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1.
A new non-singular analytical theory for the motion of near-Earth satellite orbits with the air drag effect is developed in terms of uniformly regular KS canonical elements. Diurnally varying oblate atmosphere is considered with variation in density scale height dependent on altitude. The series expansion method is utilized to generate the analytical solutions and terms up to fourth-order terms in eccentricity and c (a small parameter dependent on the flattening of the atmosphere) are retained. Only two of the nine equations are solved analytically to compute the state vector and change in energy at the end of each revolution, due to symmetry in the equations of motion. The important drag perturbed orbital parameters: semi-major axis and eccentricity are obtained up to 500 revolutions, with the present analytical theory and by numerical integration over a wide range of perigee height, eccentricity and inclination. The differences between the two are found to be very less. A comparison between the theories generated with terms up to third- and fourth-order terms in c and e shows an improvement in the computation of the orbital parameters semi-major axis and eccentricity, up to 9%. The theory can be effectively used for the re-entry of the near-Earth objects, which mainly decay due to atmospheric drag.  相似文献   

2.
The theory of velocity dependent inertial induction, based upon extended Mach’s principle, has been able to generate many interesting results related to celestial mechanics and cosmological problems. Because of the extremely minute magnitude of the effect its presence can be detected through the motion of accurately observed bodies like Earth satellites. LAGEOS I and II are medium altitude satellites with nearly circular orbits. The motions of these satellites are accurately recorded and the past data of a few decades help to test many theories including the general theory of relativity. Therefore, it is hoped that the effect of the Earth’s inertial induction can have any detectable effect on the motion of these satellites. It is established that the semi-major axis of LAGEOS I is decreasing at the rate of 1.3 mm/d. As the atmospheric drag is negligible at that altitude, a proper explanation of the secular change has been wanting, and, therefore, this paper examines the effect of the Earth’s inertial induction effect on LAGEOS I. Past researches have established that Yarkovsky thermal drag, charged and neutral particle drag might be the possible mechanisms for this orbital decay. Inertial induction is found to generate a perturbing force that results in 0.33 mm/d decay of the semi major axis. Some other changes are also predicted and the phenomenon also helps to explain the observed changes in the orbits of a few other satellites. The results indicate the feasibility of the theory of inertial induction i.e. the dynamic gravitation phenomenon of the Earth on its satellites as a possible partial cause for orbital decay.  相似文献   

3.
The axisymmetric satellite problem including radiation pressure and drag is treated. The equations of motion of the satellite are derived. The energy-like and Laplace-like invariants of motion have been derived for a general drag force function of the polar angle, and the Laplace-like invariant is used to find the orbit equation in the case of a spherical satellite. Then using the small parameter, the orbit of the satellite is determined for an axisymmetric satellite.  相似文献   

4.
An approximate semi-analytic solution of a two-body problem with drag is presented. The solution describesnon-lifting orbital motion in a central, inverse-square gravitational field. Drag deceleration is a non-linear function of velocity relative to a rotating atmosphere due to dynamic pressure and velocity-dependent drag coefficient. Neglected are aerodynamic lift, gravitational perturbations of the inverse-square field, and kinematic accelerations due to coordinate frame rotation at earth angular rate. With these simplifications, it is shown that (i) orbital motion occurs in an earth-fixed invariable plane defined by the radius and relative velocity vectors, and (ii) the simplified equations of motion are autonomous and independent of central angle measured in the invariable plane. Consequently, reduction of the differential equations from sixth to second-order is possible. Solutions for the radial and circumferential components of relative velocity are reduced to quadratures with respect to radial distance. Since the independent variable is radial distance, the solutions are singular at zero radial velocity (e. g., for circular orbits). General atmospheric density and drag coefficient models may be used to evaluate the velocity quadratures. The central angle and time variables are recovered from two additional quadratures involving the velocity quadratures. Theoretical results are compared with numerical simulation results.Presently affiliated with AVCO Systems Division, Wilmington, MA 01887, U.S.A.  相似文献   

5.
We have analyzed the motion of an infinitesimal mass in the restricted four-body problem with solar wind drag. It is assumed that the forces which govern the motion are mutual gravitational attractions of the primaries, radiation pressure force and solar wind drag. We have derived the equations of motion and found the Jacobi integral, zero velocity surfaces, and particular solutions of the system. It is found that three collinear points are real when the radiation factor 0<β<0.1 whereas only one real point is obtained when 0.125<β<0.2. The stability property of the system is examined with the help of Poincaré surface of section (PSS) and Lyapunov characteristic exponents (LCEs). It is found that in presence of drag forces LCE is negative for a specific initial condition, hence the corresponding trajectory is regular whereas regular islands in the PSS are expanded.  相似文献   

