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1.
Satellite orbits around a central body with arbitrary zonal harmonics are considered in a relativistic framework. Our starting point is the relativistic Celestial Mechanics based upon the first post-Newtonian approximation to Einstein’s theory of gravity as it has been formulated by Damour et al. (Phys Rev D 43:3273–3307, 1991; 45:1017–1044, 1992; 47:3124–3135, 1993; 49:618–635, 1994). Since effects of order \((\mathrm{GM}/c^2R) \times J_k\) with \(k \ge 2\) for the Earth are very small (of order \( 7 \times 10^{-10}\,\times \,J_k\)) we consider an axially symmetric body with arbitrary zonal harmonics and a static external gravitational field. In such a field the explicit \(J_k/c^2\)-terms (direct terms) in the equations of motion for the coordinate acceleration of a satellite are treated first with first-order perturbation theory. The derived perturbation theoretical results of first order have been checked by purely numerical integrations of the equations of motion. Additional terms of the same order result from the interaction of the Newtonian \(J_k\)-terms with the post-Newtonian Schwarzschild terms (relativistic terms related to the mass of the central body). These ‘mixed terms’ are treated by means of second-order perturbation theory based on the Lie-series method (Hori–Deprit method). Here we concentrate on the secular drifts of the ascending node \(<\!{\dot{\Omega }}\!>\) and argument of the pericenter \(<\!{\dot{\omega }}\!>\). Finally orders of magnitude are given and discussed. 相似文献
2.
The analysis of relative motion of two spacecraft in Earth-bound orbits is usually carried out on the basis of simplifying assumptions. In particular, the reference spacecraft is assumed to follow a circular orbit, in which case the equations of relative motion are governed by the well-known Hill–Clohessy–Wiltshire equations. Circular motion is not, however, a solution when the Earth’s flattening is accounted for, except for equatorial orbits, where in any case the acceleration term is not Newtonian. Several attempts have been made to account for the \(J_2\) effects, either by ingeniously taking advantage of their differential effects, or by cleverly introducing ad-hoc terms in the equations of motion on the basis of geometrical analysis of the \(J_2\) perturbing effects. Analysis of relative motion about an unperturbed elliptical orbit is the next step in complexity. Relative motion about a \(J_2\)-perturbed elliptic reference trajectory is clearly a challenging problem, which has received little attention. All these problems are based on either the Hill–Clohessy–Wiltshire equations for circular reference motion, or the de Vries/Tschauner–Hempel equations for elliptical reference motion, which are both approximate versions of the exact equations of relative motion. The main difference between the exact and approximate forms of these equations consists in the expression for the angular velocity and the angular acceleration of the rotating reference frame with respect to an inertial reference frame. The rotating reference frame is invariably taken as the local orbital frame, i.e., the RTN frame generated by the radial, the transverse, and the normal directions along the primary spacecraft orbit. Some authors have tried to account for the non-constant nature of the angular velocity vector, but have limited their correction to a mean motion value consistent with the \(J_2\) perturbation terms. However, the angular velocity vector is also affected in direction, which causes precession of the node and the argument of perigee, i.e., of the entire orbital plane. Here we provide a derivation of the exact equations of relative motion by expressing the angular velocity of the RTN frame in terms of the state vector of the reference spacecraft. As such, these equations are completely general, in the sense that the orbit of the reference spacecraft need only be known through its ephemeris, and therefore subject to any force field whatever. It is also shown that these equations reduce to either the Hill–Clohessy–Wiltshire, or the Tschauner–Hempel equations, depending on the level of approximation. The explicit form of the equations of relative motion with respect to a \(J_2\)-perturbed reference orbit is also introduced. 相似文献
3.
