A small parameter method in problems of maneuvering space vehicles with aerodynamic efficiency

The motion of a satellite with aerodynamic efficiency along a low near-circular orbit is considered in the paper. The controls of bank angle γ and lift coefficient C_y are used as control functions. The introduction of a small parameter ($\text{? = (}\text{?}{}_0\text{· S · g}{}_0\text{2G}\text{)}$) makes it possible to integrate an adjoint system of equations and to obtain an approximate solution to the complete problem in the class of piecewise-constant control functions. Maximum values for the coordinates of heading angle η and lateral derivation from the plane of a reference orbit ?, which are connected with orbit plane angle by the relation cos i = cos ? · cos h, are used as criteria of maneuvering capability for a satellite with aerodynamic efficiency. Optimal programs for bank angle and incidence variation are derived and the influence of lift-to-drag ratio on the vehicle maneuvering capabilities has been estimated.It is shown that the process of the optimal motion is a special kind of gravitational skipping similar to the Keplerian motion but with continuous descent.     

航天器异面气动力辅助变轨大气飞行段的最优轨迹

本文研究航天器利用气动力辅助变轨实现由远地轨道向近地轨道异面转移问题，就航天器利用大气飞行实现减速及轨道平面机动提出了一种考虑过程约束存在时的优化设计方法。通过方案设计将函数优化问题转换成参数优化问题，并利用美国航天飞机数据进行了模拟计算，得到满意结果。     

On the limiting mass of a dense stellar matter

In the present work, we have derived an expression $\text{M}{}_{\mathrm{L}}\text{? 4.738 M}{}_{\mathrm{\odot }}$ (M_⊙ = 1.985 × 10³³ g = mass of the Sun) giving the “limiting” value of the mass of a dense stellar matter, by introducing the concept of nuclear size correction in the theory of relativistic Thomas Fermi model for a compressed atom. We find that $\text{M}{}_{\mathrm{L}}\text{? 5.1571 M}{}_{\mathrm{<mtext>Ch</mtext>}}\text{and}\text{=3.2750(M}{}_{\mathrm{<mtext>O</mtext>}}\text{)}{}_{\mathrm{<mtext>Prev</mtext>}}$ [M_Ch and (M_O)_Prev denote respectively the Chandrasekhar and author's “limiting” masses]. By making a comparative study with those of previous results it has been shown that our present treatment would provide satisfactory results for the density ranges from ? ? 10⁸ up to $\text{? ? 10}{}^{11}\text{g}\text{/}\text{cm}{}^3$. Other results of cognate interest in the non-relativistic regime 10³ < ? ? 10⁵ (without the nuclear size effect) are presented. The astrophysical implications of the results are mentioned.     

Analyse du fonctionnement d'un propulseur a éjection modulée

ONERA developed, for studying the response of a propellant to a pressure or velocity fluctuation, an experimental rocket engine whose nozzle throat area can be modulated by a toothed disk.The paper presents a linearized theory of the functioning of this engine in the low frequency domain, i.e. when there is no wave propagation within the combuster.To describe the functioning of this motor, the Ryazantsev-Novozhilov method, which assumes that the gas response is instantaneous, is used. This analysis takes into account the erosion and radiation effects, the combustion efficiency and the thermal losses through the walls.Two particular cases are described, for two values of the Damköhler parameter $\text{D}{}_1\text{=}\text{t}{}_{\mathrm{c}}\text{t}{}_{\mathrm{th}}$, where t_c is the residence time in the combuster and t_th the characteristic thermal time of the heat penetration into the solid propellant. These two cases correspond, one to a classical propellant D₁ > 1, the other to a particular propellant of low burning rate ($\text{J}{}_{\mathrm{b}}\text{? 0.2}\text{to}\text{0.4}\text{mm s}{}^{\mathrm{?1}}$) D₁ < 1. The stability conditions are analysed as well as the pressure amplitute and phase as a function of the nozzle throat modulation frequency.Still in linearized theory, the complete solutions of the problem are presented, using a method of numerical resolution.     

