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1.
The McGehee's study of the triple collision of the 3-body problem is here applied for the stability of an equilibrium. Let us consider the homogeneous Lagrangian: $$L = \frac{{\dot x^2 + \dot y^2 }}{2} + U(x,y)$$ whereU is polynomial, with degreek. We establish a necessary and sufficient condition onU for the stability of \(\omega (x = y = \dot x = \dot y = 0)\) .  相似文献   

2.
The planar problem of three bodies is described by means of Murnaghan's symmetric variables (the sidesa j of the triangle and an ignorable angle), which directly allow for the elimination of the nodes. Then Lemaitre's regularized variables \(\alpha _j = \sqrt {(\alpha ^2 - \alpha _j )}\) , where \(\alpha ^2 = \tfrac{1}{2}(a_1 + a_2 + a_3 )\) , as well as their canonically conjugated momenta are introduced. By finally applying McGehee's scaling transformation \(\alpha _j = r^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}} \tilde \alpha _j\) , wherer 2 is the moment of inertia a system of 7 differential equations (with 2 first integrals) for the 5-dimensional triple collision manifold \(T\) is obtained. Moreover, the zero angular momentum solutions form a 4-dimensional invariant submanifold \(N \subset T\) represented by 6 differential equations with polynomial right-hand sides. The manifold \(N\) is of the topological typeS 2×S 2 with 12 points removed, and it contains all 5 restpoint (each one in 8 copies). The flow on \(T\) is gradient-like with a Lyapounov function stationary in the 40 restpoints. These variables are well suited for numerical studies of planar triple collision.  相似文献   

3.
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4.
We examine the possibility that the observed cosmic-ray protons are of primary extragalactic origin. The present \(\bar p\) data are consistent with a primary extragalactic component having \(\bar p\) /p?3.2±0.7 x 10-4 independent of energy. Following the suggestion that most extragalactic cosmic rays are from active galaxies, we propose that most of the observed \(\bar p\) 's are alos from the same sites. This would imply the possibility of destroying the corresponding \(\bar \alpha \) 'sat the source, thus leading to a flux ratio \(\bar \alpha \) /α< \(\bar p\) /p. We further predict an estimate for \(\bar \alpha \) α～10-5, within the range of future cosmic-ray detectors. the cosmological implications of this proposal are discussed.  相似文献   

5.
New photoelectric UBVRI observations of the eclipsing variable V 1016 Ori have been obtained with the AZT-11 telescope at Crimean Astrophysical Observatory and with the Zeiss-600 telescope at Mount Maidanak Observatory. Light curves are constructed from the new observations and from published and archival data. We use a total of 340, 348, 386, 185, and 62 magnitude estimates in the bands from U to I, respectively. An analysis of these data has yielded the following results. The photometric elements were refined; their new values are $Min I = JDH 2441966.820 + 65\mathop .\limits^d 4331E$ . The UBVRI magnitudes outside eclipse were found to be $5\mathop .\limits^m 95$ , $6\mathop .\limits^m 77$ , $6\mathop .\limits^m 75$ , $6\mathop .\limits^m 68$ , and $6\mathop .\limits^m 16$ , respectively. No phase effect was detected. We obtained two light-curve solutions: (1) assuming that the giant star was in front of the small one during eclipse, we determined the stellar radii, r s=0.0141 and r g=0.0228 (in fractions of the semimajor axis of the orbit); and (2) assuming that the small star was in front of the giant one, we derived r g=0.0186 and r s=0.0180 for the V band. The brightness of the primary star in the bands from U to I is L 1=0.96, 0.92, 0.90, 0.89, and 0.88, the orbital inclination is $i = 87^\circ .1$ , and the maximum eclipse phase is α0= 0.66. In both cases, we accepted the U hypothesis, assumed the orbit to be elliptical, and took into account the flux from the star Θ1 Ori E that fell within the photometer aperture. The first solution leads to a discrepancy between the primary radius determined by solving the light curve and the radial-velocity curve and its value estimated from the luminosity and temperature. This discrepancy is eliminated in the second solution, and it turns out that, by all parameters, the primary corresponds to a normal zero-age main-sequence star.  相似文献   

