Non-coplanar Orbits Research Articles

FRIENDS OF HOT JUPITERS. II. NO CORRESPONDENCE BETWEEN HOT-JUPITER SPIN-ORBIT MISALIGNMENT AND THE INCIDENCE OF DIRECTLY IMAGED STELLAR COMPANIONS

Multi-star systems are common, yet little is known about a stellar companion's influence on the formation and evolution of planetary systems. For instance, stellar companions may have facilitated the inward migration of hot Jupiters towards to their present day positions. Many observed short period gas giant planets also have orbits that are misaligned with respect to their star's spin axis, which has also been attributed to the presence of a massive outer companion on a non-coplanar orbit. We present the results of a multi-band direct imaging survey using Keck NIRC2 to measure the fraction of short period gas giant planets found in multi-star systems. Over three years, we completed a survey of 50 targets ("Friends of Hot Jupiters") with 27 targets showing some signature of multi-body interaction (misaligned or eccentric orbits) and 23 targets in a control sample (well-aligned and circular orbits). We report the masses, projected separations, and confirmed common proper motion for the 19 stellar companions found around 17 stars. Correcting for survey incompleteness, we report companion fractions of $48\%\pm9\%$, $47\%\pm12\%$, and $51\%\pm13\%$ in our total, misaligned/eccentric, and control samples, respectively. This total stellar companion fraction is $2.8\,\sigma$ larger than the fraction of field stars with companions approximately $50-2000\,$AU. We observe no correlation between misaligned/eccentric hot Jupiter systems and the incidence of stellar companions. Combining this result with our previous radial velocity survey, we determine that $72\% \pm 16\%$ of hot Jupiters are part of multi-planet and/or multi-star systems.

Analysis of Orbit Transfer Using Tracking Arc and Opportunity Arc of Spacecraft

The selection of the start time for the orbit transfer is not only directly related to its success or failure, but alse determines the fuel consumption and the time it needs. To solve this kind of problem, this paper based on the Lambert theorem studies the numerical solution of the transfer orbit whose starting point and spent time are both uncertain. Firstly, the problem of non-coplanar orbit transfer is converted into a coplanar one. Than, the concepts of the tracking arc of the chaser spacecraft and the opportunity arc of the target spacecraft are described. Meanwhile, the method of solving the tracking arc of the chaser spacecraft and the opportunity arc of the target spacecraft is put forward. Finally, through a concrete example, it’s found that there are close relationships among the tracking arc, the opportunity arc and the maneuvering capability of the chaser spacecraft. The results indicate that analyzing the problem of orbit transfer with the help of the tracking arc and the opportunity arc is easy and effective.

Near distance approximation in astrodynamical applications of Lambert’s theorem

The smallness parameter of the approximation method is defined in terms of the non-dimensional initial distance between target and chaser satellite. In the case of a circular target orbit, compact analytical expressions are obtained for the interception travel time up to third order. For eccentric target orbits, an explicit result is worked out to first order, and the tools are prepared for numerical evaluation of higher order contributions. The possible transfer orbits are examined within Lambert’s theorem. For an eventual rendezvous it is assumed that the directions of the angular momenta of the two orbits enclose an acute angle. This assumption, together with the property that the travel time should vanish with vanishing initial distance, leads to a condition on the admissible initial positions of the chaser satellite. The condition is worked out explicitly in the general case of an eccentric target orbit and a non-coplanar transfer orbit. The condition is local. However, since during a rendezvous maneuver, the chaser eventually passes through the local space, the condition propagates to non-local initial distances. As to quantitative accuracy, the third order approximation reproduces the elements of Mars, in the historical problem treated by Gauss, to seven decimals accuracy, and in the case of the International Space Station, the method predicts an encounter error of about 12 m for an initial distance of 70 km.

Вопросы минимизации затрат суммарной характеристической скорости, необходимой для обслуживания и восполнения спутниковых систем на некомпланарных круговых орбитах

Вопросы минимизации затрат суммарной характеристической скорости, необходимой для обслуживания и восполнения спутниковых систем на некомпланарных круговых орбитах

Six-impulse maneuvers for rendezvous of spacecraft in near-circular noncoplanar orbits

The problem of a rendezvous of two spacecraft in close near-circular noncoplanar orbits is considered. The angles of applying velocity impulses and their orientation are determined from necessary conditions of optimality derived using the basis vector theory. For non-degenerate six-impulse solutions the analytical formulas are found that approximate the dependence of moments of applying velocity impulses and angles determining their orientation on the rendezvous duration. The total characteristic velocity of six-impulse solutions (or five-impulse solutions derived from them) is compared to the total characteristic velocity obtained when solving the Lambert problem.

