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1.
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   

2.
The Ideal Resonance Problem, defined by the Hamiltonian $$F = B(y) + 2\mu ^2 A(y)\sin ^2 x,\mu \ll 1,$$ has been solved in Garfinkelet al. (1971). As a perturbed simple pendulum, this solution furnishes a convenient and accurate reference orbit for the study of resonance. In order to preserve the penduloid character of the motion, the solution is subject to thenormality condition, which boundsAB" andB' away from zero indeep and inshallow resonance, respectively. For a first-order solution, the paper derives the normality condition in the form $$pi \leqslant max(|\alpha /\alpha _1 |,|\alpha /\alpha _1 |^{2i} ),i = 1,2.$$ Herep i are known functions of the constant ‘mean element’y', α is the resonance parameter defined by $$\alpha \equiv - {\rm B}'/|4AB\prime \prime |^{1/2} \mu ,$$ and $$\alpha _1 \equiv \mu ^{ - 1/2}$$ defines the conventionaldemarcation point separating the deep and the shallow resonance regions. The results are applied to the problem of the critical inclination of a satellite of an oblate planet. There the normality condition takes the form $$\Lambda _1 (\lambda ) \leqslant e \leqslant \Lambda _2 (\lambda )if|i - tan^{ - 1} 2| \leqslant \lambda e/2(1 + e)$$ withΛ 1, andΛ 2 known functions of λ, defined by $$\begin{gathered} \lambda \equiv |\tfrac{1}{5}(J_2 + J_4 /J_2 )|^{1/4} /q, \hfill \\ q \equiv a(1 - e). \hfill \\ \end{gathered}$$   相似文献   

3.
The spheroidal harmonics expressions $$\left[ {P_{2k}^{2s} \left( {i\xi } \right)P_{2k - 2r}^{2s} \left( \eta \right) - P_{2k - 2r}^{2s} \left( {i\xi } \right)P_{2k}^{2s} \left( \eta \right)} \right]e^{i2s\theta } $$ and $$\left[ {\eta ^2 P_{2k}^{2s} \left( {i\xi } \right)P_{2k - 2r}^{2s} \left( \eta \right) + \xi ^2 P_{2k - 2r}^{2s} \left( {i\xi } \right)P_{2k}^{2s} \left( \eta \right)} \right]e^{i2s\theta } $$ , have ξ2+η2 as a factor. A method is presented for obtaining for these two expressions the coefficient of ξ2+η2 in the form of a linear combination of terms of the formP 2m 2s (iξ)P 2n 2s (η)e i2sθ. Explicit formulae are exhibited for the casesr=1, 2, 3 and any positive or zero integersk ands. Such identities are useful in gravitational potential theory for ellipsoidal distributions when matching Legendre function expansions are employed.  相似文献   

4.
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   

5.
The method of evaluating the photometric perturbationsB 2m of eclipsing variables in the frequency domain, developed by Kopal (1959, 1975e, 1978) for an interpretation of mutual eclipses in systems whose components are distorted by axial rotation and mutual tidal action. The aim of the present paper has been to establish explicit expressions for the photometric perturbationB 2m in such systems, regardless of the kind of eclipses and non-integral values ofm. Recently, Kopal (1978) introduced two different kinds of integrals with respect to associated α-functions andI-integrals which have been expressed in terms of certain general types of series that can be easily programmed for automatic computation within seconds of real time on highspeed computers. Following a brief introduction (Section 1) in which the need of this new approach will be expounded, in Section 3 we shall deduce the integral $$\int_0^{\theta \prime } {\tfrac{{\alpha _n^\prime }}{\delta }} d(sin^{2m} \theta )$$ in terms of a certain general type of series and also β-function, which should enable us to evaluate explicit expressions forf * (h) ,f 1 (h) ,f 2 (h) as well asB 2m .  相似文献   