Vinti’s potential is revisited for analytical propagation of the main satellite problem, this time in the context of relative motion. A particular version of Vinti’s spheroidal method is chosen that is valid for arbitrary elliptical orbits, encapsulating \(J_2\), \(J_3\), and generally a partial \(J_4\) in an orbit propagation theory without recourse to perturbation methods. As a child of Vinti’s solution, the proposed relative motion model inherits these properties. Furthermore, the problem is solved in oblate spheroidal elements, leading to large regions of validity for the linearization approximation. After offering several enhancements to Vinti’s solution, including boosts in accuracy and removal of some singularities, the proposed model is derived and subsequently reformulated so that Vinti’s solution is piecewise differentiable. While the model is valid for the critical inclination and nonsingular in the element space, singularities remain in the linear transformation from Earth-centered inertial coordinates to spheroidal elements when the eccentricity is zero or for nearly equatorial orbits. The new state transition matrix is evaluated against numerical solutions including the \(J_2\) through \(J_5\) terms for a wide range of chief orbits and separation distances. The solution is also compared with side-by-side simulations of the original Gim–Alfriend state transition matrix, which considers the \(J_2\) perturbation. Code for computing the resulting state transition matrix and associated reference frame and coordinate transformations is provided online as supplementary material. 相似文献
4.
5.
Oscar Perdomo 《Celestial Mechanics and Dynamical Astronomy》2017,129(1-2):89-104
We will show that the period T of a closed orbit of the planar circular restricted three body problem (viewed on rotating coordinates) depends on the region it encloses. Roughly speaking, we show that, \(2 T=k\pi +\int _\Omega g\) where k is an integer, \(\Omega \) is the region enclosed by the periodic orbit and \(g:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) is a function that only depends on the constant C known as the Jacobian constant; it does not depend on \(\Omega \). This theorem has a Keplerian flavor in the sense that it relates the period with the space “swept” by the orbit. As an application we prove that there is a neighborhood around \(L_4\) such that every periodic solution contained in this neighborhood must move clockwise. The same result holds true for \(L_5\). 相似文献
6.
Giovanni Mingari Scarpello Daniele Ritelli 《Celestial Mechanics and Dynamical Astronomy》2018,130(6):42
The present study highlights the dynamics of a body moving about a fixed point and provides analytical closed form solutions. Firstly, for the symmetrical heavy body, that is the Lagrange–Poisson case, we compute the second (precession, \(\psi \)) and third (spin, \(\varphi \)) Euler angles in explicit and real form by means of multiple hypergeometric (Lauricella) functions. Secondly, releasing the weight assumption but adding the complication of the asymmetry, by means of elliptic integrals of third kind, we provide the precession angle \(\psi \) completing the treatment of the Euler–Poinsot case. Thirdly, by integrating the relevant differential equation, we reach the finite polar equation of a special motion trajectory named the herpolhode. Finally, we keep the symmetry of the first problem, but without weight, and take into account a viscous dissipation. The use of motion first integrals—adopted for the first two problems—is no longer practicable in this situation; therefore, the Euler equations, faced directly, are driving to particular occurrences of Bessel functions of order \(-\,1/2\). 相似文献
7.
