Ratio of the Core to the Extended Emissions in the Comoving Frame for Blazars

In a two-component jet model, the emissions are the sum of the core and extended emissions: \(S^{\mathrm{ob}}=S_{\mathrm{core}}^{\mathrm{ob}}+S_{\mathrm{ext}}^{\mathrm{ob}}\), with the core emissions, \(S_{\mathrm{core}}^{\mathrm{ob}}= f S_{\mathrm{ext}}^{\mathrm{ob}}\delta ^{q}\) being a function of the Doppler factor \(\delta \), the extended emission \(S_{\mathrm{ext}}^{\mathrm{ob}}\), the jet type dependent factor q, and the ratio of the core to the extended emissions in the comoving frame, f. The f is an unobservable but important parameter. Following our previous work, we collect 65 blazars with available Doppler factor \(\delta \), superluminal velocity \(\beta _{\mathrm{app}}\), and core-dominance parameter, R, and calculated the ratio, f, and performed statistical analyses. We found that the ratio, f, in BL Lacs is on average larger than that in FSRQs. We suggest that the difference of the ratio f between FSRQs and BL Lacs is one of the possible reasons that cause the difference of other observed properties between them. We also find some significant correlations between \(\log f\) and other parameters, including intrinsic (de-beamed) peak frequency, \(\log \nu _{\mathrm{p}}^{\mathrm{in}}\), intrinsic polarization, \(\log P^{\mathrm{in}}\), and core-dominance parameter, \(\log R\), for the whole sample. In addition, we show that the ratio, f, can be estimated by R.     

The Na <Emphasis Type="SmallCaps">i</Emphasis> and Sr <Emphasis Type="SmallCaps">ii</Emphasis> Resonance Lines in Solar Prominences

We estimate the electron density, \(n_{\mathrm{e}}\), and its spatial variation in quiescent prominences from the observed emission ratio of the resonance lines Na?i?5890 Å (D₂) and Sr?ii?4078 Å. For a bright prominence (\(\tau_{\alpha}\approx25\)) we obtain a mean \(n_{\mathrm{e}}\approx2\times10^{10}~\mbox{cm}^{-3}\); for a faint one (\(\tau _{\alpha }\approx4\)) \(n_{\mathrm{e}}\approx4\times10^{10}~\mbox{cm}^{-3}\) on two consecutive days with moderate internal fluctuation and no systematic variation with height above the solar limb. The thermal and non-thermal contributions to the line broadening, \(T_{\mathrm{kin}}\) and \(V_{\mathrm{nth}}\), required to deduce \(n_{\mathrm{e}}\) from the emission ratio Na?i/Sr?ii cannot be unambiguously determined from observed widths of lines from atoms of different mass. The reduced widths, \(\Delta\lambda_{\mathrm{D}}/\lambda_{0}\), of Sr?ii?4078 Å show an excess over those from Na?D₂ and \(\mbox{H}\delta\,4101\) Å, assuming the same \(T_{\mathrm{kin}}\) and \(V_{\mathrm{nth}}\). We attribute this excess broadening to higher non-thermal broadening induced by interaction of ions with the prominence magnetic field. This is suggested by the finding of higher macro-shifts of Sr?ii?4078 Å as compared to those from Na?D₂.     

