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1.
We study the capture and crossing probabilities in the 3:1 mean motion resonance with Jupiter for a small asteroid that migrates from the inner to the middle Main Belt under the action of the Yarkovsky effect. We use an algebraic mapping of the averaged planar restricted three-body problem based on the symplectic mapping of Hadjidemetriou (Celest Mech Dyn Astron 56:563–599, 1993), adding the secular variations of the orbit of Jupiter and non-symplectic terms to simulate the migration. We found that, for fast migration rates, the captures occur at discrete windows of initial eccentricities whose specific locations depend on the initial resonant angles, indicating that the capture phenomenon is not probabilistic. For slow migration rates, these windows become narrower and start to accumulate at low eccentricities, generating a region of mutual overlap where the capture probability tends to 100 %, in agreement with the theoretical predictions for the adiabatic regime. Our simulations allow us to predict the capture probabilities in both the adiabatic and non-adiabatic cases, in good agreement with results of Gomes (Celest Mech Dyn Astron 61:97–113, 1995) and Quillen (Mon Not RAS 365:1367–1382, 2006). We apply our model to the case of the Vesta asteroid family in the same context as Roig et al. (Icarus 194:125–136, 2008), and found results indicating that the high capture probability of Vesta family members into the 3:1 mean motion resonance is basically governed by the eccentricity of Jupiter and its secular variations.  相似文献   

2.
In this paper, an efficient algorithm is established for computing the maximum (minimum) angular separation ρ max(ρ min), the corresponding apparent position angles ( $\theta|_{\rho_{\rm max}}$ , $\theta|_{\rho_{\rm min}}$ ) and the individual masses of visual binary systems. The algorithm uses Reed’s formulae (1984) for the masses, and a technique of one-dimensional unconstrained minimization, together with the solution of Kepler’s equation for $(\rho_{\rm max}, \theta|_{\rho_{\rm max}})$ and $(\rho_{\rm min}, \theta|_{\rho_{\rm min}})$ . Iterative schemes of quadratic coverage up to any positive integer order are developed for the solution of Kepler’s equation. A sample of 110 systems is selected from the Sixth Catalog of Orbits (Hartkopf et al. 2001). Numerical studies are included and some important results are as follows: (1) there is no dependence between ρ max and the spectral type and (2) a minor modification of Giannuzzi’s (1989) formula for the upper limits of ρ max functions of spectral type of the primary.  相似文献   

3.
A formulation of the perturbed two-body problem that relies on a new set of orbital elements is presented. The proposed method represents a generalization of the special perturbation method published by Peláez et al. (Celest Mech Dyn Astron 97(2):131–150, 2007) for the case of a perturbing force that is partially or totally derivable from a potential. We accomplish this result by employing a generalized Sundman time transformation in the framework of the projective decomposition, which is a known approach for transforming the two-body problem into a set of linear and regular differential equations of motion. Numerical tests, carried out with examples extensively used in the literature, show the remarkable improvement of the performance of the new method for different kinds of perturbations and eccentricities. In particular, one notable result is that the quadratic dependence of the position error on the time-like argument exhibited by Peláez’s method for near-circular motion under the $J_{2}$ perturbation is transformed into linear. Moreover, the method reveals to be competitive with two very popular element methods derived from the Kustaanheimo-Stiefel and Sperling-Burdet regularizations.  相似文献   

4.
In this paper we extend the theory of close encounters of a giant planet on a parabolic orbit with a central star developed in our previous work (Ivanov and Papaloizou in MNRAS 347:437, 2004; MNRAS 376:682, 2007) to include the effects of tides induced on the central star. Stellar rotation and orbits with arbitrary inclination to the stellar rotation axis are considered. We obtain results both from an analytic treatment that incorporates first order corrections to normal mode frequencies arising from stellar rotation and numerical treatments that are in satisfactory agreement over the parameter space of interest. These results are applied to the initial phase of the tidal circularisation problem. We find that both tides induced in the star and planet can lead to a significant decrease of the orbital semi-major axis for orbits having periastron distances smaller than 5?C6 stellar radii with tides in the star being much stronger for retrograde orbits compared to prograde orbits. Assuming that combined action of dynamic and quasi-static tides could lead to the total circularisation of orbits this corresponds to observed periods up to 4?C5 days. We use the simple Skumanich law to characterise the rotational history of the star supposing that the star has its rotational period equal to one month at the age of 5 Gyr. The strength of tidal interactions is characterised by circularisation time scale, t ev , which is defined as a typical time scale of evolution of the planet??s semi-major axis due to tides. This is considered as a function of orbital period P obs , which the planet obtains after the process of tidal circularisation has been completed. We find that the ratio of the initial circularisation time scales corresponding to prograde and retrograde orbits, respectively, is of order 1.5?C2 for a planet of one Jupiter mass having P obs ~ 4 days. The ratio grows with the mass of the planet, being of order five for a five Jupiter mass planet with the same P orb . Note, however, this result might change for more realistic stellar rotation histories. Thus, the effect of stellar rotation may provide a bias in the formation of planetary systems having planets on close orbits around their host stars, as a consequence of planet?Cplanet scattering, which favours systems with retrograde orbits. The results reported in the paper may also be applied to the problem of tidal capture of stars in young stellar clusters.  相似文献   

