Coplanar Orbits Research Articles

AN M DWARF COMPANION AND ITS INDUCED SPIRAL ARMS IN THE HD 100453 PROTOPLANETARY DISK

ABSTRACT Recent VLT/SPHERE near-infrared imaging observations revealed two spiral arms with a near m = 2 rotational symmetry in the protoplanetary disk around the ∼1.7 M ⊙ Herbig star HD 100453. A ∼0.3 M ⊙ M dwarf companion, HD 100453 B, was also identified at a projected separation of 120 AU from the primary. In this Letter, we carry out hydrodynamic and radiative transfer simulations to examine the scattered light morphology of the HD 100453 disk as perturbed by the companion on a circular and coplanar orbit. We find that the companion truncates the disk at ∼45 AU in scattered light images, and excites two spiral arms in the remaining (circumprimary) disk with a near m = 2 rotational symmetry. Both the truncated disk size and the morphology of the spirals are in excellent agreement with the SPHERE observations at Y, J, H, and K1-bands, suggesting that the M dwarf companion is indeed responsible for the observed double-spiral-arm pattern. Our model suggests that the disk is close to face on (inclination angle ∼5°), and that the entire disk-companion system rotates counterclockwise on the sky. The HD 100453 observations, along with our modeling work, demonstrate that double spiral arm patterns in near-infrared scattered light images can be generically produced by companions, and support future observations to identify the companions responsible for the arms observed in the MWC 758 and SAO 206462 systems.

Secular dynamics of S-type planetary orbits in binary star systems: applicability domains of first- and second-order theories

We analyse the secular dynamics of planets on S-type coplanar orbits in tight binary systems, based on first- and second-order analytical models, and compare their predictions with full N-body simulations. The perturbation parameter adopted for the development of these models depends on the masses of the stars and on the semimajor axis ratio between the planet and the binary. We show that each model has both advantages and limitations. While the first-order analytical model is algebraically simple and easy to implement, it is only applicable in regions of the parameter space where the perturbations are sufficiently small. The second-order model, although more complex, has a larger range of validity and must be taken into account for dynamical studies of some real exoplanetary systems such as $\gamma$-Cephei and HD 41004A. However, in some extreme cases, neither of these analytical models yields quantitatively correct results, requiring either higher-order theories or direct numerical simulations. Finally, we determine the limits of applicability of each analytical model in the parameter space of the system, giving an important visual aid to decode which secular theory should be adopted for any given planetary system in a close binary.

A triple origin for the lack of tight coplanar circumbinary planets around short-period binaries

Transiting circumbinary planets are more easily detected around short-period than long-period binaries, but none have yet been observed by {\it Kepler} orbiting binaries with periods shorter than seven days. In triple systems, secular Kozai-Lidov cycles and tidal friction (KLCTF) have been shown to reduce the inner orbital period from $\sim 10^4$ to a few days. Indeed, the majority of short-period binaries are observed to possess a third stellar companion. Using secular evolution analysis and population synthesis, we show that KLCTF makes it unlikely for circumbinary transiting planets to exist around short-period binaries. We find the following outcomes. (1) Sufficiently massive planets in tight and/or coplanar orbits around the inner binary can quench the KL evolution because they induce precession in the inner binary. The KLCTF process does not take place, preventing the formation of a short-period binary. (2) Secular evolution is not quenched and it drives the planetary orbit into a high eccentricity, giving rise to an unstable configuration, in which the planet is most likely ejected from the system. (3) Secular evolution is not quenched but the planet survives the KLCTF evolution. Its orbit is likely to be much wider than the currently observed inner binary orbit, and is likely to be eccentric and inclined with respect to the inner binary. These outcomes lead to two main conclusions: (1) it is unlikely to find a massive planet on a tight and coplanar orbit around a short-period binary, and (2) the properties of circumbinary planets in short-period binaries are constrained by secular evolution.

