Radially accelerated optimal feedback orbits in central gravity field with linear drag

The solution of a feedback optimal control problem arising in orbital mechanics is addressed in this paper. The dynamics is that of a massless body moving in a central gravitational force field subject also to a drag and a radial modulated force. The drag is linearly proportional to the velocity and inversely proportional to the square of the distance from the center of attraction. The problem is tackled by exploiting the properties of a suitably devised linearizing map that transforms the nonlinear dynamics into an inhomogeneous linear system of differential equations supplemented by a quadratic objective function. The generating function method is then applied to this new system, and the solution is back transformed in the old variables. The proposed technique, in contrast to the classical optimal control problem, allows us to derive analytic closed-loop solutions without solving any two-point boundary value problem. Applications are discussed.     

Radiation pressure and dust particle dynamics

The dynamics of small dust grains orbiting a planet are investigated when solar radiation pressure forces are added to the planet's gravitational central field. In the first part a set of differential equations is derived in a reference frame linked to the solar motion. The complete solution of these equations is given for particles lying in the planet's orbital plane, and we show that the orbital eccentricity may undergo considerable variation. At the same time the pericenter longitude librates or circulates according to initial conditions. With this result we establish a criterion for any orbiting particle (because of its highly eccentric orbit) to collide with its planet's atmosphere. The case of inclined orbit is studied through a numerical integration and allows us to draw conclusions related to the stability of the orbital plane. All solutions are periodic, with the period being independent of the initial conditions. This last point permits us to investigate the different time scales involved in that problem. Finally, the Poynting-Robertson drag is included, along with the radial radiation pressure forces, and the secular trend is considered. A coupling effect between the two components is ascertained, yielding a systematic behavior in the eccentricity and thus in the pericenter distance. Our solutions generalize the results of S. J. Peale (1966, J. Geophys. Res.71, 911–933) and J. A. Burns, P. Lamy, and S. Soter (1979, Icarus40, 1–48) by allowing eccentricities to be large (of order 1) and inclinations to be nonzero and by considering Poynting-Robertson drag.     

Approximate analysis for relative motion of satellite formation flying in elliptical orbits

This paper studies the relative motion of satellite formation flying in arbitrary elliptical orbits with no perturbation. The trajectories of the leader and follower satellites are projected onto the celestial sphere. These two projections and celestial equator intersect each other to form a spherical triangle, in which the vertex angles and arc-distances are used to describe the relative motion equations. This method is entitled the reference orbital element approach. Here the dimensionless distance is defined as the ratio of the maximal distance between the leader and follower satellites to the semi-major axis of the leader satellite. In close formations, this dimensionless distance, as well as some vertex angles and arc-distances of this spherical triangle, and the orbital element differences are small quantities. A series of order-of-magnitude analyses about these quantities are conducted. Consequently, the relative motion equations are approximated by expansions truncated to the second order, i.e. square of the dimensionless distance. In order to study the problem of periodicity of relative motion, the semi-major axis of the follower is expanded as Taylor series around that of the leader, by regarding relative position and velocity as small quantities. Using this expansion, it is proved that the periodicity condition derived from Lawden’s equations is equivalent to the condition that the Taylor series of order one is zero. The first-order relative motion equations, simplified from the second-order ones, possess the same forms as the periodic solutions of Lawden’s equations. It is presented that the latter are further first-order approximations to the former; and moreover, compared with the latter more suitable to research spacecraft rendezvous and docking, the former are more suitable to research relative orbit configurations. The first-order relative motion equations are expanded as trigonometric series with eccentric anomaly as the angle variable. Except the terms of order one, the trigonometric series’ amplitudes are geometric series, and corresponding phases are constant both in the radial and in-track directions. When the trajectory of the in-plane relative motion is similar to an ellipse, a method to seek this ellipse is presented. The advantage of this method is shown by an example.     

球状星团M79的自行测定和轨道运动

利用上海天文台相隔29年的两期天体测量底片，测量了球状星团M79的绝对自行，采用Harris给出的这个星团离开太阳的距离和视向速度数据，计算了星团当前的空间运动速度；根据银河系引力势模型，进一步计算了该星团在银河系中的轨道参数，还对利用自行数据所作的球状星团运动学研究的不确定性作了讨论。     

Satellite motion in a Manev potential with drag

In this paper, we consider a satellite orbiting in a Manev gravitational potential under the influence of an atmospheric drag force that varies with the square of velocity. Using an exponential atmosphere that varies with the orbital altitude of the satellite, we examine a circular orbit scenario. In particular, we derive expressions for the change in satellite radial distance as a function of the drag force parameters and obtain numerical results. The Manev potential is an alternative to the Newtonian potential that has a wide variety of applications, in astronomy, astrophysics, space dynamics, classical physics, mechanics, and even atomic physics.     

Theory of the motion of an artificial Earth satellite

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 0^o or the critical inclination. Approximately ninety percent of the satellites currently in orbit satisfy all these restrictions.     

