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1.
The motion of a point mass in the J 2 problem has been generalized to that of a rigid body in a J 2 gravity field for new high-precision applications in the celestial mechanics and astrodynamics. Unlike the original J 2 problem, the gravitational orbit-rotation coupling of the rigid body is considered in the generalized problem. The existence and properties of both the classical and non-classical relative equilibria of the rigid body are investigated in more details in the present paper based on our previous results. We nondimensionalize the system by the characteristic time and length to make the study more general. Through the study, it is found that the classical relative equilibria can always exist in the real physical situation. Numerical results suggest that the non-classical relative equilibria only can exist in the case of a negative J 2, i.e., the central body is elongated; they cannot exist in the case of a positive J 2 when the central body is oblate. In the case of a negative J 2, the effect of the orbit-rotation coupling of the rigid body on the existence of the non-classical relative equilibria can be positive or negative, which depends on the values of J 2 and the angular velocity Ω e . The bifurcation from the classical relative equilibria, at which the non-classical relative equilibria appear, has been shown with different parameters of the system. Our results here have given more details of the relative equilibria than our previous paper, in which the existence conditions of the relative equilibria are derived and primarily studied. Our results have also extended the previous results on the relative equilibria of a rigid body in a central gravity field by taking into account the oblateness of the central body.  相似文献   

2.
We say that a planet is Earth-like if the coefficient of the second order zonal harmonic dominates all other coefficients in the gravity field. This paper concerns the zonal problem for satellites around an Earth-like planet, all other perturbations excluded. The potential contains all zonal coefficientsJ 2 throughJ 9. The model problem is averaged over the mean anomaly by a Lie transformation to the second order; we produce the resulting Hamiltonian as a Fourier series in the argument of perigee whose coefficients are algebraic functions of the eccentricity — not truncated power series. We then proceed to a global exploration of the equilibria in the averaged problem. These singularities which aerospace engineers know by the name of frozen orbits are located by solving the equilibria equations in two ways, (1) analytically in the neighborhood of either the zero eccentricity or the critical inclination, and (2) numerically by a Newton-Raphson iteration applied to an approximate position read from the color map of the phase flow. The analytical solutions we supply in full to assist space engineers in designing survey missions. We pay special attention to the manner in which additional zonal coefficients affect the evolution of bifurcations we had traced earlier in the main problem (J 2 only). In particular, we examine the manner in which the odd zonalJ 3 breaks the discrete symmetry inherent to the even zonal problem. In the even case, we find that Vinti's problem (J 4+J 2 2 =0) presents a degeneracy in the form of non-isolated equilibria; we surmise that the degeneracy is a reflection of the fact that Vinti's problem is separable. By numerical continuation we have discovered three families of frozen orbits in the full zonal problem under consideration; (1) a family of stable equilibria starting from the equatorial plane and tending to the critical inclination; (2) an unstable family arising from the bifurcation at the critical inclination; (3) a stable family also arising from that bifurcation and terminating with a polar orbit. Except in the neighborhood of the critical inclination, orbits in the stable families have very small eccentricities, and are thus well suited for survey missions.  相似文献   

3.
The motion of a point mass in the J 2 problem is generalized to that of a rigid body in a J 2 gravity field. The linear and nonlinear stability of the classical type of relative equilibria of the rigid body, which have been obtained in our previous paper, are studied in the framework of geometric mechanics with the second-order gravitational potential. Non-canonical Hamiltonian structure of the problem, i.e., Poisson tensor, Casimir functions and equations of motion, are obtained through a Poisson reduction process by means of the symmetry of the problem. The linear system matrix at the relative equilibria is given through the multiplication of the Poisson tensor and Hessian matrix of the variational Lagrangian. Based on the characteristic equation of the linear system matrix, the conditions of linear stability of the relative equilibria are obtained. The conditions of nonlinear stability of the relative equilibria are derived with the energy-Casimir method through the projected Hessian matrix of the variational Lagrangian. With the stability conditions obtained, both the linear and nonlinear stability of the relative equilibria are investigated in details in a wide range of the parameters of the gravity field and the rigid body. We find that both the zonal harmonic J 2 and the characteristic dimension of the rigid body have significant effects on the linear and nonlinear stability. Similar to the classical attitude stability in a central gravity field, the linear stability region is also consisted of two regions that are analogues of the Lagrange region and the DeBra-Delp region respectively. The nonlinear stability region is the subset of the linear stability region in the first quadrant that is the analogue of the Lagrange region. Our results are very useful for the studies on the motion of natural satellites in our solar system.  相似文献   

