To the description of long-term variations in the solar magnetic flux: The sunspot area index

We show that the Wolf sunspot numbers W and the group sunspot numbers GSN are physically different indices of solar activity and that it is improper to compare them. Based on the approach of the so-called “primary” indices from the observational series of W(t) and GSN(t), we suggest series of yearly mean sunspot areas beginning in 1610 and monthly mean sunspot areas beginning in 1749.     

Solar cycles during the Maunder minimum

We have obtained new consistent versions of the 400-yr time series of the Wolf sunspot number W, the sunspot group number G, and the total sunspot area S (or the total sunspot magnetic flux Φ). We show that the 11-yr cycle did not cease during the Maunder minimum of solar activity. The characteristics of the extrema of individual 11-yr cycles in 1600–2005 have been determined in terms of the total sunspot area index. We provide arguments for using alternating (“magnetic”) time series of indices in investigating the solar cyclicity.     

Systematic error of the Gaia DR1 TGAS parallaxes from data for the red giant clump

Based on the Gaia DR1 TGAS parallaxes and photometry from the Tycho-2, Gaia, 2MASS, andWISE catalogues, we have produced a sample of ~100 000 clump red giants within ~800 pc of the Sun. The systematic variations of the mode of their absolute magnitude as a function of the distance, magnitude, and other parameters have been analyzed. We show that these variations reach 0.7 mag and cannot be explained by variations in the interstellar extinction or intrinsic properties of stars and by selection. The only explanation seems to be a systematic error of the Gaia DR1 TGAS parallax dependent on the square of the observed distance in kpc: 0.18R ² mas. Allowance for this error reduces significantly the systematic dependences of the absolute magnitude mode on all parameters. This error reaches 0.1 mas within 800 pc of the Sun and allows an upper limit for the accuracy of the TGAS parallaxes to be estimated as 0.2 mas. A careful allowance for such errors is needed to use clump red giants as “standard candles.” This eliminates all discrepancies between the theoretical and empirical estimates of the characteristics of these stars and allows us to obtain the first estimates of the modes of their absolute magnitudes from the Gaia parallaxes: mode(M _H ) = ?1.49 ^m ± 0.04 ^m , mode(M _Ks ) = ?1.63 ^m ± 0.03 ^m , mode(M _W1) = ?1.67 ^m ± 0.05 ^m mode(M _W2) = ?1.67 ^m ± 0.05 ^m , mode(M _W3) = ?1.66 ^m ± 0.02 ^m , mode(M _W4) = ?1.73 ^m ± 0.03 ^m , as well as the corresponding estimates of their de-reddened colors.     

Series of classical solar activity indices: Kislovodsk data

We present data on the series of solar activity indices, Wolf sunspot numbers W and total sunspot areas S, obtained at the Kislovodsk high-altitude station of the Pulkovo Observatory. The problem of properly extending the 133-year-long Zürich series of W and the 102-year-long Greenwich series of S, which were discontinued in 1980 and 1976, respectively, is emphasized. We stress that the Kislovodsk data have retained mutual homogeneity with the classical series until now and that they are preferred for extension. The question under consideration is of fundamental importance in studying the solar activity variations on long time scales and related processes in the Sun-Earth system.     

Optimal prediction of the peak of the next 11-year activity cycle and of the peaks of several succeeding cycles on the basis of long-term variations in the solar radius or solar constant

We propose a new technique for the optimal prediction of the peak of the next 11-year activity cycle prior to the cycle beginning and of the peaks of several succeeding cycles on the basis of long-term variations in the solar radius or solar constant. The method is based on the already established fact that the long-term cyclic variations of the activity, radius, and solar constant are correlated in both phase and amplitude, since they are caused by some common processes in the Sun. The peak of the succeeding cycle 24 is expected to have the height W _max = 70 ± 10 (in units of relative sunspot number). The subsequent cycles 25 and 26, which will be formed during the descent of the current secular cycle, will have still lower peaks with the heights W _max = 50 ± 15 and W _max = 35 ± 20.     