Contribution to the Solar Mean Magnetic Field from Different Solar Regions

Seven-year-long seeing-free observations of solar magnetic fields with the Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO) were used to study the sources of the solar mean magnetic field, SMMF, defined as the net line-of-sight magnetic flux divided over the solar disk area. To evaluate the contribution of different regions to the SMMF, we separated all the pixels of each SDO/HMI magnetogram into three subsets: weak (\(B^{\mathrm{W}}\)), intermediate (\(B^{\mathrm{I}}\)), and strong (\(B^{\mathrm{S}}\)) fields. The \(B^{\mathrm{W}}\) component represents areas with magnetic flux densities below the chosen threshold; the \(B^{\mathrm{I}}\) component is mainly represented by network fields, remains of decayed active regions (ARs), and ephemeral regions. The \(B^{\mathrm{S}}\) component consists of magnetic elements in ARs. To derive the contribution of a subset to the total SMMF, the linear regression coefficients between the corresponding component and the SMMF were calculated. We found that i) when the threshold level of 30 Mx?cm^?2 is applied, the \(B^{\mathrm{I}}\) and \(B^{\mathrm{S}}\) components together contribute from 65% to 95% of the SMMF, while the fraction of the occupied area varies in a range of 2?–?6% of the disk area; ii) as the threshold magnitude is lowered to 6 Mx?cm^?2, the contribution from \(B^{\mathrm{I}}+B^{\mathrm{S}}\) grows to 98%, and the fraction of the occupied area reaches a value of about 40% of the solar disk. In summary, we found that regardless of the threshold level, only a small part of the solar disk area contributes to the SMMF. This means that the photospheric magnetic structure is an intermittent inherently porous medium, resembling a percolation cluster. These findings suggest that the long-standing concept that continuous vast unipolar areas on the solar surface are the source of the SMMF may need to be reconsidered.     

Evaluating (and Improving) Estimates of the Solar Radial Magnetic Field Component from Line-of-Sight Magnetograms

Although for many solar physics problems the desirable or meaningful boundary is the radial component of the magnetic field \(B_{\mathrm {r}}\), the most readily available measurement is the component of the magnetic field along the line of sight to the observer, \(B_{\mathrm {los}}\). As this component is only equal to the radial component where the viewing angle is exactly zero, some approximation is required to estimate \(B_{\mathrm {r}}\) at all other observed locations. In this study, a common approximation known as the “\(\mu\)-correction”, which assumes all photospheric field to be radial, is compared to a method that invokes computing a potential field that matches the observed \(B_{\mathrm {los}}\), from which the potential field radial component, \(B_{\mathrm {r}}^{\mathrm {pot}}\) is recovered. We demonstrate that in regions that are truly dominated by a radially oriented field at the resolution of the data employed, the \(\mu\)-correction performs acceptably if not better than the potential-field approach. However, it is also shown that for any solar structure that includes horizontal fields, i.e. active regions, the potential-field method better recovers both the strength of the radial field and the location of magnetic neutral line.     

Estimation of Reconnection Flux Using Post-eruption Arcades and Its Relevance to Magnetic Clouds at 1 AU

We report on a new method to compute the flare reconnection (RC) flux from post-eruption arcades (PEAs) and the underlying photospheric magnetic fields. In previous works, the RC flux has been computed using the cumulative flare ribbon area. Here we obtain the RC flux as the flux in half of the area underlying the PEA in EUV imaged after the flare maximum. We apply this method to a set of 21 eruptions that originated near the solar disk center in Solar Cycle 23. We find that the RC flux from the arcade method (\(\Phi_{\mathrm{rA}}\)) has excellent agreement with the flux from the flare-ribbon method (\(\Phi_{\mathrm{rR}}\)) according to \(\Phi_{\mathrm{rA}} = 1.24(\Phi_{\mathrm{rR}})^{0.99}\). We also find \(\Phi_{\mathrm{rA}}\) to be correlated with the poloidal flux (\(\Phi_{\mathrm{P}}\)) of the associated magnetic cloud at 1 AU: \(\Phi_{\mathrm{P}} = 1.20(\Phi_{\mathrm{rA}})^{0.85}\). This relation is nearly identical to that obtained by Qiu et al. (Astrophys. J. 659, 758, 2007) using a set of only 9 eruptions. Our result supports the idea that flare reconnection results in the formation of the flux rope and PEA as a common process.     