4.
We hereby study the stability of a massless probe orbiting around an oblate central body (planet or planetary satellite) perturbed by a third body, assumed to lay in the equatorial plane (Sun or Jupiter for example) using a Hamiltonian formalism. We are able to determine, in the parameters space, the location of the frozen orbits, namely orbits whose orbital elements remain constant on average, to characterize their stability/unstability and to compute the periods of the equilibria. The proposed theory is general enough, to be applied to a wide range of probes around planet or natural planetary satellites. The BepiColombo mission is used to motivate our analysis and to provide specific numerical data to check our analytical results. Finally, we also bring to the light that the coefficient J 2 is able to protect against the increasing of the eccentricity due to the Kozai-Lidov effect and the coefficient J 3 determines a shift of the equilibria.  相似文献   

5.
We describe a collection of results obtained by numerical integration of orbits in the main problem of artificial satellite theory (theJ 2 problem). The periodic orbits have been classified according to their stability and the Poincaré surfaces of section computed for different values ofJ 2 andH (whereH is thez-component of angular momentum). The problem was scaled down to a fixed value (–1/2) of the energy constant. It is found that the pseudo-circular periodic solution plays a fundamental role. They are the equivalent of the Poincaré first-kind solutions in the three-body problem. The integration of the variational equations shows that these pseudo-circular solutions are stable, except in a very narrow band near the critical inclincation. This results in a sequence of bifurcations near the critical inclination, refining therefore some known results on the critical inclination, for instance by Izsak (1963), Jupp (1975, 1980) and Cushman (1983). We also verify that the double pitchfork bifurcation around the critical inclination exists for large values ofJ 2, as large as |J 2|=0.2. Other secondary (higher-order) bifurcations are also described. The equations of motion were integrated in rotating meridian coordinates.  相似文献   

6.
We study numerically the photogravitational version of the problem of four bodies, where an infinitesimal particle is moving under the Newtonian gravitational attraction of three bodies which are finite, moving in circles around their center of mass fixed at the origin of the coordinate system, according to the solution of Lagrange where they are always at the vertices of an equilateral triangle. The fourth body does not affect the motion of the three bodies (primaries). We consider that the primary body m 1 is dominant and is a source of radiation while the other two small primaries m 2 and m 3 are equal. In this case (photogravitational) we examine the linear stability of the Lagrange triangle solution. The allowed regions of motion as determined by the zero-velocity surface and corresponding equipotential curves, as well as the positions of the equilibrium points on the orbital plane are given. The existence and the number of the collinear and the non-collinear equilibrium points of the problem depends on the mass parameters of the primaries and the radiation factor q 1. Critical masses m 3 and radiation q 1 associated with the existence and the number of the equilibrium points are given. The stability of the relative equilibrium solutions in all cases are also studied. In the last section we investigate the existence and location of the out of orbital plane equilibrium points of the problem. We found that such critical points exist. These points lie in the (x,z) plane in symmetrical positions with respect to (x,y) plane. The stability of these points are also examined.  相似文献   

7.
The total energy E of a star as a function of its angular momentum J and mass M in the Newtonian theory, E=E(J, M) [in general relativity, the gravitational mass of a star as a function of its angular momentum J and rest mass m, M=M(J, m)], is used to determine the remaining parameters (angular velocity, chemical potential, etc.) in the case of rigid rotation. Expressions are derived for the energy release during accretion onto a cool (with constant entropy), rapidly rotating neutron star (NS) in the Newtonian theory and in general relativity. A separate analysis is performed for the cases where the NS equatorial radius is larger and smaller than the radius of the marginally stable orbit in the disk plane. An approximate formula is proposed for the NS equatorial radius for an arbitrary equation of state, which matches the exact equation of state at J=0.  相似文献   