6.
Using γ-ray data detected by Fermi Large Area Telescope (LAT) and multi-wave band data for 35 TeV blazars sample, we have studied the possible correlations between different broad band spectral indices ( $\alpha_{\rm r.ir}$ , $\alpha_{\rm{r.o}}$ , $\alpha_{\rm r.x}$ , $\alpha_{\rm r.\gamma}$ , $\alpha_{\rm{ir.o}}$ , $\alpha_{\rm ir.x}$ , $\alpha_{\rm ir.\gamma}$ , $\alpha_{\rm o.x}$ , $\alpha_{\rm o.\gamma}$ , $\alpha_{\rm r.x}$ , $\alpha_{\rm x.\gamma}$ ) in all states (average/high/low). Our results are as follows: (1) For our TeV blazars sample, the strong positive correlations were found between $\alpha_{\rm r.ir}$ and $\alpha_{\rm{r.o}}$ , between $\alpha_{\rm r.ir}$ and $\alpha_{\rm r.x}$ , between $\alpha_{\rm r.ir}$ and $\alpha_{\rm r.\gamma}$ in all states (average/high/low); (2) For our TeV blazars sample, the strong anti-correlations were found between $\alpha_{\rm r.ir}$ and $\alpha_{\rm x.\gamma}$ , between $\alpha_{\rm{r.o}}$ and $\alpha_{\rm ir.\gamma}$ , between $\alpha_{\rm{r.o}}$ and $\alpha_{\rm o.\gamma}$ , between $\alpha_{\rm{r.o}}$ and $\alpha_{\rm x.\gamma}$ , between $\alpha_{\mathrm{ir.o}}$ and $\alpha_{\rm o.\gamma}$ , between $\alpha_{\rm r.x}$ and $\alpha_{\rm x.\gamma}$ , between $\alpha_{\rm ir.x}$ and $\alpha_{\rm x.\gamma}$ in all states (average/high/low). The results suggest that the synchrotron self-Compton radiation (SSC) is the main mechanism of high energy γ-ray emission and the inverse Compton scattering of circum-nuclear dust is likely to be a important complementary mechanism for TeV blazars. Our results also show that the possible correlations vary from state to state in the same pair of indices, Which suggest that there may exist differences in the emitting process and in the location of the emitting region for different states.  相似文献   

7.
In 1982 and 1993, we carried out highly accurate photoelectric WBVR measurements for the close binary IT Cas. Based on these measurements and on the observations of other authors, we determined the apsidal motion $\left[ {\dot \omega _{obs} = {{(11\mathop .\limits^ \circ 0 \pm 2\mathop .\limits^ \circ 5)} \mathord{\left/ {\vphantom {{(11\mathop .\limits^ \circ 0 \pm 2\mathop .\limits^ \circ 5)} {100 years}}} \right. \kern-0em} {100 years}}} \right]$ . This value is in agreement with the theoretically calculated apsidal motion for these stars $\left[ {\dot \omega _{th} = {{(14^\circ \pm 3^\circ )} \mathord{\left/ {\vphantom {{(14^\circ \pm 3^\circ )} {100 years}}} \right. \kern-0em} {100 years}}} \right]$ .  相似文献   

8.
We present our photometric observations of an early B supergiant with an infrared excess, the protoplanetary object LSIV-12°111, and the previously suspected variable star NSV 24971. We confirm its photometric variability. During two observing seasons (2000–2001), the star exhibited rapid irregular light variations with amplitudes $\Delta V \sim 0\mathop .\limits^m 3$ , $\Delta B \sim 0\mathop .\limits^m 3$ , and $\Delta U \sim 0\mathop .\limits^m 4$ and a time scale of ～1d. There is no correlation between the colors and magnitudes of the star. The variability patterns of LSIV-12°111 and two other hot post-AGB stars, V886 Her and V1853 Cyg, are shown to be similar.  相似文献   

9.
We present photoelectric and photographic observations of the supergiant HD 179821 with a large infrared excess, a candidate for protoplanetary objects. Over, ten years of our UBV observations, the star exhibited semiregular light variations with amplitudes $\Delta V = 0\mathop .\limits^m 10$ , $\Delta B = 0\mathop .\limits^m 15$ , and $\Delta U = 0\mathop .\limits^m 25$ , as well as systematic color and light variations. From 1990 until 1996, the yearly mean U-B and B-V color indices decreased by 0.25 and 0.15, respectively. After 1996, the motion of the star in the two-color (B-V)-(U-B) diagram upward and to the left slowed down. The color excess that we derived from our observations, by assuming that the star’s spectral type was F3 I in the 1990s, is E(B-V)=1.0. The photographic observations of HD 179821 from 1899 until 1989 show that its brightness m pg generally increased while significantly fluctuating. An analysis of the observational data suggests that HD 179821 is most likely a post-AGB star of intermediate or low mass.  相似文献   

