our new numerical technique.     

Non‐holonomic filamentary dynamos as generalized Arnold's maps in Riemannian space

A filamentary non‐holonomic dynamo solution of self‐induction magnetic field equation is found by considering highly conducting filaments. It is shown that planar filaments cannot support dynamo action since the flow along the filament vanishes for torsion‐free filaments. This is a generalization of the Zeldovich theorem for linear magnetic dynamo filaments. The flow of filament is proportionally to the product between Frenet torsion and curvature. This shows that filamentary dynamos must possess Frenet torsion. A well‐known example of this result is the α ‐dynamo in solar physics. Magnetic helicity and magnetic energy for this filamentary dynamo are computed. Magnetic helicity vanishes by construction and the magnetic field decays with torsion energy in helicoidal dynamos. The approach considered here is useful for the investigation of anisotropic turbulent cascades. As a particular simple example it is shown that under certain constraints the solution can be reduced to the Arnold cat dynamo map solution where the non‐holonomic directional mixed derivative, would play the role of the Lyapunov exponent which appears on stretching the magnetic field in Riemannian space. The solution seems to describe marginal slow dynamos when the velocities involved in the dynamo flows are constants. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Formation pumping of molecular hydrogen in the Messier 17 photodissociation region

We have imaged the emission from the near-infrared   v =1–0  S(1), 1–0 S(7), 2–1 S(1) and 6–4 O(3) lines of molecular hydrogen in the Northern and South Western Bars of M17, together with the hydrogen Br γ and Br10 lines. This includes the first emission-line image ever to be obtained of a line from the highly excited   v =6  level of molecular hydrogen. In both Bars, the H₂ emission is generally distributed in clumps along filamentary features. The 1–0 S(1) and 2–1 S(1) images have similar morphologies. Together with their relative line ratios, this supports a fluorescent origin for their emission, within a photodissociation region. The SW-Bar contains a clumpy medium, but in the N-Bar the density is roughly constant. The 1–0 S(7) line image is also similar to the 1–0 S(1) image, but the 6–4 O(3) image is significantly different from it. Since the emission wavelengths of these two lines are similar (1.748 to 1.733 μm), this cannot be due to differential extinction between the   v =6  and the   v =1  lines. We attribute the difference to the pumping of newly formed H₂ into the   v =6  , or to a nearby, level. However, this also requires a time-dependent photodissociation region (where molecule formation does not balance dissociation), rather than it to be in steady state, and/or for the formation spectrum to vary with position in the source. If this interpretation of formation pumping of molecular hydrogen is correct, it is the first clear signature from this process to be seen.     

Flux linkages of bipolar sunspot groups: A computer study

Past studies of the structure of solar magnetic fields have used magnetograph data to compute selected field lines for comparison with the morphology of structures seen in various spectral wavelengths. While those analyses examine one of the integral properties of magnetic fields (field lines), they are not complete since they fail to determine the other important integral property: the boundaries of the flux of field lines of given connectivity. In the present analysis we determine such a system of boundaries, called separatrices, for the current free field of two p-f spot pairs so as to exhibit the line of self-intersection, called the separator. The analysis is compared with previous analytical work. These computer results, confirming earlier studies carried out using iron fillings, show that the separatrix has the form of two intersecting ovoids, defining four flux cells. New features which have emerged from this study include the observation that the projections of the separatrix in a plane perpendicular to the separator at its highest point do not intersect at 90° as has been widely believed, but rather closer to 60° in the case studied. The separator is very nearly circular over most of its length. The two neutral points (B = 0) which appear at the photospheric ends of the separator have the mixed radial-hyperbolic form as expected, a feature requiring every field line lying on the separatrix to connect with at least one of the two neutral points. The rotation of line direction with height (shear) is graphically illustrated in the potential field case studied here. We also exhibit a magnetic arcade.     

Non-Gaussian cosmic microwave background temperature fluctuations from peculiar velocities of clusters

We use numerical simulations of a (480 Mpc  h ⁻¹)³ volume to show that the distribution of peak heights in maps of the temperature fluctuations from the kinematic and thermal Sunyaev–Zeldovich (SZ) effects will be highly non-Gaussian, and very different from the peak-height distribution of a Gaussian random field. We then show that it is a good approximation to assume that each peak in either SZ effect is associated with one and only one dark matter halo. This allows us to use our knowledge of the properties of haloes to estimate the peak-height distributions. At fixed optical depth, the distribution of peak heights resulting from the kinematic effect is Gaussian, with a width that is approximately proportional to the optical depth; the non-Gaussianity comes from summing over a range of optical depths. The optical depth is an increasing function of halo mass and the distribution of halo speeds is Gaussian, with a dispersion that is approximately independent of halo mass. This means that observations of the kinematic effect can be used to put constraints on how the abundance of massive clusters evolves, and on the evolution of cluster velocities. The non-Gaussianity of the thermal effect, on the other hand, comes primarily from the fact that, on average, the effect is larger in more massive haloes, and the distribution of halo masses is highly non-Gaussian. We also show that because haloes of the same mass may have a range of density and velocity dispersion profiles, the relation between halo mass and the amplitude of the thermal effect is not deterministic, but has some scatter.     

Compound diffusion of energetic particles: a Kappa model for the parallel distribution function

One can assume that energetic particles follow magnetic field lines while they propagate through a magnetized plasma. The latter scenario is usually described by the so-called field line random walk limit. This limit, however, is only valid if parallel diffusion is suppressed. As soon as the latter effect is taken into account, perpendicular transport becomes subdiffusive. This physical scenario is usually called compound diffusion or compound subdiffusion and can be described by a Chapman-Kolmogorov equation. In the latter equation the parallel distribution function is an essential ingredient. In the present paper we replace the standard Gaussian model by a Kappa distribution to compute distribution functions and mean square displacements across the field.