A Helicity-Based Method to Infer the CME Magnetic Field Magnitude in Sun and Geospace: Generalization and Extension to Sun-Like and M-Dwarf Stars and Implications for Exoplanet Habitability

Patsourakos et al. (Astrophys. J. 817, 14, 2016) and Patsourakos and Georgoulis (Astron. Astrophys. 595, A121, 2016) introduced a method to infer the axial magnetic field in flux-rope coronal mass ejections (CMEs) in the solar corona and farther away in the interplanetary medium. The method, based on the conservation principle of magnetic helicity, uses the relative magnetic helicity of the solar source region as input estimates, along with the radius and length of the corresponding CME flux rope. The method was initially applied to cylindrical force-free flux ropes, with encouraging results. We hereby extend our framework along two distinct lines. First, we generalize our formalism to several possible flux-rope configurations (linear and nonlinear force-free, non-force-free, spheromak, and torus) to investigate the dependence of the resulting CME axial magnetic field on input parameters and the employed flux-rope configuration. Second, we generalize our framework to both Sun-like and active M-dwarf stars hosting superflares. In a qualitative sense, we find that Earth may not experience severe atmosphere-eroding magnetospheric compression even for eruptive solar superflares with energies \({\approx}\, 10^{4}\) times higher than those of the largest Geostationary Operational Environmental Satellite (GOES) X-class flares currently observed. In addition, the two recently discovered exoplanets with the highest Earth-similarity index, Kepler 438b and Proxima b, seem to lie in the prohibitive zone of atmospheric erosion due to interplanetary CMEs (ICMEs), except when they possess planetary magnetic fields that are much higher than that of Earth.     

Differential Rotation of the Solar Corona from Magnetic Field Data

A method for investigating the differential rotation of the solar corona using the coronal magnetic field as a tracer is proposed. The magnetic field is calculated in the potential approximation from observational data at the photospheric level. The time interval from June 24, 1976, to December 31, 2004, is considered. The magnetic field has been calculated for all latitudes from the equator to ±75? with a 5? step at distances from the base of the corona 1.0 R_⊙ to 2.45 R_⊙ near the source surface. The coronal rotation periods at 14 distances from the solar center have been determined by the method of periodogram analysis. The coronal rotation is shown to become progressively less differential with increasing heliocentric distance; it does not become rigid even near the source surface. The change in the coronal rotation periods with time is considered. At the cycleminimumthe rotation has been found to bemost differential, especially at small distances from the solar center. The change in coronal rotation with time is consistent with the tilt of the solar magnetic equator. The results from the magnetic field are compared with those obtained from the brightness of the green coronal Fe XIV 530.3 nm line. The consistency between these results confirms the reliability of the proposed method for studying the coronal rotation. Studying the rotation of the coronal magnetic field gives hope for the possibility of using this method to diagnose the differential rotation in subphotospheric layers.     

Relation of CME Speed and Magnetic Helicity in CME Source Regions on the Sun during the Early Phase of Solar Cycles 23 and 24

To investigate the relations between coronal mass ejection (CME) speed and magnetic field properties measured in the photospheric surface of CME source regions, we selected 22 disk CMEs in the rising and early maximum phases of the current Solar Cycle 24. For the CME speed, we used two-dimensional (2D) projected speed observed by the Large Angle and Spectroscopic Coronagraph onboard the Solar and Heliospheric Observatory (SOHO/LASCO), as well as a 3D speed calculated from the triangulation method using multi-point observations. Two magnetic parameters of CME source regions were considered: the average of magnetic helicity injection rate and the total unsigned magnetic flux. We then classified the selected CMEs into two groups, showing: i) a monotonically increasing pattern with one sign of helicity (group A: 16 CMEs) and ii) a pattern of significant helicity injection followed by its sign reversal (group B: 6 CMEs). We found that: 1) 3D speed generally shows better correlations with the magnetic parameters than the 2D speed for 22 CME events in Solar Cycle 24; 2) 2D speed and the magnetic parameters of 22 CME events in this solar cycle have lower values than those of 47 CME events in Solar Cycle 23; 3) all events of group B in Solar Cycle 24 occur only after the beginning of the maximum phase, a trend well consistent with that shown in Solar Cycle 23; 4) the 2D speed and the helicity parameter of group B events continue to increase in the declining phase of Solar Cycle 23, while those of group A events abruptly decrease in the same period. Our results indicate that the two CME groups have a different tendency in the solar cycle variations of CME speed and the helicity parameters. Active regions that show a complex helicity evolution pattern tend to appear in the maximum and declining phases, while active regions with a relatively simple helicity evolution pattern appear throughout the whole solar cycle.     