10.
We investigate the dynamics of two satellites with masses $\mu _s$ and $\mu '_s$ orbiting a massive central planet in a common plane, near a first order mean motion resonance $m+1{:}m$ (m integer). We consider only the resonant terms of first order in eccentricity in the disturbing potential of the satellites, plus the secular terms causing the orbital apsidal precessions. We obtain a two-degrees-of-freedom system, associated with the two critical resonant angles $\phi = (m+1)\lambda ' -m\lambda - \varpi $ and $\phi '= (m+1)\lambda ' -m\lambda - \varpi '$ , where $\lambda $ and $\varpi $ are the mean longitude and longitude of periapsis of $\mu _s$ , respectively, and where the primed quantities apply to $\mu '_s$ . We consider the special case where $\mu _s \rightarrow 0$ (restricted problem). The symmetry between the two angles $\phi $ and $\phi '$ is then broken, leading to two different kinds of resonances, classically referred to as corotation eccentric resonance (CER) and Lindblad eccentric Resonance (LER), respectively. We write the four reduced equations of motion near the CER and LER, that form what we call the CoraLin model. This model depends upon only two dimensionless parameters that control the dynamics of the system: the distance $D$ between the CER and LER, and a forcing parameter $\epsilon _L$ that includes both the mass and the orbital eccentricity of the disturbing satellite. Three regimes are found: for $D=0$ the system is integrable, for $D$ of order unity, it exhibits prominent chaotic regions, while for $D$ large compared to 2, the behavior of the system is regular and can be qualitatively described using simple adiabatic invariant arguments. We apply this model to three recently discovered small Saturnian satellites dynamically linked to Mimas through first order mean motion resonances: Aegaeon, Methone and Anthe. Poincaré surfaces of section reveal the dynamical structure of each orbit, and their proximity to chaotic regions. This work may be useful to explore various scenarii of resonant capture for those satellites.  相似文献   

11.
A linear analysis of the asymmetries in Stokes profiles of magnetic lines is performed. The asymmetries in the linear and circular polarization profiles are characterized by suitable quantities, \(\delta \tilde Q\) and \(\delta \tilde V\) , strictly related to observed profiles. The response functions of \(\delta \tilde Q\) and \(\delta \tilde V\) to velocity fields are introduced and computed for various configurations of the magnetic field vector in a Milne-Eddington atmosphere. Some conclusions are drawn as to the importance of the asymmetries in Stokes profiles for recovering the velocity gradients from observations.  相似文献   

12.
In the now classical Lindblad-Lin density-wave theory, the linearization of the collisionless Boltzmann equation is made by assuming the potential functionU expressed in the formU=U 0 + \(\tilde U\) +... WhereU 0 is the background axisymmetric potential and \(\tilde U<< U_0 \) . Then the corresponding density distribution is \(\rho = \rho _0 + \tilde \rho (\tilde \rho<< \rho _0 )\) and the linearized equation connecting \(\tilde U\) and the component \(\tilde f\) of the distribution function is given by $$\frac{{\partial \tilde f}}{{\partial t}} + \upsilon \frac{{\partial \tilde f}}{{\partial x}} - \frac{{\partial U_0 }}{{\partial x}} \cdot \frac{{\partial \tilde f}}{{\partial \upsilon }} = \frac{{\partial \tilde U}}{{\partial x}}\frac{{\partial f_0 }}{{\partial \upsilon }}.$$ One looks for spiral self-consistent solutions which also satisfy Poisson's equation $$\nabla ^2 \tilde U = 4\pi G\tilde \rho = 4\pi G\int {\tilde f d\upsilon .} $$ Lin and Shu (1964) have shown that such solutions exist in special cases. In the present work, we adopt anopposite proceeding. Poisson's equation contains two unknown quantities \(\tilde U\) and \(\tilde \rho \) . It could be completelysolved if a second independent equation connecting \(\tilde U\) and \(\tilde \rho \) was known. Such an equation is hopelesslyobtained by direct observational means; the only way is to postulate it in a mathematical form. In a previouswork, Louise (1981) has shown that Poisson's equation accounted for distances of planets in the solar system(following to the Titius-Bode's law revised by Balsano and Hughes (1979)) if the following relation wasassumed $$\rho ^2 = k\frac{{\tilde U}}{{r^2 }} (k = cte).$$ We now postulate again this relation in order to solve Poisson's equation. Then, $$\nabla ^2 \tilde U - \frac{{\alpha ^2 }}{{r^2 }}\tilde U = 0, (\alpha ^2 = 4\pi Gk).$$ The solution is found in a classical way to be of the form $$\tilde U = cte J_v (pr)e^{ - pz} e^{jn\theta } $$ wheren = integer,p =cte andJ v (pr) = Bessel function with indexv (v 2 =n 2 + α2). By use of the Hankel function instead ofJ v (pr) for large values ofr, the spiral structure is found to be given by $$\tilde U = cte e^{ - pz} e^{j[\Phi _v (r) + n\theta ]} , \Phi _v (r) = pr - \pi /2(v + \tfrac{1}{2}).$$ For small values ofr, \(\tilde U\) = 0: the center of a galaxy is not affected by the density wave which is onlyresponsible of the spiral structure. For various values ofp,n andv, other forms of galaxies can be taken into account: Ring, barred and spiral-barred shapes etc. In order to generalize previous calculations, we further postulateρ 0 =kU 0/r 2, leading to Poisson'sequation which accounts for the disc population $$\nabla ^2 U_0 - \frac{{\alpha ^2 }}{{r^2 }}U_0 = 0.$$ AsU 0 is assumed axisymmetrical, the obvious solution is of the form $$U_0 = \frac{{cte}}{{r^v }}e^{ - pz} , \rho _0 = \frac{{cte}}{{r^{2 + v} }}e^{ - pz} .$$ Finally, Poisson's equation is completely solvable under the assumptionρ =k(U/r 2. The general solution,valid for both disc and spiral arm populations, becomes $$U = cte e^{ - pz} \left\{ {r^{ - v} + } \right.\left. {cte e^{j[\Phi _v (r) + n\theta ]} } \right\},$$ The density distribution along the O z axis is supported by Burstein's (1979) observations.  相似文献   