Analysis and implementation of in-plane stationkeeping of continuously perturbed Walker constellations

The stationkeeping of symmetric Walker constellations is analyzed by considering the perturbations arising from a high order and degree Earth gravity field and the solar radiation pressure. These perturbations act differently on each group of spacecraft flying in a given orbital plane, causing a differential drift effect that would disrupt the initial symmetry of the constellation. The analysis is based on the consideration of a fictitious set of rotating reference frames that move with the spacecraft in the mean sense, but drift at a rate equal to the average drift rate experienced by all the vehicles over an extended period. The frames are also allowed to experience the $J_2$-precession such that each vehicle is allowed to drift in 3D relative to its frame. A two-impulse rendezvous maneuver is then constructed to bring each vehicle to the center of its frame as soon as a given tolerance deadband is about to be violated. This paper illustrates the computations associated with the stationkeeping of a generic Walker constellation by maneuvering each leading spacecraft within an orbit plane and calculating the associated velocity changes required for controlling the in-plane motions in an exacting sense, at least for the first series of maneuvers. The analysis can be easily extended to lower flying constellations, which experience additional perturbations due to drag.     

Order of magnitude analysis of convective effects in dendritic solidification

The different types of convective phenomena which may occur during the dendritic solidification of metallic alloys are discussed from an order of magnitude analysis. Bulk thermal convection and/or interdendritic solutal convection have to be considered according to the values of the experimental data. Scaling laws for the solute boundary layer resulting from bulk thermal convection have already been derived. It is shown here that the interdendritic flow depends on a solutal Grashof number Gr based on the horizontal density gradient and a characteristic length L_s which is of the order of the liquid channels width. For Gr < 1, which is generally verified in practical cases, the interdendritic flow velocity U_r is proportional to the Grashof number. This a priori law compares favorably with the results of horizontal solidification experiments where the mean interdendritic flow velocity has been estimated from the resulting measured macrosegregation. In these experiments, as well as for most horizontal dendritic solidifications of metallic alloys at 1 g, the ratio $\text{U}{}_{\mathrm{r}}\text{R}$ (R is the growth rate) is of order one. In order to cancel the interdendritic flow effects, this ratio has to be lowered by one order of magnitude. According to our analysis, this can be obtained by performing the experiments either at a slightly reduced g level (～10^?1 g), or at 1 g in a vertical stable configuration with a sufficiently low residual horizontal thermal gradient.     

Small cryogenic propulsion unit for upper stage application

The feasibility study was conducted to use the cryogenic propulsion system for the third stage of the future H-1 vehicle. While the $\text{LO}{}_2\text{LH}{}_2$ third-stage mass fraction is less than the current solid propellant third stage, the 50% higher Isp results in a significantly higher payload. Two basic configurations of the new propulsion system were proposed: one pressure-fed system and two pump-fed systems. The first is a pressure-fed system providing a 700 kg thrust at an Isp of 441 sec with restart capability. The second is a pump-fed system, operating on an expander cycle principle. A midget turbopump with a 90 000 rpm shaft speed feeds the thrust chamber which delivers 1 ton of thrust at an Isp of 471 sec. The third proposed system is also a pump-fed design using a unique expander bleed cycle, and delivers a 1 ton thrust at an Isp of 470 sec with a turbopump speed of 80 000 rpm. The results of engine testing predict the performance feasibility of respective propulsion system designs.     

Approximate solutions to minimax optimal control problems for aeroassisted orbital transfer

