Error analysis regarding the calculation of nonlinear force-free field

Magnetic field extrapolation is an alternative method to study chromospheric and coronal magnetic fields. In this paper, two semi-analytical solutions of force-free fields (Low and Lou in Astrophys. J. 352:343, 1990) have been used to study the errors of nonlinear force-free (NLFF) fields based on force-free factor α. Three NLFF fields are extrapolated by approximate vertical integration (AVI) Song et al. (Astrophys. J. 649:1084, 2006), boundary integral equation (BIE) Yan and Sakurai (Sol. Phys. 195:89, 2000) and optimization (Opt.) Wiegelmann (Sol. Phys. 219:87, 2004) methods. Compared with the first semi-analytical field, it is found that the mean values of absolute relative standard deviations (RSD) of α along field lines are about 0.96–1.19, 0.63–1.07 and 0.43–0.72 for AVI, BIE and Opt. fields, respectively. While for the second semi-analytical field, they are about 0.80–1.02, 0.67–1.34 and 0.33–0.55 for AVI, BIE and Opt. fields, respectively. As for the analytical field, the calculation error of 〈|RSD|〉 is about 0.1∼0.2. It is also found that RSD does not apparently depend on the length of field line. These provide the basic estimation on the deviation of extrapolated field obtained by proposed methods from the real force-free field.     

Calculating and Testing Nonlinear Force-Free Fields

Improvements to an existing method for calculating nonlinear force-free magnetic fields (Wheatland, Solar Phys. 238, 29, 2006) are described. In particular a solution of the 3-D Poisson equation using 2-D Fourier transforms is presented. The improved nonlinear force-free method is demonstrated in application to linear force-free test cases with localized nonzero values of the normal component of the field in the boundary. These fields provide suitable test cases for nonlinear force-free calculations because the boundary conditions involve localized nonzero values of the normal components of the field and of the current density, and because (being linear force-free fields) they have more direct numerical solutions. Despite their simplicity, fields of this kind have not been recognized as test cases for nonlinear methods before. The examples illustrate the treatment of the boundary conditions on current in the nonlinear force-free method, and in particular the limitations imposed by field lines that connect outside of the boundary region.     

An Improved Approach to Non-Force-Free Coronal Magnetic Field Extrapolation

We develop an approach to deriving the three-dimensional non-force-free coronal magnetic field from vector magnetograms. Based on the principle of minimum dissipation rate, a general non-force-free magnetic field is expressed as the superposition of one potential field and two constant-α (linear) force-free fields. Each is extrapolated from its bottom boundary data, providing the normal component only. The constant-α parameters are distinct and determined by minimizing the deviations between the numerically computed and measured transverse magnetic field at the bottom boundary. The boundary conditions required are at least two layers of vector magnetograms, one at the photospheric level and the other at the chromospheric level, presumably. We apply our approach to a few analytic test cases, especially to two nonlinear force-free cases examined by Schrijver et al. (Solar Phys. 235, 161, 2006). We find that for one case with small α parameters, the quantitative measures of the quality of our result are better than the median values of those from a set of nonlinear force-free methods. The reconstructed magnetic-field configuration is valid up to a vertical height of the transverse scale. For the other cases, the results remain valid to a lower vertical height owing to the limitations of the linear force-free-field solver. Because our method is based on the fast-Fourier-transform algorithm, it is much faster and easy to implement. We discuss the potential usefulness of our method and its limitations.     

Numerical Simulation of a Solar Active Region. I: Bastille Day Flare

We present three-dimensional unsteady modeling and numerical simulations of a coronal active region, carried out within the compressible single-fluid MHD approximation. We focus on AR 9077 on 14 July 2000, and the triggering of the X5.7 GOES X-ray class “Bastille Day” flare. We simulate only the lower corona, although we include a virtual photosphere and chromosphere below. The boundary conditions at the base of this layer are set using temperature maps from line intensities and line-of-sight magnetograms (SOHO/MDI). From the latter, we generate vector magnetograms using the force-free approximation; these vector magnetograms are then used to produce the boundary condition on the velocity field using a minimum energy principle (Longcope, Astrophys. J. 612, 1181, 2004). The reconnection process is modeled through a dynamical hyper-resistivity which is activated when the current exceeds a critical value (Klimas et al., J. Geophys. Res. 109, 2218, 2004). Comparing the time series of X-ray fluxes recorded by GOES with modeled time series of various mean physical variables such as current density, Poynting energy flux, or radiative loss inside the active region, we can demonstrate that the model properly captures the evolution of an active region over a day and, in particular, is able to explain the initiation of the flare at the observed time.     

