Application of Hamiltonian structure-preserving control to formation flying on a J 2-perturbed mean circular orbit

The bounded quasi-periodic relative trajectories are investigated in this paper for on-orbit surveillance, inspection or repair, which requires rapid changes in formation configuration for full three-dimensional imaging and unpredictable evolutions of relative trajectories for non-allied spacecraft. A linearized differential equation for modeling J ₂ perturbed relative dynamics is derived without any simplified treatment of full short-period effects. The equation serves as a nominal reference model for stationkeeping controller to generate the quasi-periodic trajectories near the equilibrium, i.e., the location of the chief. The developed model exhibits good numerical accuracy and is applicable to an elliptic orbit with small eccentricity inheriting from the osculating conversion of orbital elements. A Hamiltonian structure-preserving controller is derived for the three-dimensional time-periodic system that models the J ₂-perturbed relative dynamics on a mean circular orbit. The equilibrium of the system has time-varying topological types and no fixed-dimensional unstable/stable/center manifolds, which are quite different from the two-dimensional time-independent system with a permanent pair of hyperbolic eigenvalues and fixed-dimensions of unstable/stable/ center manifolds. The unstable and stable manifolds are employed to change the hyperbolic equilibrium to elliptic one with the poles assigned on the imaginary axis. The detailed investigations are conducted on the critical controller gain for Floquet stability and the optimal gain for the fuel cost, respectively. Any initial relative position and velocity leads to a bounded trajectory around the controlled elliptic equilibrium. The numerical simulation indicates that the controller effectively stabilizes motions relative to the perturbed elliptic orbit with small eccentricity and unperturbed elliptic orbit with arbitrary eccentricity. The developed controller stabilizes the quasi-periodic relative trajectories involved in six foundational motions with different frequencies generated by the eigenvectors of the Floquet multipliers, rather than to track a reference relative configuration. Only the relative positions are employed for the feedback without the information from the direct measurement or the filter estimation of relative velocity. So the current controller has potential applications in formation flying for its less computation overload for on-board computer, less constraint on the measurements, and easily-achievable quasi-periodic relative trajectories.     

Dust ring formation due to ice sublimation of radially drifting dust particles under the Poynting-Robertson effect in debris disks

In a disk with a low optical depth, dust particles drift radially inward by the Poynting-Robertson (P-R) drag rather than are blown out by stellar radiation pressure following destructive collisions. We investigate the radial distribution of icy dust composed of pure ice and refractory materials in dust-debris disks taking into account the P-R drag and ice sublimation. We find that icy dust particles form a dust ring by their pile-ups at the edge of their sublimation zone, where they sublime substantially at the temperature 100-110 K. The distance of the dust ring is 20-35 AU from the central star with its luminosity L_??30L_⊙ and 65(L_?/100L_⊙)^1/2 AU for L_??30L_⊙, where L_⊙ is the solar luminosity. The effective optical depth is enhanced by a factor of 2 for L_??100L_⊙ and more than 10 for L_??100L_⊙. The optical depth of the outer icy dust disk exceeds that of the inner disk filled with refractory particles, namely, the residue of ice sublimation, which are further subjected to the P-R effect. As a result, an inner hole is formed inside the sublimation zone together with a dust ring along the outer edge of the hole.     

Dust ring formation due to sublimation of dust grains drifting radially inward by the Poynting-Robertson drag: An analytical model

Dust particles exposed to the stellar radiation and wind drift radially inward by the Poynting-Robertson (P-R) drag and pile up at the zone where they begin to sublime substantially. The reason they pile up or form a ring is that their inward drifts due to the P-R drag are suppressed by stellar radiation pressure when the ratio of radiation pressure to stellar gravity on them increases during their sublimation phases. We present analytic solutions to the orbital and mass evolution of such subliming dust particles, and find their drift velocities at the pileup zone are almost independent of their initial semimajor axes and masses. We derive analytically an enhancement factor of the number density of the particles at the outer edge of the sublimation zone from the solutions. We show that the formula of the enhancement factor reproduces well numerical simulations in the previous studies. The enhancement factor for spherical dust particles of silicate and carbon extends from 3 to more than 20 at stellar luminosities L_?=0.8-500L_⊙, where L_⊙ is solar luminosity. Although the enhancement factor for fluffy dust particles is smaller than that for spherical particles, sublimating particles inevitably form a dust ring as long as their masses decrease faster than their surface areas during sublimation. The formulation is applicable to dust ring formation for arbitrary shape and material of dust in dust-debris disks as well as in the Solar System.     