Relating Solar Energetic Particle Event Fluences to Peak Intensities

Recently we (Kahler and Ling, Solar Phys.292, 59, 2017: KL) have shown that time–intensity profiles [\(I(t)\)] of 14 large solar energetic particle (SEP) events can be fitted with a simple two-parameter fit, the modified Weibull function, which is characterized by shape and scaling parameters [\(\alpha\) and \(\beta\)]. We now look for a simple correlation between an event peak energy intensity [\(I_{\mathrm{p}}\)] and the time integral of \(I(t)\) over the event duration: the fluence [\(F\)]. We first ask how the ratio of \(F/I_{\mathrm{p}}\) varies for the fits of the 14 KL events and then examine that ratio for three separate published statistical studies of SEP events in which both \(F\) and \(I_{\mathrm{p}}\) were measured for comparisons of those parameters with various solar-flare and coronal mass ejection (CME) parameters. The three studies included SEP energies from a 4?–?13 MeV band to \(E > 100~\mbox{MeV}\). Within each group of SEP events, we find a very robust correlation (\(\mathrm{CC} > 0.90\)) in log–log plots of \(F\)versus\(I_{\mathrm{p}}\) over four decades of \(I_{\mathrm{p}}\). The ratio increases from western to eastern longitudes. From the value of \(I_{\mathrm{p}}\) for a given event, \(F\) can be estimated to within a standard deviation of a factor of \({\leq}\,2\). Log–log plots of two studies are consistent with slopes of unity, but the third study shows plot slopes of \({<}\,1\) and decreasing with increasing energy for their four energy ranges from \(E > 10~\mbox{MeV}\) to \({>}\,100~\mbox{MeV}\). This difference is not explained.     

Dark energy cosmological models with cosmic string

In this paper we have studied the anisotropic Kantowski-Sachs, locally rotationally symmetric (LRS) Bianchi type-I and LRS Bianchi type-III geometries filled with dark energy and one dimensional cosmic string in the Saez-Ballester theory of gravitation. To get physically valid solution we take hybrid expansion law of the average scale factor which is a product of power and exponential type of functions that results in time dependent deceleration parameter (\(q\)). The equation of state parameter of dark energy (\(\omega _{\mathit{de}}\)) has been discussed and we have observed that for the three models it crosses the phantom divide line (\(\omega _{\mathit{de}} = -1\)) and shows quintom like behavior. The density of dark energy (\(\rho _{\mathit{de}}\)) is an increasing function of redshift and remains positive throughout the evolution of the universe for the three models. Moreover in Kantowski-Sachs and LRS Bianchi type-I geometries the dark energy density dominates the string tension density (\(\lambda \)) and proper density (\(\rho \)) throughout the evolution of the universe. The physical and geometrical aspects of the statefinder parameters (\(r,s\)), squared speed of sound (\(v_{s}^{2} \)) and \(\omega _{\mathit{de}}\)–\(\omega ^{\prime }_{\mathit{de}}\) plane are also discussed.     

Nature of Coherent Radio Emission from Pulsars

The pulsar radio emission originates from regions below 10% of the light cylinder radius. This requires a mechanism where coherent emission is excited in relativistic pair plasma with frequency \(\nu _{\mathrm{cr}}\) which is below the plasma frequency \(\nu _{\circ }\) i.e. \(\nu _{\mathrm{cr}} < \nu _{\circ }\). A possible model for the emission mechanism is charged bunches (charged solitons) moving relativistically along the curved open dipolar magnetic field lines capable of exciting coherent curvature radio emission. In this article, we review the results from high quality observations in conjunction with theoretical models to unravel the nature of coherent curvature radio emission in pulsars.     

Kelvin–Helmholtz Instability in a Cool Solar Jet in the Framework of Hall Magnetohydrodynamics: A Case Study