6.
Revised equations of motion are formulated on more general assumptions than hitherto making allowance for some reflection of sunlight by a dust-particle, and from these the secular rates of change of the orbital elements of the particle are obtained. The equation for the eccentricity yields numerical results for the time taken for given changes in this element to occur. Other elements turn out to be expressible in terms of the eccentricity and thence are effectively also known in terms of the time. More general forms of earlier results are found, and some new mathematical results in the theory of the process are derived. The time of infall to the Sun associated with almost circular initial motion of a particle is calculated, and also the time from an orbit of initially high eccentricity. In this latter case, infall takes place much more rapidly than from a circular orbit of radius comparable with the average distance in the eccentric orbit. The effect on a particle of a long-period comet during a single return is negligible compared with the change in its binding-energy to the Sun that will in general result from planetary action. The possible history of a dust-particle from original capture by the Sun to final infall to the solar surface is briefly considered.  相似文献   

7.
A practical and important problem encountered during the atmospheric re-entry phase is to determine analytical solutions for the space vehicle dynamical equations of motion. The author proposes new solutions for the equations of trajectory and flight-path angle of the space vehicle during the re-entry phase in Earth’s atmosphere. Explicit analytical solutions for the aerodynamic equations of motion can be effectively applied to investigate and control the rocket flight characteristics. Setting the initial conditions for the speed, re-entering flight-path angle, altitude, atmosphere density, lift and drag coefficients, the nonlinear differential equations of motion are linearized by a proper choice of the re-entry range angles. After integration, the solutions are expressed with the Exponential Integral, and Generalized Exponential Integral functions. Theoretical frameworks for proposed solutions as well as, several numerical examples, are presented.  相似文献   

8.
The Kelperian motion of dust particles in the solar system is mainly influenced by the electromagnetic and plasma Poynting-Robertson drag. The first force is isotropic while the second one shows latitudinal variations due to the observed differences of the solar wind parameters in the ecliptic plane and over the solar poles. Close to the Sun other effects become important, e.g. sublimation and sputtering, as well as for submicron particles Lorentz scattering has to be taken into account. These forces are very weak for dust grains of moderate size (10–100 µ) not too close (>0.03 AU) to the Sun and are neglected here. Assuming that the general form of the latidudinally dependent force is a series expansion in Legendre polynomials, we have studied the averaged equations of motion for the classical elements and found the first integral of them. The general character of motion is the same as for the classical Poynting-Robertson drag: particles spiral towards the Sun. The new features in the orbital evolution under the latitudinally dependent force as compared with the isotropic Poynting-Robertson drag are:
  1. not only the semimajor axisa and the eccentricity ε but also the argument of the perihelion ω varies with time,
  2. the rate of change ofa, ε, ω depends on the inclination.
An example of particle trajectories in the phase space of elements is presented.  相似文献   

9.
Theory of the motion of an artificial Earth satellite   总被引：1，自引：0，他引：1  
An improved analytical solution is obtained for the motion of an artificial Earth satellite under the combined influences of gravity and atmospheric drag. The gravitational model includes zonal harmonics throughJ 4, and the atmospheric model assumes a nonrotating spherical power density function. The differential equations are developed through second order under the assumption that the second zonal harmonic and the drag coefficient are both first-order terms, while the remaining zonal harmonics are of second order.Canonical transformations and the method of averaging are used to obtain transformations of variables which significantly simplify the transformed differential equations. A solution for these transformed equations is found; and this solution, in conjunction with the transformations cited above, gives equations for computing the six osculating orbital elements which describe the orbital motion of the satellite. The solution is valid for all eccentricities greater than 0 and less than 0.1 and all inclinations not near 0o or the critical inclination. Approximately ninety percent of the satellites currently in orbit satisfy all these restrictions.  相似文献   

