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.
    

     

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.     

Geomagnetic Effects of Corotating Interaction Regions

We present an analysis of the geoeffectiveness of corotating interaction regions (CIRs), employing the data recorded from 25 January to 5 May 2005 and throughout 2008. These two intervals in the declining phase of Solar Cycle 23 are characterised by a particularly low number of interplanetary coronal mass ejections (ICMEs). We study in detail how four geomagnetic-activity parameters (the Dst, Ap, and AE indices, as well as the Dst time derivative, \(\mathrm{dDst}/\mathrm{d}t\)) are related to three CIR-related solar wind parameters (flow speed, \(V\), magnetic field, \(B\), and the convective electric field based on the southward Geocentric solar magnetospheric (GSM) magnetic field component, \(\mathit{VB}_{s}\)) on a three-hour time resolution. In addition, we quantify statistical relationships between the mentioned geomagnetic indices. It is found that Dst is correlated best to \(V\), with a correlation coefficient of \(\mathrm{cc}\approx0.6\), whereas there is no correlation between \(\mathrm{dDst}/\mathrm{d}t\) and \(V\). The Ap and AE indices attain peaks about half a day before the maximum of \(V\), with correlation coefficients ranging from \(\mathrm{cc}\approx0.6\) to \(\mathrm{cc}\approx0.7\), depending on the sample used. The best correlations of Ap and AE are found with \(\mathit{VB}_{s}\) with a delay of 3 h, being characterised by \(\mathrm{cc}\gtrsim 0.6\). The Dst derivative \(\mathrm{dDst}/\mathrm{d}t\) is also correlated with \(\mathit{VB}_{s}\), but the correlation is significantly weaker \(\mathrm{cc}\approx 0.4\)?–?0.5, with a delay of 0?–?3 h, depending on the employed sample. Such low values of correlation coefficients indicate that there are other significant effects that influence the relationship between the considered parameters. The correlation of all studied geomagnetic parameters with \(B\) are characterised by considerably lower correlation coefficients, ranging from \(\mathrm{cc}=0.3\) in the case of \(\mathrm{dDst}/\mathrm{d}t\) up to \(\mathrm{cc}=0.56\) in the case of Ap. It is also shown that peak values of geomagnetic indices depend on the duration of the CIR-related structures. The Dst is closely correlated with Ap and AE (\(\mathrm{cc}=0.7\)), Dst being delayed for about 3 h. On the other hand, \(\mathrm{dDst}/\mathrm{d}t\) peaks simultaneously with Ap and AE, with correlation coefficients of 0.48 and 0.56, respectively. The highest correlation (\(\mathrm{cc}=0.81\)) is found for the relationship between Ap and AE.     

Spectroscopic Variability of Supergiant Star HD14134, B3Ia

Profile variations in the \(\hbox {H}\alpha \) and \(\hbox {H}\beta \) lines in the spectra of the star HD14134 are investigated using observations carried out in 2013–2014 and 2016 with the 2-m telescope at the Shamakhy Astrophysical Observatory. The absorption and emission components of the \(\hbox {H}\alpha \) line are found to disappear on some observational days, and two of the spectrograms exhibit inverse P-Cyg profile of \(\hbox {H}\alpha \). It was revealed that when the \(\hbox {H}\alpha \) line disappeared or an inversion of the P-Cyg-type profile is observed in the spectra, the \(\hbox {H}\beta \) line is displaced to the longer wavelengths, but no synchronous variabilities were observed in other spectral lines (CII \( \lambda \) 6578.05 Å, \( \lambda \) 6582.88 Å  and HeI \( \lambda \) 5875.72 Å) formed in deeper layers of the stellar atmosphere. In addition, the profiles of the \(\hbox {H}\alpha \) and \(\hbox {H}\beta \) lines have been analysed, as well as their relations with possible expansion, contraction and mixed conditions of the atmosphere of HD14134. We suggest that the observational evidence for the non-stationary atmosphere of HD14134 can be associated in part with the non-spherical stellar wind.     