6.
If \(T = \sum\nolimits_{i = 1}^\infty {\varepsilon ^i } T_i\) and \(W = \sum\nolimits_{n = 1}^\infty {n\varepsilon ^{n - 1} } W^{\left( n \right)}\) are respectively the generators of Giorgilli-Galgani's and Deprit's transformations, we show that the change of variables generated byT is the inverse of the one generated byW, ifT i =W (i) for anyi. The method used is to show that the recurrence which defines the first algorithm can also be obtained with the second one.  相似文献   

7.
We compute the ultra-high energy (UHE) neutrino fluxes from plausible accreting supermassive black holes closely linking to the 377 active galactic nuclei (AGNs). They have well-determined black hole masses collected from the literature. The neutrinos are produced via simple or modified URCA processes, even after the neutrino trapping, in superdense proto-matter medium. The resulting fluxes are ranging from: (1) (quark reactions)— $J^{q}_{\nu\varepsilon}/(\varepsilon_{d}\ \mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1})\simeq8.29\times 10^{-16}$ to 3.18×10?4, with the average $\overline{J}^{q}_{\nu\varepsilon}\simeq5.53\times 10^{-10}\varepsilon_{d}\ \mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}$ , where ε d ～10?12 is the opening parameter; (2) (pionic reactions)— $J^{\pi}_{\nu\varepsilon} \simeq0.112J^{q}_{\nu\varepsilon}$ , with the average $J^{\pi}_{\nu\varepsilon} \simeq3.66\times 10^{-11}\varepsilon_{d}\ \mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}$ ; and (3) (modified URCA processes)— $J^{URCA}_{\nu\varepsilon}\simeq7.39\times10^{-11} J^{q}_{\nu\varepsilon}$ , with the average $\overline{J}^{URCA}_{\nu\varepsilon} \simeq2.41\times10^{-20} \varepsilon_{d}\ \mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}$ . We conclude that the AGNs are favored as promising pure neutrino sources, because the computed neutrino fluxes are highly beamed along the plane of accretion disk, peaked at high energies and collimated in smaller opening angle θ～ε d .  相似文献   

8.
A spherically-symmetric static scalar field in general relativity is considered. The field equations are defined by $$\begin{gathered} R_{ik} = - \mu \varphi _i \varphi _k ,\varphi _i = \frac{{\partial \varphi }}{{\partial x^i }}, \varphi ^i = g^{ik} \varphi _k , \hfill \\ \hfill \\ \end{gathered} $$ where ?=?(r,t) is a scalar field. In the past, the same problem was considered by Bergmann and Leipnik (1957) and Buchdahl (1959) with the assumption that ?=?(r) be independent oft and recently by Wyman (1981) with the assumption ?=?(r, t). The object of this paper is to give explicit results with a different approach and under a more general condition $$\phi _{;i}^i = ( - g)^{ - 1/2} \frac{\partial }{{\partial x^i }}\left[ {( - g)^{1/2} g^{ik} \frac{\partial }{{\partial x^k }}} \right] = - 4\pi ( -g )^{ - 1/2} \rho $$ where ?=?(r, t) is the mass or the charge density of the sources of the field.  相似文献   