Abnormal Stokes Profiles of the Photospheric Lines in the Region of Chromospheric Dual Flows in the Surroundings of a Solar Pore: 1. Observations

The results of the analysis of the full Stokes profiles of the photospheric lines Fe I λ 630.15 nm and Fe I λ 630.25 nm in a region of chromospheric dual flows appearance in the vicinity of a small pore are presented. The analysis is based on the spectropolarimetric observations of the active region NOAA 11024 with the THEMIS French–Italian telescope (Tenerife Island, Spain). The temporal variations in the high-resolution Stokes parameters I, Q, U, and V were considered for each pixel. It was found that the dual chromospheric flows appeared in the region of the abnormal Stokes profiles of the photospheric lines. Most of the Stokes profiles Q, U, and V have a complex shape and vary greatly from pixel to pixel, which indicates strong inhomogeneities in the structure of the magnetic field in that region. The amplitude and shape of the Stokes profiles were rapidly changing during the observations. A change in the polarity of the photospheric magnetic field took place during the observations in the region of a bright chromospheric point. The evidence of the emergence of a new small-scale magnetic flux of the opposite polarity is obtained; this could lead to magnetic reconnections, appearance of dual chromospheric flows, and occurrence of a microflare.     

Probing the Quiet Solar Atmosphere from the Photosphere to the Corona

We investigate the morphology and temporal variability of a quiet-Sun network region in different solar layers. The emission in several extreme ultraviolet (EUV) spectral lines through both raster and slot time-series, recorded by the EUV Imaging Spectrometer (EIS) on board the Hinode spacecraft is studied along with \(\mbox{H}\upalpha\) observations and high-resolution spectropolarimetric observations of the photospheric magnetic field. The photospheric magnetic field is extrapolated up to the corona, showing a multitude of large- and small-scale structures. We show for the first time that the smallest magnetic structures at both the network and internetwork contribute significantly to the emission in EUV lines, with temperatures ranging from \(8\times 10^{4}~\mbox{K}\) to \(6\times 10^{5}~\mbox{K}\). Two components of transition region emission are present, one associated with small-scale loops that do not reach coronal temperatures, and another component that acts as an interface between coronal and chromospheric plasma. Both components are associated with persistent chromospheric structures. The temporal variability of the EUV intensity at the network region is also associated with chromospheric motions, pointing to a connection between transition region and chromospheric features. Intensity enhancements in the EUV transition region lines are preferentially produced by \(\mbox{H}\upalpha\) upflows. Examination of two individual chromospheric jets shows that their evolution is associated with intensity variations in transition region and coronal temperatures.     

Understanding the Internal Magnetic Field Configurations of ICMEs Using More than 20 Years of <Emphasis Type="Italic">Wind</Emphasis> Observations

