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.     

Variability of hot supergiant IRAS 19336-0400 in the early phase of planetary nebula ionization

We present photoelectric and spectral observations of a hot candidate proto-planetary nebula—early B-type supergiant with emission lines in spectrum—IRAS 19336-0400. The light and color curves display fast irregular brightness variations with maximum amplitudes \(\Delta V = 0_ \cdot ^m 30\), \(\Delta B = 0_ \cdot ^m 35\), \(\Delta U = 0_ \cdot ^m 40\) and color-brightness correlations. By the variability characteristics IRAS 19336-0400 appears similar to other hot proto-planetary nebulae. Based on low-resolution spectra in the range λ4000–7500 Å we have derived absolute intensities of the emission lines Hα, Hβ, Hγ, [S II], [N II], physical conditions in gaseous nebula: n _e = 10⁴ cm^?3, T _e = 7000 ± 1000 K. The emission line Hα, Hβ equivalent widths are found to be considerably variable and related to light changes. By UBV-photometry and spectroscopy the color excess has been estimated: E _B-V = 0.50–0.54. Joint photometric and spectral data analysis allows us to assume that the star variability is caused by stellar wind variations.     

Laplace series for the level ellipsoid of revolution

The outer gravitational potential V of the level ellipsoid of revolution T is uniquely determined by two quantities: the eccentricity \(\varepsilon \) of the ellipsoid and Clairaut parameter q, proportional to the angular velocity of rotation squared and inversely proportional to the mean density of the ellipsoid. Quantities \(\varepsilon \) and q are independent, though they lie in a rather strict two-dimensional domain. It follows that Stokes coefficients \(I_n\) of Laplace series representing the outer potential of T are uniquely determined by \(\varepsilon \) and q. In this paper, we have found explicit expressions for Stokes coefficients via \(\varepsilon \) and q, as well as their asymptotics when \(n\rightarrow \infty \). If T does not coincide with a Maclaurin ellipsoid, then \(|I_n|\sim B\varepsilon ^n/n\) with a certain constant B. Let us compare this asymptotics with one of \(I_n\) for ellipsoids constrained by the only condition of increasing (even nonstrict) of oblateness from the centre to the periphery: \(|I_n|\sim \bar{B}\varepsilon ^n/(n^2)\). Hence, level ellipsoids with ellipsoidal equidensites do not exist. The only exception represents Maclaurin ellipsoids. It should be recalled that we confine ourselves by ellipsoids of revolution.     

Instability of radial modes at the Wolf-Rayet evolutionary stage

The evolution of a Population-I star with an initial mass M _ZAMS = 60 M _⊙ has been calculated. At the stage when a red giant turns into an early-type helium star, the vast bulk of the stellar mass is concentrated in a compact core surrounded by an extended envelope that is unstable with respect to radial oscillations. The range of effective temperatures within which the instability arises extends to T _eff ? 10⁵ K. For the models corresponding to the Wolf-Rayet evolutionary stage (5 × 10⁴ K ≤ T _eff ≤ 1.05 × 10⁵ K), hydrodynamic calculations of self-exciting radial stellar pulsations have been performed. The pulsational instability develops in a time interval comparable to the dynamic timescale. Once the amplitude has ceased to grow, the pulsational motions are nonlinear traveling waves propagating from the core boundary to the stellar surface. The velocity amplitude of the outer layers is 500 km s^?1 < ΔU < 10³ km s^?1, depending on the effective temperature. During the evolution of a helium star, the mean ratio of the maximum expansion velocity of the outer layers to the local escape velocity decreases and lies within the range 0.25 < U _max/v _esc < 0.6 for the models considered. The nonlinearity of the stellar pulsations is responsible for the increase in the mean radius \(\bar r\) of the Lagrangian layers compared to the equilibrium radius r _eq. The effect of the increase in mean radius decreases with rising effective temperature from\(\bar r\)/r ～ 10 at T _eff = 7 × 10⁴ K to \(\bar r\)/r ≈ 2 at T _eff = 10⁵ K. The radial pulsation periods for the models considered lie within the range 0.1 day ≤ Π ≤ 1.6 day and the amplitude of the bolometric magnitude variations does not exceed 0 _. ^m 2.     