Magnetofrictional Extrapolations of Low and Lou’s Force-Free Equilibria

We present a careful investigation of the magnetofrictional relaxation and extrapolation technique applied to the reconstruction of two test fields. These fields are taken from the family of nonlinear force-free magnetic equilibria constructed by Low and Lou (Astrophys. J. 352, 343, 1990), which have emerged as standard tests for extrapolation techniques in recent years. For the practically relevant case that only the field values in the bottom plane of the considered volume (vector magnetogram) are used as input information (i.e., not including the knowledge about the test field at the side and top boundaries), the test field is reconstructed to a higher accuracy than obtained previously. Detailed diagnostics of the reconstruction accuracy show that the implementation of fourth-order spatial discretization was essential to reach this accuracy for the given test fields and to achieve near machine precision in satisfying the solenoidal condition. Different variants of boundary conditions are tested, which all yield comparable accuracy. In its present implementation, the technique yields a scaling of computing time with total number of grid points only slightly below N ^5/3, which is too steep for applications to large (≥1024²) magnetograms, except on supercomputers. Directions for improvement are outlined.     

A New Look at Mode Conversion in a Stratified Isothermal Atmosphere

Recent numerical investigations of wave propagation near coronal magnetic null points (McLaughlin and Hood: Astron. Astrophys. 459, 641, 2006) have indicated how a fast MHD wave partially converts into a slow MHD wave as the disturbance passes from a low-β plasma to a high-β plasma. This is a complex process and a clear understanding of the conversion mechanism requires the detailed investigation of a simpler model. An investigation of mode conversion in a stratified, isothermal atmosphere with a uniform, vertical magnetic field is carried out, both numerically and analytically. In contrast to previous investigations of upward-propagating waves (Zhugzhda and Dzhalilov: Astron. Astrophys. 112, 16, 1982a; Cally: Astrophys. J. 548, 473, 2001), this paper studies the downward propagation of waves from a low-β to high-β environment. A simple expression for the amplitude of the transmitted wave is compared with the numerical solution.     

Electron Densities in Quiescent Prominences Derived from Eclipse Observations

We develop a diagnostic tool for determination of the electron densities in solar prominences using eclipse data. The method is based on analysis of the hydrogen Balmer-line intensities (namely Hα and Hβ) and the white-light emission due to Thomson scattering on the prominence electrons. Our approach represents a generalization of the ratio method already used by Koutchmy, Lebecq, and Stellmacher (Astron. Astrophys. 119, 261, 1983). In this paper we use an extended grid of non-LTE prominence models of Gouttebroze, Heinzel, and Vial (Astron. Astrophys. Suppl. Ser. 99, 513, 1993) and derive various useful relations between prominence radiation properties and electron densities. Simultaneously, an effective geometrical thickness of the prominence can also be obtained. As an example we apply our general technique to original eclipse data of Koutchmy, Lebecq, and Stellmacher (Astron. Astrophys. 119, 261, 1983). Finally, we use our results to determine the color of prominences as it should be seen during total eclipses.     