The magnetic topology, structure, and geometry of the magnetic obstacles embedded within interplanetary coronal mass ejections (ICMEs) are not yet fully and consistently described by in situ models and reconstruction techniques. The main goal of this work is to better understand the status of the internal magnetic field of ICMEs and to explore in situ signatures to identify clues to develop a more accurate and reliable in situ analytical models. We take advantage of more than 20 years of Wind observations of transients at 1 AU to compile a comprehensive database of ICMEs through three solar cycles, from 1995 to 2015. The catalog is publicly available at wind.gsfc.nasa.gov and is fully described in this article. We identify and collect the properties of 337 ICMEs, of which 298 show organized magnetic field signatures. To allow for departures from idealized magnetic configurations, we introduce the term “magnetic obstacle” (MO) to signify the possibility of more complex configurations. To quantify the asymmetry of the magnetic field strength profile within these events, we introduce the distortion parameter (DiP) and calculate the expansion velocity within the magnetic obstacle. Circular-cylindrical geometry is assumed when the magnetic field strength displays a symmetric profile. We perform a statistical study of these two parameters and find that only 35% of the events show symmetric magnetic profiles and a low enough expansion velocity to be compatible with the assumption of an idealized cylindrical static flux rope, and that 41% of the events do not show the expected relationship between expansion and magnetic field compression in the front, with the maximum magnetic field closer to the first encounter of the spacecraft with the magnetic obstacle; 18% show contractions (i.e. apparent negative expansion velocity), and 30% show magnetic field compression in the back. We derive an empirical relation between DiP and expansion velocity that is the first step toward improving reconstructions with possible applications to space weather studies. In summary, our main results demonstrate that the assumed correlation between expanding structure and asymmetric magnetic field is not always valid. Although 59% of the cases could be described by circular-cylindrical geometry, with or without expansion, the remaining cases show significant in situ signatures of departures from circular-cylindrical geometry. These results will aid in the development of more accurate in situ models to reconcile image.     

High-Resolution Vector Magnetograms of the Sun’s Poles from Hinode: Flux Distributions and Global Coronal Modeling

The Sun’s polar fields play a leading role in structuring the large-scale solar atmosphere and in determining the interplanetary magnetic field. They are also believed to supply the seed field for the subsequent solar activity cycle. However, present-day synoptic observations do not have sufficient spatial resolution or sensitivity to diagnose accurately the high-latitude magnetic vector field. The high spatial resolution and sensitivity of the full-Stokes observations from the Hinode Solar Optical Telescope Spectro-Polarimeter, observing the poles long-term, allows us to build up a detailed picture of the Cycle 24 polar field reversal, including the changing latitude distribution of the high-latitude flux, and to study the effect on global coronal field models. The Hinode observations provide detailed information on the dominant facular-scale magnetic structure of the polar fields, and their field inclination and flux distribution. Hybrid synoptic magnetograms are constructed from Hinode polar measurements and full-disk magnetograms from the Synoptic Optical Long-term Investigations of the Sun (SOLIS) Vector Spectro-Magnetograph (VSM), and coronal potential field models are calculated. Loss of effective spatial resolution at the highest latitudes presents complications. Possible improvements to synoptic polar data are discussed.     

Diagnostics of Coronal Bright Points using IRIS,AIA, and HMI Observations

We perform the detailed imaging and spectroscopic analysis of two coronal bright points (CBPs). These CBPs are dominated by bright dots or elongated bright features. Their rapid temporal variations lead to a continuous change in their overall morphology at chromospheric and transition-region (TR) temperatures. A 3D potential magnetic field extrapolation predicts the dominance of magnetic loops in the extent of both CBPs, which are clearly visible at the Si iv 1393.75 Å line formation temperature. Short, low-lying magnetic loops or loop segments are the integral parts of these CBPs at TR temperature. A correlation between the various parameters of Mg ii resonance lines (e.g. intensity, Doppler velocity, velocity gradient) is present in the region of magnetic loops or loop segments. However, a quiet-Sun (QS) region does not show any correlation. Doppler velocities as well as the full width at half maximum (FWHM) of these lines are very prominent in the magnetic loops and loop segments compared to the Doppler velocities and FWHM in the QS region. Higher red-shifts and FWHM at TR temperatures are directly related to the dominance of the energy release process in these regions in the framework of the nanoflare model. A magnetogram from the Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamics Observatory (SDO) reveals the existence of two opposite magnetic polarities in the extent of both CBPs, which is a very well established result. We find that one CBP is formed by the convergence of two opposite magnetic polarities, while the other is triggered by the emergence of a new magnetic field prior to the onset of this CBP.     

