Satellite attitude acquisition by momentum transfer-the controlled wheel speed method

An implementation of the momentum transfer method for spacecraft attitude acquisition of momentum wheel stabilized geostationary satellites is presented, in which the wheel speed is varied in a predeterminable manner to reduce the nutation usually associated with the method. The implementation is found to be capable of achieving the transfer to the desired zero nutation end point with 5° to 20° of residual nutation in practical situations without additional nutation damping. The transfer time is typically 10–30 min. The implementation is described in terms of the momentum sphere and energy ellipsoid. The detailed functional dynamics and parametric relationships are given in terms of phase planes and elliptic integral solutions. Feasibility of the implementation is shown to be dependent on the moment of inertia configuration and the degree to which tolerances on moments of inertia, satellite spin rate, and wheel speed are predeterminable.Nomenclature A, B, C Principal moments of inertia of the total spacecraft - b ₁,b ₂,b ₃ Unit vectors parallel to principal axes - E, E(t) Energy of the spacecraft body (excludes the spin energy of momentum wheel) - E ₀ Initial energy of the spacecraft (wheel not spinning) - E _c A constant arbitrary value ofE(t) - E _i Energy after an impulse which causes to equal zero, Equation (13) - E ₂ The constant value ofE during Stage 2 - E ₃ Energy at the end of Stage 3 - E _s The separatrix value ofE, i.e.E when =0 - H ₀,H ₀ Angular momentum vector of the total spacecraft - H _x ,H _y ,H _z Variable components ofH ₀ resolved with respect tob ₁,b ₂, andb ₃ - J Moment of inertia of the momentum wheel - L The slope of the wheel speed profile, Equation (31) - L _y Momentum wheel torque - m Direction cosine of the angle, , betweenH ₀ andb ₂ - m ₁,m ₂ Constant values ofm - O The centre of the momentum sphere - Oxyz Spacecraft coordinate axes - P Represents a state on the momentum sphere and energy ellipsoid - s Momentum wheel speed - s ₂ Constant value ofs during Stage 2 - s _i Wheel speed after an impulse which causes to be zero - t Time - t ₁, t ₂, t ₃ Elapsed time for Stages 1, 2, and 3 respectively - ₁ to ₄ Roots of Equation (19) (poles on the phase plane) - Parameter defined in Equation (9) - cos^–1 m - ₂ The value of at the termination of Stage 2 - _min The minimum of the periodic function (t) (during Stage 2) - _x , _y , _z Angular rates of spacecraft body - ₀ Initial value of _z - K _m Curvature of the momentum sphere - K _xy ,K _yz Curvatures of the principal lines of the energy ellipsoid at (O,H ₀,O), in theOH _x H _y ) and (OH _y H _z ) planes respectively     

Some properties of the dumbbell satellite attitude dynamics

The dumbbell satellite is a simple structure consisting of two point masses connected by a massless rod. We assume that it moves around the planet whose gravity field is approximated by the field of the attracting center. The distance between the point masses is assumed to be much smaller than the distance between the satellite’s center of mass and the attracting center, so that we can neglect the influence of the attitude dynamics on the motion of the center of mass and treat it as an unperturbed Keplerian one. Our aim is to study the satellite’s attitude dynamics. When the center of mass moves on a circular orbit, one can find a stable relative equilibrium in which the satellite is permanently elongated along the line joining the center of mass with the attracting center (the so called local vertical). In case of elliptic orbits, there are no stable equilibrium positions even for small values of the eccentricity. However, planar periodic motions are determined, where the satellite oscillates around the local vertical in such a way that the point masses do not leave the orbital plane. We prove analytically that these planar periodic motions are unstable with respect to out-of-plane perturbations (a result known from numerical investigations cf. Beletsky and Levin Adv Astronaut Sci 83, 1993). We provide also both analytical and numerical evidences of the existence of stable spatial periodic motions.     