The radio and $\gamma$-ray variability analysis of S5 0716+714

We have studied the variability of S5 0716+714 at radio 15 GHz and \(\gamma\)-ray band using three different methods. A possible periodicity of \(P_{15~\text{GHz}}=266.0\pm11.5\) and \(P_{\gamma}=344.0 \pm16.4\) days are obtained for radio 15 GHz and \(\gamma\)-ray light curves, respectively. The variability may be related to the intrinsically emission mechanism. The difference between the variability timescales of radio 15 GHz and \(\gamma \)-ray may be due to that the emission of radio 15 GHz is produced via the synchrotron process, while the \(\gamma\)-ray is produced by both the SSC and EC processes.     

Nowcasting Solar Energetic Particle Events Using Principal Component Analysis

We perform a principal component analysis (PCA) on a set of six solar variables (i.e. width/size (\(s\)) and velocity (\(u\)) of a coronal mass ejection, logarithm of the solar flare (SF) magnitude (\(\log\mathit{SXRs}\)), SF longitude (\(\mathit{lon}\)), duration (\(\mathit{DT}\)), and rise time (\(\mathit{RT}\))). We classify the solar energetic particle (SEP) event radiation impact (in terms of the National Oceanic and Atmospheric Administration scales) with respect to the characteristics of their parent solar events. We further attempt to infer the possible prediction of SEP events. In our analysis, we use 126 SEP events with complete solar information, from 1997 to 2013. Each SEP event is a vector in six dimensions (corresponding to the six solar variables used in this work). The PCA transforms the input vectors into a set of orthogonal components. By mapping the characteristics of the parent solar events, a new base defined by these components led to the classification of the SEP events. We furthermore applied logistic regression analysis with single, as well as multiple explanatory variables, in order to develop a new index (\(I\)) for the nowcasting (short-term forecasting) of SEP events. We tested several different schemes for \(I\) and validated our findings with the implementation of categorical scores (probability of detection (POD) and false-alarm rate (FAR)). We present and interpret the obtained scores, and discuss the strengths and weaknesses of the different implementations. We show that \(I\) holds prognosis potential for SEP events. The maximum POD achieved is 77.78% and the relative FAR is 40.96%.     

New holographic dark energy model with constant bulk viscosity in modified $$ gravity theory

The aim of this paper is to study new holographic dark energy (HDE) model in modified \(f(R,T)\) gravity theory within the framework of a flat Friedmann-Robertson-Walker model with bulk viscous matter content. It is thought that the negative pressure caused by the bulk viscosity can play the role of dark energy component, and drive the accelerating expansion of the universe. This is the motive of this paper to observe such phenomena with bulk viscosity. In the specific model \(f(R,T)=R+\lambda T\), where \(R\) is the Ricci scalar, \(T\) the trace of the energy-momentum tensor and \(\lambda \) is a constant, we find the solution for non-viscous and viscous new HDE models. We analyze new HDE model with constant bulk viscosity, \(\zeta =\zeta _{0}= \text{const.}\) to explain the present accelerated expansion of the universe. We classify all possible scenarios (deceleration, acceleration and their transition) with possible positive and negative ranges of \(\lambda \) over the constraint on \(\zeta _{0}\) to analyze the evolution of the universe. We obtain the solutions of scale factor and deceleration parameter, and discuss the evolution of the universe. We observe the future finite-time singularities of type I and III at a finite time under certain constraints on \(\lambda \). We also investigate the statefinder and \(\mathit{Om}\) diagnostics of the viscous new HDE model to discriminate with other existing dark energy models. In late time the viscous new HDE model approaches to \(\varLambda \mathit{CDM}\) model. We also discuss the thermodynamics and entropy of the model and find that it satisfies the second law of thermodynamics.     

Average Magnetic Field Magnitude Profiles of <Emphasis Type="Italic">Wind</Emphasis> Magnetic Clouds as a Function of Closest Approach to the Clouds’ Axes and Comparison to Model