An Early Diagnostics of the Geoeffectiveness of Solar Eruptions from Photospheric Magnetic Flux Observations: The Transition from SOHO to SDO

In our previous articles (Chertok et al. in Solar Phys. 282, 175, 2013; Chertok et al. in Solar Phys. 290, 627, 2015), we presented a preliminary tool for the early diagnostics of the geoeffectiveness of solar eruptions based on the estimate of the total unsigned line-of-sight photospheric magnetic flux in accompanying extreme ultraviolet (EUV) arcades and dimmings. This tool was based on the analysis of eruptions observed during 1996?–?2005 with the Extreme-ultraviolet Imaging Telescope (EIT) and the Michelson Doppler Imager (MDI) onboard the Solar and Heliospheric Observatory (SOHO). Empirical relationships were obtained to estimate the probable importance of upcoming space weather disturbances caused by an eruption, which just occurred, without data on the associated coronal mass ejections. In particular, it was possible to estimate the intensity of a non-recurrent geomagnetic storm (GMS) and Forbush decrease (FD), as well as their onset and peak times. After 2010?–?2011, data on solar eruptions are obtained with the Atmospheric Imaging Assembly (AIA) and the Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamics Observatory (SDO). We use relatively short intervals of overlapping EIT–AIA and MDI–HMI detailed observations, and additionally, a number of large eruptions over the next five years with the 12-hour cadence EIT images to adapt the SOHO diagnostic tool to SDO data. We show that the adopted brightness thresholds select practically the same areas of arcades and dimmings from the EIT 195 Å and AIA 193 Å image, with a cross-calibration factor of 3.6?–?5.8 (5.0?–?8.2) for the AIA exposure time of 2.0 s (2.9 s). We also find that for the same photospheric areas, the MDI line-of-sight magnetic flux systematically exceeds the HMI flux by a factor of 1.4. Based on these results, the empirical diagnostic relationships obtained from SOHO data are adjusted to SDO instruments. Examples of a post-diagnostics based on SDO data are presented. As before, the tool is applicable to non-recurrent GMSs and FDs caused by nearly central eruptions from active regions, provided that the southern component of the interplanetary magnetic field near the Earth is predominantly negative, which is not predicted by this tool.     

Determining the Intrinsic CME Flux Rope Type Using Remote-sensing Solar Disk Observations

A key aim in space weather research is to be able to use remote-sensing observations of the solar atmosphere to extend the lead time of predicting the geoeffectiveness of a coronal mass ejection (CME). In order to achieve this, the magnetic structure of the CME as it leaves the Sun must be known. In this article we address this issue by developing a method to determine the intrinsic flux rope type of a CME solely from solar disk observations. We use several well-known proxies for the magnetic helicity sign, the axis orientation, and the axial magnetic field direction to predict the magnetic structure of the interplanetary flux rope. We present two case studies: the 2 June 2011 and the 14 June 2012 CMEs. Both of these events erupted from an active region, and despite having clear in situ counterparts, their eruption characteristics were relatively complex. The first event was associated with an active region filament that erupted in two stages, while for the other event the eruption originated from a relatively high coronal altitude and the source region did not feature a filament. Our magnetic helicity sign proxies include the analysis of magnetic tongues, soft X-ray and/or extreme-ultraviolet sigmoids, coronal arcade skew, filament emission and absorption threads, and filament rotation. Since the inclination of the post-eruption arcades was not clear, we use the tilt of the polarity inversion line to determine the flux rope axis orientation and coronal dimmings to determine the flux rope footpoints, and therefore, the direction of the axial magnetic field. The comparison of the estimated intrinsic flux rope structure to in situ observations at the Lagrangian point L1 indicated a good agreement with the predictions. Our results highlight the flux rope type determination techniques that are particularly useful for active region eruptions, where most geoeffective CMEs originate.     