We investigate the conditions under which the magnetohydrodynamic (MHD) modes in a cylindrical magnetic flux tube moving along its axis become unstable against the Kelvin–Helmholtz (KH) instability. We use the dispersion relations of MHD modes obtained from the linearized Hall MHD equations for cool (zero beta) plasma by assuming real wave numbers and complex angular wave frequencies/complex wave phase velocities. The dispersion equations are solved numerically at fixed input parameters and varying values of the ratio \(l_{\mathrm{Hall}}/a\), where \(l_{\mathrm{Hall}} = c/\omega_{\mathrm{pi}}\) (\(c\) being the speed of light, and \(\omega_{\mathrm{pi}}\) the ion plasma frequency) and \(a\) is the flux tube radius. It is shown that the stability of the MHD modes depends upon four parameters: the density contrast between the flux tube and its environment, the ratio of external and internal magnetic fields, the ratio \(l_{\mathrm{Hall}}/a\), and the value of the Alfvén Mach number defined as the ratio of the tube axial velocity to Alfvén speed inside the flux tube. It is found that at high density contrasts, for small values of \(l_{\mathrm{Hall}}/a\), the kink (\(m = 1\)) mode can become unstable against KH instability at some critical Alfvén Mach number (or equivalently at critical flow speed), but a threshold \(l_{\mathrm{Hall}}/a\) can suppress the onset of the KH instability. At small density contrasts, however, the magnitude of \(l_{\mathrm{Hall}}/a\) does not affect noticeably the condition for instability occurrence – even though it can reduce the critical Alfvén Mach number. It is established that the sausage mode (\(m = 0\)) is not subject to the KH instability.     

Advective accretion flow properties around rotating black holes – application to GRO J1655-40

We examine the properties of the viscous dissipative accretion flow around rotating black holes in the presence of mass loss. Considering the thin disc approximation, we self-consistently calculate the inflow-outflow solutions and observe that the mass outflow rates decrease with the increase in viscosity parameter (\(\alpha \)). Further, we carry out the model calculation of quasi-periodic oscillation frequency (\(\nu _{\mathrm{QPO}}\)) that is frequently observed in black hole sources and observe that \(\nu ^\mathrm{max}_{\mathrm{QPO}}\) increases with the increase of black hole spin (\(a_k\)). Then, we employ our model in order to explain the High Frequency Quasi-Periodic Oscillations (HFQPOs) observed in black hole source GRO J1655-40. While doing this, we attempt to constrain the range of \(a_k\) based on observed HFQPOs (\(\sim \)300 Hz and \(\sim \)450 Hz) for the black hole source GRO J1655-40.     

Photospheric Shear Flows in Solar Active Regions and Their Relation to Flare Occurrence

Solar active regions (ARs) that produce major flares typically exhibit strong plasma shear flows around photospheric magnetic polarity inversion lines (MPILs). It is therefore important to quantitatively measure such photospheric shear flows in ARs for a better understanding of their relation to flare occurrence. Photospheric flow fields were determined by applying the Differential Affine Velocity Estimator for Vector Magnetograms (DAVE4VM) method to a large data set of 2548 coaligned pairs of AR vector magnetograms with 12-min separation over the period 2012?–?2016. From each AR flow-field map, three shear-flow parameters were derived corresponding to the mean (\(\langle S\rangle \)), maximum (\(S_{\mathrm{max}}\)) and integral (\(S_{\mathrm{sum}}\)) shear-flow speeds along strong-gradient, strong-field MPIL segments. We calculated flaring rates within 24 h as a function of each shear-flow parameter and we investigated the relation between the parameters and the waiting time (\(\tau \)) until the next major flare (class M1.0 or above) after the parameter observation. In general, it is found that the larger \(S_{\mathrm{sum}}\) an AR has, the more likely it is for the AR to produce flares within 24 h. It is also found that among ARs which produce major flares, if one has a larger value of \(S_{\mathrm{sum}}\) then \(\tau \) generally gets shorter. These results suggest that large ARs with widespread and/or strong shear flows along MPILs tend to not only be more flare productive, but also produce major flares within 24 h or less.     

A Solar Stationary Type IV Radio Burst and Its Radiation Mechanism

A stationary Type IV (IVs) radio burst was observed on September 24, 2011. Observations from the Nançay RadioHeliograph (NRH) show that the brightness temperature (\(T_{\mathrm{B}}\)) of this burst is extremely high, over \(10^{11}\) K at 150 MHz and over \(10^{8}\) K in general. The degree of circular polarization (\(q\)) is between \(-60\% \sim -100\%\), which means that it is highly left-handed circularly polarized. The flux–frequency spectrum follows a power-law distribution, and the spectral index is considered to be roughly \(-3 \sim -4\) throughout the IVs. Radio sources of this event are located in the wake of the coronal mass ejection and are spatially dispersed. They line up to present a formation in which lower-frequency sources are higher. Based on these observations, it is suggested that the IVs was generated through electron cyclotron maser emission.     