Z. F. Bostancı T. Yontan S. Bilir T. Ak T. Güver S. Ak E. Paunzen Ç. S. Başaran E. Vurgun B. A. Akti M. Çelebi H. Ürgüp 《Astrophysics and Space Science》2018,363(7):143
In this study, we present CCD UBV photometry of poorly studied open star clusters, Dolidze 36, NGC 6728, NGC 6800, NGC 7209, and Platais 1, located in the first and second Galactic quadrants. Observations were obtained with T100, the 1-m telescope of the TÜB?TAK National Observatory. Using photometric data, we determined several astrophysical parameters such as reddening, distance, metallicity and ages and from them, initial mass functions, integrated magnitudes and colours. We took into account the proper motions of the observed stars to calculate the membership probabilities. The colour excesses and metallicities were determined independently using two-colour diagrams. After obtaining the colour excesses of the clusters Dolidze 36, NGC 6728, NGC 6800, NGC 7209, and Platais 1 as \(0.19\pm0.06\), \(0.15\pm0.05\), \(0.32\pm0.05\), \(0.12\pm 0.04\), and \(0.43\pm0.06\) mag, respectively, the metallicities are found to be \(0.00\pm0.09\), \(0.02\pm0.11\), \(0.03\pm0.07\), \(0.01\pm0.08\), and \(0.01\pm0.08\) dex, respectively. Furthermore, using these parameters, distance moduli and age of the clusters were also calculated from colour-magnitude diagrams simultaneously using PARSEC theoretical models. The distances to the clusters Dolidze 36, NGC 6728, NGC 6800, NGC 7209, and Platais 1 are \(1050\pm90\), \(1610\pm190\), \(1210\pm150\), \(1060\pm90\), and \(1710\pm250\) pc, respectively, while corresponding ages are \(400\pm100\), \(750\pm150\), \(400\pm100\), \(600\pm100\), and \(175\pm50\) Myr, respectively. Our results are compatible with those found in previous studies. The mass function of each cluster is derived. The slopes of the mass functions of the open clusters range from 1.31 to 1.58, which are in agreement with Salpeter’s initial mass function. We also found integrated absolute magnitudes varying from ?4.08 to ?3.40 for the clusters. 相似文献
8.
A. Elipe J. I. Montijano L. Rández M. Calvo 《Celestial Mechanics and Dynamical Astronomy》2017,129(4):415-432
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
9.
We investigate the conditions under which the magnetohydrodynamic (MHD) modes in a cylindrical magnetic flux tube moving along its axis become unstable against the Kelvin–Helmholtz (KH) instability. We use the dispersion relations of MHD modes obtained from the linearized Hall MHD equations for cool (zero beta) plasma by assuming real wave numbers and complex angular wave frequencies/complex wave phase velocities. The dispersion equations are solved numerically at fixed input parameters and varying values of the ratio \(l_{\mathrm{Hall}}/a\), where \(l_{\mathrm{Hall}} = c/\omega_{\mathrm{pi}}\) (\(c\) being the speed of light, and \(\omega_{\mathrm{pi}}\) the ion plasma frequency) and \(a\) is the flux tube radius. It is shown that the stability of the MHD modes depends upon four parameters: the density contrast between the flux tube and its environment, the ratio of external and internal magnetic fields, the ratio \(l_{\mathrm{Hall}}/a\), and the value of the Alfvén Mach number defined as the ratio of the tube axial velocity to Alfvén speed inside the flux tube. It is found that at high density contrasts, for small values of \(l_{\mathrm{Hall}}/a\), the kink (\(m = 1\)) mode can become unstable against KH instability at some critical Alfvén Mach number (or equivalently at critical flow speed), but a threshold \(l_{\mathrm{Hall}}/a\) can suppress the onset of the KH instability. At small density contrasts, however, the magnitude of \(l_{\mathrm{Hall}}/a\) does not affect noticeably the condition for instability occurrence – even though it can reduce the critical Alfvén Mach number. It is established that the sausage mode (\(m = 0\)) is not subject to the KH instability. 相似文献
10.
Alessandro Margheri Rafael Ortega Carlota Rebelo 《Celestial Mechanics and Dynamical Astronomy》2017,127(1):35-48
In this work we consider the Kepler problem with linear drag, and prove the existence of a continuous vector-valued first integral, obtained taking the limit as \(t\rightarrow +\infty \) of the Runge–Lenz vector. The norm of this first integral can be interpreted as an asymptotic eccentricity \(e_{\infty }\) with \(0\le e_{\infty } \le 1\). The orbits satisfying \(e_{\infty } <1\) approach the singularity by an elliptic spiral and the corresponding solutions \(x(t)=r(t)e^{i\theta (t)}\) have a norm r(t) that goes to zero like a negative exponential and an argument \(\theta (t)\) that goes to infinity like a positive exponential. In particular, the difference between consecutive times of passage through the pericenter, say \(T_{n+1} -T_n\), goes to zero as \(\frac{1}{n}\). 相似文献
11.