Sunspot as an isolated magnetic structure: Stability and oscillations

A simple energy model of a sunspot as a compact magnetic feature is described where the main energy contribution is provided by the coolest and most compressed part of the magnetic force tube of the spot at depths ranging from Wilson’s depression level (300–500 km) down to 2–3 thousand km. The equilibrium and stability conditions for such a system are analyzed using the variation principle, and oscillations of the system as a whole about the inferred equilibrium position are studied. The sunspot is shown to be stable in the magnetic field strength interval from 0.8–1 to 4–5 kG. The dependence of the eigenfrequency on magnetic field strength ω(B) is computed for the main oscillatory mode, where only the umbra of the sunspot takes part in oscillations, ω = ω ₁ (B). Lower subharmonics may appear in the case where penumbra too becomes involved in the oscillatory process: ω ₂ = ω ₁/2, ω ₃ = ω ₁/3. Theoretical curves agree well with the observational data obtained in Pulkovo using various independent methods: from temporal variations of sunspot magnetic field and from line-of-sight-velocity measurements. The periods of oscillations found range from 40 to 200 minutes.     

Kinematics of B-F Stars as a Function of Their Dereddened Color from Gaia and PCRV Data

Parallaxes with an accuracy better than 10% and proper motions from the Gaia DR1 TGAS catalogue, radial velocities from the Pulkovo Compilation of Radial Velocities (PCRV), accurate Tycho-2 photometry, theoretical PARSEC, MIST, YaPSI, BaSTI isochrones, and the most accurate reddening and interstellar extinction estimates have been used to analyze the kinematics of 9543 thin-disk B-F stars as a function of their dereddened color. The stars under consideration are located on the Hertzsprung–Russell diagram relative to the isochrones with an accuracy of a few hundredths of a magnitude, i.e., at the level of uncertainty in the parallax, photometry, reddening, extinction, and the isochrones themselves. This has allowed us to choose the most plausible reddening and extinction estimates and to conclude that the reddening and extinction were significantly underestimated in some kinematic studies of other authors. Owing to the higher accuracy of TGAS parallaxes than that of Hipparcos ones, the median accuracy of the velocity components U, V, W in this study has improved to 1.7 km s^?1, although outside the range ?0.1 ^m < (B _T ? V _T )₀ < 0.5 ^m the kinematic characteristics are noticeably biased due to the incompleteness of the sample. We have confirmed the variations in the mean velocity of stars relative to the Sun and the stellar velocity dispersion as a function of their dereddened color known from the Hipparcos data. Given the age estimates for the stars under consideration from the TRILEGAL model and the Geneva–Copenhagen survey, these variations may be considered as variations as a function of the stellar age. A comparison of our results with the results of other studies of the stellar kinematics near the Sun has shown that selection and reddening underestimation explain almost completely the discrepancies between the results. The dispersions and mean velocities from the results of reliable studies fit into a ±2 km s^?1 corridor, while the ratios σ _V /σ _U and σ _W /σ _U fit into ±0.05. Based on all reliable studies in the range ?0.1 ^m < (B _T ? V _T )₀ < 0.5^m, i.e., for an age from 0.23 to 2.4 Gyr, we have found: W_⊙ = 7.15 km s^?1, \({\sigma _U} = 16.0{e^{1.29({B_T} - {V_T})o}}\), \({\sigma _V} = 10.9{e^{1.11({B_T} - {V_T})o}}\), \({\sigma _W} = 6.8{e^{1.46({B_T} - {V_T})o}}\), the stellar velocity dispersions in km s^?1 are proportional to the age in Gyr raised to the power β _U = 0.33, β _V = 0.285, and β _W = 0.37.     