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.     

Radius of the neutron star magnetosphere during disk accretion

The dependence of the spin frequency derivative \(\dot \nu \) of accreting neutron stars with a strongmagnetic field (X-ray pulsars) on the mass accretion rate (bolometric luminosity, L_bol) has been investigated for eight transient pulsars in binary systems with Be stars. Using data from the Fermi/GBM and Swift/BAT telescopes, we have shown that for seven of the eight systems the dependence \(\dot \nu \) (L_bol) can be fitted by the model of angular momentum transfer through an accretion disk, which predicts the relation \(\dot \nu \) ～ L^6/7_bol. Hysteresis in the dependence \(\dot \nu \) (L_bol) has been confirmed in the system V 0332+53 and has been detected for the first time in the systems KS 1947+300, GRO J1008-57, and 1A 0535+26. Estimates for the radius of the neutron star magnetosphere in all of the investigated systems have been obtained. We show that this quantity varies from pulsar to pulsar and depends strongly on the analytical model and the estimates for the neutron star and binary system parameters.     

Dynamics of Large-Scale Solar-Wind Streams Obtained by the Double Superposed Epoch Analysis: 3. Deflection of the Velocity Vector

This work is a continuation of our previous articles (Yermolaev et al. in J. Geophys. Res.120, 7094, 2015 and Yermolaev et al. in Solar Phys.292, 193, 2017), which describe the average temporal profiles of interplanetary plasma and field parameters in large-scale solar-wind (SW) streams: corotating interaction regions (CIRs), interplanetary coronal mass ejections (ICMEs, including both magnetic clouds (MCs) and ejecta), and sheaths as well as interplanetary shocks (ISs). Changes in the longitude angle, \(\varphi\), in CIRs from ?2 to \(2^{\circ}\) agree with earlier results (e.g. Gosling and Pizzo, 1999). We have also analyzed the average temporal profiles of the bulk velocity angles in sheaths and ICMEs. We have found that the angle \(\varphi\) in ICMEs changes from 2 to \(-2^{\circ}\), while in sheaths it changes from ?2 to \(2^{\circ}\) (similar to the change in CIRs), i.e. the angle in CIRs and sheaths deflects in the opposite sense to ICMEs. When averaging the latitude angle \(\vartheta\) on all the intervals of the chosen SW types, the angle \(\vartheta\) is almost constant at \({\sim}\,1^{\circ}\). We made for the first time a selection of SW events with increasing and decreasing \(\vartheta\) and found that the average \(\vartheta\) temporal profiles in the selected events have the same “integral-like” shape as for \(\varphi\). The difference in \(\varphi\) and \(\vartheta\) average profiles is explained by the fact that most events have increasing profiles for the angle in the ecliptic plane as a result of solar rotation, while for the angle in the meridional plane, the numbers of events with increasing and decreasing profiles are equal.     

STEREO observations of hybrid stars V775 Tau and V483 Tau

In this study, pulsational and physical characteristics of two \(\delta\) Scuti stars, V775?Tau and V483?Tau, are analysed by using four-year high-precision photometric data of the STEREO satellite. Thus, it is aimed to gain new insights into behaviours of these pulsators and evolution of \(\delta\) Scuti, \(\gamma\) Dor and Am type stars. The data are taken between 2007–2011 and examined with the help of the Lomb–Scargle method. The detection precision in the four-year combined data is around \(10^{-5}\) cd^?1 in frequency and \(10^{-5}\) mag in amplitude. It is revealed that V775?Tau exhibits weak pulsation characteristic which is interpreted as the existence of the interaction between the helium loss in the partial ionization zone and pulsation intensities. It is also considered that the absence of strong pulsations is also related to the evolution status of the star. Further, its periodogram shows low-frequency peaks. If these oscillations are g-modes, V775?Tau can be thought to be one of the rare stars that show all \(\gamma\) Dor, \(\delta\) Scuti and Am type variations. V483?Tau is comparatively more luminous, hotter and has higher rotational velocity. Therefore, although it shares the same region with V775?Tau in the H–R diagram, it is not considered to be an Am star. Yet, it exactly overlaps with the \(\gamma\) Dor stars. These clues as well as g-modes detected in its periodogram indicate that V483?Tau is a hybrid star. Finally, both V775?Tau and V483?Tau display period changes whose rates are between \(10^{-3}\) and \(10^{-4}\) yr^?1. Considering the \(\delta\) Scuti nature, it may be speculated that these changes are non-evolutionary.     