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
12.
We aim to probe the dynamic structure of the extended Solar neighborhood by calculating the radial metallicity gradients from orbit properties, which are obtained for axisymmetric and non-axisymmetric potential models, of red clump (RC) stars selected from the RAdial Velocity Experiment’s Fourth Data Release. Distances are obtained by assuming a single absolute magnitude value in near-infrared, i.e. \(M_{Ks}=-1.54\pm0.04\) mag, for each RC star. Stellar orbit parameters are calculated by using the potential functions: (i) for the MWPotential2014 potential, (ii) for the same potential with perturbation functions of the Galactic bar and transient spiral arms. The stellar age is calculated with a method based on Bayesian statistics. The radial metallicity gradients are evaluated based on the maximum vertical distance (\(z_{max}\)) from the Galactic plane and the planar eccentricity (\(e_{p}\)) of RC stars for both of the potential models. The largest radial metallicity gradient in the \(0< z_{max} \leq0.5\) kpc distance interval is \(-0.065\pm0.005~\mbox{dex}\,\mbox{kpc}^{-1}\) for a subsample with \(e_{p}\leq0.1\), while the lowest value is \(-0.014\pm0.006~\mbox{dex}\,\mbox{kpc}^{-1}\) for the subsample with \(e_{p}\leq0.5\). We find that at \(z_{max}>1\) kpc, the radial metallicity gradients have zero or positive values and they do not depend on \(e_{p}\) subsamples. There is a large radial metallicity gradient for thin disc, but no radial gradient found for thick disc. Moreover, the largest radial metallicity gradients are obtained where the outer Lindblad resonance region is effective. We claim that this apparent change in radial metallicity gradients in the thin disc is a result of orbital perturbation originating from the existing resonance regions. 相似文献
13.
In extending the analysis of the four secular resonances between close orbits in Li and Christou (Celest Mech Dyn Astron 125:133–160, 2016) (Paper I), we generalise the semianalytical model so that it applies to both prograde and retrograde orbits with a one-to-one map between the resonances in the two regimes. We propose the general form of the critical angle to be a linear combination of apsidal and nodal differences between the two orbits \( b_1 \Delta \varpi + b_2 \Delta \varOmega \), forming a collection of secular resonances in which the ones studied in Paper I are among the strongest. Test of the model in the orbital vicinity of massive satellites with physical and orbital parameters similar to those of the irregular satellites Himalia at Jupiter and Phoebe at Saturn shows that \({>}20\) and \({>}40\%\) of phase space is affected by these resonances, respectively. The survivability of the resonances is confirmed using numerical integration of the full Newtonian equations of motion. We observe that the lowest order resonances with \(b_1+|b_2|\le 3\) persist, while even higher-order resonances, up to \(b_1+|b_2|\ge 7\), survive. Depending on the mass, between 10 and 60% of the integrated test particles are captured in these secular resonances, in agreement with the phase space analysis in the semianalytical model. 相似文献
14.
Vineet K. Srivastava Jai Kumar Padmdeo Mishra Badam Singh Kushvah 《Journal of Astrophysics and Astronomy》2018,39(5):63
In this paper, computation of the halo orbit for the KS-regularized photogravitational circular restricted three-body problem is carried out. This work extends the idea of Srivastava et al. (Astrophys. Space Sci. 362: 49, 2017) which only concentrated on the (i) regularization of the 3D-governing equations of motion, and (ii) validation of the modeling for small out-of-plane amplitude (\(A_z =110000\) km) assuming the third-order analytical approximation as an initial guess with and without differential correction. This motivated us to compute the halo orbits for the large out-of-plane amplitudes and to study their stability analysis for the regularized motion. The stability indices are described as a function of out-of-plane amplitude, mass reduction factor and oblateness coefficient. Three different Sun–planet systems: the Sun–Earth, Sun–Mars and the Sun–Jupiter are chosen in this study. Stable halo orbits do not exist around the \(L_{1}\) point, however, around the \(L_{2}\) point stable halo orbits are found for the considered systems. 相似文献
15.
