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1.
In this note a study of the convergence properties of some starters \( E_0 = E_0(e,M)\) in the eccentricity–mean anomaly variables for solving the elliptic Kepler’s equation (KE) by Newton’s method is presented. By using a Wang Xinghua’s theorem (Xinghua in Math Comput 68(225):169–186, 1999) on best possible error bounds in the solution of nonlinear equations by Newton’s method, we obtain for each starter \( E_0(e,M)\) a set of values \( (e,M) \in [0, 1) \times [0, \pi ]\) that lead to the q-convergence in the sense that Newton’s sequence \( (E_n)_{n \ge 0}\) generated from \( E_0 = E_0(e,M)\) is well defined, converges to the exact solution \(E^* = E^*(e,M)\) of KE and further \( \vert E_n - E^* \vert \le q^{2^n -1}\; \vert E_0 - E^* \vert \) holds for all \( n \ge 0\). This study completes in some sense the results derived by Avendaño et al. (Celest Mech Dyn Astron 119:27–44, 2014) by using Smale’s \(\alpha \)-test with \(q=1/2\). Also since in KE the convergence rate of Newton’s method tends to zero as \( e \rightarrow 0\), we show that the error estimates given in the Wang Xinghua’s theorem for KE can also be used to determine sets of q-convergence with \( q = e^k \; \widetilde{q} \) for all \( e \in [0,1)\) and a fixed \( \widetilde{q} \le 1\). Some remarks on the use of this theorem to derive a priori estimates of the error \( \vert E_n - E^* \vert \) after n Kepler’s iterations are given. Finally, a posteriori bounds of this error that can be used to a dynamical estimation of the error are also obtained.  相似文献   

2.
Based on the kinetic theory, Landau damping of dust acoustic waves (DAWs) propagating in a dusty plasma composed of hybrid nonthermal nonextensive distributed electrons, Maxwellian distributed ions and negatively charged dust grains is investigated using Vlasov-Poisson’s equations. The characteristics of the DAWs Landau damping are discussed. It is found that the wave frequency increases by decreasing (increasing) the value of nonextensive (nonthermal) parameter, \(q\) (\(\alpha \)). It is recognized that \(\alpha \) plays a significant role in observing damping or growing DAW oscillations. For small values of \(\alpha \), damping modes have been observed until reaching a certain value of \(\alpha \) at which \(\omega _{i}\) vanishes, then a growing mode appears in the case of superextensive electrons. However, only damping DAW modes are observed in case of subextensive electrons. The present study is useful in the space situations where such distribution exists.  相似文献   

3.
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.  相似文献   

4.
X-ray stars have been studied since the beginning of X-ray astronomy. Investigating and studying the chromospheric activity from X-ray stellar optical spectra is highly significant in providing insights into stellar magnetic activity. The big data of LAMOST survey provides an opportunity for researching stellar optical spectroscopic properties of X-ray stars. We inferred the physical properties of X-ray stellar sources from the analysis of LAMOST spectra. First, we cross-matched the X-ray stellar catalogue (12254 X-ray stars) from ARXA with LAMOST data release 3 (DR3), and obtained 984 good spectra from 713 X-ray sources. We then visually inspected and assigned spectral type to each spectrum and calculated the equivalent width (EW) of H\(\alpha\) line using the Hammer spectral typing facility. Based on the EW of H\(\alpha\) line, we found 203 spectra of 145 X-ray sources with H\(\alpha\) emission above the continuum. For these spectra we also measured the EWs of H\(\beta\), H\(\gamma\), H\(\delta\) and Ca ii IRT lines of these spectra. After removing novae, planetary nebulae and OB-type stars, we found there are 127 X-ray late-type stars with H\(\alpha\) line emission. By using our spectra and results from the literature, we found 53 X-ray stars showing H\(\alpha\) variability; these objects are Classical T Tauri stars (CTTs), cataclysmic variables (CVs) or chromospheric activity stars. We also found 18 X-ray stars showing obvious emissions in the Ca ii IRT lines. Of the 18 X-ray stars, 16 are CTTs and 2 are CVs. Finally, we discussed the relationships between the EW of H\(\alpha\) line and X-ray flux.  相似文献   