Tangent Orbit Technique in Three Dimensions

T HE two-body two-point boundary value problem is well known as Lambert’s problem when the transfer time is assigned. The transfer time can be written as a function of different transfer orbital parameters as independent variables. Classic methods for Lambert’s problem are to solve the semimajor axis in Lagrange’s equation of transfer time [1]. Other orbital parameters, including the flight-path angle [2] and the transverse eccentricity vector component [3], are proposed to solve Lambert’s problem. The multiple-revolution Lambert’s problem (MRLP) can also be solved with these orbital parameters [4–6]. There are some other results for MRLP by using the semimajor axis as an independent variable [7–9]. If the velocity directions of the transfer orbit at the boundary points, instead of the assigned transfer time, are required to be specified, the two-point boundary value problem can be solved as the tangent transfer orbit problem. For two coplanar tangent orbits, the velocity vectors at their single common point are collinear. If two spacecraft move in the same direction, the task of nulling the relative velocity would be dramatically simplified for an orbital rendezvous problem. By investigating the geometric characteristics of twoimpulse cotangential transfers between coplanar elliptic orbits, Adamyan et al. [10] derived the transfer orbital parameters and the magnitudes of the velocity impulses in explicit form. Moreover, based on the orbital hodograph theory, Thompson et al. [11] solved the tangent transfer orbit problem, but a numerical iterative technique was implemented for the cotangent transfer. Furthermore, Zhang et al. [12] provided closed-form solutions for the tangent transfer orbit and its solution-existence conditions. In addition, Zhang et al. [13] studied a tangent orbit rendezvous problem which requires the same flight time and the same direction of terminal velocities. All the preceding methods for the tangent transfer orbit are only valid for the coplanar case, where two orbits are tangent to each other at the common point if and only if the flight-direction angles are equal. Thus, the tangent to initial/final orbit problem can be transformed into the specified departure/arrival flight-direction angle problem whose flight-direction angle is equal to that of the initial/ final orbit. The velocity directions with the same flight-direction angle for two orbits at their common point, which is the tangency condition for noncoplanar orbits defined in [11], produce a cone such that they cannot be viewed as tangency in geometry. Actually, two different curves are tangent at the common point if and only if they share the same tangent unit vector. With this definition, if the initial and final orbits are noncoplanar, there is no transfer orbit tangent to the initial/final orbit. Thus, a new definition of “tangent” orbits in three dimensions (3D) is required. In this Note, based on a definition of tangent orbits in 3D, the flight-direction angle of the transfer orbit tangent to the initial/final orbit will be obtained. Then the tangent to initial/final orbit problem in 3D can be solved. A novel and simple relationship between the transfer angle and the initial true anomaly is provided. Then the cotangent transfer problem in 3D will be solved by the secant method.

Optimization of low-thrust orbit transfers with constraints

Approximate numerical methods of optimization are presented for multi-orbit noncoplanar orbit transfers of low-thrust spacecraft. The linear representation of derivatives of boundary parameters is used in the vicinity of a reference trajectory with its discretization into small segments. For each segment a set of pseudo-impulses is introduced, representing possible directions of the thrust vector. In order to solve essentially nonlinear problems, the iterative process of upgrading the reference trajectory is used. At each iteration the linear programming problem of high dimensionality is solved, its boundary conditions being represented in the form of a linear matrix equation. Interval constraints are considered in the form of blocking the propulsion system operation on shadow segments of the orbit, as well as cycling constraints, and constraints on total duration of maneuvers at certain trajectory segments. The results of comparison with solutions obtained by other methods are presented together with examples illustrating the convergence of iterative processes. Optimizations of the trajectories for launching geosynchronous satellites to their orbits and of the trajectories of a noncoplanar transfer from low to high-elliptic Molniya orbit are considered under these constraints.