We examine the average magnetic field magnitude (\(| \boldsymbol{B} | \equiv B\)) within magnetic clouds (MCs) observed by the Wind spacecraft from 1995 to July 2015 to understand the difference between this \(B\) and the ideal \(B\)-profiles expected from using the static, constant-\(\alpha\), force-free, cylindrically symmetric model for MCs of Lepping, Jones, and Burlaga (J. Geophys. Res. 95, 11957, 1990, denoted here as the LJB model). We classify all MCs according to an assigned quality, \(Q_{0}\) (\(= 1, 2, 3\), for excellent, good, and poor). There are a total of 209 MCs and 124 when only \(Q_{0} = 1\), 2 cases are considered. The average normalized field with respect to the closest approach (\(\mathit{CA}\)) is stressed, where we separate cases into four \(\mathit{CA}\) sets centered at 12.5 %, 37.5 %, 62.5 %, and 87.5 % of the average radius; the averaging is done on a percentage-duration basis to treat all cases the same. Normalized \(B\) means that before averaging, the \(B\) for each MC at each point is divided by the LJB model-estimated \(B\) for the MC axis, \(B_{0}\). The actual averages for the 209 and 124 MC sets are compared to the LJB model, after an adjustment for MC expansion (e.g. Lepping et al. in Ann. Geophys. 26, 1919, 2008). This provides four separate difference-relationships, each fitted with a quadratic (Quad) curve of very small \(\sigma\). Interpreting these Quad formulae should provide a comprehensive view of the variation in normalized \(B\) throughout the average MC, where we expect external front and rear compression to be part of its explanation. These formulae are also being considered for modifying the LJB model. This modification will be used in a scheme for forecasting the timing and magnitude of magnetic storms caused by MCs. Extensive testing of the Quad formulae shows that the formulae are quite useful in correcting individual MC \(B\)-profiles, especially for the first \({\approx\,}1/3\) of these MCs. However, the use of this type of \(B\) correction constitutes a (slight) violation of the force-free assumption used in the original LJB MC model.     

A study of the geomagnetic indices asymmetry based on the interplanetary magnetic field polarities

Data of geomagnetic indices (aa, Kp, Ap, and Dst) recorded near 1 AU over the period 1967–2016, have been studied based on the asymmetry between the interplanetary magnetic field (IMF) directions above and below of the heliospheric current sheet (HCS). Our results led to the following conclusions: (i) Throughout the considered period, 31 random years (62%) showed apparent asymmetries between Toward (\(\mathbf{T}\)) and Away (\(\mathbf{A}\)) polarity days and 19 years (38%) exhibited nearly a symmetrical behavior. The days of \(\mathbf{A}\) polarity predominated over the \(\mathbf{T}\) polarity days by 4.3% during the positive magnetic polarity epoch (1991–1999). While the days of \(\mathbf{T}\) polarity exceeded the days of \(\mathbf{A}\) polarity by 5.8% during the negative magnetic polarity epoch (2001–2012). (ii) Considerable yearly North–South (N–S) asymmetries of geomagnetic indices observed throughout the considered period. (iii) The largest toward dominant peaks for \(aa\) and \(Ap\) indices occurred in 1995 near to minimum of solar activity. Moreover, the most substantial away dominant peaks for \(aa\) and \(Ap\) indices occurred in 2003 (during the descending phase of the solar cycle 23) and in 1991 (near the maximum of solar activity cycle) respectively. (iv) The N–S asymmetry of \(Kp\) index indicated a most significant away dominant peak occurred in 2003. (v) Four of the away dominant peaks of Dst index occurred at the maxima of solar activity in the years 1980, 1990, 2000, and 2013. The largest toward dominant peak occurred in 1991 (at the reversal of IMF polarity). (vi) The geomagnetic indices (aa, Ap, and \(Kp\)) all have northern dominance during positive magnetic polarity epoch (1971–1979), while the asymmetries shifts to the southern solar hemisphere during negative magnetic polarity epoch (2001–2012).     

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.     

High-Reflectivity Coatings for a Vacuum Ultraviolet Spectropolarimeter

Precise polarization measurements in the vacuum ultraviolet (VUV) region are expected to be a new tool for inferring the magnetic fields in the upper atmosphere of the Sun. High-reflectivity coatings are key elements to achieving high-throughput optics for precise polarization measurements. We fabricated three types of high-reflectivity coatings for a solar spectropolarimeter in the hydrogen Lyman-\(\upalpha \) (Ly\(\upalpha \); 121.567 nm) region and evaluated their performance. The first high-reflectivity mirror coating offers a reflectivity of more than 80 % in Ly\(\upalpha \) optics. The second is a reflective narrow-band filter coating that has a peak reflectivity of 57 % in Ly\(\upalpha \), whereas its reflectivity in the visible light range is lower than 1/10 of the peak reflectivity (\(\sim 5~\%\) on average). This coating can be used to easily realize a visible light rejection system, which is indispensable for a solar telescope, while maintaining high throughput in the Ly\(\upalpha \) line. The third is a high-efficiency reflective polarizing coating that almost exclusively reflects an s-polarized beam at its Brewster angle of 68° with a reflectivity of 55 %. This coating achieves both high polarizing power and high throughput. These coatings contributed to the high-throughput solar VUV spectropolarimeter called the Chromospheric Lyman-Alpha SpectroPolarimeter (CLASP), which was launched on 3 September, 2015.     