Going with the flow: using gas clouds to probe the accretion flow feeding Sgr A*

The massive black hole in our galactic center, Sgr A*, accretes only a small fraction of the gas available at its Bondi radius. The physical processes determining this accretion rate remain unknown, partly due to a lack of observational constraints on the gas at distances between ~10 and ~10$^5$ Schwarzschild radii (Rs) from the black hole. Recent infrared observations identify low-mass gas clouds, G1 and G2, moving on highly eccentric, nearly co-planar orbits through the accretion flow around Sgr A*. Although it is not yet clear whether these objects contain embedded stars, their extended gaseous envelopes evolve independently as gas clouds. In this paper we attempt to use these gas clouds to constrain the properties of the accretion flow at ~10$^3$ Rs. Assuming that G1 and G2 follow the same trajectory, we model the small differences in their orbital parameters as evolution resulting from interaction with the background flow. We find evolution consistent with the G-clouds originating in the clockwise disk. Our analysis enables the first unique determination of the rotation axis of the accretion flow: we localize the rotation axis to within 20 degrees, finding an orientation consistent with the parsec-scale jet identified in x-ray observations and with the circumnuclear disk, a massive torus of molecular gas ~1.5 pc from Sgr A*. This suggests that the gas in the accretion flow comes predominantly from the circumnuclear disk, rather than the winds of stars in the young clockwise disk. This result will be tested by the Event Horizon Telescope within the next year. Our model also makes testable predictions for the orbital evolution of G1 and G2, falsifiable on a 5-10 year timescale.

Extremal Control and Guidance Solutions for Orbital Transfer with Intermediate Thrust

Mayer’s variational problem of determining optimal trajectories of motion with constant exhaust velocity and limited mass-flow rate in a Newtonian field is considered. Analytical solutions for extremal transfers between two coplanar elliptical orbits via one and two intermediate-thrust arcs are presented. It is shown that in the cases of transfers via one and two intermediate-thrust arcs, the orbital parameters have to satisfy respectively two constraints and one constraint imposed by the continuity conditions formed at the junctions. This means that respectively two and one of these orbital parameters must not be arbitrary. Furthermore, by utilizing the analytical solutions for the intermediate-thrust arcs, the corresponding extremal guidance laws for a transfer between two points with arbitrary position vectors and for a transfer between elliptical orbits are presented. The guidance commands are formulated in terms of commanded thrust angle and commanded mass-flow rate. A numerical example illustrating the proposed trajectory and guidance solutions is discussed. The results of the computations are compared to those of similar one- and two- impulsive transfers, and a low-thrust orbit raising maneuver. It is demonstrated that for some eccentricities of the terminal orbits, the fuel consumption for a transfer via an intermediate-thrust arc is comparable to that of the two-impulsive transfers. Analyses show that a maneuver with such an arc may be fuel-efficient with longer duration and higher thrust than that of the low-thrust maneuver. The trajectory computations and guidance performances are illustrated by graphical relationships.

Terrestrial planet formation constrained by Mars and the structure of the asteroid belt

Reproducing the large Earth/Mars mass ratio requires a strong mass depletion in solids within the protoplanetary disk between 1 and 3 AU. The Grand Tack model invokes a specific migration history of the giant planets to remove most of the mass initially beyond 1 AU and to dynamically excite the asteroid belt. However, one could also invoke a steep density gradient created by inward drift and pile-up of small particles induced by gas-drag, as has been proposed to explain the formation of close-in super Earths. Here we show that the asteroid belt's orbital excitation provides a crucial constraint against this scenario for the Solar System. We performed a series of simulations of terrestrial planet formation and asteroid belt evolution starting from disks of planetesimals and planetary embryos with various radial density gradients and including Jupiter and Saturn on nearly circular and coplanar orbits. Disks with shallow density gradients reproduce the dynamical excitation of the asteroid belt by gravitational self-stirring but form Mars analogs significantly more massive than the real planet. In contrast, a disk with a surface density gradient proportional to $r^{-5.5}$ reproduces the Earth/Mars mass ratio but leaves the asteroid belt in a dynamical state that is far colder than the real belt. We conclude that no disk profile can simultaneously explain the structure of the terrestrial planets and asteroid belt. The asteroid belt must have been depleted and dynamically excited by a different mechanism such as, for instance, in the Grand Tack scenario.