On the attitude dynamics of perturbed triaxial rigid bodies

Attitude dynamics of perturbed triaxial rigid bodies is a rather involved problem, due to the presence of elliptic functions even in the Euler equations for the free rotation of a triaxial rigid body. With the solution of the Euler–Poinsot problem, that will be taken as the unperturbed part, we expand the perturbation in Fourier series, which coefficients are rational functions of the Jacobian nome. These series converge very fast, and thus, with only few terms a good approximation is obtained. Once the expansion is performed, it is possible to apply to it a Lie-transformation. An application to a tri-axial rigid body moving in a Keplerian orbit is made.     

A unified state model of orbital trajectory and attitude dynamics

The trajectory and attitude dynamics of an orbital spacecraft are defined by a unified state model, which enables efficient and rapid machine computation for mission analysis, orbit determination and prediction, satellite geodesy and reentry analysis. The state variables are momenta — a general form for attitude, and a parametric form for orbital motion. The orbital parameters are the velocity state characteristics of the orbital hodograph. The coordinate variables are sets of four Euler parameters, which define the rotation transformation by the quaternion algebra. The unified state model possesses many analytical properties which are invaluable for dynamical system synthesis, numerical analysis and machine solution: regularization, unified matrix algebra, state graphs and transforms. The analytic partials of position and velocity with the state and coordinate variables are presented, as well as representative perturbation functions such as air drag, gravitational potential harmonics, and propulsion thrust.     

Investigation of periodic orbits in elliptical galaxies using the method of implicit functions

Existence of periodic orbits inside elliptical galaxies has been investigated. Necessary conditions for regular, small amplitude periodic motion around the center of galaxy have been derived using implicit functions and solved by approximating through Taylor's series. The solution procedure requires to obtain functions of partial derivatives of dependent variables with respect to initial conditions. Derivation of these functions can be accomplished through solving a set of ordinary differential equations by proper choices of associated initial conditions. The results obtained show complete agreement with those obtained through the application of Poincaré-Lindstedt's method.     

An improved version of the implicit integral method to solving radiative transfer problems

Radiative transfer (RT) problems in which the source function includes a scattering-like integral are typical two-points boundary problems. Their solution via differential equations implies making hypotheses on the solution itself, namely the specific intensity I (τ; n) of the radiation field. On the contrary, integral methods require making hypotheses on the source function S(τ). It seems of course more reasonable to make hypotheses on the latter because one can expect that the run of S(τ) with depth is smoother than that of I (τ; n). In previous works we assumed a piecewise parabolic approximation for the source function, which warrants the continuity of S(τ) and its first derivative at each depth point. Here we impose the continuity of the second derivative S′′(τ). In other words, we adopt a cubic spline representation to the source function, which highly stabilizes the numerical processes.     

A method of astronomical autonomous orbit and attitude determinations for satellites

A method of realtime autonomous orbit determination for earth satellites using the extended Kalman filtering is proposed. The observed quantities are: the satellite-sun direction vector measured by a sun sensor, the satellite-earth and satellite-moon direction vectors measured by an ultraviolet sensor, and the geocentric distance measured by a radar altimeter. At the same time the satellite attitude to the earth is also determined. Results of our simulation of the autonomous orbit determination show that the precision of the orbit determinations is better than 200 m. The effects of the sampling period, orbital inclination, orbital eccentricity and orbital altitude on the precision of orbit determination are analyzed and compared, and certain principles helpful for improving the precision of orbit determination are suggested.     

Long-term attitude dynamics of space debris in Sun-synchronous orbits: Cassini cycles and chaotic stabilization

Comprehensive analysis of space debris rotational dynamics is vital for active debris removal missions that require physical capture or detumbling of a target. We study the attitude motion of large space debris objects that admittedly pose an immediate danger to space operations in low Earth orbits. Particularly, we focus on Sun-synchronous orbits (SSO) with altitude range 600–800 km, where the density of space debris is maximal. Our mathematical model takes into account the gravity-gradient torque and the torque due to eddy currents induced by the interaction of conductive materials with the geomagnetic field. Using perturbation techniques and numerical methods, we examine the deceleration of the initial fast rotation and the subsequent transition to a relative equilibrium with respect to the local vertical. A better understanding of the latter phase is achieved owing to a more accurate model of the eddy-current torque than in most prior research. We show that SSO precession is also a crucial factor influencing the motion properties. One of its effects is manifested at the deceleration stage as oscillations of the angular momentum vector about the direction to the south celestial pole.     