Do Long-Lived Features Really Exist in the Solar Photosphere? II. Contrast of Time-Averaged Granulation Images

The decrease in the rms contrast of time-averaged images with the averaging time is compared between four data sets: (1) a series of solar granulation images recorded at La Palma in 1993, (2) a series of artificial granulation images obtained in numerical simulations by Rieutord et al. (Nuovo Cimento 25, 523, 2002), (3) a similar series computed by Steffen and his colleagues (see Wedemeyer et al. in Astron. Astrophys. 44, 1121, 2004), (4) a random field with some parameters typical of the granulation, constructed by Rast (Astron. Astrophys. 392, L13, 2002). In addition, (5) a sequence of images was obtained from real granulation images by using a temporal and spatial shuffling procedure, and the contrast of the average of n images from this sequence as a function of n is analysed. The series (1) of real granulation images exhibits a considerably slower contrast decrease than do both the series (3) of simulated granulation images and the series (4) of random fields. Starting from some relatively short averaging times t, the behaviour of the contrast in series (3) and (4) resembles the t ^−1/2 statistical law, whereas the shuffled series (5) obeys the n ^−1/2 law from n=2 on. Series (2) demonstrates a peculiarly slow decline of contrast, which could be attributed to particular properties of the boundary conditions used in the simulations. Comparisons between the analysed contrast-variation laws indicate quite definitely that the brightness field of solar granulation contains a long-lived component, which could be associated with locally persistent dark intergranular holes and/or with the presence of quasi-regular structures. The suggestion that the random field (4) successfully reproduces the contrast-variation law for the real granulation (Rast in Astron. Astrophys. 392, L13, 2002) can be dismissed.     

Spike-like Bursts as Fine Structure of Zebras

We studied the characteristics of the zebra-associated spike-like bursts that were recorded with high time resolution at 1420 MHz in four intervals (from 12:45 to 12:48 UT) during 5 August 2003. Our detailed analysis is based on the selection of more than 500 such spike-like bursts and it is, at least to our knowledge, the first study devoted to such short-lived bursts. Their characteristics are different from those pertinent to “normal” spike bursts, as presented in the paper by Güdel and Benz (Astron. Astrophys. 231, 202, 1990); in particular, their duration (about 7.4 ms at half power) is shorter, so they should be members of the SSS (super short structures) family (Magdalenić et al., Astrophys. J. 642, L77, 2006). The bursts were generally strongly R-polarized; however, during the decaying part of interval I a low R-polarized and L-polarized bursts were also present. This change of polarization shows a trend that resembles the peculiar form of the zebra lines in the spectral dominion (“V” like). A global statistical analysis on the bursts observed in the two polarimetric channels shows that the highest cross-correlation coefficient (about 0.5) was pertinent to interval I. The zebras and the bursts can be interpreted by the same double plasma resonance process as proposed by Bárta and Karlicky (Astron. Astrophys. 379, 1045, 2001) and Karlicky et al. (Astron. Astrophys. 375, 638, 2001); in particular, the spikes are generated by the interruption of this process by assumed turbulence (density or magnetic field variations). This process should be present in the region close to the reconnection site (e.g., in the plasma reconnection outflows) where the density and the magnetic field vary strongly.     

On Signatures of Twisted Magnetic Flux Tube Emergence

Recent studies of NOAA active region 10953, by Okamoto et al. (Astrophys. J. Lett. 673, 215, 2008; Astrophys. J. 697, 913, 2009), have interpreted photospheric observations of changing widths of the polarities and reversal of the horizontal magnetic field component as signatures of the emergence of a twisted flux tube within the active region and along its internal polarity inversion line (PIL). A filament is observed along the PIL and the active region is assumed to have an arcade structure. To investigate this scenario, MacTaggart and Hood (Astrophys. J. Lett. 716, 219, 2010) constructed a dynamic flux emergence model of a twisted cylinder emerging into an overlying arcade. The photospheric signatures observed by Okamoto et al. (2008, 2009) are present in the model although their underlying physical mechanisms differ. The model also produces two additional signatures that can be verified by the observations. The first is an increase in the unsigned magnetic flux in the photosphere at either side of the PIL. The second is the behaviour of characteristic photospheric flow profiles associated with twisted flux tube emergence. We look for these two signatures in AR 10953 and find negative results for the emergence of a twisted flux tube along the PIL. Instead, we interpret the photospheric behaviour along the PIL to be indicative of photospheric magnetic cancellation driven by flows from the dominant sunspot. Although we argue against flux emergence within this particular region, the work demonstrates the important relationship between theory and observations for the successful discovery and interpretation of signatures of flux emergence.     