KDG218, a nearby ultra-diffuse galaxy

We present properties of the low-surface-brightness galaxy KDG218 observed with the HST/ACS. The galaxy has a half-light (effective) diameter of a _e = 47″ and a central surface brightness of SB _V (0) = 24.^m4/□″. The galaxy remains unresolved with the HST/ACS, which implies its distance of D > 13.1 Mpc and linear effective diameter of A _e > 3.0 kpc. We notice that KDG218 is most likely associated with a galaxy group around the massive lenticular NGC4958 galaxy at approximately 22 Mpc, or with the Virgo Southern Extension filament at approximately 16.5 Mpc. At these distances, the galaxy is classified as an ultra-diffuse galaxy (UDG) similar to those found in the Virgo, Fornax, and Coma clusters. We also present a sample of 15 UDG candidates in the Local Volume. These sample galaxies have the following mean parameters: 〈D〉 = 5.1 Mpc, 〈A _e 〉 = 4.8 kpc, and 〈SB _B (e)〉 = 27.^m4/□″. All the local UDG candidates reside near massive galaxies located in the regions with the mean stellar mass density (within 1 Mpc) about 50 times greater than the average cosmic density. The local fraction of UDGs does not exceed 1.5% of the Local Volume population. We notice that the presented sample of local UDGs is a heterogeneous one containing irregular, transition, and tidal types, as well as objects consisting of an old stellar population.     

On Estimating the Dissipative Factor of the Martian Interior

An attempt is made to construct a trial Q_μ(l) distribution in the silicate mantle of Mars. With the allowance for the fact that on the P–T plane the Earth’s geotherm is close to the distribution of areotherms, it was concluded that Q_μ(l) should be distributed in the Martian interior topologically close to the Q_μ(l) distribution in the Earth. The initial distribution was specified by the four-layer piecewise-constant distribution from the QML9 model. An important step was to select the power index in the frequency dependence of Q_μ. Based on the laboratory data and on the experience of studying this problem for the Earth, n was specified in the interval 0.1–0.3. It was found that with the conversion of the initial distribution to the orbital period of Phobos around Mars, which is the only constraint for the problem derived from the observations, this distribution agrees reasonably well with the observational data at n = 0.1.     

Evolutionary studies of the young star clusters: NGC 1960, NGC 2453 and NGC 2384

We report the analysis of the young star clusters NGC 1960, NGC 2453 and NGC 2384 observed in the J (1.12 μm), H (1.65 μm) and K′ (2.2 μm) bands. Estimates of reddening, distance and age as E(B?V)=0.25, d=1380 pc and t=31.6 to 125 Myr for NGC 1960, E(B?V)=0.47, d=3311 pc and t=40 to 200 Myr for NGC 2453 and E(B?V)=0.25, d=3162 pc and t=55 to 125 Myr for NGC 2384 have been obtained. Also, we have extended the color–magnitude diagrams of these clusters to the fainter end and thus extended the luminosity functions to fainter magnitudes. The evolution of the main sequence and luminosity functions of these clusters have been compared with themselves as well as Lyngå 2 and NGC 1582.     

The matter in the Big-Bang model is dust and not any arbitrary perfect fluid!

We consider a spherically symmetric general relativistic perfect fluid in its comoving frame. It is found that, by integrating the local energy momentum conservation equation, a general form of g ₀₀ can be obtained. During this study, we get a cue that an adiabatically evolving uniform density isolated sphere having ρ(r,t)=ρ ₀(t), should comprise “dust” having p ₀(t)=0; as recently suggested by Durgapal and Fuloria (J. Mod. Phys. 1:143, 2010) In fact, we offer here an independent proof to this effect. But much more importantly, we find that for the homogeneous and isotropic Friedmann-Robertson-Walker (FRW) metric having p(r,t)=p ₀(t) and ρ(r,t)=ρ ₀(t), \(g_{00} = e^{-2p_{0}/(p_{0} +\rho_{0})}\). But in general relativity (GR), one can choose an arbitrary t→t _?=f(t) without any loss of generality, and thus set g ₀₀(t _?)=1. And since pressure is a scalar, this implies that p ₀(t _?)=p ₀(t)=0 in the Big-Bang model based on the FRW metric. This result gets confirmed by the fact the homogeneous dust metric having p(r,t)=p ₀(t)=0 and ρ(r,t)=ρ ₀(t) and the FRW metric are exactly identical. In other words, both the cases correspond to the same Einstein tensor \(G^{a}_{b}\) because they intrinsically have the same energy momentum tensor \(T^{a}_{b}=\operatorname {diag}[\rho_{0}(t), 0,0, 0]\).     