The contour of an orbit and a stable periodic orbit

We derive an equation that relates the contour of an orbit and a stable periodic orbit.     

Tetrahedron formation of nanosatellites with single-input control

The present paper studies the formation flight of four nanosatellites forming a tetrahedron. The main goal of this research is to find the relative orbits of these satellites that, at least in the linear Hill–Clohessy–Wiltshire model, ensure finite relative motion and keep the volume and shape of the tetrahedron configuration. Since real motions of these satellites will differ from the linear ones, especially under the influence of the \(J_{2}\) perturbation, active control is necessary. In addition, the limited size of the satellites does not allow us to use a complex 3-axis attitude control system. In the present paper we consider the passive magnetic attitude control system and suppose that the thrust direction is always aligned with the local geomagnetic field. In order to increase mission lifetime the control algorithm that minimizes the propellant consumption and keeps the tetrahedron volume and shape is investigated.     

Long-term effects of main-body's obliquity on satellite formation perturbed by third-body gravity in elliptical and inclined orbit

A new non-simplified model of formation flying is derived in the presence of an oblate mainbody and third-body perturbation.In the proposed model,considering the perturbation of the thirdbody in an inclined orbit,the effect of obliquity(axial tilt) of the main-body is becoming important and has been propounded in the absolute motion of a reference satellite and the relative motion of a follower satellite.From a new point of view,J2 perturbed relative motion equations and considering a disturbing body in an elliptic inclined three dimensional orbit,are derived using Lagrangian mechanics based on accurate introduced perturbed reference satellite motion.To validate the accuracy of the model presented in this study,an auxiliary model was constructed as the Main-body Center based Relative Motion(MCRM) model.Finally,the importance of the main-body's obliquity is demonstrated by several examples related to the Earth-Moon system in relative motion and lunar satellite formation keeping.The main-body's obliquity has a remarkable effect on formation keeping in the examined in-track and projected circular orbit(PCO) formations.     

The formation of an eccentric gap in a gas disc by a planet in an eccentric orbit

We investigate the effect of a planet on an eccentric orbit on a two-dimensional low-mass gaseous disc. At a planet eccentricity above the planet's Hill radius divided by its semimajor axis, we find that the disc morphology differs from that exhibited by a disc containing a planet in a circular orbit. An eccentric gap is created with eccentricity that can exceed the planet's eccentricity and precesses with respect to the planet's orbit. We find that a more massive planet is required to open a gap when the planet is on an eccentric orbit. We attribute this behaviour to spiral density waves excited at corotation resonances by the eccentric planet. These act to increase the disc's eccentricity and exert a torque opposite in sign to that exerted by the Lindblad resonances. The reduced torque makes it more difficult for waves driven by the planet to overcome viscous inflow in the disc.     

Energetics of the aligned pulsar magnetosphere

An energy analysis is performed on two explicit models, due to Jackson, of a pulsar with aligned magnetic and rotational axes. The unknown parameters of these models are determined by calculating the minimum total energy states of the models. It is found that the minimum energy analysis favors states with extended, dynamically active magnetospheres with a high degrees of corotation. By calculating total power input to the magnetosphere via collisions in the stellar crust, and the total power radiated due to azimuthal drift motion, it is determined that the minimum energy states are the only states where a power balance can be achieved. Consideration of a local power balance condition and dissipative flows in the magnetosphere shows that neither model is completely self-consistent, but one is considerably better than the other. Properties of both models and implications for other models are discussed.     