Constructing Semi-Empirical Sunspot Models for Helioseismology

One goal of helioseismology is to determine the subsurface structure of sunspots. In order to do so, it is important to understand first the near-surface effects of sunspots on solar waves, which are dominant. Here we construct simplified, cylindrically-symmetric sunspot models that are designed to capture the magnetic and thermodynamics effects coming from about 500 km below the quiet-Sun τ ₅₀₀₀=1 level to the lower chromosphere. We use a combination of existing semi-empirical models of sunspot thermodynamic structure (density, temperature, pressure): the umbral model of Maltby et al. (1986, Astrophys. J. 306, 284) and the penumbral model of Ding and Fang (1989, Astron. Astrophys. 225, 204). The OPAL equation-of-state tables are used to derive the sound-speed profile. We smoothly merge the near-surface properties to the quiet-Sun values about 1 Mm below the surface. The umbral and penumbral radii are free parameters. The magnetic field is added to the thermodynamic structure, without requiring magnetostatic equilibrium. The vertical component of the magnetic field is assumed to have a Gaussian horizontal profile, with a maximum surface field strength fixed by surface observations. The full magnetic-field vector is solenoidal and determined by the on-axis vertical field, which, at the surface, is chosen such that the field inclination is 45° at the umbral – penumbral boundary. We construct a particular sunspot model based on SOHO/MDI observations of the sunspot in active region NOAA 9787. The helioseismic signature of the model sunspot is studied using numerical simulations of the propagation of f, p ₁, and p ₂ wave packets. These simulations are compared against cross-covariances of the observed wave field. We find that the sunspot model gives a helioseismic signature that is similar to the observations.     

Enhanced Joule Heating in Umbral Dots

We present a study of magnetic fields in umbral dots (UDs) and its consequences on the Joule heating of the UDs. Hamedivafa (Astron. Astrophys. 407, 761, 2003) studied the Joule heating using the vertical component of the magnetic field. In this paper the magnetic field profile in the UDs is investigated by including a new azimuthal component of the magnetic field that might explain a relatively large enhancement of Joule heating causing higher brightness near the circumference of the UDs.     

Neutrino radiation of the AGN black holes

In the framework of ‘microscopic’ theory of black holes (J. Phys. Soc. Jpn. Suppl. B 70, 84, 2001; Astrophys. USSR 4, 659, 1996; 35, 335, 1991, 33, 143, 1990, 31, 345, 1989a; Astrophys. Space Sci. 1, 1992; Dokl. Akad. Nauk USSR 309, 97, 1989b), and references therein, we address the ‘pre-radiation time’ (PRT) of neutrinos from black holes, which implies the lapse of time from black hole’s birth till radiation of an extremely high energy neutrinos. For post-PRT lifetime, the black hole no longer holds as a region of spacetime that cannot communicate with the external universe. We study main features of spherical accretion onto central BH and infer a mass accretion rate onto it, and, further, calculate the resulting PRT versus bolometric luminosity due to accretion onto black hole. We estimate the PRTs of AGN black holes, with the well-determined masses and bolometric luminosities, collected from the literature by Woo Jong-Hak and Urry (Astrophys. J. 579, 530, 2002) on which this paper is partially based. The simulations for the black holes of masses M _BH ≃(1.1⋅10⁶ ÷4.2⋅10⁹) M _⊙ give the values of PRTs varying in the range of about T _BH ≃(4.3⋅10⁵ ÷5.6⋅10¹¹) yr. The derived PRTs for the 60 AGN black holes are longer than the age of the universe (∼13.7 Gyr) favored today. At present, some of remaining 174 BHs may radiate neutrinos. However, these results would be underestimated if the reservoir of gas for accretion in the galaxy center is quite modest, and no obvious way to feed the BHs with substantial accretion.     