8.
The aim of Dawn mission is the acquisition of data from orbits around two bodies (4) Vesta and (1) Ceres, the two most massive asteroids.Due to the low thrust propulsion, Dawn will slowly cross and transit through ground-track resonances, where the perturbations on Dawn orbit may be significant. In this context, to safety go the Dawn mission from the approach orbit to the lowest science orbit, it is essential to know the properties of the crossed resonances.This paper analytically investigates the properties of the major ground-track resonances (1:1, 1:2, 2:3 and 3:2) appearing for Vesta orbiters: location of the equilibria, aperture of the resonances and period at the stable equilibria. We develop a general method using an averaged Hamiltonian formulation with a spherical harmonic approximation of the gravity field. If the values of the gravity field coefficient change, our method stays correct and applicable. We also discuss the effect of one uncertainty on the C20 and C22 coefficients on the properties of the 1:1 resonance. These results are checked by numerical tests. We determine that the increase of the eccentricity appearing in the 2:3 resonance is due to the C22 and S22 coefficients.Our method can be easily adapted to missions similar to Dawn because, contrarily to the numerical results, the analytical formalism stays the same and is valid for a wide range of physical parameters of the asteroid (namely the shape and the mass) as well as for different spacecraft orbits.Finally we numerically study the probability of the capture in resonance 1:1. Our paper reproduces, explains and supplements the results of Tricarico and Sykes (2010).  相似文献   

9.
Ming L. Xue  James Chen 《Solar physics》1983,84(1-2):119-124
A study is made of equilibrium and stability properties of a semi-toroidal current loop imbedded in a high temperature plasma. The loop carries a toroidal current density J t and poloidal current density J p. By explicity including the global curvature of the loop, the net Lorentz and pressure forces acting along the major radius are calculated. Requirement of equilibrium force-balance gives rise to conditions that must be satisfied by the physical parameters and geometry. On the basis of these conditions, we deduce a class of equilibrium semi-toroidal current loops satisfying c #X2212;1 J × B ? ▽p = 0. It is found that the averge pressure inside the loop is less than the ambient coronal pressure in equilibrium. Furthermore, this class of equilibria is shown to be stable to a number of destructive MHD modes. The theoretical results are discussed in the context of solar bipolar current loops.  相似文献   

10.
This paper analyses three types of artificial orbits around Mars pushed by continuous low-thrust control: artificial frozen orbits, artificial Sun-Synchronous orbits and artificial Sun-Synchronous frozen orbits. These artificial orbits have similar characteristics to natural frozen orbits and Sun-Synchronous orbits, and their orbital parameters can be selected arbitrarily by using continuous low-thrust control. One control strategy to achieve the artificial frozen orbit is using both the transverse and radial continuous low-thrust control, and another to achieve the artificial Sun-Synchronous orbit is using the normal continuous low-thrust control. These continuous low-thrust control strategies consider J 2, J 3, and J 4 perturbations of Mars. It is proved that both control strategies can minimize characteristic velocity. Relevant formulas are derived, and numerical results are presented. Given the same initial orbital parameters, the control acceleration and characteristic velocity taking into account J 2, J 3, and J 4 perturbations are similar to those taking into account J 2 perturbations for both Mars and the Earth. The control thrust of the orbit around Mars is smaller than that around the Earth. The magnitude of the control acceleration of ASFOM-4 (named as Artificial Sun-Synchronous Frozen Orbit Method 4) is the lowest among these strategies and the characteristic velocity within one orbital period is only 0.5219 m/s for the artificial Sun-Synchronous frozen orbit around Mars. It is evident that the relationship among the control thrusts and the primary orbital parameters of Martian artificial orbits is always similar to that of the Earth. Simulation shows that the control scheme extends the orbital parameters’ selection range of three types of orbits around Mars, compared with the natural frozen orbit and Sun-Synchronous orbit.  相似文献   