5.
G. J. D. Petrie 《Solar physics》2014,289(10):3663-3680
It is shown that expressions for the global Lorentz force associated with a flaring active region derived by Fisher et al. (Solar Phys. 277, 59, 2012) can be used to estimate the Lorentz-force changes for strong fields in large structures over photospheric subdomains within active regions. Gary’s (Solar Phys. 203, 71, 2001) model for the stratified solar atmosphere is used to demonstrate that in large-scale structures with typical horizontal magnetic length scale ??300 km and with strong magnetic fields (≥?1 kG at the τ=1 opacity layer at 5000 Å), the Lorentz force acting on the photosphere may be approximated by a surface integral based on photospheric boundary data alone. These conditions cover many of the sunspot fields and major neutral lines that have been studied using Fisher et al.’s (2012) expressions over the past few years. The method gives a reasonable estimate of flare-related Lorentz-force changes based on photospheric magnetogram observations provided that the Lorentz-force changes associated with the flare have a lasting effect on the observed fields, and they are not immediately erased by post-flare equilibration processes.  相似文献   

6.
A Hamiltonian model is constructed for the spin axis of a planet perturbed by a nearby planet with both planets in orbit about a star. We expand the planet–planet gravitational potential perturbation to first order in orbital inclinations and eccentricities, finding terms describing spin resonances involving the spin precession rate and the two planetary mean motions. Convergent planetary migration allows the spinning planet to be captured into spin resonance. With initial obliquity near zero, the spin resonance can lift the planet’s obliquity to near 90\(^\circ \) or 180\(^\circ \) depending upon whether the spin resonance is first or zeroth order in inclination. Past capture of Uranus into such a spin resonance could give an alternative non-collisional scenario accounting for Uranus’s high obliquity. However, we find that the time spent in spin resonance must be so long that this scenario cannot be responsible for Uranus’s high obliquity. Our model can be used to study spin resonance in satellite systems. Our Hamiltonian model explains how Styx and Nix can be tilted to high obliquity via outward migration of Charon, a phenomenon previously seen in numerical simulations.  相似文献   

7.
This paper presents a new theory of the dynamical tides of celestial bodies. It is founded on a Newtonian creep instead of the classical delaying approach of the standard viscoelastic theories and the results of the theory derive mainly from the solution of a non-homogeneous ordinary differential equation. Lags appear in the solution but as quantities determined from the solution of the equation and are not arbitrary external quantities plugged in an elastic model. The resulting lags of the tide components are increasing functions of their frequencies (as in Darwin’s theory), but not small quantities. The amplitudes of the tide components depend on the viscosity of the body and on their frequencies; they are not constants. The resulting stationary rotations (often called pseudo-synchronous) have an excess velocity roughly proportional to $6ne^2/(\chi ^2+\chi ^{-2})$ ( $\chi $ is the mean-motion in units of one critical frequency—the relaxation factor—inversely proportional to the viscosity) instead of the exact $6ne^2$ of standard theories. The dissipation in the pseudo-synchronous solution is inversely proportional to $(\chi +\chi ^{-1})$ ; thus, in the inviscid limit, it is roughly proportional to the frequency (as in standard theories), but that behavior is inverted when the viscosity is high and the tide frequency larger than the critical frequency. For free rotating bodies, the dissipation is given by the same law, but now $\chi $ is the frequency of the semi-diurnal tide in units of the critical frequency. This approach fails, however, to reproduce the actual tidal lags on Earth. In this case, to reconcile theory and observations, we need to assume the existence of an elastic tide superposed to the creeping tide. The theory is applied to several Solar System and extrasolar bodies and currently available data are used to estimate the relaxation factor $\gamma $ (i.e. the critical frequency) of these bodies.  相似文献   