Monte Carlo Simulation of Solar Active-Region Energy

A Monte Carlo approach to solving a stochastic-jump transition model for active-region energy (Wheatland and Glukhov: Astrophys. J. 494, 858, 1998; Wheatland: Astrophys. J. 679, 1621, 2008) is described. The new method numerically solves the stochastic differential equation describing the model, rather than the equivalent master equation. This has the advantages of allowing more efficient numerical solution, the modeling of time-dependent situations, and investigation of details of event statistics. The Monte Carlo approach is illustrated by application to a Gaussian test case and to the class of flare-like models presented in Wheatland (Astrophys. J. 679, 1621, 2008), which are steady-state models with constant rates of energy supply, and power-law distributed jump transition rates. These models have two free parameters: an index (δ), which defines the dependence of the jump transition rates on active-region energy, and a nondimensional ratio ( ) of total flaring rate to rate of energy supply. For the nondimensional mean energy of the active-region satisfies , resulting in a power-law distribution of flare events over many decades of energy. The Monte Carlo method is used to explore the behavior of the waiting-time distributions for the flare-like models. The models with δ≠0 are found to have waiting times that depart significantly from simple Poisson behavior when . The original model from Wheatland and Glukhov (Astrophys. J. 494, 858, 1998), with δ=0 (i.e., no dependence of transition rates on active-region energy), is identified as being most consistent with observed flare statistics.     

A new class of LRS Bianchi type-II dark energy models with variable EoS parameter

A new class of dark energy models in a Locally Rotationally Symmetric Bianchi type-II (LRS B-II) space-time with variable equation of state (EoS) parameter and constant deceleration parameter have been investigated in the present paper. The Einstein’s field equations have been solved by applying a variation law for generalized Hubble’s parameter given by Berman: Nuovo Cimento 74:182 (1983) which generates two types of solutions for the average scale factor, one is of power-law type and other is of the exponential-law form. Using these two forms, Einstein’s field equations are solved separately that correspond to expanding singular and non-singular models of the universe respectively. The dark energy EoS parameter ω is found to be time dependent and its existing range for both models is in good agreement with the three recent observations of (i) SNe Ia data (Knop et al.: Astrophys. J. 598:102 (2003)), (ii) SNe Ia data collaborated with CMBR anisotropy and galaxy clustering statistics (Tegmark et al.: Astrophys. J. 606:702 (2004)) and latest (iii) a combination of cosmological datasets coming from CMB anisotropies, luminosity distances of high redshift type Ia supernovae and galaxy clustering (Hinshaw et al.: Astrophys. J. Suppl. 180:225 (2009); Komatsu et al. Astrophys. J. Suppl. 180:330 (2009)). The cosmological constant Λ is found to be a positive decreasing function of time and it approaches a small positive value at late time (i.e. the present epoch) which is corroborated by results from recent supernovae Ia observations. The physical and geometric behaviour of the universe have also been discussed in detail.     

Fir continuum and line emissions from interstellar medium in galaxies

A self-consistent method has been evolved to infer physical parameters like density, radiation field and abundances using line and continuum radiations as diagnostics. For that purpose, we first calculate the temperatures of graphite and silicate grains using the model of Li and Draine (Astrophys. J. 554:778, 2001) by solving self-consistently the energy balance for G ₀ (1–10⁴) times the radiation field following Weingartner and Draine (Astrophys. J. Suppl. Ser. 134:263, 2001). Consequently, infrared emission fluxes are also obtained. To keep it simple, this is presented in the empirical form of parameters T _D and wavelength. The same model of the grain is adopted for photoelectric heating of gas using the formalism of Weingartner and Draine (Astrophys. J. Suppl. Ser. 134:263, 2001) (hereafter referred to as WD) and Bakes and Tielens (Astrophys. J. 427:822, 1994) (hereafter referred to as BT) for radiation field cited above in the range (6<hν≤13.6 eV). Temperature and abundances are determined using our own code for PDR very similar to cloudy code. All the possible sources of heating and cooling are considered for setting up the thermal balance. For the gas phase abundances that vary with depth in the cloud due to dust, self- and mutual shielding, chemical balance is solved. Most of the photoionization, photodissociation or chemical reaction rates are taken from UMIST database. We present an analysis of the cooling lines of singly ionized carbon [CII] at 158 μm and neutral oxygen [OI], at 63 μm and far infrared (FIR) continuum for a variety of star forming galaxies. Method of analysis of observational data is different from that of Malhotra et al. (Astrophys. J. 561:766, 2001). The radiation field G ₀, density N _h and abundance of carbon are obtained through best fit of observed and calculated intensities for lines and continuum radiations.     