On Flare-CME Characteristics from Sun to Earth Combining Remote-Sensing Image Data with <Emphasis Type="Italic">In Situ</Emphasis> Measurements Supported by Modeling

We analyze the well-observed flare and coronal mass ejection (CME) from 1 October 2011 (SOL2011-10-01T09:18) covering the complete chain of effects – from Sun to Earth – to better understand the dynamic evolution of the CME and its embedded magnetic field. We study in detail the solar surface and atmosphere associated with the flare and CME using the Solar Dynamics Observatory (SDO) and ground-based instruments. We also track the CME signature off-limb with combined extreme ultraviolet (EUV) and white-light data from the Solar Terrestrial Relations Observatory (STEREO). By applying the graduated cylindrical shell (GCS) reconstruction method and total mass to stereoscopic STEREO-SOHO (Solar and Heliospheric Observatory) coronagraph data, we track the temporal and spatial evolution of the CME in the interplanetary space and derive its geometry and 3D mass. We combine the GCS and Lundquist model results to derive the axial flux and helicity of the magnetic cloud (MC) from in situ measurements from Wind. This is compared to nonlinear force-free (NLFF) model results, as well as to the reconnected magnetic flux derived from the flare ribbons (flare reconnection flux) and the magnetic flux encompassed by the associated dimming (dimming flux). We find that magnetic reconnection processes were already ongoing before the start of the impulsive flare phase, adding magnetic flux to the flux rope before its final eruption. The dimming flux increases by more than 25% after the end of the flare, indicating that magnetic flux is still added to the flux rope after eruption. Hence, the derived flare reconnection flux is most probably a lower limit for estimating the magnetic flux within the flux rope. We find that the magnetic helicity and axial magnetic flux are lower in the interplanetary space by ～?50% and 75%, respectively, possibly indicating an erosion process. A CME mass increase of 10% is observed over a range of \({\sim}\,4\,\mbox{--}\,20~\mathrm{R}_{\odot }\). The temporal evolution of the CME-associated core-dimming regions supports the scenario that fast outflows might supply additional mass to the rear part of the CME.     

Spectropolarimetric investigation of an Ellerman bomb: 2. Photospheric models

Semiempirical models of the photosphere of an Ellerman bomb in the NOAA 11024 active region were obtained using profiles of Stokes parameters I, Q, U, and V of photospheric lines. Spectropolarimetric observations were conducted using the French–Italian THEMIS telescope (Tenerife, Spain). The SIR inversion code [28] was used in the modeling. The models have two components: a magnetic flux tube and nonmagnetic surroundings. The dependences of temperature, magnetic field strength, inclination of the magnetic field vector, and line-of-sight velocity in the tube on the optical depth were obtained. The models demonstrate that the thermodynamic parameters of the Ellerman bomb photosphere differ considerably from those of the quiet photosphere. The temperature in the tube model varied nonmonotonically with height and deviated by up to 700–900 K from its values for the quiet photosphere. Downflows were observed in the lower and the upper photospheric layers. The line-of-sight velocity in the upper layers of the photosphere was as high as 17 km/s. The magnetic field strength in the models varied from 0.1–0.13 T in the lower photospheric layers to 0.04–0.07 T in the upper ones. The physical state of the photosphere did change in the course of observations.     

Coronal Holes and Open Magnetic Flux over Cycles 23 and 24

As the observational signature of the footprints of solar magnetic field lines open into the heliosphere, coronal holes provide a critical measure of the structure and evolution of these lines. Using a combination of Solar and Heliospheric Observatory/Extreme ultraviolet Imaging Telescope (SOHO/EIT), Solar Dynamics Observatory/Atmospheric Imaging Assembly (SDO/AIA), and Solar Terrestrial Relations Observatory/Extreme Ultraviolet Imager (STEREO/EUVI A/B) extreme ultraviolet (EUV) observations spanning 1996?–?2015 (nearly two solar cycles), coronal holes are automatically detected and characterized. Coronal hole area distributions show distinct behavior in latitude, defining the domain of polar and low-latitude coronal holes. The northern and southern polar regions show a clear asymmetry, with a lag between hemispheres in the appearance and disappearance of polar coronal holes.     