This paper considers minimax problems of optimal control arising in the study of aeroassisted orbital transfer. The maneuver considered involves the coplanar transfer from a high planetary orbit to a low planetary orbit. An example is the HEO-to-LEO transfer of a spacecraft, where HEO denotes high Earth orbit and LEO denotes low Earth orbit. In particular, HEO can be GEO, a geosynchronous Earth orbit.The basic idea is to employ the hybrid combination of propulsive maneuvers in space and aerodynamic maneuvers in the sensible atmosphere. Hence, this type of flight is also called synergetic space flight. With reference to the atmospheric part of the maneuver, trajectory control is achieved by means of lift modulation. The presence of upper and lower bounds on the lift coefficient is considered.The following minimax problems of optimal control are investigated: (i) minimize the peak heating rate, problem P1; and (ii) minimize the peak dynamic pressure, problem P2. It is shown that problems P1 and P2 are approximately equivalent to the following minimax problem of optimal control: (iii) minimize the peak altitude drop occurring in the atmospheric portion of the trajectory, problem P3.Problems P1–P3 are Chebyshev problems of optimal control, which can be converted into Bolza problems by suitable transformations. However, the need for these transformations can be bypassed if one reformulates problem P3 as a two-subarc problem of optimal control, in which the first subarc connects the initial point and the point where the path inclination is zero, and the second subarc connects the point where the path inclination is zero and the final point: (iv) minimize the altitude drop achieved at the point of junction between the first subarc and the second subarc, problem P4. Note that problem P4 is a Bolza problem of optimal control.Numerical solutions for problems P1–P4 are obtained by means of the sequential gradient-restoration algorithm for optimal control problems. Numerical examples are presented, and their engineering implications are discussed. In particular, it is shown that, from an engineering point of view, it is desirable to solve problem P3 or P4, rather than problems P1 and P2.     

Theoretical study about the Marangoni convection in a liquid column in zero-gravity

The thermal Marangoni effect on the surface of a liquid bridge induces a convection inside the liquid. For an imposed arbitrary periodic axial circumferential temperature distribution on the liquid surface the velocity distributions in radial-, angular- and axial direction are determined theoretically by solving the linearized Navier-Stokes equations. Of particular interest is the effect of the viscosity parameter $\text{v}\text{a}{}^2$ and axial wave length to diameter ratio $\text{l}\text{a}$. It was found that the increase of viscosity decreases the magnitude of the velocity distributions and that for small axial wave length to diameter ratios the radial- and axial velocities exhibit peak values close to the free surface of the liquid. This is in a less pronounced way also true for the angular velocity, which shows for increasing moderate values $\text{l}\text{a}\text{(0 ≤}\text{l}\text{a}\text{≤ 2}$) a strong increase in magnitude and for larger axial wavelength a decrease again. For increasing axial wavelength the peak value of the radial- and axial velocity shifts towards the center of the liquid bridge, of which for a further increase a decrease of the magnitude appears.     

Collisions of stars by oscillating orbits of the second kind

Ordinary estimations of the number of star collisions in our galaxy—by simple kinematic considerations—lead to a very small number of such collisions: about one or even less every millions of years. However star collisions can occur through the following indirect way which has a much higher probability. (a) Binary stars are very common in our galaxy, about 30–50% of the stars. (b) If two binary stars meet a triple system can be formed by an ordinary exchange type motion. (c) A triple system is generally decomposed into the “inner orbit” (i.e. the relative orbit of the two nearest stars) and the “outer orbit” (i.e. the relative orbit of the third star with respect to the center of mass of the two nearest stars). The major axes of these two orbits have generally small perturbations and it is the same for the eccentricity of the outer orbit. On the contrary, if the relative inclination of the two orbits is large, the perturbations of the eccentricity of the inner orbit are important and can even in some cases lead to an eccentricity equal to one, that is to a collision of the two stars of the inner orbit.Such orbits can be called “oscillating orbits of the second kind”, indeed the first oscillating orbits—conceived by Khilmi and described for the first time in an example by Sitnikov—have unbounded mutual distances r_ij, but the system always come back to small sizes, it has an infinite number of very large expansions followed by strong contractions and, in the three-body case, an upper bound of lim inf (r_1.2 + r_1.3 + r_2.3) can be given in terms of the three masses and the integrals of motion. For the oscillating orbits of the second kind the mutual distances r_ij are bounded, but the velocities are unbounded (i.e. lim inf r_ij = 0 for at least one r_ij) and the system goes to a collision if the bodies have non-zero radius even small. The analytical study of the oscillating orbits of the second kind is a part of the general analytical study of the three-body problem, a part which must be valid for large eccentricities and large inclinations. The use of Delaunay's variables and of a Von Zeipel transformation lead to a first order integrable approximation, valid for any eccentricities and any inclinations, and giving the following results: (a) The oscillating orbits of the second kind occur when the angular momentum of the outer orbit has a modulus sufficiently close to the modulus of the total angular momentum of the three-body system. Hence these orbits occur for inclinations in the vicinity of 90°. (b) The oscillating orbits represent a set of positive measure of phase space and the first order study allows to give a rough estimation of the probability of collisions—even for stars of infinitely small radius. This probability, for given initial major axes and eccentricities and for isotropic arbitrary initial orientations, is generally of the order of m₃$\text{R}\text{M}$ (m₃ being the mass of the outer star, M the total mass and R the ratio of the period of the inner orbit to that of outer orbit).One question remains to be solved: how many collisions of stars are due to that phenomenon? That question is difficult because the probability of formation of a triple system by a random meeting of two binaries is very uneasy to estimate. However it seems that, compared to the usual evaluations based on pure kinematic considerations without gravitational effects, the number of collisions must be multiplied by a factor between one thousand and one million.     