A study of optimal low-thrust limited-power transfers between coplanar orbits with small eccentricities

In this paper, an analytical solution, which includes the short periodic terms, for the problem of optimal time-fixed low-thrust limited power transfers (no rendezvous), in an inverse-square force field, between coplanar orbits with small eccentricities is described. This solution is derived through canonical transformation theory. Some numerical results based on the analytical solution are presented for a preliminary analysis of interplanetary missions.

Bounds on maximal and minimal distances for coplanar satellite relative motion under given initial conditions

Maximal and minimal distances are the key parameters for a typical leader–follower satellite formation. For the mission designers, the definition of the initial relative orbital elements that are constrained by extreme distances is a difficult but meaningful problem. In this paper, the analytic distance bounds for the coplanar relative motion are obtained. These solutions are accurate for those periodic close formations in which the follower and leader satellite have equal semi-major axis. Besides, the assumption that the leader satellite has a circular orbit is adopted. Five kinds of special positions that are located on a single axis and generated the given extremum distances are derived and discussed. The simulation results are also provided to corroborate the analytic developments.

THE STATISTICAL MECHANICS OF PLANET ORBITS

The final "giant-impact" phase of terrestrial planet formation is believed to begin with a large number of planetary "embryos" on nearly circular, coplanar orbits. Mutual gravitational interactions gradually excite their eccentricities until their orbits cross and they collide and merge; through this process the number of surviving bodies declines until the system contains a small number of planets on well-separated, stable orbits. In this paper we explore a simple statistical model for the orbit distribution of planets formed by this process, based on the sheared-sheet approximation and the ansatz that the planets explore uniformly all of the stable region of phase space. The model provides analytic predictions for the distribution of eccentricities and semimajor axis differences, correlations between orbital elements of nearby planets, and the complete N-planet distribution function, in terms of a single parameter, the "dynamical temperature", that is determined by the planetary masses. The predicted properties are generally consistent with N-body simulations of the giant-impact phase and with the distribution of semimajor axis differences in the Kepler catalog of extrasolar planets. A similar model may apply to the orbits of giant planets if these orbits are determined mainly by dynamical evolution after the planets have formed and the gas disk has disappeared.

Optimal low-thrust transfers between coplanar orbits with small eccentricities

In this paper, a first-order analytical solution, which includes the short periodic terms, for the problem of optimal time-fixed low-thrust limited-power transfers (no rendezvous), in an inverse-square force field, between coplanar orbits with small eccentricities is obtained through canonical transformation theory. Short periodic terms are eliminated from the maximum Hamiltonian, expressed in non-singular orbital elements, through an infinitesimal canonical transformation built through the Hori method. Closed-form analytical solutions are obtained for the average canonical system by solving the Hamilton–Jacobi equation through the separation of variables technique. For long duration maneuvers, the existence of conjugate points is investigated through the Jacobi condition.