5.
In extremely dense neutrino environments like in supernova core, the neutrino-neutrino refraction may give rise to self-induced flavor conversion. These neutrino flavor oscillations are well understood from the idea of the exponentially growing modes of the interacting oscillators in the flavor space. Until recently, the growth rates of these modes were found to be of the order of the vacuum oscillation frequency \(\Delta m^2/2E\) [\(\mathcal {O}(1~\mathrm{km}^{-1})\)] and were considered slow growing. However, in the last couple of years it was found that if the system was allowed to have different zenith-angle distributions for the emitted \(\nu _e\) and \(\bar{\nu }_e\) beams then the fastest growing modes of the interacting oscillators grew at the order of \(\mu =\sqrt{2} G_\mathrm{F}n_{\nu }\), a typical \(\nu \)–\(\nu \) interaction energy [\(\mathcal {O}(10^5~\mathrm{km}^{-1})\)]. Thus the growth rates are very large in comparison to the so-called ‘slow oscillations’ and can result in neutrino flavor conversion on a much faster scale. In fact, the point that the growth rates are no longer dependent on the vacuum oscillation frequency \(\Delta m^2/2E\), makes these ‘fast flavor conversions’ independent of \(\Delta m^2\) (thus mass) and energy. This is a surprising result as neutrino flavor conversions are considered to be the ultimate proof of massive neutrinos. However, the importance of this effect in the realistic astrophysical scenarios still remains to be understood.  相似文献   

6.
We use the data for the \(\text{H}\beta\) emission-line, far-ultraviolet (FUV) and mid-infrared 22 μm continuum luminosities to estimate star formation rates \(\langle \mbox{SFR} \rangle \) averaged over the galaxy lifetime for a sample of about 14000 bursting compact star-forming galaxies (CSFGs) selected from the Data Release 12 (DR12) of the Sloan Digital Sky Survey (SDSS). The average coefficient linking \(\langle \mbox{SFR} \rangle \) and the star formation rate \(\mbox{SFR}_{0}\) derived from the \(\text{H}\beta\) luminosity at zero starburst age is found to be 0.04. We compare \(\langle \mbox{SFR} \rangle \mbox{s}\) with some commonly used SFRs which are derived adopting a continuous star formation during a period of \({\sim}\,100~\mbox{Myr}\), and find that the latter ones are 2–3 times higher. It is shown that the relations between SFRs derived using a geometric mean of two star-formation indicators in the UV and IR ranges and reduced to zero starburst age have considerably lower dispersion compared to those with single star-formation indicators. We suggest that our relations for \(\langle \mbox{SFR} \rangle \) determination are more appropriate for CSFGs because they take into account a proper temporal evolution of their luminosities. On the other hand, we show that commonly used SFR relations can be applied for approximate estimation within a factor of \({\sim}\,2\) of the \(\langle \mbox{SFR} \rangle \) averaged over the lifetime of the bursting compact galaxy.  相似文献   

7.
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.  相似文献   

8.
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.  相似文献   

9.
This paper deals with the photo-gravitational restricted four-body problem (PR4BP) with variable mass. Following the procedure given by Gascheau (C. R. 16:393–394, 1843) and Routh (Proc. Lond. Math. Soc. 6:86–97, 1875), the conditions of linear stability of Lagrange triangle solution in the PR4BP are determined. The three radiating primaries having masses \(m_{1}\), \(m_{2}\) and \(m_{3}\) in an equilateral triangle with \(m_{2}=m_{3}\) will be stable as long as they satisfy the linear stability condition of the Lagrangian triangle solution. We have derived the equations of motion of the mentioned problem and observed that there exist eight libration points for a fixed value of parameters \(\gamma (\frac{m \ \text{at time} \ t}{m \ \text{at initial time}}, 0<\gamma\leq1 )\), \(\alpha\) (the proportionality constant in Jeans’ law (Astronomy and Cosmogony, Cambridge University Press, Cambridge, 1928), \(0\leq\alpha\leq2.2\)), the mass parameter \(\mu=0.005\) and radiation parameters \(q_{i}, (0< q_{i}\leq1, i=1, 2, 3)\). All the libration points are non-collinear if \(q_{2}\neq q_{3}\). It has been observed that the collinear and out-of-plane libration points also exist for \(q_{2}=q_{3}\). In all the cases, each libration point is found to be unstable. Further, zero velocity curves (ZVCs) and Newton–Raphson basins of attraction are also discussed.  相似文献   