Broad-search algorithms for the spacecraft trajectory design of Callisto–Ganymede–Io triple flyby sequences from 2024 to 2040, Part II: Lambert pathfinding and trajectory solutions

Triple-satellite-aided capture employs gravity-assist flybys of three of the Galilean moons of Jupiter in order to decrease the amount of ΔV$\mathrm{\Delta }V$ required to capture a spacecraft into Jupiter orbit. Similarly, triple flybys can be used within a Jupiter satellite tour to rapidly modify the orbital parameters of a Jovicentric orbit, or to increase the number of science flybys. In order to provide a nearly comprehensive search of the solution space of Callisto–Ganymede–Io triple flybys from 2024 to 2040, a third-order, Chebyshev's method variant of the p-iteration solution to Lambert's problem is paired with a second-order, Newton–Raphson method, time of flight iteration solution to the _V∞$V_{\infty }$-matching problem. The iterative solutions of these problems provide the orbital parameters of the Callisto–Ganymede transfer, the Ganymede flyby, and the Ganymede–Io transfer, but the characteristics of the Callisto and Io flybys are unconstrained, so they are permitted to vary in order to produce an even larger number of trajectory solutions. The vast amount of solution data is searched to find the best triple-satellite-aided capture window between 2024 and 2040.     

Analytical solutions for the relative motion of spacecraft subject to Lorentz-force perturbations

A spacecraft capable of producing higher-than-natural electrostatic charges may achieve propellantless orbital maneuvering via the Lorentz-force interaction with a planetary magnetic field. Development of maneuver strategies for these propellantless vehicles is complicated by the fact that the perturbative Lorentz force acts along only a single line of action at any instant. Relative-motion dynamical models are developed that lead to approximate analytical solutions for the motion of charged spacecraft subject to the Lorentz force. These solutions indicate that the principal effects of the Lorentz force on a spacecraft in a circular orbit are to change the intrack position and to change the orbit plane. A rendezvous example is presented in which a spacecraft with a specific charge of ?3.81 $\times$ 10^?4 C/kg reaches a target vehicle initially 10 km away (on the same equatorial low-Earth orbit) in 1 day. Fly-around maneuvers may be achieved in low-Earth orbit with specific charges on the order of 0.001 C/kg.     

基于序优化理论的晕轨道转移轨道设计

利用晕轨道的稳定流形可以设计从地球到晕轨道的转移轨道。但由于小幅度晕轨道的稳 定流形与地球停泊轨道无法相交,因此需采用两脉冲转移。微分修正法是求解两脉冲转移常 用的优化方法,虽然收敛速度快,但很难获取全局最优解,而且收敛半径小,如果初始猜想 与最优解相差很远,该方法可能会不收敛。将序优化理论与微分修正法相结合,利用序优化 思想缩小搜索空间,得到足够好的初始猜想,然后利用微分修正法快速收敛到满足终端精度 要求的解。仿真结果表明该方法有很好的收敛性,且计算量小。