9.
A plane-wave analysis on a simplified scheme based on the Boussinesq approximation and shallow convection is used to establish the necessary conditions for stability of a differentiallyrotating, compressible flow between two coaxial cylinders subject to non-axisymmetric perturbations. To test the adequateness of this simplification, the sufficient conditions for stability are again established which agree with those obtained by a normal-mode analysis on an exact scheme in an earlier paper by the author. This model is applicable to stellar models with rotation Ω=Ω(ω), where ω is the radial distance from the axis of rotation (thez-axis). A necessary condition for stability, in the non-dissipative case, is found to be that $$\frac{1}{\varrho }G_\varpi S_\varpi + \frac{{k_z^2 }}{M}\Phi - \frac{1}{4}\frac{{m^2 }}{M}\left( {D\Omega } \right)^2 \geqslant 0$$ everywhere. Here,m andk z are the wave numbers in the ø- andz-direction, \(M \equiv k_z^2 + m^2 /\varpi ^2 ,D \equiv d/d\varpi ,G_\varpi \equiv - \varrho ^{ - 1} Dp,\varrho \) the density,p the pressure,S ω and Φ the Schwarzschild and the Rayleigh discriminants defined as \(S_\varpi \equiv \left( {\gamma p/\varrho } \right)^{ - 2} Dp - D\varrho and \Phi \equiv ^{ - 3} d\left( {\varpi ^4 \Omega ^2 } \right)/d\varpi \) respectively, γ the ratio of specific heats. This condition is also a sufficient one. Some conjectures regarding the stabilizing influence of uniform rotation and the destabilizing influence of differential rotation are also verified. The most striking instability mechanism introduced by shear forces and by radiative dissipation is the excitation of the stable motion of small oscillations into that of oscillations with growing amplitude, i. e., overstability. In the case of radiative dissipation and axisymmetric perturbations, the Goldreich-Schubert criterion is only necessary but not sufficient for stability. Instability sets in as soon as the Schwarzschild criterion is violated. When the perturbations are non-axisymmetric, instability always sets in as overstability as long as rotation is differential. This may explain the convective turbulence in the upper atmosphere where the radiation is active.  相似文献   

10.
It is shown that the fractional increase in binding energy of a galaxy in a fast collision with another galaxy of the same size can be well represented by the formula $$\xi _2 = 3({G \mathord{\left/ {\vphantom {G {M_2 \bar R}}} \right. \kern-\nulldelimiterspace} {M_2 \bar R}}) ({{M_1 } \mathord{\left/ {\vphantom {{M_1 } {V_p }}} \right. \kern-\nulldelimiterspace} {V_p }})^2 e^{ - p/\bar R} = \xi _1 ({{M_1 } \mathord{\left/ {\vphantom {{M_1 } {M_2 }}} \right. \kern-\nulldelimiterspace} {M_2 }})^3 ,$$ whereM 1,M 2 are the masses of the perturber and the perturbed galaxy, respectively,V p is the relative velocity of the perturber at minimum separationp, and \(\bar R\) is the dynamical radius of either galaxy.  相似文献   

11.
The equilibrium points and the curves of zero-velocity (Roche varieties) are analyzed in the frame of the regularized circular restricted three-body problem. The coordinate transformation is done with Levi-Civita generalized method, using polynomial functions of n degree. In the parametric plane, five families of equilibrium points are identified: \(L_{i}^{1}, L_{i}^{2}, \ldots, L_{i}^{n}\) , \(i\in\{ 1,2,\ldots,5 \}, n \in\mathbb{N}^{*}\) . These families of points correspond to the five equilibrium points in the physical plane L 1,L 2,…,L 5. The zero-velocity curves from the physical plane are transformed in Roche varieties in the parametric plane. The properties of these varieties are analyzed and the Roche varieties for n∈{1,2,…,6} are plotted. The equation of the asymptotic variety is obtained and its shape is analyzed. The slope of the Roche variety in \(L_{1}^{1}\) point is obtained. For n=1 the slope obtained by Plavec and Kratochvil (1964) in the physical plane was found.  相似文献   

12.
The fact that the energy density ρg of a static spherically symmetric gravitational field acts as a source of gravity, gives us a harmonic function \(f\left( \varphi \right) = e^{\varphi /c^2 } \) , which is determined by the nonlinear differential equation $$\nabla ^2 \varphi = 4\pi k\rho _g = - \frac{1}{{c^2 }}\left( {\nabla \varphi } \right)^2 $$ Furthermore, we formulate the infinitesimal time-interval between a couple of events measured by two different inertial observers, one in a position with potential φ-i.e., dt φ and the other in a position with potential φ=0-i.e., dt 0, as $${\text{d}}t_\varphi = f{\text{d}}t_0 .$$ When the principle of equivalence is satisfied, we obtain the well-known effect of time dilatation.  相似文献   