Variable-time-domain neighboring optimal guidance applied to space trajectories

This research describes and applies the recently introduced, general-purpose variable-time-domain neighboring optimal guidance scheme, which is capable of driving a space vehicle in the proximity of a specified nominal, optimal path. This goal is achieved by minimizing the second differential of the objective function (related to fuel consumption) along the perturbed trajectory. This minimization principle leads to deriving all the corrective maneuvers, in the context of a closed-loop guidance scheme. Several time-varying gain matrices, referring to the nominal trajectory, are defined, computed offline, and stored in the onboard computer. Original analytical developments, based on optimal control theory and adoption of a variable time domain, constitute the theoretical foundation for three relevant features that characterize the guidance algorithm used in this work: (i) a new, efficient law for the real-time update of the time of flight (usually referred to as time-to-go), (ii) a new, effective termination criterion, and (iii) a new formulation of the sweep method. As a first application, this paper considers minimum-time lunar ascent and descent paths, under the flat Moon approximation. The second application is represented by the minimum-time, continuous-thrust orbit transfer between two coplanar circular orbits. In both cases, the nominal trajectories are two-dimensional, while the corresponding perturbed paths are three-dimensional. Specifically, perturbations arising from the imperfect knowledge of the propulsive parameters and from errors in the initial conditions are included in the dynamical simulations. Extensive Monte Carlo tests are performed, and definitely prove the effectiveness and accuracy of the variable-time-domain neighboring optimal guidance algorithm.

Global optimization of fuel consumption in rendezvous scenarios by the method of interval analysis

To reduce the optimal but large Δv of the fixed-short-time two impulse Lambert rendezvous between two spacecrafts along two coplanar circular orbits, the three-impulse Lambert rendezvous optimized via the optimization algorithm-interval analysis (IA) is proposed in this paper. The purpose of optimization is to minimize the velocity increment of the fixed-short-time three-impulse Lambert rendezvous. The optimization algorithm IA is given for solving the rendezvous optimization problem with multiple uncertain variables, and strong nonlinearity and nonconvexity. Numerical examples of the time-open, coplanar-circular-orbit, multiple-revolution Lambert rendezvous with a parking time optimized via the optimization algorithm IA are firstly undertaken to validate the feasibility of the optimization algorithm IA by comparing the optimization results with those of the globally optimal Hohmann transfer. The results indicate that the globally optimal parameters of the time-open coplanar-circular-orbit multiple-revolution Lambert rendezvous can be obtained by the optimization algorithm IA, and the initial separation angle of two spacecrafts with different orbit radius can be adjusted to obtain the globally optimal and small Δv by distributing an optimal parking time. After that, for the fixed-short-time two-impulse Lambert rendezvous problem without sufficient time to adjust the separation angle by distributing a parking time like the open-time Lambert rendezvous problem, three-impulse Lambert rendezvous involving multiple optimization variables is given and the variables are optimized by the optimization algorithm IA to obtain an optimal and small Δv. Numerical simulation indicates that the optimal and small Δv of the fixed short time, three-impulse Lambert rendezvous can be obtained using the optimization algorithm IA.

A method of coverage analysis for multipoint coupling plasma detection on space processes

针对空间物理过程时空协同探测任务的定量化探测覆盖分析需求, 在采用时空相关函数方法描述和分析空间物理时空过程的基础上, 提出一种基于<em>θ</em>-<em>t</em>_R覆盖图的探测覆盖分析方法. 通过分析不同设计方案在一定累计时间周期内的<em>θ</em>-<em>t</em>_R覆盖图, 根据其覆盖范围的不同选择所需要的设计方案. 采用该方法对两条共轨道面椭圆轨道的时空协同探测过程进行了定量的覆盖分析, 验证了该方法的有效性.

Two-impulse transfer between coplanar elliptic orbits using along-track thrust

The two-body two-impulse orbit transfer problem between two coplanar elliptic orbits using only along-track thrust is studied in this paper. Both impulse vectors are constrained to be on the along-track directions at the impulse points. First, the relationship between transfer angle and final true anomaly is derived. For a given final true anomaly, the range of the transfer angle is obtained for different cases. Then, the transfer angle is solved by the secant method in its feasible range. With the transfer angle and the transfer semilatus rectum, the velocity vectors of the transfer orbit at impulse points are obtained. Two impulse points and the corresponding impulse magnitudes for a 180\(^{\circ }\) transfer are also analyzed. Finally, the minimum-energy and minimum-transit-time solutions are obtained by optimizing the final true anomaly. Numerical tests demonstrate the effectiveness of the proposed method for two-impulse along-track orbit transfers. The results show that the minimum-energy transfer with two along-track impulses is close to the “true” minimum-energy transfer with two free-direction impulses.