10.
Exact solutions have been found to the Fokker-Planck equations, incorporating stochastic velocity changes and modelling particles moving in an inverse square central force field under an inverse square collision frequency. The solutions for the velocity distribution contain a combination of collisional and dynamical (reversible) heating. At a general position, there are two populations each with three distinct temperatures, one normal to the orbital plane and the others closely parallel and perpendicular to the mean orbit. Collisional heating is strong and most readily detected in the secondary component of gas which reaches upstream directions along indirect orbits (attractive central force). For interplanetary helium gas reaching 1 a.u., the collisional heating ranges from effective transverse increase of 200 K and radial increase of 1500 K in the downstream wake, to several thousand K increase in radial temperature of the secondary component transverse to the initial gas stream. In interpreting 584 Å sky background radiation observations, the dynamical changes in the velocity spread have to be taken into account for helium gas that is initially hot, when Doppler shifts relative to the solar emission line are significant; the present solutions being the thermal approximations to the distribution function reveal the appropriate radial temperature as a function of space.  相似文献   

11.
The orbit-averaged differential equations of motion of dust particles under gravity, radiation pressure and Poynting-Robertson drag were given by Wyatt and Whipple (1950). An integral of motion enables the system of two equations in semi-major axis a and eccentricity e to be reduced to one equation, the solution of which is presented here in terms of analytical formulae. An efficient numerical algorithm to compute the solution is given. Listings of two FORTRAN routines are included.  相似文献   

12.
An approximate solution of the encounter problem of two small satellites describing initially elliptical orbits around a massive oblate primary is obtained. The equations of motion of the center of mass of the two masses are developed in the most general form without any restrictions on the orbital elements. The method of multiple scales which seeks a solution whose behavior depends on several time scales is used. To overcome the singularity the equations of motion are transformed to the Struble variables. An analytical second order theory of the evolution dynamics is obtained. A MATHEMATICA program is constructed. The evolution dynamics of the orbital parameters between the perturbed and the unperturbed cases are plotted. The effect of changing eccentricity and changing inclination on the orbital parameters are highlighted.  相似文献   

13.
Unified analytical solutions to two-body problems with drag   总被引：3，自引：0，他引：3  
The two-body problem with a generalized Stokes drag is discussed. The drag force is proportional to the product of the velocity vector and the inverse square of the distance. The generalization consists of allowing two different proportionality constants for the radial and the transverse components of the force. Under the 'generalized Robertson transformation', the equation of the orbit takes the form of the Lommel equation and admits solutions in terms of Bessel and Lommel functions. The exact, analytical solutions for this type of drag reveal a paradoxical effect of increasing eccentricity for all trajectories. The Poynting–Robertson drag and Poynting–Plummer–Danby problems are discussed as particular cases of the general solution.  相似文献   

14.
We investigate the response of dust particles in the mid-plane of a protoplanetary disc to the turbulent velocity field of long-lived, large-scale vortical circulation. The dynamical problem is studied through numerical integrations of the equations of motion for individual particles (the sizes of which range from centimetres to metres) subject to the solar gravity and the friction drag of the nebular gas. It is found, neglecting the thickness of the disc, that the particles do not drift inwards to the central star as occurs in a standard symmetrical nebula. Vortices tend to capture a large number of the particles. The effectiveness of this size-selective concentration mechanism depends not only on the value of the drag and the distance from the Sun, but also on the elongation of the vortex and its characteristic lifetime. Typical anticyclonic vortices with exponential decay times of 30 orbital periods and semi-axis ratios of 4 can increase the local surface density by a factor of 4 in a lifetime and accumulate 0.03–0.3 Earth masses. If the elongation is significant (>7), the vortex cannot concentrate any significant amount of solid material. Vortices with an elongation of about 2 are the most effective as regards trapping of dust. We have also found analytical expressions for the capture time as well as capture constraints as a function of the friction parameter, the elongation of the vortex and the impact parameter. By increasing the lifetime and the surface density of the solid particles, this confining mechanism can make the agglomeration of the solid material of the nebula (through collisional aggregation or gravitational instabilities) much more efficient than previously believed. This offers new possibilities for the formation of the planetesimals and the giant planet cores, and may explain the rapid formation of extrasolar giant planets.  相似文献   

15.
A new nonsingular analytical theory for the motion of near Earth satellite orbits with the air drag effect is developed for long term motion in terms of the KS uniformly regular canonical elements by a series expansion method, by assuming the atmosphere to be symmetrically spherical with constant density scale height. The series expansions include up to third order terms in eccentricity. Only two of the nine equations are solved analytically to compute the state vector and change in energy at the end of each revolution, due to symmetry in the equations of motion. Numerical comparisons of the important orbital parameters semi major axis and eccentricity up to 1000 revolutions, obtained with the present solution, with KS elements analytical solution and Cook, King-Hele and Walker's theory with respect to the numerically integrated values, show the superiority of the present solution over the other two theories over a wide range of eccentricity, perigee height and inclination.  相似文献   