Numerical simulations of aligned neutron star magnetospheres

We present detailed numerical simulations of the magnetosphere of an isolated neutron star in which the spin and magnetic dipole axes of the star are aligned. We demonstrate that stable charge distributions are always found, rather than particle outflows. A stable magnetosphere consists of a dome above the polar cap containing plasma of one charge and an equatorial belt containing plasma of the other sign: E · B =0 inside both of these. These are separated by a vacuum gap in which E · B ≠0 ( ρ =0 instead). We show that the charge distribution used in the 'standard' Goldreich–Julian pulsar model is inherently unstable: it collapses to a stable configuration that is very similar to the others illustrated here. An instructive video of this collapse is available at http://spacsun.rice.edu/~ian/. For typical pulsars, the stable solution has no particles near to the light cylinder, and if there were any there then their loss from the system would not lead to a replacement from the star (in contradiction to the explicit assumption used in the Goldreich–Julian model). We discuss the generic effects of pair creation, in particular as an additional source of ionization in the vacuum gap. The overall effect is simply to reduce the value of E · B in the vacuum gap so that the pair-production rate drops towards zero. A dome, disc and gap geometry is still the resulting solution. In conclusion, we confirm previous studies that the aligned rotator cannot make an active pulsar.     

The angular momentum problem and magnetic braking during star formation: Exact solutions for an aligned and a perpendicular rotator

The problems of fragmentation, angular momentum, and magnetic flux during star formation are reviewed briefly. Then the resolution of the angular momentum problem through magnetic braking is studied rigorously.A disk-like interstellar cloud of uniform density _cl is given an initial angular velocity _o about its axis of symmetry, which isaligned with an initially uniform, frozen-in magnetic field. Torsional Alfvén waves transport angular momentum from the cloud to the external medium, which has a uniform density _ext . The angular velocity of the cloud ( _cl ) is determined analytically as a function of space and time for different ratios _cl / _ext (the only free parameter in the equations), representing different stages of contraction. Despite dissimilar transient response of the cloud (or fragment) structure to different initial conditions, the characteristic time for magnetic braking of the rotation of the cloud (or fragment) as a whole is remarkably insensitive to the initial conditions and independent of the stage of contraction. The latter conclusion is in agreement with an approximate result obtained recently (Mouschovias, 1978; 1979a).A cylindrical cloud (or fragment) of uniform density is also imparted an initial angular velocity about its axis of symmetry with respect to the external medium. The frozen-in magnetic field is now initially radial andperpendicular to the axis of symmetry. In this case magnetic braking becomes more efficient upon contraction. It is more efficient than the aligned rotator case typically by one order of magnitude. The angular momentum problem can be resolved in about 10⁶ yr during the early stages of cloud contraction. Planetary systems, such as the Sun-Jupiter pair, become dynamically possible. A stage exists in which a cloud (or fragment) is in retrograde rotation with respect to its surroundings. This provides the first and only observable prediction of magnetic braking in action. It also constitutes a natural explantation of retrograde rotation in stellar and planetary systems.This work was supported in part by the National Science Foundation under grant NSF AST-77-23568.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratorio di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.     

The orbit of the Dhajala meteorite

Observations of the trail caused by the meteorite which fell around Dhajala, Gujarat (India), on 28 January 1976 have been used to compute the probable orbit of the meteoroid in space. The cosmic ray effects in the meteorite fragments indicate high mass ablation (?90%), suggesting a high velocity (?20 km/sec) of entry into the Earth's atmosphere. The atmospheric trajectory is reasonably well documented and its deviation from the projected ground fallout can be understood in terms of the ambient wind pattern. The apparent radiant of the trail was at a point in the sky with right ascension 165°, declination +60°. Considering the errors in estimating the radiant, we get a range of orbits with a = 2.3 ± 0.8 AU, e = 0.6 ± 0.1, and i = 28 ± 4° with the constraints of a ? 1.5 AU and V_∞ < 25 km/sec (which causes nearly complete evaporation of the meteoroid). Taking V_∞ = 21.5 lm/sec as indicated by the measured mass ablation of the meteorite, the orbital elements are deduced to be $\text{a = 1.8}\text{AU}\text{, e = 0.59, i = 27°.6, ω = 109°.1, Ω = 307°.8,}\text{and}\text{q = 0.74}$.     