Pulse phase-resolved analysis of SMC X-3 during its 2016–2017 super-Eddington outburst

The Be X-ray pulsar SMC X-3 underwent an extra long and ultraluminous giant outburst from 2016 August to 2017 March. The peak X-ray luminosity is up to \(\sim10^{39}~\mbox{erg/s}\), suggesting a mildly super-Eddington accretion onto the strongly magnetized neutron star. It therefore bridges the gap between the Galactic Be/X-ray binaries (\(L_{\mathrm{X}}^{\mathrm{peak}} \leq10^{38}~\mbox{erg/s}\)) and the ultraluminous X-ray pulsars (\(L_{\mathrm{X}}^{\mathrm{peak}} \geq10^{40}~\mbox{erg/s}\)) found in nearby galaxies. A number of observations were carried out to observe the outburst. In this paper, we perform a comprehensive phase-resolved analysis on the high quality data obtained with the Nustar and XMM-Newton, which were observed at a high and intermediate luminosity levels. In order to get a better understanding on the evolution of the whole extreme burst, we take the Swift results at the low luminosity state into account as well. At the early stage of outburst, the source shows a double-peak pulse profile, the second main peak approaches the first one and merges into the single peak at the low luminosity. The second main peak vanishes beyond 20 keV, and its radiation becomes much softer than that of the first main peak. The line widths of fluorescent iron line vary dramatically with phases, indicating a complicated geometry of accretion flows. In contrast to the case at low luminosity, the pulse fraction increases with the photon energy. The significant small pulse fraction detected below 1 keV can be interpreted as the existence of an additional thermal component located at far away from the central neutron star.     

Accretion flow dynamics during 1999 outburst of XTE J1859+226—modeling of broadband spectra and constraining the source mass

We examine the dynamical behavior of accretion flow around XTE J1859+226 during the 1999 outburst by analyzing the entire outburst data (～166 days) from RXTE Satellite. Towards this, we study the hysteresis behavior in the hardness intensity diagram (HID) based on the broadband (3–150 keV) spectral modeling, spectral signature of jet ejection and the evolution of Quasi-periodic Oscillation (QPO) frequencies using the two-component advective flow model around a black hole. We compute the flow parameters, namely Keplerian accretion rate (\({\dot{m}}_{d}\)), sub-Keplerian accretion rate (\({\dot{m}}_{h}\)), shock location (\(r_{s}\)) and black hole mass (\(M_{\mathit{bh}}\)) from the spectral modeling and study their evolution along the q-diagram. Subsequently, the kinetic jet power is computed as \(L^{\mathrm{obs}}_{\mathrm{jet}} \sim3\mbox{--}6 \times10^{37}~\mbox{erg}\,\mbox{s}^{-1}\) during one of the observed radio flares which indicates that jet power corresponds to 8–16% mass outflow rate from the disc. This estimate of mass outflow rate is in close agreement with the change in total accretion rate (～14%) required for spectral modeling before and during the flare. Finally, we provide a mass estimate of the source XTE J1859+226 based on the spectral modeling that lies in the range of 5.2–7.9 \(M_{\odot}\) with 90% confidence.     

A New Solar Imaging System for Observing High-Speed Eruptions: <Emphasis Type="Italic">Solar Dynamics Doppler Imager</Emphasis> (SDDI)

A new solar imaging system was installed at Hida Observatory to observe the dynamics of flares and filament eruptions. The system (Solar Dynamics Doppler Imager; SDDI) takes full-disk solar images with a field of view of \(2520~\mbox{arcsec} \times 2520~\mbox{arcsec}\) at multiple wavelengths around the \(\mathrm{H}\alpha\) line at 6562 Å. Regular operation was started in May 2016, in which images at 73 wavelength positions spanning from \(\mathrm{H}\alpha -9~\mathring{\mathrm{A}}\) to \(\mathrm{H}\alpha +9~\mathring{\mathrm{A}}\) are obtained every 15 seconds. The large dynamic range of the line-of-sight velocity measurements (\({\pm}\,400~\mbox{km}\,\mbox{s}^{-1}\)) allows us to determine the real motions of erupting filaments in 3D space. It is expected that SDDI provides unprecedented datasets to study the relation between the kinematics of filament eruptions and coronal mass ejections (CME), and to contribute to the real-time prediction of the occurrence of CMEs that cause a significant impact on the space environment of the Earth.     