New characteristics of the solar cycle and dynamo theory

Based on an analysis of the observational data for solar cycles 12–23 (Royal Greenwich Observatory-USAF/NOAA Sunspot Data), we have studied various parameters of the “Maunder butterflies.” Based on the observational data for cycles 16–23, we have found that BT/Land S depend linearly on each other, where B is the mean magnetic field of the cycle, T is the cycle duration, S is the cycle strength, and L is the mean sunspot latitude in the cycle (the arithmetic mean of the absolute values of the mean latitudes in the north and south). The connection of the observed quantities with the α-ω-dynamo theory is discussed.     

Multi-product splitting and Runge-Kutta-Nyström integrators

The splitting of e^h(A+B) into a single product of e ^{h A} and e ^hB results in symplectic integrators when A and B are classical Lie operators. However, at high orders, a single product splitting, with exponentially growing number of operators, is very difficult to derive. This work shows that, if the splitting is generalized to a sum of products, then a simple choice of the basis product reduces the problem to that of extrapolation, with analytically known coefficients and only quadratically growing number of operators. When a multi-product splitting is applied to classical Hamiltonian systems, the resulting algorithm is no longer symplectic but is of the Runge-Kutta-Nyström (RKN) type. Multi-product splitting, in conjunction with a special force-reduction process, explains why at orders p = 4 and 6, RKN integrators only need p ? 1 force evaluations.     

Rotation curve and mass distribution in the Galaxy from the velocities of objects at distances up to 200 kpc

Three three-component (bulge, disk, halo) model Galactic gravitational potentials differing by the expression for the dark matter halo are considered. The central (bulge) and disk components are described by the Miyamoto–Nagai expressions. The Allen–Santillán (I), Wilkinson–Evans (II), and Navarro–Frenk–White (III) models are used to describe the halo. A set of present-day observational data in the range of Galactocentric distances R from 0 to 200 kpc is used to refine the parameters of thesemodels. For the Allen–Santillán model, a dimensionless coefficient γ has been included as a sought-for parameter for the first time. In the traditional and modified versions, γ = 2.0 and 6.3, respectively. Both versions are considered in this paper. The model rotation curves have been fitted to the observed velocities by taking into account the constraints on the local matter density ρ _⊙ = 0.1 M _⊙ pc^?3 and the force K _z =1.1/2πG = 77 M _⊙ pc^?2 acting perpendicularly to the Galactic plane. The Galactic mass within a sphere of radius 50 kpc, M _G (R ≤ 50 kpc) ≈ (0.41 ± 0.12) × 10¹² M _⊙, is shown to satisfy all three models. The differences between the models become increasingly significant with increasing radius R. In model I, the Galactic mass within a sphere of radius 200 kpc at γ = 2.0 turns out to be greatest among the models considered, M _G (R ≤ 200 kpc) = (1.45 ±0.30)× 10¹² M _⊙, M _G (R ≤ 200 kpc) = (1.29± 0.14)× 10¹² M _⊙ at γ = 6.3, and the smallest value has been found in model II, M _G (R ≤ 200 kpc) = (0.61 ± 0.12) × 10¹² M _⊙. In our view, model III is the best one among those considered, because it ensures the smallest residual between the data and the constructed model rotation curve provided that the constraints on the local parameters hold with a high accuracy. Here, the Galactic mass is M _G (R ≤ 200 kpc) = (0.75 ± 0.19) × 10¹² M _⊙. A comparative analysis with the models by Irrgang et al. (2013), including those using the integration of orbits for the two globular clusters NGC 104 and NGC 1851 as an example, has been performed. The third model is shown to have subjected to a significant improvement.     