10.
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.  相似文献   

11.
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.  相似文献   

12.
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.  相似文献   

13.
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.  相似文献   

14.
In this paper we address an \(n+1\)-body gravitational problem governed by the Newton’s laws, where n primary bodies orbit on a plane \(\varPi \) and an additional massless particle moves on the perpendicular line to \(\varPi \) passing through the center of mass of the primary bodies. We find a condition for the described configuration to be possible. In the case when the primaries are in a rigid motion, we classify all the motions of the massless particle. We study the situation when the massless particle has a periodic motion with the same minimal period as the primary bodies. We show that this fact is related to the existence of a certain pyramidal central configuration.  相似文献   

15.
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.  相似文献   

16.
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.  相似文献   

17.
ALMA is now fully operational, and has been observing in early science mode since 2011. The millimetric (mm) and sub-mm domain is ideal to tackle galaxies at high redshift, since the emission peak of the dust at 100 \(\upmu \)m is shifted in the ALMA bands (0.3–1 mm) for \(z=\) 2–9, and the CO lines, stronger at the high-J levels of the ladder, are found all over the 0.3–3 mm range. Pointed surveys and blind deep fields have been observed, and the wealth of data collected reveal a drop at high redshifts (\(z>6\)) of dusty massive objects, although surprisingly active and gas-rich objects have been unveiled through gravitational lensing. The window of the reionization epoch is now wide open, and ALMA has detected galaxies at \(z=8\)–9 mainly in continuum, [CII] and [OIII] lines. Galaxies have a gas fraction increasing steeply with redshift, as \((1+z)^2\), while their star formation efficiency increases also but more slightly, as \((1+z)^{0.6}\) to \((1+z)^1\). Individual object studies have revealed luminous quasars, with black hole masses much higher than expected, clumpy galaxies with resolved star formation rate compatible with the Kennicutt–Schmidt relation, extended cold and dense gas in a circumgalactic medium, corresponding to Lyman-\(\alpha \) blobs, and proto-clusters, traced by their brightest central galaxies.  相似文献   

18.
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.  相似文献   

19.
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.  相似文献   

20.
We use a formulation of the N-body problem in spaces of constant Gaussian curvature, \({\kappa }\in \mathbb {R}\), as widely used by A. Borisov, F. Diacu and their coworkers. We consider the restricted three-body problem in \(\mathbb {S}^2\) with arbitrary \({\kappa }>0\) (resp. \(\mathbb {H}^2\) with arbitrary \({\kappa }<0\)) in a formulation also valid for the case \({\kappa }=0\). For concreteness when \({\kappa }>0\) we restrict the study to the case of the three bodies at the upper hemisphere, to be denoted as \(\mathbb {S}^2_+\). The main goal is to obtain the totality of relative equilibria as depending on the parameters \({\kappa }\) and the mass ratio \(\mu \). Several general results concerning relative equilibria and its stability properties are proved analytically. The study is completed numerically using continuation from the \({\kappa }=0\) case and from other limit cases. In particular both bifurcations and spectral stability are also studied. The \(\mathbb {H}^2\) case is similar, in some sense, to the planar one, but in the \(\mathbb {S}^2_+\) case many differences have been found. Some surprising phenomena, like the coexistence of many triangular-like solutions for some values \(({\kappa },\mu )\) and many stability changes will be discussed.  相似文献   

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