A comprehensive photometric study of the eclipsing binary EP Aurigae

We present new observations for the eclipsing binary EP Aurigae, which were performed by using three small telescopes in China from 2003 December to 2014 January. With the updated 2003 version of the Wilson–Devinney code, the photometric elements were deduced from three sets of light curves. Based on all available eclipsing times, the orbital period changes were investigated. It is discovered that the (O–C) curve may show an existence of light-time effect due to an unseen third body, which was weakly identified by the photometric solution. The modulated period and amplitude of the cyclic variation are P3=71.2(±8.0)yr and A=0.0101(±0.0008)day, respectively. In the co-planar orbit with the binary system, the mass of the third body is M3=0.18(±0.02)M⊙. The photometric results imply that EP Aur is an Algol-type binary with a mass ratio of q=0.831(±0.004). Its primary component almost fills its Roche lobe. Therefore, EP Aur may consist of a normal main-sequence star and a cool Roche-lobe filling subgiant, which may be undergoing rapid mass transfer.

Fast solar sail rendezvous mission to near Earth asteroids

The concept of fast solar sail rendezvous missions to near Earth asteroids is presented by considering the hyperbolic launch excess velocity as a design parameter. After introducing an initial constraint on the hyperbolic excess velocity, a time optimal control framework is derived and solved by using an indirect method. The coplanar circular orbit rendezvous scenario is investigated first to evaluate the variational trend of the transfer time with respect to different hyperbolic excess velocities and solar sail characteristic accelerations. The influence of the asteroid orbital inclination and eccentricity on the transfer time is studied in a parametric way. The optimal direction and magnitude of the hyperbolic excess velocity are identified via numerical simulations. The found results for coplanar circular scenarios are compared in terms of fuel consumption to the corresponding bi-impulsive transfer of the same flight time, but without using a solar sail. The fuel consumption tradeoff between the required hyperbolic excess velocity and the achievable flight time is discussed. The required total launch mass for a particular solar sail is derived in analytical form. A practical mission application is proposed to rendezvous with the asteroid 99942 Apophis by using a solar sail in combination with the provided hyperbolic excess velocity.

Planet formation in a triple stellar system: implications of the third star's orbital inclination

AbstractPlanets have been revealed both in binary and triple stellar systems. Although there have been several studies of the late stages of planet formation in binary stars this process does not appear to have been studied in triple stellar systems. To understand how the late stage of planetary accretion is affected by a third companion, in this work we have numerically investigated the formation of planets in a hypothetical triple stellar system. The system is composed by an inner binary formed by two half-solar-mass components orbited by a solar-mass star. In our experiments, lunar and Mars-sized planetary embryos are distributed around the centre of mass of the inner binary system. Our main goal is to analyse how the formation of planets evolves depending on the orbital configuration of the massive distant companion. We have performed an extensive number of numerical simulations considering different orbital configurations for the third star. All simulations were numerically integrated for at least 107 years. The results show that when the protoplanetary disc and the stars are initially on coplanar orbits, one or two planets are quickly formed between 6 and 8 AU. In general such planets have also small eccentricities with values about 10−2. On the other hand, when the third star is considered initially on inclined orbits (even tiny values), there tends to occur a significant increase in the inclination of bodies of protoplanetary disc, which prevents the collisions between these objects and their growth. As a result, in this latter case we do not evidence the formation of planets during the timescale of our integrations but note the existence of several leftover objects that can survive for longer than 10 Myr, moving in orbits with semi-major axes ranging between ~6 and 8 AU. Thus, our results do not rule out the planet formation in this kind of stellar arrangements at all, but they indicate that, if planetary bodies keep stable orbits, the late stage of planet formation in systems with a highly inclined third star can be a very long process and many of these triple hierarchical systems might not have had time to form planets and planetary systems. They could be harbouring only debris discs, fragments or planetesimals.