13.
The ratio between the Earth's perihelion advance (Δθ) E and the solar gravitational red shift (GRS) (Δø s e)a 0/c 2 has been rewritten using the assumption that the Newtonian constant of gravitationG varies seasonally and is given by the relationship, first found by Gasanalizade (1992b) for an aphelion-perihelion difference of (ΔG)a?p . It is concluded that $$\begin{gathered} (\Delta \theta )_E = \frac{{3\pi }}{e}\frac{{(\Delta \phi _{sE} )_{A_0 } }}{{c^2 }}\frac{{(\Delta G)_{a - p} }}{{G_0 }} = 0.038388 \sec {\text{onds}} {\text{of}} {\text{arc}} {\text{per}} {\text{revolution,}} \hfill \\ \frac{{(\Delta G)_{a - p} }}{{G_0 }} = \frac{e}{{3\pi }}\frac{{(\Delta \theta )_E }}{{(\Delta \phi _{sE} )_{A_0 } /c^2 }} = 1.56116 \times 10^{ - 4} . \hfill \\ \end{gathered} $$ The results obtained here can be readily understood by using the Parametrized Post-Newtonian (PPN) formalism, which predicts an anisotropy in the “locally measured” value ofG, and without conflicting with the general relativity.  相似文献   

14.
The periodic solutions for an Hamiltonian system with $$H = \frac{1}{2}\mathop \Sigma \limits_1^3 (\dot x_\alpha ^2 + \omega _\alpha ^2 x_\alpha ^2 ) - \varepsilon x_1 x_\alpha ^2 - \eta x_2 x_\alpha ^2 $$ are investigated analytically. The frequencies ωα, α=1, 2, 3 are assumed near the ratio 4—4—1. We find different families of periodic solutions whose periods are in the vicinity of the period T′=2π/ω3=2π/ω′. As in the case of the problem with two degrees of liberty, for particular values of ω1, ω2, ω3 and ε, η, we find that the families near the x3-axis are discontinuous. These families are periodic with periods near the period T′ in a region for ε, η, approximatively [0; 0.4] if we choose \(\omega ' = \sqrt {0.1} \) and h=0.00765.  相似文献   

15.
The disk positions for galaxies of various morphological and nuclear-activity types (normal galaxies, QSO, Sy, E/S0, low-surface-brightness galaxies, etc.) on the μ0-h (central surface brightness-exponential disk scale) plane are considered. The stellar disks are shown to form a single sequence on this plane $(SB_0 = 10^{ - 0.4\mu _0 } \propto h^{ - 1} )$ over a wide range of surface brightnesses (μ0(I)≈12–25) and sizes (h≈10–100 kpc). The existence of this observed sequence can probably be explained by a combination of three factors: a disk-stability requirement, a limited total disk luminosity, and observational selection. The model by Mo et al. (1998) for disk formation in the CDM hierarchical-clustering scenario is shown to satisfactorily reproduce the salient features of the galaxy disk distribution on the μ0-h plane.  相似文献   

16.
We present the results of polarimetric observations of the icymoons of Uranus (Ariel, Titania, Oberon, and Umbriel) performed at the 6-m BTA telescope of the SAO RAS with the SCORPIO-2 focal reducer within the phase angle range of $0_.^ \circ 06 - 2_.^ \circ 37$ . The parameters of the negative polarization branch (referred to the scattering plane) are obtained in the V filter: for Ariel the maximum branch depth of P min ≈ ?1.4% is reached at the phase angle of α min ≈ 1°; for Titania P min ≈ ?1.2%, $\alpha _{\min } \approx 1_.^ \circ 4$ ; for Oberon P min ≈ ?1.1%, $\alpha _{\min } \approx 1_.^ \circ 8$ . For Umbriel the polarization minimum was not reached: for the last measurement point at $\alpha _{\min } \approx 2_.^ \circ 4$ , polarization amounts to ?1.7%. The declining P min and shifting αmin towards larger phase angles correlate with a decrease of the geometric albedo of the Uranian moons. There is no longitudinal dependence of polarization for the moons within the observational errors which indicates a similarity in the physical properties of the leading and trailing hemispheres. The phase-angle dependences of polarization for the major moons of Uranus are quite close to those observed in the group of small trans-Neptunian objects (Ixion, Huya, Varuna, 1999 DE9, etc.), which are characterized by a large gradient of negative polarization, about ?1% per degree in the phase-angle range of $0_.^ \circ 1 - 1^ \circ$ .  相似文献   

