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.     

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.     

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.     

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.     

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.     

Addendum to: Strength of the Solar Coronal Magnetic Field – A Comparison of Independent Estimates Using Contemporaneous Radio and White-Light Observations

This addendum uses an alternate fit for the electron density distribution \(N(r)\) (see Figure 1) and estimates the coronal magnetic field using the new model. We find that the estimates of the magnetic field are in close agreement using both the models.

We have fit the \(N(r)\) distribution obtained from STEREO-A/COR1 and SOHO/LASCO-C2 using a fifth-order polynomial (see Figure 1). The expression can be written as

$$\begin{aligned} N_{\text{cor}}(r) &= 1.43 \times 10^{9} r^{-5} - 1.91 \times 10^{9} r^{-4} + 1.07 \times 10^{9} r^{-3} - 2.87 \times 10^{8} r^{-2} \\ &\quad {} + 3.76 \times 10^{7} r^{-1} - 1.91 \times 10^{6} , \end{aligned}$$

(1)

where \(N_{\text{cor}}(r)\) is in units of cm^?3 and \(r\) is in units of \(\mathrm{R}_{\odot}\). The background coronal electron density is enhanced by a factor of 5.5 at 2.63 \(\mathrm{R}_{\odot}\) during the coronal mass ejection (CME). The estimated coronal magnetic field strength (\(B\)) using radio data indicates that \(B(r) \approx(0.51\text{\,--\,}0.48) \pm 0.02\ \mathrm{G}\) in the range \(r \approx2.65\text{\, --\,}2.82\ \mathrm{R}_{\odot}\). The field strengths for STEREO-A/COR1 and SOHO/LASCO-C2 are ≈?0.32 G at \(r \approx 3.11\ \mathrm{R}_{\odot}\) and ≈?0.12 G at \(r \approx 4.40\ \mathrm{R}_{\odot}\), respectively.

     

Statistical approach to triple systems in three-dimensional motion

This paper considers disruption of triple close approaches with low initial velocities and equal masses in the framework of statistical escape theory in a three-dimensional space. The statistical escape theory is based on the assumption that the phase trajectory of a triple system is quasi-ergodic. This system is described by allowing for both energy and angular momentum conservation in the phase space. In this paper, “possibility of escape” is derived with the formation of a binary on the basis of relative distances of the participating bodies. The complete statistical solutions (i.e. the semi-major axis \(a\), the distributions of eccentricity \(e\) of the binary, binary energy \({E}_{{b}}\), escape energy \({E}_{{s}}\) of escaper, and its escape velocity \({v}_{{s}}\)) of the system are derived from the allowable phase space volumes and are in good agreement with the numerical results in the range of perturbing velocities \({v}_{{i}}\)(\(10^{ - 1} \le {v}_{{i}} \le 10^{ - 10}\)) and directions of \({v}_{{i}}(0 \le \alpha _{{i}},\beta _{{i}},\gamma _{{i}} \le \pi )\), \({i} = 1,2,3\). In this paper, the double limit process has been applied to approximate the escape probability. Through this process, it is observed that the perturbing velocity \({v}_{{i}} \to 0^{ +} \), as the product of the semi-major axis \(a\) of the final binary and the square of the escape velocity \({v}_{{s}}\) approach 2/3, i.e. \({a} {v}_{{s}}^{2} \to 2 / 3\), whatever direction of \(\mathbf{v}_{{i}}\) may be.     

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.     

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.     