Radial drift of particles in the solar nebula: implications for planetesimal formation

For standard cosmic abundances of heavy elements, a layer of small particles in the central plane of the solar nebula cannot attain the critical density for gravitational instability. Youdin and Shu (2002, Astrophys. J. 580, 494-505) suggest that the local surface density of solids can be enhanced by radial migration of particles due to gas drag. However, they consider only motions of individual particles. Collective motion due to turbulent stress on the particle layer acts to inhibit such enhancement and may prevent gravitational instability.     

Radial diffusion rate of planetesimals due to gravitational encounters

We study the rate of radial diffusion of planetesimals due to mutual gravitational encounters under Hill’s approximations in the three-body problem. Planetesimals orbiting a central star radially migrate inward and outward as a result of mutual gravitational encounters and transfer angular momentum. We calculate the viscosity in a disk of equal-sized planetesimals due to their mutual gravitational encounters using three-body orbital integrations, and obtain a semianalytic expression that reproduces the numerical results. We find that the viscosity is independent of the velocity dispersion of planetesimals when the velocity dispersion is so small that Kepler shear dominates planetesimals’ relative velocities. On the other hand, in high-velocity cases where random velocities dominate the relative velocities, the viscosity is a decreasing function of the velocity dispersion, and is found to agree with previous estimates under the two-body approximation neglecting the solar gravity. We also calculate the rate of radial diffusion of planetesimals due to gravitational scattering by a massive protoplanet. Using these results, we discuss a condition for formation of nonuniform radial surface density distribution of planetesimals by gravitational perturbation of an embedded protoplanet.     

Models of Motion in a Central Force Field with Quadratic Drag

Some general properties for the motion of a particle in a central force field with a general power law drag are derived. Exact and approximate solutions of the equations of motion are found in various cases. Emphasis is placed on inverse square gravitation and drag that varies with the square of the speed and inversely with the distance from the center of attraction. For this model two results stand out. The first is a particular solution in closed form that demonstrates the decay of an initially circular orbit under drag. The second, found from an approximation of the equations of motion when the radial speed is small compared with the tangential speed, demonstrates the decay of an initially elliptic orbit that is not highly eccentric. Formulas for calculation of the time of flight are presented for the two principal results.     

Multiple resonance in one problem of the stability of the motion of a satellite relative to the center of mass

We investigate the stability of the periodic motion of a satellite, a rigid body, relative to the center of mass in a central Newtonian gravitational field in an elliptical orbit. The orbital eccentricity is assumed to be low. In a circular orbit, this periodic motion transforms into the well-known motion called hyperboloidal precession (the symmetry axis of the satellite occupies a fixed position in the plane perpendicular to the radius vector of the center of mass relative to the attractive center and describes a hyperboloidal surface in absolute space, with the satellite rotating around the symmetry axis at a constant angular velocity). We consider the case where the parameters of the problem are close to their values at which a multiple parametric resonance takes place (the frequencies of the small oscillations of the satellite’s symmetry axis are related by several second-order resonance relations). We have found the instability and stability regions in the first (linear) approximation at low eccentricities.     

Effect of solar radiation pressure on the motion and stability of the system of two inter-connected satellites when their center of mass moves in a circular orbit

The effect of the perturbative force, solar pressure, on the motion and stability of two cableconnected satellites in the Earth's central gravitational field of force has been studied. The osculating plane of the orbit of the centre of mass has been supposed to be inclined at a constant angle with the plane of the ecliptic. The particular solutions of the nonlinear, non-homogeneous and non-autonomous equations of relative motion in three-dimensional cases have been obtained.The particular solution in which the system lies wholly along the radius vector joining the attracting centre and the centre of mass of the system under the central attracting force alone was found to be stable (cf. Singh, 1973). It has been established in our case too that the same particular solution remains stable if the parameter of the solar pressure lies within a certain limit.     

Restriction of motions in wide pairs in the Galactic field

The motions of the components of wide binary stars in the solar neighborhood in the regular Galactic gravitational field on time scales ～10¹⁰ yr have been studied numerically. The regions of restricted motions of the components in wide pairs have been found depending on the initial conditions: the magnitude of the relative velocity of the components, their mutual distance, and the inclination of the relative velocity vector to the Galactic plane. The size of the main part of the region of restricted motions is approximately equal to the tidal radius. Profound changes in the eccentricity of the binary orbit occur at inclinations close to 90°, which can lead to close approaches of the stars with a pericenter distance less than 1 AU. In the case of retrograde motions (the binary rotates in a direction opposite to the Galactic rotation), there is a region of restricted motions extending at least to 10 pc. Examples of the trajectories of relative motion of the stars and the change in osculating orbital elements are given for systems with restricted motions.     