8.
We use thermal radiometry and visible photometry to constrain the size, shape, and albedo of the large Kuiper belt object Haumea. The correlation between the visible and thermal photometry demonstrates that Haumea’s high amplitude and quickly varying optical light curve is indeed due to Haumea’s extreme shape, rather than large scale albedo variations. However, the well-sampled high precision visible data we present does require longitudinal surface heterogeneity to account for the shape of lightcurve. The thermal emission from Haumea is consistent with the expected Jacobi ellipsoid shape of a rapidly rotating body in hydrostatic equilibrium. The best Jacobi ellipsoid fit to the visible photometry implies a triaxial ellipsoid with axes of length 1,920 × 1,540 × 990 km and density $2.6$  g cm $^{-3}$ , as found by Lellouch et al. (A&A, 518:L147, 2010. doi:10.1051/0004-6361/201014648). While the thermal and visible data cannot uniquely constrain the full non-spherical shape of Haumea, the match between the predicted and measured thermal flux for a dense Jacobi ellipsoid suggests that Haumea is indeed one of the densest objects in the Kuiper belt.  相似文献   

9.
Subdwarf B stars (sdBs) can significantly change the ultraviolet spectra of populations at age t～1 Gyr, and have been even included in the evolutionary population synthesis (EPS) models by Han et al. (Mon. Not. R. Astron. Soc. 380:1098, 2007). In this study we present the spectral energy distributions (SEDs) of binary stellar populations (BSPs) by combining the EPS models of Han et al. (Mon. Not. R. Astron. Soc. 380:1098, 2007) and those of the Yunnan group (Zhang et al. in Astron. Astrophys. 415:117, 2004; Mon. Not. R. Astron. Soc. 357:1088, 2005), which have included various binary interactions (except sdBs) in EPS models. This set of SEDs is available upon request from the authors. Using this set of SEDs of BSPs we build the spectra of Burst, E, S0–Sd and Irr types of galaxies by using the package of Bruzual and Charlot (Mon. Not. R. Astron. Soc. 344:1000, 2003). Combined with the photometric data (filters and magnitudes), we obtain the photometric redshifts and morphologies of 1502 galaxies by using the Hyperz code of Bolzonella et al. (Astron. Astrophys. 363:476, 2000). This sample of galaxies is obtained by removing those objects, mismatched with the SDSS/DR7 and GALEX/DR4, from the catalogue of Fukugita et al. (Astron. J. 134:579, 2007). By comparison the results with the SDSS spectroscopic redshifts and the morphological index of Fukugita et al. (Astron. J. 134:579, 2007), we find that the photo-z fluctuate with the SDSS spectroscopic redshifts, while the Sa–Sc galaxies in the catalogue of Fukugita et al. (Astron. J. 134:579, 2007) are classified earlier as Burst-E galaxies.  相似文献   

10.
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   

11.
In a previous paper, Hayliet al. (1983), two families of periodic orbits in the three-dimensional potential $$U = \frac{1}{2}(Ax^2 + By^2 + Cz^2 ) - \varepsilon xz^2 - nyz^2 $$ with \(\sqrt A :\sqrt B :\sqrt C = 6:4:3\) and ?=0.5 were described. It was found empirically that the characteristic curves of the two families intersect in the space (x0, y0, η) for |η|?0.2. This property is demonstrated in the present paper by writing explicitely the Poincaré mapping and by giving an approximation directly comparable with the numerical results obtained in Hayliet al. (1983). It is thus shown that one family bifurcates off the other.  相似文献   

12.
The majority of confirmed terrestrial exoplanets orbits close to their host stars and their evolution was likely altered by tidal interaction. Nevertheless, due to their viscoelastic properties on the tidal frequencies, their response cannot be described exactly by standardly employed constant-lag models. We therefore introduce a tidal model based on the numerical evaluation of a continuum mechanics problem describing the deformation of viscoelastic (Maxwell or Andrade) planetary mantles subjected to external force. We apply the method on a model Earth-size planet orbiting a low-mass star and study the effect of the orbital eccentricity, the mantle viscosity and the chosen rheology on the tidal dissipation, the complex Love numbers and the tidal torque. The number of stable spin states (i.e., zero tidal torque) grows with increasing mantle viscosity, similarly to the analytical model of Correia et al. (Astron Astrophys 571:A50, 2014) for homogeneous bodies. This behavior is only slightly influenced by the rheology used. Similarly, the Love numbers do not distinctly depend on the considered rheological model. The increase in viscosity affects the amplitude of their variations. The tidal heating described by the Maxwell rheology attains local minima associated with low spin-orbit resonances, with depth and shape depending on both the eccentricity and the viscosity. For the Andrade rheology, the minima at low resonances are very shallow and the tidal heating for all viscosities resembles a “fluid limit.” The tidal heating is the quantity influenced the most by the rheology, having thus possible impact on the internal thermal evolution.  相似文献   