The theory of canonical perturbations applied to attitude dynamics and to the Earth rotation. Osculating and nonosculating Andoyer variables

In the method of variation of parameters we express the Cartesian coordinates or the Euler angles as functions of the time and six constants. If, under disturbance, we endow the “constants” with time dependence, the perturbed orbital or angular velocity will consist of a partial time derivative and a convective term that includes time derivatives of the “constants”. The Lagrange constraint, often imposed for convenience, nullifies the convective term and thereby guarantees that the functional dependence of the velocity on the time and “constants” stays unaltered under disturbance. “Constants” satisfying this constraint are called osculating elements. Otherwise, they are simply termed orbital or rotational elements. When the equations for the elements are required to be canonical, it is normally the Delaunay variables that are chosen to be the orbital elements, and it is the Andoyer variables that are typically chosen to play the role of rotational elements. (Since some of the Andoyer elements are time-dependent even in the unperturbed setting, the role of “constants” is actually played by their initial values.) The Delaunay and Andoyer sets of variables share a subtle peculiarity: under certain circumstances the standard equations render the elements nonosculating. In the theory of orbits, the planetary equations yield nonosculating elements when perturbations depend on velocities. To keep the elements osculating, the equations must be amended with extra terms that are not parts of the disturbing function [Efroimsky, M., Goldreich, P.: J. Math. Phys. 44, 5958–5977 (2003); Astron. Astrophys. 415, 1187–1199 (2004); Efroimsky, M.: Celest. Mech. Dyn. Astron. 91, 75–108 (2005); Ann. New York Acad. Sci. 1065, 346–374 (2006)]. It complicates both the Lagrange- and Delaunay-type planetary equations and makes the Delaunay equations noncanonical. In attitude dynamics, whenever a perturbation depends upon the angular velocity (like a switch to a noninertial frame), a mere amendment of the Hamiltonian makes the equations yield nonosculating Andoyer elements. To make them osculating, extra terms should be added to the equations (but then the equations will no longer be canonical). Calculations in nonosculating variables are mathematically valid, but their physical interpretation is not easy. Nonosculating orbital elements parameterise instantaneous conics not tangent to the orbit. (A nonosculating i may differ much from the real inclination of the orbit, given by the osculating i.) Nonosculating Andoyer elements correctly describe perturbed attitude, but their interconnection with the angular velocity is a nontrivial issue. The Kinoshita–Souchay theory tacitly employs nonosculating Andoyer elements. For this reason, even though the elements are introduced in a precessing frame, they nevertheless return the inertial velocity, not the velocity relative to the precessing frame. To amend the Kinoshita–Souchay theory, we derive the precessing-frame-related directional angles of the angular velocity relative to the precessing frame. The loss of osculation should not necessarily be considered a flaw of the Kinoshita–Souchay theory, because in some situations it is the inertial, not the relative, angular velocity that is measurable [Schreiber, K. U. et al.: J. Geophys. Res. 109, B06405 (2004); Petrov, L.: Astron. Astrophys. 467, 359–369 (2007)]. Under these circumstances, the Kinoshita–Souchay formulae for the angular velocity should be employed (as long as they are rightly identified as the formulae for the inertial angular velocity).     

Solution to the rotation of the elastic earth by method of rigid dynamics

Hamiltonian mechanics is applied to the problem of the rotation of the elastic Earth. We first show the process for the formulation of the Hamiltonian for rotation of a deformable body and the derivation of the equations of motion from it. Then, based on a simple model of deformation, the solution is given for the period of Euler motion, UT1 and the nutation of the elastic Earth. In particular it is shown that the elasticity of the Earth acts on the nutation so as to decrease the Oppolzer terms of the nutation of the rigid Earth by about 30 per cent. The solution is in good agreement with results which have been obtained by other, different approaches.     