Pulsation and orbit of AU Pegasi

AU Pegasi is a pulsating star in a spectroscopic binary system with an orbital period of 53.26 days. Between 1960 and 1990 an extremely rapid period increase was observed in the value of the pulsation period, but in the last 15 years the observation show that the period set in 2.411 days. Fourier analysis of photometric data obtained during the ASAS project and those taken at the Piszkéstető Mountain Station of the Konkoly Observatory during 1994–2005 indicate that AU Pegasi is pulsating in two modes simultaneously, and the ratio of the frequencies of the two modes is 0.706, a value common for double‐mode classical Cepheids. A careful analysis of other photometric observations obtained during the era of the strong period increase also revealed existence of a second mode. This may suggest that this star is not a Type II Cepheid, despite its galactic position. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Symbolic analytical solution of N‐dimensional radially symmetric polytropes

In this paper, accelerated power series solutions are developed for N‐dimensional symmetric radially polytropes. The solutions are valid for any geometric and polytropic index. The implementation of Padé technique and changing of the independent variable give us identical polytrope solutions to the numerical one. Physical parameters are presented for Ndimensional radially polytropes. In case of the isothermal sphere, the density profile is accurate to 10^–5 per cent whatever the value of ξ is. All the computations are performed using Mathematica software. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The orbit of comet Tsuchinshan 1977q

Comet Tsuchinshan (1977q) was discovered at Purple Mountain Observatory on Nov. 3, 1977. The osculating orbital elements (referred to the barycenter of the sun and Mercury) given in table 1 are derived from 42 observations Nov. 3, 1977 to Jan. 10, 1978 at our observatory and abroad, taking into account the perturbations of all major planets. In all the observations the rms residual is 1.″78. Our observations are also published in table 2. With the elements thus obtained, the appropriate ephemerides in recent years have been calculated and listed in table 4. And then, in consideration of accurate perturbations, we compute the original and future orbits (referred to the center of mass of solar system) while the comet is about 60 A. U. from the sun. The results that both the original and future orbits are hyperbolicaare given in table 5 and may be compared with the osculating elements. Nevertheless, the two months observed interval for determining orbit is short. In order to obtain the more reliable conclusion, it is necessary to make some efforts for increasing the observed arc.     

On the history of the lunar orbit

A frequency-dependent model of tidal friction is used in the determination of the time rate of change of the lunar orbital elements and the angular velocity of the Earth. The variational equations consider eccentricity, the solar tide on the Earth, Earth oblateness, and higher-order terms in the Earth's tidal potential. A linearized solution of the equations governing the precission of the Earth's rotational angular momentum and the lunar ascending node is found. This allows the analytical averaging of the variational equations over the period of relative precession which, though large, is necessarily small in comparison to the time step of the numerical integrator that yields the system history over geological time. Results for this history are presented and are identified as consistent with origin of the Moon by capture. This model may be applied to any planet-satellite system where evolution under tidal friction is of interest.     

The Coulomb Interaction Energy of Plasmas with Superconducting Particles

We derive a partial differential equation for the determination of the electric potential in a fully ionized plasma. Using the time independent solutions for electrostatic potential we calculate the Coulomb interaction energy of the particles in a superconducting plasma. We show that, when electrons become superconducting, the energy change corresponding to the Coulomb interaction part is positive, while the correlation part is negative. The same phenomenon occurs in the core of the neutron stars when protons become super-conducting.__________Published in Astrofizika, Vol. 48, No. 3, pp. 431–437 (August 2005).     

库仑耦合效应对类太阳恒星模型的影响

物态方程一直是恒星模型理论研究中的最重要的物理方程之一，对于日震学等高精度要求的研究领域来讲，只包含理想效应的物态方程并不能满足研究需要，而要更细致、包含非理想效应的物态方程对系统作更精确和完善的描述，在弱耦合及弱电子简并的系统中，为研究各种带电粒子之间的库仑耦合效应，以化学图像为基础，分别用具有硬核改正的Debye—Hueckel扩展理论来处理离子-电子作用、多体作用的经典点粒子相互作用理论描述离子-离子作用和量子统计方法处理电子-电子作用，并把此3种作用对物态方程自由能的贡献用半解析的数学公式给出，为适应日震学研究的要求，特别选取一组类太阳的恒星模型作为研究对象，从实际模型计算出发，对库仑耦合中的各种效应进行了比较分析，讨论并检验了所作的理论改进。     

The spectroscopic and interferometric orbit of Gliese 150.2

We determined the spectroscopic-interferometric orbit of the binary red dwarf Gliese 150.2 with a period of 13.84 yr and a semimajor axis of 0.257 arcsec. Based on the orbital elements and on accurate measurements of the magnitude difference at several wavelengths, we estimated the spectral types and masses of the components (K0 V and M0 V, 0.79 and 0.55M _⊙) and the dynamical parallax of the binary (40.4 mas).