17.
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   

18.
New theoretical electron-density-sensitive Fe xii emission line ratios $$R_1 = I(3s^2 3p^3 {}^4S_{3/2} - 3s3p^4 {}^4P_{5/2} )/I(3s^2 3p^3 {}^2P_{3/2} - 3s3p^4 D_{5/2} )$$ and $$R_2 = I(3s^2 3p^3 {}^2P_{3/2} - 3s3p^4 {}^2D_{5/2} )/I(3s^2 3p^3 {}^4S_{3/2} - 3s3p^2 P_{3/2} )$$ are derived using R-matrix electron impact excitation rate calculations. We have identified the Fexii \(3s^2 3p^3 {}^4S_{3/2} - 3s3p^4 {}^4P_{5/2} ,{\text{ }}3s^2 3p^3 {}^2P_{3/2} - 3s^3 3p^4 {}^2D_{5/2} ,{\text{ }}3s^2 3p^3 S_{3/2} - 3s^2 3p^3 P_{3/2} \) and \(3s^2 3p^3 {}^4S_{3/2} - 3s^2 3p^3 {}^2P_{1/2}\) transitions in an active region spectrum obtained with the Harvard S-055 spectrometer on board Skylab at wavelengths of 364.0, 382.8, 1241.7, and 1349.4 Å, respectively. Electron densities determined from the observed values of R 1 (log N e ? 11.0) and R 2(log N e ? 11.4) are significantly larger than the typical active region measurements, but are similar to those derived from some active region spectra observed with the Skylab 2082A instrument, which provides observational support for the atomic data adopted in the line ratio calculations, and also for the identification of the Fe xii transitions in the S-055 spectrum. However the observed value of R 3 = I(1349.4 Å)/I(1241.7 Å) is approximately a factor of two larger than one would expect from theory which, considering that the 1349.4 Å line lies at the edge of the S-055 wavelength coverage, may reflect errors in the instrument efficiency curve. Another possibility is that the 1349.4 Å transition is blended, probably with Si ii 1350.1 Å.  相似文献   

19.
The Ideal Resonance Problem is defined by the Hamiltonian $$F = B(y) + 2\varepsilon A(y) \sin ^2 x,\varepsilon \ll 1.$$ The classical solution of the Problem, expanded in powers of ε, carries the derivativeB′ as a divisor and is, therefore, singular at the zero ofB′, associated with resonance. With α denoting theresonance parameter, defined by $$\alpha \equiv - B'/|4AB''|^{1/2} \mu ,\mu = \varepsilon ^{1/2} ,$$ it is shown here that the classical solution is valid only for $$\alpha ^2 \geqslant 0(1/\mu ).$$ In contrast, the global solution (Garfinkelet al., 1971), expanded in powers ofμ=ε1/2, removes the classical singularity atB′=0, and is valid for all α. It is also shown here that the classical solution is an asymptotic approximation, for largeα 2, of the global solution expanded in powers ofα ?2. This result leads to simplified expressions for resonancewidth and resonantamplification. The two solutions are compared with regard to their general behavior and their accuracy. It is noted that the global solution represents a perturbed simple pendulum, while the classical solution is the limiting case of a pendulum in a state offast circulation.  相似文献   

20.
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   

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