Particle swarm optimization applied to impulsive orbital transfers

The particle swarm optimization (PSO) technique is a population-based stochastic method developed in recent years and successfully applied in several fields of research. It mimics the unpredictable motion of bird flocks while searching for food, with the intent of determining the optimal values of the unknown parameters of the problem under consideration. At the end of the process, the best particle (i.e. the best solution with reference to the objective function) is expected to contain the globally optimal values of the unknown parameters. The central idea underlying the method is contained in the formula for velocity updating. This formula includes three terms with stochastic weights. This research applies the particle swarm optimization algorithm to the problem of optimizing impulsive orbital transfers. More specifically, the following problems are considered and solved with the PSO algorithm: (i) determination of the globally optimal two- and three-impulse transfer trajectories between two coplanar circular orbits; (ii) determination of the optimal transfer between two coplanar, elliptic orbits with arbitrary orientation; (iii) determination of the optimal two-impulse transfer between two circular, non-coplanar orbits; (iv) determination of the globally optimal two-impulse transfer between two non-coplanar elliptic orbits. Despite its intuitiveness and simplicity, the particle swarm optimization method proves to be capable of effectively solving the orbital transfer problems of interest with great numerical accuracy.

Collisions in young triple systems

We perform N-body simulations of young triple systems consisting of two low-mass companions orbiting around a significantly more massive primary. We find that, when the orbits of the companions are coplanar and not too widely separated, the chance of a collision between the two companions can be as high as 20 per cent. Collisions between one of the companions (always the less massive) and the primary can also occur in systems with unequal-mass companions. The chance of collisions is a few per cent in systems with more realistic initial conditions, such as with slightly non-coplanar orbits and unequal-mass companions. If the companions start widely separated then collision are very rare except in some cases when the total mass of the companions is large. We suggest that collisions between members of young multiple systems may explain some unusual young multiple systems such as apparently non-coeval companions.

An educated search for transiting habitable planets:

Because the planets of a system form in a flattened disk, they are expected to share similar orbital inclinations at the end of their formation. The high-precision photometric monitoring of stars known to host a transiting planet could thus reveal the transits of one or more other planets. We investigate here the potential of this approach for the M dwarf GJ 1214 that hosts a transiting super-Earth. For this system, we infer the transit probabilities as a function of orbital periods. Using Monte-Carlo simulations we address both the cases for fully coplanar and for non-coplanar orbits, with three different choices of inclinations distribution for the non-coplanar case. GJ 1214 reveals to be a very promising target for the considered approach. Because of its small size, a ground-based photometric monitoring of this star could detect the transit of a habitable planet as small as the Earth, while a space-based monitoring could detect any transiting habitable planet down to the size of Mars. The mass measurement of such a small planet would be out of reach for current facilities, but we emphasize that a planet mass would not be needed to confirm the planetary nature of the transiting object. Furthermore, the radius measurement combined with theoretical arguments would help us to constrain the structure of the planet.

PHOTOMETRIC PHASE VARIATIONS OF LONG-PERIOD ECCENTRIC PLANETS

The field of exoplanetary science has diversified rapidly over recent years as the field has progressed from exoplanet detection to exoplanet characterization. For those planets known to transit, the primary transit and secondary eclipse observations have a high yield of information regarding planetary structure and atmospheres. The current restriction of these information sources to short-period planets may be abated in part through refinement of orbital parameters. This allows precision targeting of transit windows and phase variations which constrain the dynamics of the orbit and the geometric albedo of the atmosphere. Here we describe the expected phase function variations at optical wavelengths for long-period planets, particularly those in the high-eccentricity regime and multiple systems in resonant and non-coplanar orbits. We apply this to the known exoplanets and discuss detection prospects and how observations of these signatures may be optimized by refining the orbital parameters.

Modulation of circumstellar extinction in a young binary system with a low-mass companion in a noncoplanar orbit

We consider a model of cyclic brightness variations in a young star with a low-mass (q = M 2/M 1 ≤ 0.1) companion that accretes matter from the remnants of a protostellar cloud (circumbinary disk). We assume that the orbit of the companion is circular and that its plane does not coincide with the disk plane. We have computed grids of hydrodynamic models for such a binary by the SPH method based on which we have investigated the circumstellar extinction variations produced by the streams of matter and density waves excited in the circumbinary disk by the orbital motion of the companion. We show that, depending on the inclination and orientation of the binary’s line of nodes relative to the observer, the brightness of the primary component can undergo various (in shape and depth) oscillations with a period equal to the orbital one. In contrast to the models with coplanar circular orbits, the accretion rate onto the components of a binary with a noncoplanar orbit depends on the orbital phase. The results of our computations can be used to study the cyclic activity of UX Ori stars and young eclipsing binaries with anomalously long eclipses.