Application of an Improved Method of Image Segmentation and Some Considerations on Identification and Tracking of Umbral Dots

An improved method of image segmentation is introduced. The object-tracking algorithm, originally developed by Sobotka, Brandt, and Simon (Astron. Astrophys. 328, 682, 1997) is modified with special attentions on splitting and merging of umbral dots (UDs), definition of the umbral boundary, and the birth-frames and the death-frames of UDs. By applying the new method of image segmentation and the object-tracking algorithm on a 67-min series of white-light images of a large pore (Sobotka et al., Astrophys. J. 511, 436, 1999), the physical characteristics of 20 “resolved” UDs with umbral origin were recorded. The most probable lifetime of the UDs is between 7 and 10 min. Umbral dots show a typical size of about 230 km. Their mean speeds are smaller than 2 km s⁻¹ with a distribution around a value less than 1 km s⁻¹. However, their average velocities are less than 0.8 km s⁻¹. Brighter (fainter) UDs are formed in the brighter (dimmer) region of the pore. There is no correlation between time-averaged area or time-averaged speeds and lifetimes. Also, the time-averaged peak intensities of UDs do not show any well-defined dependence on the corresponding time-averaged areas. It seems that there is a relation between average velocities of UDs and their time-averaged peak intensities, with brighter UDs moving more slowly.     

Are There Rings Around the Sun?

The Theory of Alfven drag (Drell et al. in J Geophys Res 70: 3131–3145 1965; Anselmo and Farinella in Icarus, 58, 182–185 1983) is applied here to show that the existence of a possible solar ring structure at a radial distance of 0.02 AU (~4R _⊙ , R _⊙  = radius of the sun) predicted by earlier authors (Brecher et al. in Nature 282, 50–52 1979; Rawal in Bull. Astr. Soc. India 6, 92–95 1978, Moon Planets 24, 407–414 1981, Moon Planets 31, 175–182 1984, J Astrophys Astr 10, 257–259 1989) may not survive Alfven drag produced during even moderate solar magnetic storms which take place from time to time through the age of the sun, but a possible solar ring structure at a radial distance of 0.13 AU (~27R _⊙ ) (Brecher et al. in Nature 282, 50–52 1979; Rawal in Bull. Astr. Soc. India 6, 92–95 1978, Moon Planets 24, 407–414 1981, Moon Planets 31, 175–182 1984, J Astrophys Astr 10, 257–259 1989) may survive intense Alfven drag produced during even strong magnetic storms of magnetic field value up to 1,000 G.     

String cosmological models in five dimensional bimetric theory of gravitation

Five-dimensional spherically symmetric space-time is considered in bimetric theory of gravitation formulated by Rosen (Gen. Rel. Grav. 4, 435, 1973) in the presence of cosmic string dust cloud. Exact cosmological models which represent geometric (Nambu) string, p-string (Takabayasi string) and Reddy string (Astrophys. Space Sci. 301, 2006) are obtained in the static and non-static cases. Some physical properties of the models are also discussed.     

Force-free magnetic fields in the solar atmosphere

Reliable measurements of the solar magnetic field are restricted to the level of the photosphere. For about half a century attempts have been made to calculate the field in the layers above the photosphere, i.e. in the chromosphere and in the corona, from the measured photospheric field. The procedure is known as magnetic field extrapolation. In the superphotospheric parts of active regions the magnetic field is approximately force-free, i.e. electric currents are aligned with the magnetic field. The practical application to solar active regions has been largely confined to constant-α or linear force-free fields, with a spatially constant ratio, α, between the electric current and the magnetic field. We review results obtained from extrapolations with constant-α force-free fields, in particular on magnetic topologies favourable for flares and on magnetic and current helicities. Presently, different methods are being developed to calculate non-constant-α or nonlinear force-free fields from photospheric vector magnetograms. We also briefly discuss these methods and present a comparison of a linear and a nonlinear force-free magnetic field extrapolation applied to the same photospheric boundary data. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)