Yu Cheng Gerard Gómez Josep J. Masdemont Jianping Yuan 《Celestial Mechanics and Dynamical Astronomy》2017,128(4):409-433
This paper is devoted to the study of the transfer problem from a libration point orbit of the Earth–Moon system to an orbit around the Moon. The transfer procedure analysed has two legs: the first one is an orbit of the unstable manifold of the libration orbit and the second one is a transfer orbit between a certain point on the manifold and the final lunar orbit. There are only two manoeuvres involved in the method and they are applied at the beginning and at the end of the second leg. Although the numerical results given in this paper correspond to transfers between halo orbits around the \(L_1\) point (of several amplitudes) and lunar polar orbits with altitudes varying between 100 and 500 km, the procedure we develop can be applied to any kind of lunar orbits, libration orbits around the \(L_1\) or \(L_2\) points of the Earth–Moon system, or to other similar cases with different values of the mass ratio. 相似文献
16.
In this paper, we study the invariant manifold and its application in transfer trajectory problem from a low Earth parking orbit to the Sun-Earth \(L_{1}\) and \(L_{2}\)-halo orbits with the inclusion of radiation pressure and oblateness. Invariant manifold of the halo orbit provides a natural entrance to travel the spacecraft in the solar system along some specific paths due to its strong hyperbolic character. In this regard, the halo orbits near both collinear Lagrangian points are computed first. The manifold’s approximation near the nominal halo orbit is computed using the eigenvectors of the monodromy matrix. The obtained local approximation provides globalization of the manifold by applying backward time propagation to the governing equations of motion. The desired transfer trajectory well suited for the transfer is explored by looking at a possible intersection between the Earth’s parking orbit of the spacecraft and the manifold. 相似文献
17.
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
18.
J. Lino Cornelio M. Álvarez–Ramírez Josep M. Cors 《Celestial Mechanics and Dynamical Astronomy》2017,129(3):321-328
We study planar central configurations of the five-body problem where three bodies, \(m_1, m_2\) and \(m_3\), are collinear and ordered from left to right, while the other two, \(m_4\) and \(m_5\), are placed symmetrically with respect to the line containing the three collinear bodies. We prove that when the collinear bodies form an Euler central configuration of the three-body problem with \(m_1=m_3\), there exists a new family, missed by Gidea and Llibre (Celest Mech Dyn Astron 106:89–107, 2010), of stacked five-body central configuration where the segments \(m_4m_5\) and \(m_1m_3\) do not intersect. 相似文献
19.
Donald V. Reames 《Solar physics》2018,293(10):144
We find that element abundances in energetic ions accelerated by shock waves formed at corotating interaction regions (CIRs) mirror the abundances of the solar wind modified by a decreasing power-law dependence on the mass-to-charge ratio \(A\)/\(Q\) of the ions. This behavior is similar in character to the well-known power-law dependence on \(A\)/\(Q\) of abundances in large gradual solar energetic particles (SEP). The CIR ions reflect the pattern of \(A\)/\(Q\), with \(Q\) values of the source plasma temperature or freezing-in temperature of 1.0?–?1.2 MK typical of the fast solar wind in this case. Thus the relative ion abundances in CIRs are of the form \((A\mbox{/}Q)^{a}\), where \(a\) is nearly always negative and evidently decreases with distance from the shocks, which usually begin beyond 1 AU. For one unusual historic CIR event where \(a \approx 0\), the reverse shock wave of the CIR seems to occur at 1 AU, and these abundances of the energetic ions become a direct proxy for the abundances of the fast solar wind. 相似文献
20.
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