Analysis of the exoplanet containing system Kepler-13

We have applied the close binary system analysis program WinFitter, with its physically detailed fitting function, to an intensive study of the complex multiple system Kepler-13 using photometry data from all 13 short cadence quarters downloaded from the NASA Exoplanet Archive (NEA) (http://exoplanetarchive.ipac.caltech.edu). The data-point error of our normalized, phase-sequenced and binned (380 points per bin: 0.00025 phase interval) flux values, at 14 ppm, allows the model’s specification for the mean reference flux level of the system to a precision better than 1 ppm. Our photometrically derived values for the mass and radius of KOI13.01 are \(6.8\pm0.6~\mbox{M}_{\mathrm{J}}\) and \(1.44\pm0.04~\mbox{R}_{\mathrm{J}}\). The star has a radius of \(1.67\pm0.05~\mbox{R}_{\odot}\). Our modelling sets the mean of the orbital inclination \(i\) at \(94.35\pm0.14^{\circ}\), with the star’s mean precession angle \(\phi_{p}\)—\(49.1\pm5.0^{\circ}\) and obliquity \(\theta_{o}\)\(67.9 \pm 3.0^{\circ}\), though there are known ambiguities about the sense in which such angles are measured.Our findings did not confirm secular variation in the transit modelling parameters greater than their full correlated errors, as argued by previous authors, when each quarter’s data was best-fitted with a determinable parameter set without prejudice. However, if we accept that most of the parameters remain the same for each transit, then we could confirm a small but steady diminution in the cosine of the orbital inclination over the 17 quarter timespan. This is accompanied by a slight increase of the star’s precession angle (less negative), but with no significant change in the obliquity of its spin axis. There are suggestions of a history of strong dynamical interaction with a highly distorted planet rotating in a 3:2 resonance with its revolution, together with a tidal lag of \(\sim30~\mbox{deg}\). The mean precessional period is derived to be about 1000 y, but at the present time the motion of the star’s rotation axis appears to be supporting the gravitational torque, rather than providing the balance against it that would be expected over long periods of time.The planet has a small but detectable backwarming effect on the star, which helps to explain the difference in brightness just after transit and just before occultation eclipses. In assessing these findings it is recognized that sources of uncertainty remain, notably with possible inherent micropulsational effects, variations from other components of the multiple star, stellar activity, differential rotation and the neglect of higher order terms (than \(r_{1}^{5}\)) in the fitting function, where \(r_{1}\) is the ratio of the radius of the star to the mean orbital separation of planet and host star.     

Establishing the Galactic Centre distance using VVV Bulge RR Lyrae variables

This study’s objective was to exploit infrared VVV (VISTA Variables in the Via Lactea) photometry for high latitude RRab stars to establish an accurate Galactic Centre distance. RRab candidates were discovered and reaffirmed (\(n=4194\)) by matching \(K_{s}\) photometry with templates via \(\chi ^{2}\) minimization, and contaminants were reduced by ensuring targets adhered to a strict period-amplitude (\(\Delta K_{s}\)) trend and passed the Elorietta et al. classifier. The distance to the Galactic Centre was determined from a high latitude Bulge subsample (\(|b|>4^{\circ}\), \(R_{\mathit{GC}}=8.30 \pm 0.36\) kpc, random uncertainty is relatively negligible), and importantly, the comparatively low color-excess and uncrowded location mitigated uncertainties tied to the extinction law, the magnitude-limited nature of the analysis, and photometric contamination. Circumventing those problems resulted in a key uncertainty being the \(M_{K_{s}}\) relation, which was derived using LMC RRab stars (\(M_{K_{s}}=-(2.66\pm 0.06) \log {P}-(1.03\pm 0.06)\), \((J-K_{s})_{0}=(0.31\pm 0.04) \log {P} + (0.35\pm 0.02)\), assuming \(\mu _{0,\mathit{LMC}}=18.43\)). The Galactic Centre distance was not corrected for the cone-effect. Lastly, a new distance indicator emerged as brighter overdensities in the period-magnitude-amplitude diagrams analyzed, which arise from blended RRab and red clump stars. Blending may thrust faint extragalactic variables into the range of detectability.     