13.
We develop an analytical Hamiltonian formalism adapted to the study of the motion of two planets in co-orbital resonance. The Hamiltonian, averaged over one of the planetary mean longitudes, is expanded in power series of eccentricities and inclinations. The model, which is valid in the entire co-orbital region, possesses an integrable approximation modeling the planar and quasi-circular motions. First, focusing on the fixed points of this approximation, we highlight relations linking the eigenvectors of the associated linearized differential system and the existence of certain remarkable orbits like the elliptic Eulerian Lagrangian configurations, the anti-Lagrange (Giuppone et al. in MNRAS 407:390–398, 2010) orbits and some second sort orbits discovered by Poincaré. Then, the variational equation is studied in the vicinity of any quasi-circular periodic solution. The fundamental frequencies of the trajectory are deduced and possible occurrence of low order resonances are discussed. Finally, with the help of the construction of a Birkhoff normal form, we prove that the elliptic Lagrangian equilateral configurations and the anti-Lagrange orbits bifurcate from the same fixed point $L_4$ L 4 .  相似文献   

14.
The analytical techniques of the Nekhoroshev theorem are used to provide estimates on the coefficient of Arnold diffusion along a particular resonance in the Hamiltonian model of Froeschlé et al. (Science 289:2108–2110, 2000). A resonant normal form is constructed by a computer program and the size of its remainder ||R opt || at the optimal order of normalization is calculated as a function of the small parameter ${\epsilon}$ . We find that the diffusion coefficient scales as ${D \propto ||R_{opt}||^3}$ , while the size of the optimal remainder scales as ${||R_{opt}|| \propto {\rm exp}(1/\epsilon^{0.21})}$ in the range ${10^{-4} \leq \epsilon \leq 10^{-2}}$ . A comparison is made with the numerical results of Lega et al. (Physica D 182:179–187, 2003) in the same model.  相似文献   

15.
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16.
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   

17.
We study the secular dynamics of lunar orbiters, in the framework of high-degree gravity models. To achieve a global view of the dynamics, we apply a frequency analysis (FA) technique which is based on Prony’s method. This allows for an extensive exploration of the eccentricity ( $e$ )—inclination ( $i$ ) space, based on short-term integrations ( $\sim $ 8 months) over relatively high-resolution grids of initial conditions. Different gravity models are considered: 3rd, 7th and 10th degree in the spherical harmonics expansion, with the main perturbations from the Earth being added. Since the dynamics is mostly regular, each orbit is characterised by a few parameters, whose values are given by the spectral decomposition of the orbital elements time series. The resulting frequency and amplitude maps in ( $e_0,i_0$ ) are used to identify the dominant perturbations and deduce the “minimum complexity” model necessary to capture the essential features of the long-term dynamics. We find that the 7th degree zonal harmonic ( $J_7$ term) is of profound importance at low altitudes as, depending on the initial secular phases, it can lead to collision with the Moon’s surface within a few months. The 3rd-degree non-axisymmetric terms are enough to describe the deviations from the 1 degree-of-freedom zonal problem; their main effect is to modify the equilibrium value of the argument of periselenium, $\omega $ , with respect to the “frozen” solution ( $\omega =\pm 90^{\circ }, \forall \Omega $ , where $\Omega $ is the nodal longitude). Finally, we show that using FA on a fine grid of initial conditions, set around a suitably chosen ‘first guess’, one can compute an accurate approximation of the initial conditions of a periodic orbit.  相似文献   