11.
We have studied a modified version of the classical restricted three-body problem (CR3BP) where both primaries are considered as oblate spheroids and are surrounded by a homogeneous circular planar cluster of material points centered at the mass center of the system. In this dynamical model we have examined the effects of oblateness of both primaries up to zonal harmonic J 4; together with gravitational potential from the circular cluster of material points on the existence and linear stability of the triangular equilibrium points. It is found that, the triangular points are stable for 0<μ<μ c and unstable for $\mu_{c} \le \mu \le \frac{1}{2}$ , where μ c is the critical mass ratio affected by the oblateness up to J 4 of the primaries and potential from the circular cluster of material points. The coefficient J 4 has stabilizing tendency, while J 2 and the potential from the circular cluster of material points have destabilizing tendency. A practical application of this model could be the study of the motion of a dust particle near oblate bodies surrounded by a circular cluster of material points.  相似文献   

12.
An understanding of the rates of frost grain growth is essential to the goal of relating spectral data on surface mineralogy to the physical history of a planetary surface. Models of grain growth kinetics have been constructed for various frosts based on their individual thermodynamic properties and on the difference in binding energy between molecules on plane vs curved faces. A steady state situation can occur on planetary surfaces in which thermal elimination of small grains competes with their creation, usually by meteorite impact. We utilize predicted grain growth rates to explain telescopic spectral data on condensate surfaces throughout the solar system. On Pluto, predicted CH4 ice grain growth rates are very high despite the low temperature, resulting in a multicentimeter optical path. This explains the strong CH4 absorption band depths, which otherwise would require large amounts of CH4 gas. On the Uranian and Saturnian satellites, extremely slow grain growth rates are predicted because of the low vapor pressure of H2O at the existing average surface temperatures. This may explain evidence for fine grain size and peculiar microstructure. On Io, ordinary thermal exchange is more effective than sputtering in promoting grain growth because of the properties of SO2. Over much of Io's disk, submicron size grains of SO2 could plausibly reconfigure into a surface glaze on a timescale comparable to the resurfacing rate. This may explain the relatively strong SO2 signature in Io's infrared absorption spectrum as opposed to its weaker manifestation in the visible spectrum. In spite of lower sputtering fluxes, sputtering plays a more important role in grain growth for Europa, Ganymede, and Callisto than on Io. This is a result of high rates of thermally activated grain growth and resurfacing on Io. The sequence of H2O-ice absorption band depths (related to the mean grain size) is J2(T) ～ J3(T) > J2(L) > J3(L) ～ J4(T) ～ J4(L), where L = leading and T = trailing. This is to be expected if sputtering were dominant. The calculations show, however, that neither thermalized exchange fluxes nor sputtering exchange fluxes can produce the implied grain growth or the ordering by ice absorption band depths of the six satellite hemispheres. Only sputtering control by simple ejection of H2O from the satellites, as the dominant cause of shorter mean lifetimes for smaller exposed grains, can satisfactorily explain the data. Some observations, which suggest that there are vertical grain size gradients, may result from a steady state balance between intense near surface production of fine frost by comminution, coupled with ongoing ubiquitous grain growth in the vertical column. In certain cases, e.g., Europa and Enceladus, the possibility exists that endogenic activity as well as comminution could affect grain size—at least locally. It is concluded that not only ice identification and mapping, but ice grain size mapping is an important experiment to be conducted on future missions.  相似文献   

13.
Most existing satellite relative motion theories utilize mean elements, and therefore cannot be used for calculating long-term bounded perturbed relative orbits. The goal of the current paper is to find an integrable approximation for the relative motion problem under the J 2 perturbation, which is adequate for long-term prediction of bounded relative orbits with arbitrary inclinations. To that end, a radial intermediary Hamiltonian is utilized. The intermediary Hamiltonian retains the original structure of the full J 2 Hamiltonian, excluding the latitude dependence. This formalism provides integrability via separation, a fact that is utilized for finding periodic relative orbits in a local-vertical local-horizontal frame and determine an initialization scheme that yields long-term boundedness of the relative distance. Numerical experiments show that the intermediary-based computation of orbits provides long-term bounded orbits in the full J 2 problem for various inclinations. In addition, a test case is shown in which the radial intermediary-based initial conditions of the chief and deputy satellites yield bounded relative distance in a high-precision orbit propagator.  相似文献   