Statistical Properties of Ribbon Evolution and Reconnection Electric Fields in Eruptive and Confined Flares

A statistical study of the chromospheric ribbon evolution in H\(\alpha\) two-ribbon flares was performed. The data set consists of 50 confined (62%) and eruptive (38%) flares that occurred from June 2000 to June 2015. The flares were selected homogeneously over the H\(\alpha\) and Geostationary Operational Environmental Satellite (GOES) classes, with an emphasis on including powerful confined flares and weak eruptive flares. H\(\alpha\) filtergrams from the Kanzelhöhe Observatory in combination with Michelson Doppler Imager (MDI) and Helioseismic and Magnetic Imager (HMI) magnetograms were used to derive the ribbon separation, the ribbon-separation velocity, the magnetic-field strength, and the reconnection electric field. We find that eruptive flares reveal statistically larger ribbon separation and higher ribbon-separation velocities than confined flares. In addition, the ribbon separation of eruptive flares correlates with the GOES SXR flux, whereas no clear dependence was found for confined flares. The maximum ribbon-separation velocity is not correlated with the GOES flux, but eruptive flares reveal on average a higher ribbon-separation velocity (by ≈?10 km?s^?1). The local reconnection electric field of confined (\(cc=0.50 \pm0.02\)) and eruptive (\(cc=0.77 \pm0.03\)) flares correlates with the GOES flux, indicating that more powerful flares involve stronger reconnection electric fields. In addition, eruptive flares with higher electric-field strengths tend to be accompanied by faster coronal mass ejections.     

Fine Structure of the Core of the Blazar OJ 287. II. Wavelength 2 cm

We have continued our studies of the fine structure of the active region in the blazar OJ 287 at wavelength λ = 2cm with a resolution of 20 μas, the epochs of 1995–2017. We have identified fragments of two arms along which the surrounding plasma comes to the nozzle. The brightness temperature of the flows rises as the nozzle is approached to T_b ? 10¹² K. The high-velocity bipolar outflow surrounded by lowvelocity components carries away an excess angular momentum as it is accumulated. The high collimation and helicity of the flows are determined by rotation and precession, respectively. Ring currents responsible for the longitudinal magnetic fields are excited in the flows. The jet and counterjet are a mirror reflection of each other; the difference in sizes is determined by the acceleration/deceleration of the flows along/opposite to the magnetic field. The velocity of the high-velocity outflow is v ? 0.06 c. The brightness temperature of the nozzle reaches T_b ? 10¹⁴ K. The spectral index of the southern and northern nozzles is α ≈ 0.66 and ≈0.4, respectively; the difference is determined by absorption in the bulge. The separation between the nozzles is 12 μas or 0.05 pc. The central region of reduced brightness with a diameter ? ≈ 3.6 pc corresponds to the bulge inclined toward the jet at an angle of 65° to the plane of the sky. The counterjet is ejected toward the observer; the jet is ejected in the opposite direction and is visible outside the bulge from a distance of 1.5 pc. The structure and kinematics of the bulge correspond to a vortex nature. An enhanced supply of matter from the northern arm in the middle of 2000 increased the activity of the low-velocity nozzle. A secondary vortex located at a distance of 0.28 mas (1.3 pc) was formed. The high-velocity flow is ejected in a direction of ?110°.     

Stars with Discrepant <Emphasis Type="Italic">v</Emphasis> sin <Emphasis Type="Italic">i</Emphasis> as Derived from the Ca II 3933 and Mg II 4481 Å Lines. VII. HD9531 (SB), HD31592 (SB2), HD129174 (SB?)