18.
Limits are placed on the range of orbits and masses of possible moons orbiting extrasolar planets which orbit single central stars. The Roche limiting radius determines how close the moon can approach the planet before tidal disruption occurs; while the Hill stability of the star–planet–moon system determines stable orbits of the moon around the planet. Here the full three-body Hill stability is derived for a system with the binary composed of the planet and moon moving on an inclined, elliptical orbit relative the central star. The approximation derived here in Eq. (17) assumes the binary mass is very small compared with the mass of the star and has not previously been applied to this problem and gives the criterion against disruption and component exchange in a closed form. This criterion was applied to transiting extrasolar planetary systems discovered since the last estimation of the critical separations (Donnison in Mon Not R Astron Soc 406:1918, 2010a) for a variety of planet/moon ratios including binary planets, with the moon moving on a circular orbit. The effects of eccentricity and inclination of the binary on the stability of the orbit of a moon is discussed and applied to the transiting extrasolar planets, assuming the same planet/moon ratios but with the moon moving with a variety of eccentricities and inclinations. For the non-zero values of the eccentricity of the moon, the critical separation distance decreased as the eccentricity increased in value. Similarly the critical separation decreased as the inclination increased. In both cases the changes though very small were significant.  相似文献   

19.
We have applied the close binary system analysis program WinFitter, with its physically detailed fitting function, to an intensive study of the complex multiple system Kepler-13 using photometry data from all 13 short cadence quarters downloaded from the NASA Exoplanet Archive (NEA) (http://exoplanetarchive.ipac.caltech.edu). The data-point error of our normalized, phase-sequenced and binned (380 points per bin: 0.00025 phase interval) flux values, at 14 ppm, allows the model’s specification for the mean reference flux level of the system to a precision better than 1 ppm. Our photometrically derived values for the mass and radius of KOI13.01 are \(6.8\pm0.6~\mbox{M}_{\mathrm{J}}\) and \(1.44\pm0.04~\mbox{R}_{\mathrm{J}}\). The star has a radius of \(1.67\pm0.05~\mbox{R}_{\odot}\). Our modelling sets the mean of the orbital inclination \(i\) at \(94.35\pm0.14^{\circ}\), with the star’s mean precession angle \(\phi_{p}\)—\(49.1\pm5.0^{\circ}\) and obliquity \(\theta_{o}\)\(67.9 \pm 3.0^{\circ}\), though there are known ambiguities about the sense in which such angles are measured.Our findings did not confirm secular variation in the transit modelling parameters greater than their full correlated errors, as argued by previous authors, when each quarter’s data was best-fitted with a determinable parameter set without prejudice. However, if we accept that most of the parameters remain the same for each transit, then we could confirm a small but steady diminution in the cosine of the orbital inclination over the 17 quarter timespan. This is accompanied by a slight increase of the star’s precession angle (less negative), but with no significant change in the obliquity of its spin axis. There are suggestions of a history of strong dynamical interaction with a highly distorted planet rotating in a 3:2 resonance with its revolution, together with a tidal lag of \(\sim30~\mbox{deg}\). The mean precessional period is derived to be about 1000 y, but at the present time the motion of the star’s rotation axis appears to be supporting the gravitational torque, rather than providing the balance against it that would be expected over long periods of time.The planet has a small but detectable backwarming effect on the star, which helps to explain the difference in brightness just after transit and just before occultation eclipses. In assessing these findings it is recognized that sources of uncertainty remain, notably with possible inherent micropulsational effects, variations from other components of the multiple star, stellar activity, differential rotation and the neglect of higher order terms (than \(r_{1}^{5}\)) in the fitting function, where \(r_{1}\) is the ratio of the radius of the star to the mean orbital separation of planet and host star.  相似文献   

20.
We develop analytical series representing the main part of corrections to the geopotential coefficients caused by the solid Earth tides, where Love numbers are assumed to be frequency-independent. The series are compact, precise and valid over 1800 A.D.–2200 A.D. The maximum difference between the corrections given by the analytical series and their numerical values, obtained with use of the DE/LE-423 planetary/lunar ephemerides, does not exceed $0.7\times 10^{-12}$ . A new algorithm is proposed for calculating amplitudes of the additional variations of the geopotential coefficients for frequency dependence of Love numbers. It uses the representation of the Earth tide-generating potential in the standard HW95 format and takes into account the phase of tidal waves. Corrections of up to $2\times 10^{-12}$ to the published by the IERS Conventions (2010) amplitudes of the additional variations of the geopotential coefficients are suggested. Examples of use of the obtained series in analytical theories of motion of low-altitude STARLETTE and high-altitude ETALON-1 satellites are given.  相似文献   

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