14.
We derive astrophysical and structural parameters of the poorly studied open clusters NGC 6866, NGC 7062, and NGC 2360 based on filtered 2MASS (J, J ? H) diagrams, and stellar radial density profiles. The field star decontamination technique is utilised for selecting high-probability cluster members. The E(B ? V) reddening values of the three clusters derived from 2MASS JHKs agree with those inferred from UBV and uvby ? β photometries. We find that the core mass function slopes are flatter than the halo’s for the three clusters. The large core and cluster radii of NGC 6866 and NGC 2360 indicate an expanded core, which may suggest the presence of stellar mass black-holes. NGC 2360 is located in the third quadrant (? = 229°.80), where Giant Molecular Clouds are scarce that, together with its relatively large mass (～1800 m⊙), might explain its longevity (～1.8 Gyr) in the Galaxy.  相似文献   

15.
A method of analysis has been developed making it possible to obtain electrojet parameters from vertical profiles of the POGO electrojet data. This has provided an unprecedented quantity of electrojet parameters—about 500 each of its peak eastward current intensity J0, its total eastward current I+, and its half width w—spanning through 360° longitude round the Earth, 5 h of daytime and September equinoctial months of the years 1967, 1968 and 1969. The daytime all-longitude averages of the parameters for the 3 years are 235 ± 14 km for w, 232 ± 63A km?1 for J0 and (54 ± 11) × 103A for I+. This first coverage round the globe shows that J0 and I+ vary considerably with longitude unlike w. While confirming the expected maximum of J0 and I+ at about 280°E longitude, we find that each of them also has an unexpected major maximum at about 100°E and a subsidiary maximum at 190°E longitude. The major maxima are found to be prominent only at hours near local noon. It is argued that the surprising longitudinal variations of J0 and I+ cannot be explained satisfactorily by variations in the upper mantle conductivity along the dip equator but possibly by longitudinal variations of wind systems and gradients at electrojet altitudes.  相似文献   

16.
Jeremy Bailey  Linda Ahlsved 《Icarus》2011,213(1):218-232
We have obtained spatially resolved spectra of Titan in the near-infrared J, H and K bands at a resolving power of ∼5000 using the near-infrared integral field spectrometer (NIFS) on the Gemini North 8 m telescope. Using recent data from the Cassini/Huygens mission on the atmospheric composition and surface and aerosol properties, we develop a multiple-scattering radiative transfer model for the Titan atmosphere. The Titan spectrum at these wavelengths is dominated by absorption due to methane with a series of strong absorption band systems separated by window regions where the surface of Titan can be seen. We use a line-by-line approach to derive the methane absorption coefficients. The methane spectrum is only accurately represented in standard line lists down to ∼2.1 μm. However, by making use of recent laboratory data and modeling of the methane spectrum we are able to construct a new line list that can be used down to 1.3 μm. The new line list allows us to generate spectra that are a good match to the observations at all wavelengths longer than 1.3 μm and allow us to model regions, such as the 1.55 μm window that could not be studied usefully with previous line lists such as HITRAN 2008. We point out the importance of the far-wing line shape of strong methane lines in determining the shape of the methane windows. Line shapes with Lorentzian, and sub-Lorentzian regions are needed to match the shape of the windows, but different shape parameters are needed for the 1.55 μm and 2 μm windows. After the methane lines are modeled our observations are sensitive to additional absorptions, and we use the data in the 1.55 μm region to determine a D/H ratio of 1.77 ± 0.20 × 10−4, and a CO mixing ratio of 50 ± 11 ppmv. In the 2 μm window we detect absorption features that can be identified with the ν5 + 3ν6 and 2ν3 + 2ν6 bands of CH3D.  相似文献   