Active Region Photospheric Magnetic Properties Derived from Line-of-Sight and Radial Fields

The effect of using two representations of the normal-to-surface magnetic field to calculate photospheric measures that are related to the active region (AR) potential for flaring is presented. Several AR properties were computed using line-of-sight (\(B_{\mathrm{los}}\)) and spherical-radial (\(B_{r}\)) magnetograms from the Space-weather HMI Active Region Patch (SHARP) products of the Solar Dynamics Observatory, characterizing the presence and features of magnetic polarity inversion lines, fractality, and magnetic connectivity of the AR photospheric field. The data analyzed correspond to \({\approx\,}4{,}000\) AR observations, achieved by randomly selecting 25% of days between September 2012 and May 2016 for analysis at 6-hr cadence. Results from this statistical study include: i) the \(B_{r}\) component results in a slight upwards shift of property values in a manner consistent with a field-strength underestimation by the \(B_{\mathrm{los}}\) component; ii) using the \(B_{r}\) component results in significantly lower inter-property correlation in one-third of the cases, implying more independent information as regards the state of the AR photospheric magnetic field; iii) flaring rates for each property vary between the field components in a manner consistent with the differences in property-value ranges resulting from the components; iv) flaring rates generally increase for higher values of properties, except the Fourier spectral power index that has flare rates peaking around a value of \(5/3\). These findings indicate that there may be advantages in using \(B_{r}\) rather than \(B_{\mathrm{los}}\) in calculating flare-related AR magnetic properties, especially for regions located far from central meridian.     

The luminosity correlation analysis for Fermi blazars

In this paper, multiwavelength chromatic luminosity at radio (\(\log L _{\mathrm{R}}\)), optical (\(\log L_{\mathrm{O}}\)), X-ray (\(\log L _{\mathrm{X}}\)), and \(\gamma \)-rays (\(\log L_{\gamma }\)) for a sample of 442 Fermi blazars with known redshifts are collected from Fan et al. (2016), to study the correlations between the \(\gamma \)-rays and the low-energy bands using a multiple linear regression analysis. In this way, we can see which band is more important for the \(\gamma \)-ray emissions. Mutual correlation analysis is also used to discuss the correlations between the \(\gamma \)-ray and the low energy bands for the whole sample and subclasses. We come to following conclusions:

1. 1.
   The multiple linear correlation indicates that the \(\gamma \)-rays are correlated with the radio, optical and the X-ray emissions for the whole sample and the subclasses of flat spectrum radio quasars (FSRQs) and BL Lac objects (BL Lacs), the correlation between the \(\gamma \)-rays and the radio emissions is the strongest one.
    
2. 2.
   For BL Lacs, the optical emissions are more important than the X-rays for the \(\gamma \)-rays, while the X-ray emissions are more important than optical ones in FSRQs.
    
3. 3.
   The \(\gamma \)-ray emissions in HBL are from an synchrotron self-Compton, while those in FSRQs may be from external Compton and synchrotron self-Compton as well.
    

     

<Emphasis Type="Italic">GALEX</Emphasis> far-ultraviolet observations of metal-poor subdwarfs