An efficient method for studying the stability and dynamics of the rotational motions of celestial bodies

We use the alternative MEGNO (Mean Exponential Growth of Nearby Orbits) technique developed by Cincotta and Simó to study the stability of orbital—rotational motions for plane oscillations and three-dimensional rotations. We present a detailed numerical—analytical study of a rigid body in the case where the proper rotation of the body is synchronized with its orbital motion as 3: 2(Mercurian—type synchronism). For plane rotations, the loss of stability of the periodic solution that corresponds to a 3: 2 resonance is shown to be soft, which should be taken into account to estimate the upper limit for the ellipticity of Mercury. In studying stable and chaotic translational—rotational motions, we point out that the MEGNO criterion can be effectively used. This criterion gives a clear picture of the resonant structures and allows the calculations to be conveniently presented in the form of the corresponding MEGNO stability maps for multidimensional systems. We developed an appropriate software package.     

非坐标方法确定卫星时间信号的传递时延

在卫星电视授时系统方案中，为确定标准秒信号从发射站经卫星到用户接收站的传播时延值，需要知道收、发地面站及卫星的坐标，所以，在高精度的卫星电视授时系统中，需要建立高精度的同步卫星定位同。本文提出在没有卫星定位同的条件下，用非坐标方法来确定卫星时间信号到用户的传递时延，其授时精度为几μｓ。     

The estimation of distances to Hyades and Praesepe clusters by a geometrical method

Die Tauglichkeit einer geometrischen Methode zur Entfernungsbestimmung wird am Praesepehaufen überprüft. Die mit der Hertzsprungmethode berechnete Entfernung beträgt r = 175 ± 47 pc. Die gleiche Methode ergibt für die Hyaden r = 48.4 ± 2.2 pc.     

Automated estimation of stellar fundamental parameters from low resolution spectra: the PLS method

PLS (Partial Least Squares regression) is introduced into an automatic esti-mation of fundamental stellar spectral parameters. It extracts the most correlative spec-tral component to the parameters (Teff, log g and [Fe/H]), and sets up a linear regres-sion function from spectra to the corresponding parameters. Considering the properties of stellar spectra and the PLS algorithm, we present a piecewise PLS regression method for estimation of stellar parameters, which is composed of one PLS model for Teff, and seven PLS models for log g and [Fe/H] estimation. Its performance is investigated by large experiments on flux calibrated spectra and continuum normalized spectra at dif-ferent signal-to-noise ratios (SNRs) and resolutions. The results show that the piecewise PLS method is robust for spectra at the medium resolution of 0.23 nm. For low resolu-tion 0.5 nm and 1 nm spectra, it achieves competitive results at higher SNR. Experiments using ELODIE spectra of 0.23 nm resolution illustrate that our piecewise PLS models trained with MILES spectra are efficient for O ～ G stars: for flux calibrated spectra, the systematic offsets are 3.8%, 0.14dex, and -0.09 dex for Teff, log g and [Fe/H], with error scatters of 5.2%, 0.44 dex and 0.38 dex, respectively; for continuum normalized spectra, the systematic offsets are 3.8%, 0.12dex, and -0.13 dex for Teff, log g and [Fe/H], with error scatters of 5.2%, 0.49 dex and 0.41 dex, respectively. The PLS method is rapid, easy to use and does not rely as strongly on the tightness of a parameter grid of templates to reach high precision as Artificial Neural Networks or minimum distance methods do.     