Kinematics of the Galaxy from OB Stars with Data from the Gaia DR2 Catalogue

We have selected and analyzed a sample of OB stars with known line-of-sight velocities determined through ground-based observations and with trigonometric parallaxes and propermotions from the Gaia DR2 catalogue. Some of the stars in our sample have distance estimates made from calcium lines. A direct comparison with the trigonometric distance scale has shown that the calcium distance scale should be reduced by 13%. The following parameters of the Galactic rotation curve have been determined from 495 OB stars with relative parallax errors less than 30%: (U, V,W)_⊙ = (8.16, 11.19, 8.55)± (0.48, 0.56, 0.48) km s^?1, Ω₀ = 28.92 ± 0.39 km s^?1 kpc^?1, Ω'₀ = ?4.087 ± 0.083 km s^?1 kpc^?2, and Ω″ ₀ = 0.703 ± 0.067 km s?1 kpc^?3, where the circular velocity of the local standard of rest is V0 = 231 ± 5 km s^?1 (for the adopted R₀ = 8.0 ± 0.15 kpc). The parameters of the Galactic spiral density wave have been found from the series of radial, V_R, residual tangential, ΔV_circ, and vertical, W, velocities of OB stars by applying a periodogram analysis. The amplitudes of the radial, tangential, and vertical velocity perturbations are f_R = 7.1± 0.3 km s^?1, f_θ = 6.5 ± 0.4 km s^?1, and f_W = 4.8± 0.8 km s^?1, respectively; the perturbation wavelengths are λ_R = 3.3 ± 0.1 kpc, λ_θ = 2.3 ± 0.2 kpc, and λ_W = 2.6 ± 0.5 kpc; and the Sun’s radial phase in the spiral density wave is (χ_⊙)_R = ?135? ± 5?, (χ_⊙)θ = ?123? ± 8?, and (χ_⊙)_W = ?132? ± 21? for the adopted four-armed spiral pattern.     

Correlation function of quasars from SDSS DR3

We calculate the parameters of the two-point correlation function of quasars w(r) = (r _c /r) ^γ on the basis of the SDSS DR3 data. The correlation functions are first determined from projected distances with the use of a special technique for compiling randomized catalogs. Next the parameters of the spatial correlation function are obtained with the assumption of local isotropy. For the quasars with redshifts z = 0.8–2.1, we obtained the estimates γ = 1.76 ± 0.14, r _c = 6.60 ± 0.85 h ^?1 Mpc in the comoving distance range 2–30 Mpc and γ = 1.90 ± 0.11, r _c = 6.95±0.57 h ^?1 Mpc in the range 2–50 Mpc. These estimates agree, within the limits of errors, with the estimates obtained for the redshifts 0.4 < z < 2.1. The original catalog shows some deficit of pairs with separations less than 1 Mpc.     

GJ 900: A new hierarchical system with low-mass components

Speckle interferometric observations made with the 6 m telescope of the Special Astrophysical Observatory of the Russian Academy of Sciences in 2000 revealed the triple nature of the nearby (π _Hip = 51.80 ± 1.74 mas) low-mass young (≈ 200 Myr) star GJ 900. The configuration of the triple system allowed it to be dynamically unstable. Differential photometry performed from 2000 through 2004 yielded I- and K-band absolute magnitudes and spectral types for the components to be I _A =6.66±0.08, I _B =9.15±0.11, I _C =10.08±0.26, K _A =4.84±0.08, K _B =6.76±0.20, K _C =7.39±0.31, Sp _A ≈K5?K7, Sp _B ≈M3?M4, Sp _C ≈M5?M6. The “mass-luminosity” relation is used to estimate the individual masses of the components: M _A ≈0.64M _⊙, M _B ≈0.21M _⊙, M _C ≈0.13M _⊙. From the observations of the components’ relative motion in the period 2000–2006, we conclude that GJ 900 is a hierarchical triple star with the possible orbital periods P_A-BC≈80 yrs and P_BC≈20 yrs. An analysis of the 2MASS images of the region around GJ 900 leads us to suggest that the system can include other very-low-mass components.     