13.
A statistical study is carried out on the photospheric magnetic nonpotentiality in solar active regions and its relationship with associated flares. We select 2173 photospheric vector magnetograms from 1106 active regions observed by the Solar Magnetic Field Telescope at Huairou Solar Observing Station, National Astronomical Observatories of China, in the period of 1988??C?2008, which covers most of the 22nd and 23rd solar cycles. We have computed the mean planar magnetic shear angle ( $\overline{\Delta\phi}$ ), mean shear angle of the vector magnetic field ( $\overline{\Delta\psi}$ ), mean absolute vertical current density ( $\overline{|J_{z}|}$ ), mean absolute current helicity density ( $\overline{|h_{\mathrm{c}}|}$ ), absolute twist parameter (|?? av|), mean free magnetic energy density ( $\overline{\rho_{\mathrm{free}}}$ ), effective distance of the longitudinal magnetic field (d E), and modified effective distance (d Em) of each photospheric vector magnetogram. Parameters $\overline{|h_{\mathrm{c}}|}$ , $\overline{\rho_{\mathrm{free}}}$ , and d Em show higher correlations with the evolution of the solar cycle. The Pearson linear correlation coefficients between these three parameters and the yearly mean sunspot number are all larger than 0.59. Parameters $\overline {\Delta\phi}$ , $\overline{\Delta\psi}$ , $\overline{|J_{z}|}$ , |?? av|, and d E show only weak correlations with the solar cycle, though the nonpotentiality and the complexity of active regions are greater in the activity maximum periods than in the minimum periods. All of the eight parameters show positive correlations with the flare productivity of active regions, and the combination of different nonpotentiality parameters may be effective in predicting the flaring probability of active regions.  相似文献   

14.
This paper provides a method for judging growth or decay of the magnetic field of pulsar by using pulse period P, or frequency ν, and its first and second derivatives $ \dot P,\ddot P $ or $ \dot v,\ddot v $ . The author uses this method to judge the growth or decay of the magnetic field of Crab pulsar. The judged result for Crab pulsar is that the magnetic field of Crab pulsar is growing now, but it is not decaying. The result corresponds with the actual case of Crab pulsar.  相似文献   