Correlations Between CME Parameters and Sunspot Activity

Smoothed monthly mean coronal mass ejection (CME) parameters (speed, acceleration, central position angle, angular width, mass, and kinetic energy) for Cycle 23 are cross-analyzed, showing that there is a high correlation between most of them. The CME acceleration (a) is highly correlated with the reciprocal of its mass (M), with a correlation coefficient r=0.899. The force (Ma) to drive a CME is found to be well anti-correlated with the sunspot number (R _z), r=?0.750. The relationships between CME parameters and R _z can be well described by an integral response model with a decay time scale of about 11 months. The correlation coefficients of CME parameters with the reconstructed series based on this model (\(\overline{r}_{\mathrm{f1}}=0.886\)) are higher than the linear correlation coefficients of the parameters with R _z (\(\overline{r}_{\mathrm{0}}=0.830\)). If a double decay integral response model is used (with two decay time scales of about 6 and 60 months), the correlations between CME parameters and R _z improve (\(\overline{r}_{\mathrm{f2}}=0.906\)). The time delays between CME parameters with respect to R _z are also well predicted by this model (19/22=86%); the average time delays are 19 months for the reconstructed and 22 months for the original time series. The model implies that CMEs are related to the accumulation of solar magnetic energy. These relationships can help in understanding the mechanisms at work during the solar cycle.     

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).     

Neutral Hydrogen and Its Emission Lines in the Solar Corona

Since the Lyman-\(\alpha\) rocket observations of Gabriel (Solar Phys. 21, 392, 1971), it has been realized that the hydrogen (H) lines could be observed in the corona and that they offer an interesting diagnostic for the temperature, density, and radial velocity of the coronal plasma. Moreover, various space missions have been proposed to measure the coronal magnetic and velocity fields through polarimetry in H lines. A necessary condition for such measurements is to benefit from a sufficient signal-to-noise ratio. The aim of this article is to evaluate the emission in three representative lines of H for three different coronal structures. The computations have been performed with a full non-local thermodynamic-equilibrium (non-LTE) code and its simplified version without radiative transfer. Since all collisional and radiative quantities (including incident ionizing and exciting radiation) are taken into account, the ionization is treated exactly. Profiles are presented at two heights (1.05 and 1.9 solar radii, from Sun center) in the corona, and the integrated intensities are computed at heights up to five solar radii. We compare our results with previous computations and observations (e.g. L\(\alpha\) from Ultraviolet Coronal Spectrometer) and find a rough (model-dependent) agreement. Since the H\(\alpha\) line is a possible candidate for ground-based polarimetry, we show that in order to detect its emission in various coronal structures, it is necessary to use a very narrow (less than 2 Å wide) bandpass filter.     

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 \(L_5\).     

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

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

Secular resonances between bodies on close orbits: a case study of the Himalia prograde group of jovian irregular satellites

The gravitational interaction between two objects on similar orbits can effect noticeable changes in the orbital evolution even if the ratio of their masses to that of the central body is vanishingly small. Christou (Icarus 174:215–229, 2005) observed an occasional resonant lock in the differential node \(\varDelta \varOmega \) between two members in the Himalia irregular satellite group of Jupiter in the N-body simulations (corresponding mass ratio \(\sim 10^{-9}\)). Using a semianalytical approach, we have reproduced this phenomenon. We also demonstrate the existence of two additional types of resonance, involving angle differences \(\varDelta \omega \) and \(\varDelta (\varOmega +\varpi )\) between two group members. These resonances cause secular oscillations in eccentricity and/or inclination on timescales \(\sim \)1 Myr. We locate these resonances in (a, e, i) space and analyse their topological structure. In subsequent N-body simulations, we confirm these three resonances and find a fourth one involving \(\varDelta \varpi \). In addition, we study the occurrence rates and the stability of the four resonances from a statistical perspective by integrating 1000 test particles for 100 Myr. We find \(\sim \)10 to 30 librators for each of the resonances. Particularly, the nodal resonance found by Christou is the most stable: 2 particles are observed to stay in libration for the entire integration.     