In this paper of the series we analyze three stars listed among stars with discrepant v sin i: HD9531 and HD31592, which also show radial velocity variations inherent to spectroscopic binaries, and HD129174 which is an Mn-type star with a possible magnetic field. In HD9531 we confirm the radial velocity derived fromthe hydrogen lines as well as fromthe Ca II line at 3933 Å as variable. The profile of the calcium line also appears variable, and with the estimated magnetic induction B_e = ?630 ± 1340 G, this suggests that the abundance of calcium possibly varies over the surface of the star. We identified the lines of the secondary component in the spectrum of HD31592 revealing thus it is an SB2 binary with B9.5V and A0V components. While the primary star rotates with v sin i = 50 km s^?1, the secondary star is faster with v sin i = 170 km s^?1. We find that only 60% of the Mn lines identified in the spectrum of HD129174 can be fitted with a unique abundance value, whereas the remaining lines are stronger or fainter. We also identified two Xe II lines at 5339.33 Å and 5419.15 Å and estimated their log g f.     

The Formation of a Small-Scale Filament After Flux Emergence on the Quiet Sun

We present observations of the formation process of a small-scale filament on the quiet Sun during 5?–?6 February 2016 and investigate its formation cause. Initially, a small dipole emerged, and its associated arch filament system was found to reconnect with overlying coronal fields accompanied by numerous extreme ultraviolet bright points. When the bright points faded, many elongated dark threads formed and bridged the positive magnetic element of the dipole and the external negative network fields. Interestingly, an anticlockwise photospheric rotational motion (PRM) set in within the positive endpoint region of the newborn dark threads following the flux emergence and lasted for more than 10 hours. Under the drive of the PRM, these dispersive dark threads gradually aligned along the north-south direction and finally coalesced into an inverse S-shaped filament. Consistent with the dextral chirality of the filament, magnetic helicity calculations show that an amount of negative helicity was persistently injected from the rotational positive magnetic element and accumulated during the formation of the filament. These observations suggest that twisted emerging fields may lead to the formation of the filament via reconnection with pre-existing fields and release of its inner magnetic twist. The persistent PRM might trace a covert twist relaxation from below the photosphere to the low corona.     

Algorithms of the Potential Field Calculation in a Three-Dimensional Box

The potential field inside a three-dimensional box with the normal magnetic field component given on all boundaries needs to be calculated to estimate important quantities related to the magnetic field, such as free energy and relative helicity. In this work we present an analysis of three methods for calculating the potential field inside a three-dimensional box. The accuracy and performance of the methods are tested on artificial models with a priori known solutions.     

Magnetic field of CP stars in the Ori OB1 association. II. HD36540, HD36668, HD36916, HD37058

We present the results of magnetic field measurements of four chemically peculiar (CP) stars with helium abundance anomalies which are the members of the Orion stellar association OB1. The stars under study were classified as magnetic by other authors earlier. The present paper contains the results of the extensive study of the stars. Magnetic field measurements allowed us to conclude that HD36540 has a weak field and the longitudinal component B _e does not exceed 500 G. The longitudinal field of HD36668 varies with the period P = 2_. ^d 11884 and the amplitude from ?2 to +2 kG. The magnetic field of HD36916 has mainly negative polarity and varies within the range from 0 to ?1 kG with the period P = 1.d 565238. HD37058 is a magnetic star, the longitudinal field of which varies from ?1.2 to +0.8 kG with the period P = 14_. ^d 659. The B _e field variability pattern for the stars HD36916 and HD37058 is of a simple harmonic type. The longitudinal field of HD36668 is best described with two combined harmonic functions (“a doublewave”). The variability period of HD36540 is still undetermined. For all the stars from this paper, we measured radial velocities V _r, axial rotation rates v _e sin i, and determined basic parameters of atmospheres (effective temperatures T _eff and gravity acceleration log g). We also estimated masses M, luminosities L, and radii R of the stars.     

Orbit Orientation of the Planet in <Emphasis Type="Italic">τ</Emphasis> Boo (HD 120136)

We determined the rotation period and the parameters of the global magnetic field of τ Boo. This allowed us to estimate the inclination of the rotational axis of the star to the line of sight, as well as to obtain estimates of the inclination of the planet’s orbital plane to the stellar equator.