Optimization of constant- and variable-thrust propulsion systems

The problem of the transfer of a spacecraft with maximum payload from a fixed circular orbit to a given noncoplanar circular orbit in a spherical gravity field is solved. The spacecraft is equipped with constant-power electric propulsion and energy storage. The cases of variable-thrust and constant-thrust propulsion are considered. The increase in the payload mass due to the energy storage is estimated in both cases. The optimal time dependence of controls and the optimal relations between the mass parameters of the propulsion system are established. The ranges of these parameters where it makes sense to store energy are identified

N-Impulse Orbit Transfer Using Genetic Algorithms

T HE orbit transfer problems using impulsive thrusters have attracted researchers for a long time [1]. One of the objectives in these problems is to find the optimal fuel orbit transfer between two orbits, generally inclined eccentric orbits. The optimal two-impulse orbit transfer problem poses multiple local optima, and classical optimization methods find only local optimum solution. McCue [2] solved the problem of optimal two-impulse orbit transfer using a combination between numerical search and steepest descent optimization procedures. Jezewaski and Rozendall [3] developed an iterative method to calculate local minima solutions for the nimpulse fixed time rendezvous problems. Genetic algorithms (GAs) have been used in the literature to search for the global optimal orbit maneuver. Reichert [4] addressed the optimum two-impulse orbit transfer problem for coplanar orbits only. The accuracy obtained using this formulation is not good unless a narrow range, around the optimal value, for each design variable is known in advance [4]. Given narrow ranges for the design variables, the solution obtained using this formulation does not guarantee that the satellite will be inserted exactly into final orbit, but rather there is a small error unless the GA finds exactly the global optimal solution. Kim and Spencer [5] introduced a different formulation to the two-impulse orbit transfer problem by using six design variables for coplanar orbits. This formulation also does not guarantee the satellite is placed exactly in the final orbit. In this note, a new formulation to the problem is introduced. This formulation is general for noncoplanar elliptical orbits. It can also implement any number of thrust impulses. For the case of twoimpulse maneuver, this formulation requires only three design variables for any noncoplanar orbit transfer. The solution obtained by this formulation is guaranteed to insert the satellite in the final orbit exactly, even if the GAs did not converge to the global optimal solution. This formulation requires solving Lambert’s problem to find the parameters of the transfer orbit for a given set of the three design variables. The next section describes the orbit maneuver algorithm. The two-impulse transfer is considered a special case and is presented separately. Validation to this formulation is performed by solving several case studies to which the optimal solution is known.

Continuous Thrust Orbit Transfer Optimization Using Large-Scale Linear Programming

A new technique is developed to optimize continuous medium- and low-thrust orbit transfers. This approach combines large-scale linear programming algorithms with discretization of the trajectory dynamics on segments and a set of pseudoimpulses or control for each segment. The set is a discrete approximation with a small mesh width for a space of possible thrust directions. Boundary conditions are presented as a linear matrix equation. A matrix inequality on the sum of characteristic velocities for the pseudoimpulses is used to transform the problem into a linear programming form. The number of decision variables is on the order of tens of thousands. In modern linear programming methods there are interior-point algorithms to solve such problems. In the general case, the continuous burns include a number of adjacent segments and a postprocessing of the linear programming solutions is needed to form a sequence of the burns. An optimal number of the burns is automatically determined in the postprocessing. A maintenance of a 24-hour elliptical orbit, long-term transfer from a geo-transfer to geostationary orbit, and a one-revolution, medium-thrust transfer between two elliptical noncoplanar orbits are considered as application examples. In the last two examples an iterative solution method was used. The presented method can be used effectively for trajectory optimization in a wide range of space missions.