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.     

Measurement of coronal properties of Seyfert galaxies from NuSTAR’s hard X-ray spectrum

Precise measurement of the coronal properties of Active Galactic Nuclei (AGN) requires the availability of high signal-to-noise ratio data covering a wide range of X-ray energies. The Nuclear Spectroscopic Telescope Array (NuSTAR) which is highly sensitive to earlier missions in its operational energy range of 3–79 keV, allows us to arrive at precise estimates of the coronal parameters such as cut-off energy (\(E_\mathrm{cut}\)), coronal temperature (\(\textit{kT}_e\)) and geometry of the corona at least for sources that have \(E_\mathrm{cut}\) within the energy range of NuSTAR. In this paper, we present our preliminary results on the spectral analysis of two Seyfert galaxies namely 3C 120 and NGC 4151 using NuSTAR observations in the 3–79 keV band. We investigated the continuum and coronal parameters, the photon index \(\Gamma \), \(E_\mathrm{cut}\) and \(\textit{kT}_{e}\). By fitting the X-ray spectrum of 3C 120 and NGC 4151 with a simple phenomenological model, we found that both the sources showed a clear cut-off in their spectrum.     

Intermittency of the Solar Magnetic Field and Solar Magnetic Activity Cycle

Small-scale solar magnetic fields demonstrate features of fractal intermittent behavior, which requires quantification. For this purpose we investigate how the observational estimate of the solar magnetic flux density \(B\) depends on resolution \(D\) in order to obtain the scaling \(\ln B_{D} = - k \ln D +a\) in a reasonably wide range. The quantity \(k\) demonstrates cyclic variations typical of a solar activity cycle. In addition, \(k\) depends on the magnetic flux density, i.e. the ratio of the magnetic flux to the area over which the flux is calculated, at a given instant. The quantity \(a\) demonstrates some cyclic variation, but it is much weaker than in the case of \(k\). The scaling obtained generalizes previous scalings found for the particular cycle phases. The scaling is typical of fractal structures. In our opinion, the results obtained trace small-scale action in the solar convective zone and its coexistence with the conventional large-scale solar dynamo based on differential rotation and mirror-asymmetric convection.     

GONG p-Mode Parameters Through Two Solar Cycles

We investigate the parameters of global solar p-mode oscillations, namely damping width \(\Gamma\), amplitude \(A\), mean squared velocity \(\langle v^{2}\rangle\), energy \(E\), and energy supply rate \(\mathrm{d}E/\mathrm{d}t\), derived from two solar cycles’ worth (1996?–?2018) of Global Oscillation Network Group (GONG) time series for harmonic degrees \(l=0\,\mbox{--}\,150\). We correct for the effect of fill factor, apparent solar radius, and spurious jumps in the mode amplitudes. We find that the amplitude of the activity-related changes of \(\Gamma\) and \(A\) depends on both frequency and harmonic degree of the modes, with the largest variations of \(\Gamma\) for modes with \(2400~\upmu\mbox{Hz}\le\nu\le3300~\upmu\mbox{Hz}\) and \(31\le l \le60\) with a minimum-to-maximum variation of \(26.6\pm0.3\%\) and of \(A\) for modes with \(2400~\upmu\mbox{Hz}\le\nu\le 3300~\upmu\mbox{Hz}\) and \(61\le l \le100\) with a minimum-to-maximum variation of \(27.4\pm0.4\%\). The level of correlation between the solar radio flux \(F_{10.7}\) and mode parameters also depends on mode frequency and harmonic degree. As a function of mode frequency, the mode amplitudes are found to follow an asymmetric Voigt profile with \(\nu_{\text{max}}=3073.59\pm0.18~\upmu\mbox{Hz}\). From the mode parameters, we calculate physical mode quantities and average them over specific mode frequency ranges. In this way, we find that the mean squared velocities \(\langle v^{2}\rangle\) and energies \(E\) of p modes are anticorrelated with the level of activity, varying by \(14.7\pm0.3\%\) and \(18.4\pm0.3\%\), respectively, and that the mode energy supply rates show no significant correlation with activity. With this study we expand previously published results on the temporal variation of solar p-mode parameters. Our results will be helpful to future studies of the excitation and damping of p modes, i.e., the interplay between convection, magnetic field, and resonant acoustic oscillations.     