A novel high precision Doppler frequency estimation method based on the third-order phase-locked loop

In deep space exploration,many engineering and scientific requirements require the accuracy of the measured Doppler frequency to be as high as possible.In our paper,we analyze the possible frequency measurement points of the third-order phase-locked loop(PLL) and find a new Doppler measurement strategy.Based on this finding,a Doppler frequency measurement algorithm with significantly higher measurement accuracy is obtained.In the actual data processing,compared with the existing engineering software,the accuracy of frequency of 1 second integration is about 5.5 times higher when using the new algorithm.The improved algorithm is simple and easy to implement.This improvement can be easily combined with other improvement methods of PLL,so that the performance of PLL can be further improved.     

NLS1 galaxies and estimation of their central black hole masses from the X-ray excess variance method

Black hole mass determination in active galaxies is a key issue in understanding various luminosity states. In the present paper, we try to generalize the mass determination method based on the X-ray excess variance, successfully used for typical broad line Seyfert 1 galaxies (BLS1) to narrow line Seyfert 1 (NLS1) galaxies. NLS1 galaxies differ from BLS1 with respect to several properties. They are generally more variable in 2–10 keV energy band so the natural expectation is the need to use a different scaling coefficient between the mass and the variance in these two types of sources. However, we find that such a simple approach is not enough. Although for majority of the 21 NLS1 galaxies in our sample a single scaling coefficient (larger by a factor 20) provided us with a satisfactory method of mass determination, in a small subset of NLS1 galaxies this approach failed. Variability of those objects appeared to be at the intermediate level between NLS1 and BLS1 galaxies. These exceptional NLS1 galaxies have much harder soft X-ray spectra than majority of NLS1 galaxies. We thus postulate that the division of Seyfert 1 galaxies into BLS1 and NLS1 according to the widths of the  Hβ  line is less generic than according to the soft X-ray slope.     

行星磁场与卫星引力及其相对运动的统计关系

从知道某些天体具有磁场起，人们就对其磁场的起源提出种种解释，例如有电池说，转子说，化石说，发电机说等等。但由于这些学说都分别与某些观测事实相抵触而未被公认，因此星球磁场的起源一直是未能解决的命题。余先河先生提出，星球的磁场起源可能与所受的引力有关，行星的磁场强度正比于其所受卫星的引力；正比于卫星与行星的引力连线转动的相对角速度。本文对这两方面的命题分别进行了相关分析，得到其相关系数分别为：ｒ＝０．８４８１和ｒ＝０．８４２５，它们都在ａ＝０．０１的信度水平上相关。结果表明余新河关于行星磁场起源的设想是有基础的。本文还对其统计结果和可能机制进行了讨论。     

The flares associated with the dynamics of the sunspots

In the present study, we consider six years data of spot groups that have well developed leading and following spots obtained from the Kodaikanal Observatory white light pictures and occurrence of Hα flares. From the daily observations, we compute the variations in rotation rates, meridional velocity, the areas and longitudinal separations. We find that among all these variations, the occurrence of abnormal rotation rates (the rotation rates that have greater than 1σ and longitudinal minimum separation during the course of their evolution eventually lead to triggering of flares. We also find that the events of abnormal rotation rates, longitudinal minimum separation and the flares occur mainly during the 50–80% of the sunspots’ life span indicating magnetic reconnection probably below (0.935R⊙) the solar surface. Relevance of these results with the conventional theory of magnetic reconnection is briefly discussed.     

Satellite resonance with longitude-dependent gravity—II Effects involving the eccentricity

For a satellite in a nominally circular orbit at arbitrary inclination whose mean motion is commensurable with the Earth's rotation, the dependence of gravity on longitude leads to a resonant variation in eccentricity as well as the long-period oscillation in longitude. Provided forces capable of processing perigee are present, it is shown that the change in eccentricity for a satellite captured in librational resonance is not secular but periodic.

There are corresponding resonance effects for a satellite in a nominally equatorial but eccentric orbit. Here the commensurability condition is that the longitudes of the apses shall be nearly repetitive relative to the rotating Earth. There will be a long-period oscillation in longitude which can take the form of either a libration (trapped) or a circulation (free), and there will also be an oscillation of the orbital plane having the same period as the precession of perigee relative to inertial space.