The nearest star of spectral type O3: a component of the multiple system HD 150136

From radial velocities determined in high signal-to-noise digital spectra, we report the discovery that the brightest component of the binary system HD 150136 is of spectral type O3. We also present the first double-lined orbital solution for this binary. Our radial velocities confirm the previously published spectroscopic orbital period of 2.6 d. He  ii absorptions appear double at quadratures, but single lines of N  v and N  iv visible in our spectra define a radial velocity orbit of higher semi-amplitude for the primary component than do the He  ii lines. From our orbital analysis, we obtain minimum masses for the binary components of 27 and  18 M_⊙  . The neutral He absorptions apparently do not follow the orbital motion of any of the binary components, thus they most probably arise in a third star in the system.     

Generalized Lagrangian orbital elements for central force problems

We consider the definitions and resulting equations of motion for the Lagrangian orbital elements associated with conventional osculating orbit theory for central forces. The analysis indicates that the definitions themselves lead to difficulties which are most apparent in the circular limit. An alternate set of defining relations is presented which eliminates the problems associated with osculating elements. The remaining equation of motion based on these new definitions is reduced to quadratures. This solution completely expresses the orbits for central force problems with no restriction on the eccentricity. Both bounded and open orbits are considered. A generalized Laplace-Runge-Lenz vector is developed and a number of example solutions are presented.     

The equations of relative motion in the orbital reference frame

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.     

The mapping of Kaula's solution into the orbital reference frame

Kaula's celebrated solution to the problem of satellite motion in the gravitational field of a rigid body is transformed to give the perturbation spectra in both position and velocity in the radial, transverse and normal directions of the orbital reference frame. This work is an extension and a refinement of the theory of orbital perturbations due to the geopotential previously published by Rosborough and Tapley (1987).     

Precession of the orbital nodes of Jupiter and Saturn triggered by the mutual perturbation: A model of two rings

The problem of the precession of the orbital planes of Jupiter and Saturn under the influence of mutual gravitational perturbations was formulated and solved using a simple dynamical model. Using the Gauss method, the planetary orbits are modeled by material circular rings, intersecting along the diameter at a small angle α. The planet masses, semimajor axes and inclination angles of orbits correspond to the rings. What is new is that each ring has an angular momentum equal to the orbital angular momentum of the planet. Contrary to popular belief, it was proved that the orbital resonance 5: 2 does not preclude the use of the ring model. Moreover, the period of averaging of the disturbing force (T ≈ 1332 yr) proves to be appreciably greater than a conventionally used period (≈900 yr). The mutual potential energy of rings and the torque of gravitational forces between the rings were calculated. We compiled and solved the system of differential equations for the spatial motion of rings. It was established that a perturbing torque causes the precession and simultaneous rotation of the orbital planes of Jupiter and Saturn. Moreover, the opposite orbit nodes on the Laplace plane coincide and perform a secular movement in retrograde direction with the same velocity of 25.6″/yr and the period T _J = T _S ≈ 50687 yr. These results are close to those obtained in the general theory (25.93″/yr), which confirms the adequacy of the developed model. It was found that the vectors of the angular velocity of orbital rings move counterclockwise over circular cones and describe circles on the celestial sphere with radii β₁ ≈ 0.8403504° (Saturn) and β₂ ≈ 0.3409296° (Jupiter) around the point which is located at an angular distance of 1.647607° from the ecliptic pole.     

Stochastic and dynamic temperature changes in the interplanetary gas

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.     

The proper motion and orbital motion of the galactic globular cluster M79

Using astrometric plates of Shanghai Observatory spanning a period of 29 years, the absolute proper motion of the Galactic globular cluster M79 was measured. Adopting the distance and radial velocity given by Harris (1999), its present space velocity was derived; then by taking the Galactic gravitational potential model proposed by Allen and Santillan (1991), its past orbital parameters in the Galactic system were derived. We also discuss the uncertainties in kinematical studies of globular clusters based on the use of proper motion data.     

Poisson equations of rotational motion for a rigid triaxial body with application to a tumbling artificial satellite

Differential equations are derived for studying the effects of either conservative or nonconservative torques on the attitude motion of a tumbling triaxial rigid satellite. These equations, which are analogous to the Lagrange planetary equations for osculating elements, are then used to study the attitude motions of a rapidly spinning, triaxial, rigid satellite about its center of mass, which, in turn, is constrained to move in an elliptic orbit about an attracting point mass. The only torques considered are the gravity-gradient torques associated with an inverse-square field. The effects of oblateness of the central body on the orbit are included, in that, the apsidal line of the orbit is permitted to rotate at a constant rate while the orbital plane is permitted to precess (either posigrade or retrograde) at a constant rate with constant inclination.A method of averaging is used to obtain an intermediate set of averaged differential equations for the nonresonant, secular behavior of the osculating elements which describe the complete rotational motions of the body about its center of mass. The averaged differential equations are then integrated to obtain long-term secular solutions for the osculating elements. These solutions may be used to predict both the orientation of the body with respect to a nonrotating coordinate system and the motion of the rotational angular momentum about the center of mass. The complete development is valid to first order in (n/w ₀)², wheren is the satellite's orbital mean motion andw ₀ its initial rotational angular speed.