Eccentricity generation in hierarchical triple systems: the planetary regime

In previous papers, we developed a technique for estimating the inner eccentricity in hierarchical triple systems, with the inner orbit being initially circular. We considered systems with well-separated components and different initial setups (e.g., coplanar and non-coplanar orbits). However, the systems we examined had comparable masses. In the present paper, the validity of some of the formulae derived previously is tested by numerically integrating the full equations of motion for systems with smaller mass ratios (from 10−3 to 103, i.e. systems with Jupiter-sized bodies). There is also discussion about HD 217107 and its planetary companions.

Eccentricity Generation in Hierarchical Triple Systems with Non-coplanar and Initially Circular Orbits

In a previous paper, we developed a technique for estimating the inner eccentricity in coplanar hierarchical triple systems on initially circular orbits, with comparable masses and with well separated components, based on an expansion of the rate of change of the Runge-Lenz vector. Now, the same technique is extended to non-coplanar orbits. However, it can only be applied to systems with ${I_{0}<39.23^{\circ}}$ or ${I_{0}>140.77^{\circ}}$, where ${I}$ is the inclination of the two orbits, because of complications arising from the so-called 'Kozai effect'. The theoretical model is tested against results from numerical integrations of the full equations of motion.

Optimal Transitions of a Spacecraft with an Energy Storage and a Low-Power Thruster

A solution is found to the problem on optimal transitions of a spacecraft with a low-power thruster and an energy storage between two close noncoplanar circular orbits. The maximum payload is chosen to be the optimality criterion. The optimal time-dependences of controls and optimal relationships among the mass parameters of the propulsion system are presented. Ranges of these parameters are found within which an energy storage would be appropriate for use. The results are compared with those for a constant-power thruster. The effect of the radial increment and the increment in the orbit inclination angle on the efficiency of the energy storage is studied

The Molecular Outflow and Possible Precessing Jet from the Massive Young Stellar Object IRAS 20126+4104

We present images of the molecular gas in the IRAS 20126)4104 massive out—ow and examine the interaction between the energetic out—owing material and the surrounding molecular cloud. Mosaic interferometric images in CO(1¨0), 13CO(1¨0), C18O(1¨0), C17O(1¨0), and millimeter continuum emis- sion are compared with mid-infrared images at 12.5 and 17.9 km, near-infrared images in the band K s (2.17 km) and line emission, and optical Ha and (S II) images. We show that the molecular out—ow H 2 is approximately 6 ) 104 yr old with a mass of about 50¨60 and mass out—ow rate M _ M0 f D 8 ) 10~4 yr~1. The driving source is located near the center of the molecular cloud, and the mass M _ Z300 M _ of the disk plus circumstellar envelope traced by millimeter continuum emission is D50 The M _ . out—ow appears to be bounded on most sides by higher density gas traced by C18O emission. Shocks identi—ed by and (S II) emission knots follow a NW-SE jet close to the young stellar object and then H 2 rotate more north-south along the edges of the CO —ow. The most likely interpretation appears to be that the knots trace the working surfaces of a collimated jet that precesses through an angle of D45i. Possible mechanisms that could produce the jet precession include tidal interactions between the disk and a companion star in a noncoplanar orbit or an anisotropic accretion event that dramatically altered the angular momentum vector of the disk. Subject headings: H II regionsISM: jets and out—owsISM: moleculesstars: formation

Optimal explicit guidance for three-dimensional launch trajectory

The paper presents an exoatmospheric near-optimal explicit guidance algorithm for a satellite launch vehicle following a two- or three-dimensional trajectory. The unified steering law has been developed in a vector form using an equivalent gravitational field in an inertial coordinate frame. The resulting steering law follows a linear tangent law in a canted plane for a three-dimensional trajectory. The principle is based on the concepts of optimal transfer between non-coplanar orbits. The guidance parameters are determined by solving three simultaneous algebraic equations envolving thrust integrals which are computed recursively. Two constant gravity related vectors are defined which account for change in velocity and position due to the spherical Earth. The solution is obtained by a differential correction method after finding the required partial derivatives analytically. A predictor corrector approach is suggested where the trajectory is predicted using the Encke's method. This also enables consideration of other effects such as oblateness. The simulation results for typical multistage launch vehicles show that the guidance algorithm is extremely accurate, robust and yet simple enough for on-board implementation. It can be used for a variety of missions.