Dust color temperature distribution of two FIR cavities at IRIS and AKARI maps

By systematically searching the region of far infrared loops, we found a number of huge cavity-like dust structures at \(60\,\mu \hbox {m}\) and \(100\,\mu \hbox {m}\) IRIS maps. By checking these with AKARI maps (\(90\,\mu \hbox {m}\) and \(140\,\mu \hbox {m}\)), two new cavity-like structures (sizes \(\sim \) \( 2.7\,\hbox {pc} \times 0.8\,\hbox {pc}\) and \(\sim \) \( 1.8\,\hbox {pc} \times 1\,\hbox {pc}\)) located at R.A. (\(\hbox {J}2000)=14^{h}41^{m}23^{s}\) and Dec. \((\hbox {J}2000)=-64^{\circ }04^{\prime }17^{{\prime }{\prime }}\) and R.A. \((\hbox {J}2000)=05^{h}05^{m}35^{s}\) and Dec. \((\hbox {J}2000)=-\,69^{\circ }35^{\prime } 25^{{\prime }{\prime }}\) were selected for the study. The difference in the average dust color temperatures calculated using IRIS and AKARI maps of the cavity candidates were found to be \(3.2\pm 0.9\,\hbox {K}\) and \(4.1\pm 1.2\,\hbox {K}\), respectively. Interestingly, the longer wavelength AKARI map gives larger values of dust color temperature than that of the shorter wavelength IRIS maps. Possible explanation of the results will be presented.     

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.     

On the period of the periodic orbits of the restricted three body problem

We will show that the period T of a closed orbit of the planar circular restricted three body problem (viewed on rotating coordinates) depends on the region it encloses. Roughly speaking, we show that, \(2 T=k\pi +\int _\Omega g\) where k is an integer, \(\Omega \) is the region enclosed by the periodic orbit and \(g:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) is a function that only depends on the constant C known as the Jacobian constant; it does not depend on \(\Omega \). This theorem has a Keplerian flavor in the sense that it relates the period with the space “swept” by the orbit. As an application we prove that there is a neighborhood around \(L_4\) such that every periodic solution contained in this neighborhood must move clockwise. The same result holds true for \(L_5\).     

Improved Determination of the Location of the Temperature Maximum in the Corona

The most used method to calculate the coronal electron temperature [\(T_{\mathrm{e}} (r)\)] from a coronal density distribution [\(n_{\mathrm{e}} (r)\)] is the scale-height method (SHM). We introduce a novel method that is a generalization of a method introduced by Alfvén (Ark. Mat. Astron. Fys. 27, 1, 1941) to calculate \(T_{\mathrm{e}}(r)\) for a corona in hydrostatic equilibrium: the “HST” method. All of the methods discussed here require given electron-density distributions [\(n_{\mathrm{e}} (r)\)] which can be derived from white-light (WL) eclipse observations. The new “DYN” method determines the unique solution of \(T_{\mathrm{e}}(r)\) for which \(T_{\mathrm{e}}(r \rightarrow \infty) \rightarrow 0\) when the solar corona expands radially as realized in hydrodynamical solar-wind models. The applications of the SHM method and DYN method give comparable distributions for \(T_{\mathrm{e}}(r)\). Both have a maximum [\(T_{\max}\)] whose value ranges between 1?–?3 MK. However, the peak of temperature is located at a different altitude in both cases. Close to the Sun where the expansion velocity is subsonic (\(r < 1.3\,\mathrm{R}_{\odot}\)) the DYN method gives the same results as the HST method. The effects of the other free parameters on the DYN temperature distribution are presented in the last part of this study. Our DYN method is a new tool to evaluate the range of altitudes where the heating rate is maximum in the solar corona when the electron-density distribution is obtained from WL coronal observations.     

The Effect of Magnetic and Density Differences on the Fast Kink Oscillations of Neighboring Coronal Loops

It is reasonable that neighboring coronal loops may obtain similar momentum during a flare. The fast kink oscillations (FKOs) between them are thus mainly influenced by their physical differences. We discuss the dependencies of FKO on the physical properties of coronal loops in a low-\(\beta \) thin-tube approximation. From the analysis, we obtain the analytic relationship between the density [\(\rho _{\mathrm{i}}\)] and magnetic field [\(B\)] of loops and the corresponding period [\(\tau \)] and amplitude [\(A\)] of FKO, which may provide us with a powerful tool to diagnose the physical differences between neighboring loops.     

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.     

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.     

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.