A Comprehensive Study of 94 Open Clusters Based on the Data from IPHAS,GAIA DR2, and Other Sky Surveys

We determine the color excesses, photometric distances, ages, astrometric parallaxes and proper motions for 94 open clusters in the northern part of the Milky Way. We estimate the color excesses and photometric distances based on the data from IPHAS photometric survey of the northern Galactic plane using individual total-to-selective extinction ratios R_r = A_r/E_r?i for each cluster computed via the color-difference method based on IPHAS r, i, and Hα-band, 2MASS J-, H-, and K_s-band, WISE W1-band, and Pan-STARRS i-, z-, and y-band data. The inferred R_r values vary significantly from cluster to cluster spanning the R_r = 3.1–5.2 interval with a mean and standard deviation equal to 〈R_r〉 = 3.99 and σR_r = 0.34, respectively.We identified cluster members using (1) absolute proper motions determined from individual-epoch positions of stars retrieved from IPHAS, 2MASS,URAT1, ALLWISE,UCAC5, and Gaia DR1 catalogs and positions of stars on individual Palomar Sky Survey plates reconstructed based on the data provided in USNO-B1.0 catalog and (2) absolute proper motions provided in Gaia DR2 catalog, and computed the averageGaia DR2 trigonometric parallaxes and propermotions of the clusters. Themean formal error of the inferred astrometric parallaxes of clusters is of about 7 μas, however, a comparison of astrometric and photometric parallaxes of our cluster sample implies that Gaia DR2 parallaxes are, on the average, systematically underestimated by 45 ± 9 μas. This result agrees with estimates obtained by other authors using other objects. At the same time, we find our photometric distance scale to be correct within the quoted errors (the inferred correction factor is equal to unity to within a standard error of 0.025).     

Inverse problem of central configurations and singular curve in the collinear 4-body problem

In this paper, we consider the inverse problem of central configurations of n-body problem. For a given \({q=(q_1, q_2, \ldots, q_n)\in ({\bf R}^d)^n}\), let S(q) be the admissible set of masses denoted \({ S(q)=\{ m=(m_1,m_2, \ldots, m_n)| m_i \in {\bf R}^+, q}\) is a central configuration for m}. For a given \({m\in S(q)}\), let S _m (q) be the permutational admissible set about m = (m ₁, m ₂, . . . , m _n ) denoted

$S_m(q)=\{m^\prime | m^\prime\in S(q),m^\prime \not=m \, {\rm and} \, m^\prime\,{\rm is\, a\, permutation\, of }\, m \}.$

The main discovery in this paper is the existence of a singular curve \({\bar{\Gamma}_{31}}\) on which S _m (q) is a nonempty set for some m in the collinear four-body problem. \({\bar{\Gamma}_{31}}\) is explicitly constructed by a polynomial in two variables. We proved:

1. (1)
   If \({m\in S(q)}\), then either ^# S _m (q) = 0 or ^# S _m (q) = 1.
    
2. (2)
   ^#S _m (q) = 1 only in the following cases:

   1. (i)
      If s = t, then S _m (q) = {(m ₄, m ₃, m ₂, m ₁)}.
       
   2. (ii)
      If \({(s,t)\in \bar{\Gamma}_{31}\setminus \{(\bar{s},\bar{s})\}}\), then either S _m (q) = {(m ₂, m ₄, m ₁, m ₃)} or S _m (q) = {(m ₃, m ₁, m ₄, m ₂)}.
       
    

     

Spectroscopic and photometric characteristics of stars in the field of the cluster vdB 130

The observational data for 24 stars toward the young cluster vdB 130 are analyzed. The spectroscopic and photometric BV RIJHK observations have been carried out with the following telescopes: 6-m at the Special Astrophysical Observatory of the Russian Academy of Sciences, 60-cm at the Southern Station of the Moscow State University, and 2.5-m at the Caucasus Observatory of the Sternberg Astronomical Institute of theMoscow State University. Nine stars previously selected as cluster members have been found to belong to different subtypes of type B. A minimum color excess toward the cluster, E(B ? V) = 0.9 mag, has been revealed for the vdB 130 stars lying outside the molecular cloud. Maximum color excesses, E(B ? V) = 1.3?1.4 mag, have been found in the spectra of cluster stars 1r and 5r observed in dust blobs. Inside the cluster R _v is shown to differ from the standard one. The overwhelming majority of the remaining investigated stars belong to late types and have minor color excesses (≤0.3) typical of close distances.