Correlation Between the Magnetic Field and Plasma Parameters at 1 AU

The physical parameters of the solar wind observed in-situ near 1 AU have been studied for several decades, and relationships between them, such as the positive correlation between the solar wind plasma temperature, \(T\), and velocity, \(V\), and the negative correlation between density, \(N\), and velocity, \(V\), are well known. However, the magnetic field intensity, \(B\), does not appear to be well correlated with any individual plasma parameter. In this article, we discuss previously under-reported correlations between \(B\) and the combined plasma parameters \(\sqrt{N V^{2}} \) as well as between \(B\) and \(\sqrt{NT}\). These two correlations are strong during periods of corotating interaction regions and high-speed streams, and moderate during intervals of slow solar wind. The results indicate that the magnetic pressure in the solar wind is well correlated both with the plasma dynamic pressure and the thermal pressure.     

GeV-TeV $\gamma $-ray energy spectral break of BL Lac objects

In this paper, we compile the very-high-energy and high-energy spectral indices of 43 BL Lac objects from the literature. Based on a simple math model, \(\Delta \Gamma_{obs}= \alpha {{{z}}}+\beta \), we present evidence for the origin of an observed spectral break that is denoted by the difference between the observed very-high-energy and high-energy spectral indices, \(\Delta \Gamma_{obs}\). We find by linear regression analysis that \(\alpha \ne 0\) and \(\beta \ne 0\). These results suggest that the extragalactic background light attenuation and the intrinsic curvature dominate on the GeV-TeV \(\gamma \)-ray energy spectral break of BL Lac objects. We argue that the extragalactic background light attenuation is an exclusive explanation for the redshift evolution of the observed spectral break.     

Toward Spectroscopically Detecting the Global Latitudinal Temperature Variation on the Solar Surface

A very slight rotation-induced latitudinal temperature variation (presumably on the order of several kelvin) on the solar surface is theoretically expected. While recent high-precision solar brightness observations reported its detection, confirmation by an alternative approach using the strengths of spectral lines is desirable, for which reducing the noise due to random fluctuation caused by atmospheric inhomogeneity is critical. Toward this difficult task, we carried out a pilot study of spectroscopically investigating the relative variation of temperature (\(T\)) at a number of points in the solar circumference region near to the limb (where latitude dependence should be detectable, if any exists) based on the equivalent widths (\(W\)) of 28 selected lines in the 5367?–?5393 Å and 6075?–?6100 Å regions. We paid special attention to i) clarifying which types of lines should be employed and ii) how much precision is attainable in practice. We found that lines with strong \(T\)-sensitivity (\(|\log W/\log T|\)) should be used and that very weak lines should be avoided because they inevitably suffer strong relative fluctuations (\(\Delta W/W\)). Our analysis revealed that a precision of \(\Delta T/T \approx 0.003\) (corresponding to ≈?15 K) can be achieved at best by a spectral line with comparatively large \(|\log W/\log T|\), although this can possibly be further improved When a number of lines are used all together. Accordingly, if many such favorable lines could be measured with subpercent precision of \(\Delta W/W\) and by averaging the resulting \(\Delta T/T\) from each line, the random noise would eventually be reduced to ??1 K and detection of a very subtle amount of global \(T\)-gradient might be possible.     