15.
We analyzed 186 binary pulsars (BPSRs) in the magnetic field versus spin period (B-P) diagram, where their relations to the millisecond pulsars (MSPs) can be clearly shown. Generally, both BPSRs and MSPs are believed to be recycled and spun-up in binary accreting phases, and evolved below the spin-up line setting by the Eddington accretion rate ( $\dot{M}{\simeq}10^{18}~\mbox{g/s}$ ). It is noticed that most BPSRs are distributed around the spin-up line with mass accretion rate $\dot{M}=10^{16}~\mbox{g/s}$ and almost all MSP samples lie above the spin-up line with $\dot{M}\sim10^{15}~\mbox{g/s}$ . Thus, we calculate that a minimum accretion rate ( $\dot{M}\sim10^{15}~\mbox{g/s}$ ) is required for the MSP formation, and physical reasons for this are proposed. In the B-P diagram, the positions of BPSRs and their relations to the binary parameters, such as the companion mass, orbital period and eccentricity, are illustrated and discussed. In addition, for the seven BPSRs located above the limit spin-up line, possible causes are suggested.  相似文献   

16.
The number of equivalence classes of central configurations of $n \le 4$ bodies of positive mass is known to be finite, but it remains to be shown if this is true for $n \ge 5$ . By allowing one mass to be negative, Gareth Roberts constructed a continuum of inequivalent planar central configurations of $n = 5$ bodies. We reinterpret Roberts’ example and generalize the construction of his continuum to produce a family of continua of central configurations, each with a single negative mass. These new continua exist in even dimensional spaces $\mathbb R ^k$ for $k \ge 4$ .  相似文献   

17.
Multi-colourWBVR photoelectric observations of the eclipsing binary AS Cam have been carried out and the photometric elements, absolute dimensions, and the angular velocity of a periastron motion ( \(\mathop \omega \limits^ \cdot _{obs}\) ) are determined. The obtained value of \(\mathop \omega \limits^ \cdot _{obs}\) is almost three times smaller than that theoretically predicted.  相似文献   

18.
We obtain an approximate solution $\tilde{E}=\tilde{E}(e,M)$ of Kepler’s equation $E-e\sin (E)=M$ for any $e\in [0,1)$ and $M\in [0,\pi ]$ . Our solution is guaranteed, via Smale’s $\alpha $ -theory, to converge to the actual solution $E$ through Newton’s method at quadratic speed, i.e. the $n$ -th iteration produces a value $E_n$ such that $|E_n-E|\le (\frac{1}{2})^{2^n-1}|\tilde{E}-E|$ . The formula provided for $\tilde{E}$ is a piecewise rational function with conditions defined by polynomial inequalities, except for a small region near $e=1$ and $M=0$ , where a single cubic root is used. We also show that the root operation is unavoidable, by proving that no approximate solution can be computed in the entire region $[0,1)\times [0,\pi ]$ if only rational functions are allowed in each branch.  相似文献   

19.
We obtained U BV R photometric and spectroscopic observations during the outburst of V838 Mon. Before its outburst, the B brightness of the star had been stable ( $\tilde15.^m 85$ ) for 45 years. This was a blue star with the color index $(B - V)_0 = - 0\mathop .\limits^m 03 \pm 0\mathop .\limits^m 1$ and may have been a cataclysmic variable. In the middle of March 2002, the outburst amplitude reached $8\mathop .\limits^m 1$ in B. The star has the counterpart V 1006/7 in M 31 in luminosity at maximum and in spectrum. The unusual spectrum at the premaximum stage originated in the expanding photosphere of a cool K-type giant. The expansion velocity of the photosphere was 150 km s?1; the maximum velocity in the expanding stellar envelope reached 500 km s?1. The absorption components of neutral metal lines were enhanced by a factor of 3 or 4 compared to a normal K-type star. No overabundance of s-process elements was found. One day before the brightness peak, an intense Hα emission line with broad wings, FWZI=3100 km s?1, and numerous lines of ionized metals appeared in V838 Mon, which is characteristic of normal classical novae. We show light, color, and spectral variations of the object.  相似文献   

20.
We constrain holographic dark energy (HDE) with time varying gravitational coupling constant in the framework of the modified Friedmann equations using cosmological data from type Ia supernovae, baryon acoustic oscillations, cosmic microwave background radiation and X-ray gas mass fraction. Applying a Markov Chain Monte Carlo (MCMC) simulation, we obtain the best fit values of the model and cosmological parameters within 1σ confidence level (CL) in a flat universe as: $\varOmega_{b}h^{2}=0.0222^{+0.0018}_{-0.0013}$ , $\varOmega_{c}h^{2}=0.1121^{+0.0110}_{-0.0079}$ , $\alpha_{G}\equiv \dot{G}/(HG) =0.1647^{+0.3547}_{-0.2971}$ and the HDE constant $c=0.9322^{+0.4569}_{-0.5447}$ . Using the best fit values, the equation of state of the dark component at the present time w d0 at 1σ CL can cross the phantom boundary w=?1.  相似文献   

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