Validation of GRACE electron densities by incoherent scatter radar data and estimation of plasma scale height in the topside ionosphere

This paper presents an effort of using incoherent scatter radar data for validating electron density (Ne) measurements performed by the GRACE satellites from year 2002 to 2012. For adjusting the bias of GRACE Ne data, the observations at high latitudes from EISCAT at Tromsø and Svalbard, as well as two incoherent scatter radars (ISR) at mid- and low latitudes, Millstone Hill and Arecibo, are used. The adjusted GRACE Ne data are further compared with the observations from the four ISRs. For EISCAT observations at Tromsø and Svalbard the comparison results are quite consistent, yielding correlation coefficients as high as 0.92, and an average bias value of about 3·1010m−3 is obtained. For the radars at Millstone Hill and Arecibo the results show excellent agreement, yielding correlation coefficients as high as 0.97 and an average bias of 1·1010m−3. The scale factor of adjusted GRACE Ne data is lower by 1% and 5% compared to Millstone Hill and Arecibo readings, respectively. We consider these differences as within the uncertainty of radar measurements. Using the adjusted GRACE Ne as well as CHAMP observations during four periods of coplanar orbits between 2003 and 2008, the plasma scale heights of the topside ionosphere are determined and further compared with IRI model predictions. We find significantly larger scale heights in particular at middle and high latitudes than expected from IRI. Outstanding are the regions of the mid-latitude electron density trough.

Misaligned protoplanetary disks in a young binary star system.

Many extrasolar planets follow orbits that differ from the nearly coplanar and circular orbits found in our Solar System; their orbits may be eccentric or inclined with respect to the host star's equator, and the population of giant planets orbiting close to their host stars suggests appreciable orbital migration. There is at present no consensus on what produces such orbits. Theoretical explanations often invoke interactions with a binary companion star in an orbit that is inclined relative to the planet's orbital plane. Such mechanisms require significant mutual inclinations between the planetary and binary star orbital planes. The protoplanetary disks in a few young binaries are misaligned, but often the measurements of these misalignments are sensitive only to a small portion of the inner disk, and the three-dimensional misalignment of the bulk of the planet-forming disk mass has hitherto not been determined. Here we report that the protoplanetary disks in the young binary system HKTauri are misaligned by 60 to 68degrees, such that one or both of the disks are significantly inclined to the binary orbital plane. Our results demonstrate that the necessary conditions exist for misalignment-driven mechanisms to modify planetary orbits, and that these conditions are present at the time of planet formation, apparently because of the binary formation process.

Tour of Jupiter Galilean moons: Winning solution of GTOC6

The paper presents the trajectory designed by the Italian joint team Politecnico di Torino & Sapienza Università di Roma (Team5), winner of the 6th edition of the Global Trajectory Optimization Competition (GTOC6). In the short time available in these competitions, Team5 resorted to basic knowledge, simple tools and a powerful indirect optimization procedure. The mission concerns a 4-year tour of the Jupiter Galilean moons. The paper explains the strategy that was preliminarily devised and eventually implemented by looking for a viable trajectory. The first phase is a capture that moves the spacecraft from the arrival hyperbola to a low-energy orbit around Jupiter. Six series of flybys follow; in each one the spacecraft orbits Jupiter in resonance with a single moon; criteria to construct efficient chains of resonant flybys are presented. Transfer legs move the spacecraft from resonance with a moon to another one; precise phasing of the relevant moons is required; mission opportunities in a 11-year launch window are found by assuming ballistic trajectories and coplanar circular orbits for the Jovian satellites. The actual trajectory is found by using an indirect technique.