Far-ultraviolet photometry derived from the GALEX satellite observatory has been compiled for a sample of metal-poor subdwarfs with \(\mathrm{[Fe/H]} < -1.0\). The FUV properties of these subdwarfs are compared with those of a set of Population I dwarfs that are known to have low levels of chromospheric activity. Comparisons are made via a number of photometric plots, including an absolute FUV magnitude versus \((V-K_{s})\) diagram, two-colour diagrams involving both \((m_{ \mathrm{FUV}}-B)\) and \((m_{\mathrm{FUV}}-V)\) versus \(B-V\), and a two-colour diagram composed of \((m_{\mathrm{FUV}}-V)\) versus \((V-K_{s})\). The warmest subdwarfs with \((V-K_{s}) \sim1.2\mbox{--}1.4\) show FUV excesses ranging from \(\sim2\mbox{--}3~\mbox{mag}\) relative to the Population I dwarfs, with the amount of FUV enhancement decreasing among subdwarfs of decreasing effective temperature. The coolest dwarfs that are compared have \((V-K_{s}) \sim1.8\), and among these stars the subdwarfs with \(-2.0 \leq{\mathrm{[Fe/H]}} \leq-1.0\) approach the locus of low activity Population I dwarfs in the \((m_{\mathrm{FUV}}-V, V-K_{s})\) diagram. In the \((m_{\mathrm{FUV}}-B, B-V)\) diagram the subdwarfs in this metallicity range overlap the Population I dwarf sequence for \((B-V) > 0.6\). The behaviour of the subdwarfs is consistent with their FUV fluxes being determined by a combination of a photospheric FUV spectrum, the strength of which diminishes towards cooler effective temperatures, and a spectrum of emission lines arising from a chromosphere and/or transition region which are of comparable strength between the coolest dwarfs and subdwarfs.     

A Check on the Validity of Magnetic Field Reconstructions

We investigate a method to test whether a numerically computed model coronal magnetic field \({\boldsymbol {B}}\) departs from the divergence-free condition (also known as the solenoidality condition). The test requires a potential field \({\boldsymbol {B}}_{0}\) to be calculated, subject to Neumann boundary conditions, given by the normal components of the model field \({\boldsymbol {B}}\) at the boundaries. The free energy of the model field may be calculated using \(\frac{1}{2\mu _{0}}\int ({\boldsymbol {B}}-{\boldsymbol {B}}_{0})^{2}\mathrm{d}V\), where the integral is over the computational volume of the model field. A second estimate of the free energy is provided by calculating \(\frac{1}{2\mu _{0}}\int {\boldsymbol {B}}^{2}\,\mathrm{d}V-\frac{1}{2\mu _{0}}\int {\boldsymbol {B}}_{0}^{2}\,\mathrm{d}V\). If \({\boldsymbol {B}}\) is divergence free, the two estimates of the free energy should be the same. A difference between the two estimates indicates a departure from \(\nabla \cdot {\boldsymbol {B}}=0\) in the volume. The test is an implementation of a procedure proposed by Moraitis et al. (Solar Phys.289, 4453, 2014) and is a simpler version of the Helmholtz decomposition procedure presented by Valori et al. (Astron. Astrophys.553, A38, 2013). We demonstrate the test in application to previously published nonlinear force-free model fields, and also investigate the influence on the results of the test of a departure from flux balance over the boundaries of the model field. Our results underline the fact that, to make meaningful statements about magnetic free energy in the corona, it is necessary to have model magnetic fields that satisfy the divergence-free condition to a good approximation.     

Dependence of the Sunspot-Group Size on the Level of Solar Activity and its Influence on the Calibration of Solar Observers

We study the distribution of the sunspot-group size (area) and its dependence on the level of solar activity. We show that the fraction of small groups is not constant but decreases with the level of solar activity so that high solar activity is mainly defined by large groups. We analyze the possible influence of solar activity on the ability of a realistic observer to see and report the daily number of sunspot groups. It is shown that the relation between the number of sunspot groups as seen by different observers with different observational acuity thresholds is strongly nonlinear and cannot be approximated by the traditionally used linear scaling (\(k\)-factors). The observational acuity threshold [\(A_{\mathrm{th}}\)] is considered to quantify the quality of each observer, instead of the traditional relative \(k\)-factor. A nonlinear \(c\)-factor based on \(A_{\mathrm{th}}\) is proposed, which can be used to correct each observer to the reference conditions. The method is tested on a pair of principal solar observers, Wolf and Wolfer, and it is shown that the traditional linear correction, with the constant \(k\)-factor of 1.66 to scale Wolf to Wolfer, leads to an overestimate of solar activity around solar maxima.