16.
Auroral E region neutral winds determined from incoherent scatter radar observations at Chatanika, AK, during geomagnetic disturbances (15 May 1974) are compared with detailed theoretical calculations of neutral velocities for these conditions. The theoretical velocities are obtained by numerically solving the ion and neutral momentum equations in the ion drag approximation, including coriolis and viscous forces, using observed electric fields and electron densities. Large vertical gradients are found in the calculated velocities for altitudes below about 130 km. As a consequence of this structure and fluctuations in the electron density profiles, the data analysis procedure of Brekke et al. (1973) for obtaining neutral winds from radar data is found to underestimate the wind speed by up to 40%, but it determines the direction and temporal structure reasonably well. Comparison of observed neutral velocities with calculated values shows that ion drag alone cannot account for the observations. An equation is derived to estimate the pressure gradients required to resolve the discrepancy between calculated and observed neutral winds. Accelerations due to these pressure gradients are of the same order as those due to ion drag, but at least an order of magnitude larger than those due to solar heating. Directions of the horizontal pressure gradients are consistent with expected locations of auroral heating. During geomagnetic disturbances, ion drag and auroral heating both appear to play important roles in the generation and modification of neutral winds.  相似文献   

17.
Plasma magnetosphere surrounding rotating magnetized neutron star in the braneworld has been studied. For the simplicity of calculations Goldreich-Julian charge density is analyzed for the aligned neutron star with zero inclination between magnetic field and rotation axis. From the system of Maxwell equations in spacetime of slowly rotating star in braneworld, second-order differential equation for electrostatic potential is derived. Analytical solution of this equation indicates the general relativistic modification of an accelerating electric field and charge density along the open field lines by brane tension. The implication of this effect to the magnetospheric energy loss problem is underlined. It was found that for initially zero potential and field on the surface of a neutron star, the amplitude of the plasma mode created by Goldreich-Julian charge density will increase in the presence of the negative brane charge. Finally we derive the equations of motion of test particles in magnetosphere of slowly rotating star in the braneworld. Then we analyze particle motion in the polar cap and show that brane tension can significantly change conditions for particle acceleration in the polar cap region of the neutron star.  相似文献   

18.
We construct an idealized spherically symmetric relativistic model of an exploding object within the framework of the theory of surface layers in GR. A Vaidya solution for a radially radiating star is matched through a spherical shell of dust to a Schwarzschild solution. The (incomplete) equations for the motion of the spherical shell of dust and the radiation density of the Vaidya solution, as given by the matching conditions, are reduced to a first-order system and a general analysis of the characteristics of the motion is given. This system of differential equations is completed, adding a relation between the unknowns which represents the simplest way to avoid an unphysical singularity in the motion. The results of a numerical integration of the equations are presented in two cases which we think may have some relationship to stellar explosions. A comparative set of results for other solutions is also given, and some possible generalizations of the model are pointed out.  相似文献   

19.
The motion of a rigid body about a fixed point, under the influence of an attractive force, is investigated from a physical basis. The errors in Euler's equations were shown, and appropriate equations were derived. It was also found that by nutation in the direction of decreasing only and oscillatory precession any two bodies, assumed as oblate spheroids, move in the direction of a zero external moment, as their stable equilibrium position. This position corresponding to the minimum kinetic energy of rotation is also reached by any two bodies which were initially acted upon by an external torque.  相似文献   

20.
In order to retain separability in the Vinti theory of Earth satellite motion when a nonconservative force such as air drag is considered, a set of variational equations for the orbital elements are introduced, and expressed as functions of the transverse, radial, and normal components of the nonconservative forces acting on the system. In this approach, the Hamiltonian is preserved in form, and remains the total energy, but the initial or boundary conditions and hence the Jacobi constants of the motion advance with time through the variational equations. In particular, the atmospheric density profile is written as a fitted exponential function of the eccentric anomaly, which adheres to tabular data at all, altitudes and simultaneously reduces the variational equations to definite integrals with closed form evaluations whose limits are in terms of the eccentric anomaly. The values of the limits for any arbitrary time interval are obtained from the Vinti program.Results of this technique for the case of the intense air drag satellites San Marco-2 and Air Force Cannonball are given. These results indicate that the satellite ephemerides produced by this theory in conjunction with the Vinti program are of very high accuracy. In addition, since the program is entirely analytic, several months of ephemerides can be obtained within a few seconds of computer time.  相似文献   

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