17.
We study the various families of periodic orbits in a dynamical system representing a plane rotating barred galaxy. One can have a general view of the main resonant types of orbits by considering the axisymmetric background. The introduction of a bar perturbation produces infinite gaps along the central familyx 1 (the family of circular orbits in the axisymmetric case). It produces also higher order bifurcations, unstable regions along the familyx 1, and long period orbits aroundL 4 andL 5. The evolution of the various types of orbits is described, as the Jacobi constanth, and the bar amplitude, increase. Of special importance are the infinities of period doubling pitchfork bifurcations. The genealogy of the long and short period orbits is described in detail. There are infinite gaps along the long period orbits producing an infinity of families. All of them bifurcate from the short period family. The rules followed by these families are described. Also an infinity of higher order bridges join the short and long period families. The analogies with the restricted three body problem are stressed.  相似文献   

18.
We present a total of 289 new astrometric observations of the inner jovian satellites, Amalthea and Thebe, obtained using the Cassini ISS narrow angle camera. Observations were made using image sequences from 2000 December 11-12 (inbound) and 2001 January 15-16 (outbound), at phase angles of approximately 2° and 122°, respectively. Target distances were of order 284 RJ, giving a maximum resolution of approximately 100 km/pixel. Centroided line and sample values for 239 observations of Amalthea and 50 of Thebe are provided, together with estimated camera pointing information for each image. Orbit fitting using a uniformly precessing Keplerian ellipse model, taking into account the oblateness of Jupiter up to terms in J6, gave RMS fit residuals of 0.364 and 0.443 pixel for Amalthea and Thebe, respectively (equivalent to 0.450 and 0.547 arcsec). RMS residuals relative to the JPL JUP230 ephemeris were 0.306 and 0.604 pixel (equivalent to 0.378 and 0.746 arcsec), for Amalthea and Thebe. The fitted orbital parameters confirm the relatively high inclinations of these satellites (0.374°±0.002° and 1.076°±0.003°, respectively), equivalent to maximum vertical displacements above Jupiter's equatorial plane of 1188±6 and 4240±12 km, respectively, consistent with current estimates of the half-thicknesses of the Amalthea and Thebe gossamer rings [Ockert-Bell, M.E., Burns, J.A., Dauber, I.J., Thomas, P.C., Veverka, J., Belton, M.J.S., Klaasen, K.P., 1999. Icarus 138, 188-213].  相似文献   

19.
Having analyzed the spectrum of the quasar PKS 1232+0.82 taken by Petitjean et al. (2000), we identified HD molecular lines in an absorption system at redshift z=2.3377. We estimated the column density of HD molecules in this system, N(HD) = (1?4) × 1014cm?2. The excitation temperature of the first rotational level J=1 relative to the ground state J=0 is T ex=70 ± 7 K. As far as we know, this is the first detection of HD molecules at high redshift.  相似文献   

20.
We consider the two-body problem on surfaces of constant nonzero curvature and classify the relative equilibria and their stability. On the hyperbolic plane, for each \(q>0\) we show there are two relative equilibria where the masses are separated by a distance q. One of these is geometrically of elliptic type and the other of hyperbolic type. The hyperbolic ones are always unstable, while the elliptic ones are stable when sufficiently close, but unstable when far apart. On the sphere of positive curvature, if the masses are different, there is a unique relative equilibrium (RE) for every angular separation except \(\pi /2\). When the angle is acute, the RE is elliptic, and when it is obtuse the RE can be either elliptic or linearly unstable. We show using a KAM argument that the acute ones are almost always nonlinearly stable. If the masses are equal, there are two families of relative equilibria: one where the masses are at equal angles with the axis of rotation (‘isosceles RE’) and the other when the two masses subtend a right angle at the centre of the sphere. The isosceles RE are elliptic if the angle subtended by the particles is acute and is unstable if it is obtuse. At \(\pi /2\), the two families meet and a pitchfork bifurcation takes place. Right-angled RE are elliptic away from the bifurcation point. In each of the two geometric settings, we use a global reduction to eliminate the group of symmetries and analyse the resulting reduced equations which live on a five-dimensional phase space and possess one Casimir function.  相似文献   

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