Secular tidal changes in lunar orbit and Earth rotation

Small tidal forces in the Earth–Moon system cause detectable changes in the orbit. Tidal energy dissipation causes secular rates in the lunar mean motion n, semimajor axis a, and eccentricity e. Terrestrial dissipation causes most of the tidal change in n and a, but lunar dissipation decreases eccentricity rate. Terrestrial tidal dissipation also slows the rotation of the Earth and increases obliquity. A tidal acceleration model is used for integration of the lunar orbit. Analysis of lunar laser ranging (LLR) data provides two or three terrestrial and two lunar dissipation parameters. Additional parameters come from geophysical knowledge of terrestrial tides. When those parameters are converted to secular rates for orbit elements, one obtains dn/dt = \(-25.97\pm 0.05 ''/\)cent\(^{2}\), da/dt = 38.30 ± 0.08 mm/year, and di/dt = ?0.5 ± 0.1 \(\upmu \)as/year. Solving for two terrestrial time delays and an extra de/dt from unspecified causes gives \(\sim \) \(3\times 10^{-12}\)/year for the latter; solving for three LLR tidal time delays without the extra de/dt gives a larger phase lag of the N2 tide so that total de/dt = \((1.50 \pm 0.10)\times 10^{-11}\)/year. For total dn/dt, there is \(\le \)1 % difference between geophysical models of average tidal dissipation in oceans and solid Earth and LLR results, and most of that difference comes from diurnal tides. The geophysical model predicts that tidal deceleration of Earth rotation is \(-1316 ''\)/cent\(^{2}\) or 87.5 s/cent\(^{2}\) for UT1-AT, a 2.395 ms/cent increase in the length of day, and an obliquity rate of 9 \(\upmu \)as/year. For evolution during past times of slow recession, the eccentricity rate can be negative.     

$$M_{\bullet } - \sigma $$ relation in spherical systems

To investigate the \(M_\bullet -\sigma \) relation, we consider realistic elliptical galaxy profiles that are taken to follow a single power-law density profile given by \(\rho (r) = \rho _{0}(r/ r_{0})^{-\gamma }\) or the Nuker intensity profile. We calculate the density using Abel’s formula in the latter case by employing the derived stellar potential; in both cases. We derive the distribution function f(E) of the stars in the presence of the supermassive black hole (SMBH) at the center and hence compute the line-of-sight (LoS) velocity dispersion as a function of radius. For the typical range of values for masses of SMBH, we obtain \(M_{\bullet } \propto \sigma ^{p}\) for different profiles. An analytical relation \(p = (2\gamma + 6)/(2 + \gamma )\) is found which is in reasonable agreement with observations (for \(\gamma = 0.75{-}1.4\), \(p = 3.6{-}5.3\)). Assuming that a proportionality relation holds between the black hole mass and bulge mass, \(M_{\bullet } =f M_\mathrm{b}\), and applying this to several galaxies, we find the individual best fit values of p as a function of f; also by minimizing \(\chi ^{2}\), we find the best fit global p and f. For Nuker profiles, we find that \(p = 3.81 \pm 0.004\) and \(f = (1.23 \pm 0.09)\times 10^{-3}\) which are consistent with the observed ranges.