Density variation effect on multi-ions with kinetic Alfven wave around cusp region—a kinetic approach

The kinetic Alfven waves in the presence of homogeneous magnetic field plasma with multi-ions effect are investigated. The dispersion relation and normalised damping rate are derived for low-\(\beta\) plasma using kinetic theory. The effect of density variation of \(\text{H}^{+}\), \(\text{He}^{+}\) and \(\text{O}^{+}\) ions is observed on frequency and damping rate of the wave. The variation of frequency (\(\omega\)) and normalised damping rate (\(\gamma / \varOmega_{H^{ +}} \)) of the wave are studied with respect to \(k_{ \bot} \rho_{j}\), where \(k_{ \bot} \) is the perpendicular wave number, \(\rho_{j}\) is the ion gyroradius and \(j \) denotes \(\text{H}^{+}\), \(\text{He}^{+}\) and \(\text{O}^{+}\) ions. The variation with \(k_{ \bot} \rho_{j}\) is considered over wide range. The parameters appropriate to cusp region are used for the explanation of results. It is found that with hydrogen and helium ions gyration, the frequency of wave is influenced by the density variation of \(\text{H}^{+}\) and \(\text{He}^{+}\) ions but remains insensitive to the change in density of \(\text{O}^{+}\) ions. For oxygen ion gyration, the frequency of wave varies over a short range only for \(\text{O}^{+}\) ion density variation. The wave shows damping at lower altitude due to variation in density of lighter \(\text{H}^{+}\) and \(\text{He}^{+}\) ions whereas at higher altitude only heavy \(\text{O}^{+}\) ions contribute in wave damping. The damping of wave may be due to landau damping or energy transfer from wave to particles. The present study signifies that the both lighter and heavier ions dominate differently to change the characteristics of kinetic Alfven wave and density variation is also an important parameter to understand wave phenomena in cusp region.     

The Cadmium Zinc Telluride Imager on AstroSat

The Cadmium Zinc Telluride Imager (CZTI) is a high energy, wide-field imaging instrument on AstroSat. CZTI’s namesake Cadmium Zinc Telluride detectors cover an energy range from 20 keV to \(>200\) keV, with 11% energy resolution at 60 keV. The coded aperture mask attains an angular resolution of 17\(^\prime \) over a 4.6\(^\circ \) \(\times \) 4.6\(^\circ \)  (FWHM) field-of-view. CZTI functions as an open detector above 100 keV, continuously sensitive to GRBs and other transients in about 30% of the sky. The pixellated detectors are sensitive to polarization above \(\sim \)100 keV, with exciting possibilities for polarization studies of transients and bright persistent sources. In this paper, we provide details of the complete CZTI instrument, detectors, coded aperture mask, mechanical and electronic configuration, as well as data and products.     

Role of primordial black holes in the direct collapse scenario of supermassive black hole formation at high redshifts

In this paper, we explore the possibility of accreting primordial black holes as the source of heating for the collapsing gas in the context of the direct collapse black hole scenario for the formation of super-massive black holes (SMBHs) at high redshifts, \(z\sim \) 6–7. One of the essential requirements for the direct collapse model to work is to maintain the temperature of the in-falling gas at \(\approx \)10\(^4\) K. We show that even under the existing abundance limits, the primordial black holes of masses \(\gtrsim \)10\(^{-2}M_\odot \), can heat the collapsing gas to an extent that the \(\mathrm{H}_2\) formation is inhibited. The collapsing gas can maintain its temperature at \(10^4\) K till the gas reaches a critical density \(n_{{c}} \,{\approx }\, 10^3~\hbox {cm}^{-3}\), at which the roto-vibrational states of \(\mathrm{H}_2\) approaches local thermodynamic equilibrium and \(\mathrm{H}_2\) cooling becomes inefficient. In the absence of \(\mathrm{H}_2\) cooling, the temperature of the collapsing gas stays at \(\approx \)10\(^4\) K even as it collapses further. We discuss scenarios of subsequent angular momentum removal and the route to find collapse through either a supermassive star or a supermassive disk.     

Viscoelastic tides: models for use in Celestial Mechanics

In this note, by using Smale’s \(\alpha \)-theorem on the convergence of Newton’s method, the \(\alpha \)-sets of convergence of some starters of solving the elliptic Kepler’s equation are derived. For each starter we compute the exact \(\alpha \)-set in the eccentricity-main anomaly \((e,M)\in [0,1)\times [0,\pi ]\), showing that these sets are larger than those derived by Avendaño et al. (Celest Mech Dyn Astron 119:27–44, 2014). Further, new convergence tests based on the Newton–Kantorowitch theorem are given comparing with the derived from Smale’s \(\alpha \)-test.     

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.