The equations of relative motion in the orbital reference frame

The analysis of relative motion of two spacecraft in Earth-bound orbits is usually carried out on the basis of simplifying assumptions. In particular, the reference spacecraft is assumed to follow a circular orbit, in which case the equations of relative motion are governed by the well-known Hill–Clohessy–Wiltshire equations. Circular motion is not, however, a solution when the Earth’s flattening is accounted for, except for equatorial orbits, where in any case the acceleration term is not Newtonian. Several attempts have been made to account for the \(J_2\) effects, either by ingeniously taking advantage of their differential effects, or by cleverly introducing ad-hoc terms in the equations of motion on the basis of geometrical analysis of the \(J_2\) perturbing effects. Analysis of relative motion about an unperturbed elliptical orbit is the next step in complexity. Relative motion about a \(J_2\)-perturbed elliptic reference trajectory is clearly a challenging problem, which has received little attention. All these problems are based on either the Hill–Clohessy–Wiltshire equations for circular reference motion, or the de Vries/Tschauner–Hempel equations for elliptical reference motion, which are both approximate versions of the exact equations of relative motion. The main difference between the exact and approximate forms of these equations consists in the expression for the angular velocity and the angular acceleration of the rotating reference frame with respect to an inertial reference frame. The rotating reference frame is invariably taken as the local orbital frame, i.e., the RTN frame generated by the radial, the transverse, and the normal directions along the primary spacecraft orbit. Some authors have tried to account for the non-constant nature of the angular velocity vector, but have limited their correction to a mean motion value consistent with the \(J_2\) perturbation terms. However, the angular velocity vector is also affected in direction, which causes precession of the node and the argument of perigee, i.e., of the entire orbital plane. Here we provide a derivation of the exact equations of relative motion by expressing the angular velocity of the RTN frame in terms of the state vector of the reference spacecraft. As such, these equations are completely general, in the sense that the orbit of the reference spacecraft need only be known through its ephemeris, and therefore subject to any force field whatever. It is also shown that these equations reduce to either the Hill–Clohessy–Wiltshire, or the Tschauner–Hempel equations, depending on the level of approximation. The explicit form of the equations of relative motion with respect to a \(J_2\)-perturbed reference orbit is also introduced.     

Uncertainty propagation in orbital mechanics via tensor decomposition

Uncertainty forecasting in orbital mechanics is an essential but difficult task, primarily because the underlying Fokker–Planck equation (FPE) is defined on a relatively high dimensional (6-D) state–space and is driven by the nonlinear perturbed Keplerian dynamics. In addition, an enormously large solution domain is required for numerical solution of this FPE (e.g. encompassing the entire orbit in the \(x-y-z\) subspace), of which the state probability density function (pdf) occupies a tiny fraction at any given time. This coupling of large size, high dimensionality and nonlinearity makes for a formidable computational task, and has caused the FPE for orbital uncertainty propagation to remain an unsolved problem. To the best of the authors’ knowledge, this paper presents the first successful direct solution of the FPE for perturbed Keplerian mechanics. To tackle the dimensionality issue, the time-varying state pdf is approximated in the CANDECOMP/PARAFAC decomposition tensor form where all the six spatial dimensions as well as the time dimension are separated from one other. The pdf approximation for all times is obtained simultaneously via the alternating least squares algorithm. Chebyshev spectral differentiation is employed for discretization on account of its spectral (“super-fast”) convergence rate. To facilitate the tensor decomposition and control the solution domain size, system dynamics is expressed using spherical coordinates in a noninertial reference frame. Numerical results obtained on a regular personal computer are compared with Monte Carlo simulations.     

Study of the Cosmic-Ray Modulation During the Passage of ICMEs and CIRs

We compare the cosmic-ray response to interplanetary coronal mass ejections (ICMEs) and corotating interaction regions (CIRs) during their passage in near-Earth space. We study the relative importance of various structures/features identified during the passage of the ICMEs and CIRs observed during Cycle 23 (1995?–?2009). The identified ICME structures are the shock front, the sheath, and the CME ejecta. We isolate the shock arrival time, the passage of the sheath region, the arrival of ejecta, and the end time of their passage. Similarly, we isolate the CIR arrival, the associated forward shock, the stream interface, and the reverse shock during the passage of a CIR. For the cosmic-ray intensity, we utilize the data from high counting rate neutron monitors. In addition to neutron monitor data, we utilize near-simultaneous and same time-resolution data of interplanetary plasma and field, namely the solar-wind velocity, the interplanetary magnetic field (IMF) vector, and its variance. Further, we also utilize some derived interplanetary parameters. We apply the method of the superposed-epoch analysis. As the plasma and field properties are different during the passage of different structures, both in ICMEs and CIRs, we systematically vary the epoch time in our superposed-epoch analysis one by one. In this way, we study the role and effects of each of the identified individual structures/features during the passage of the ICMEs and CIRs. Relating the properties of various structures and the corresponding variations in plasma and field parameters with changes of the cosmic-ray intensity, we identify the relative importance of the plasma/field parameters in influencing the amplitude and time profiles of the cosmic-ray intensity variations during the passage of the ICMEs and CIRs.     

Hot deontology and cold consequentialism – an empirical exploration of ethical reasoning among climate change negotiators

Philosophers, political theorists and cognitive scientists have applied the traditional distinction between deontology and consequentialism to determine ethical responsibilities – usually of states – to take action in response to climate change. Most of this work is either purely conceptual or based on experiments with individuals, who are not part of the global political process. This paper makes two contributions to this debate. First, based on interview data I describe existing patterns of ethical reasoning among global political actors rather than groups selected for lab experiments. Integrating theories of risk perceptions, international relations and moral philosophy, I identify both deontological and consequentialist cognitive patterns, and examine their constitutive elements. My second contribution concerns the role of emotion in moral reasoning. Using the same qualitative data, I offer support for a controversial argument about the emotional nature of deontological reasoning. Further, I argue that many negotiators experience climate change not as an impersonal threat posed by the environment, but rather as an “up, close and personal” threat, over which other negotiation participants have significant control.     

A multi-proxy reconstruction of spatial and temporal variations in Asian summer temperatures over the last millennium

To investigate climate variability in Asia during the last millennium, the spatial and temporal evolution of summer (June–July–August; JJA) temperature in eastern and south-central Asia is reconstructed using multi-proxy records and the regularized expectation maximization (RegEM) algorithm with truncated total least squares (TTLS), under a point-by-point regression (PPR) framework. The temperature index reconstructions show that the late 20th century was the warmest period in Asia over the past millennium. The temperature field reconstructions illustrate that temperatures in central, eastern, and southern China during the 11th and 13th centuries, and in western Asia during the 12th century, were significantly higher than those in other regions, and comparable to levels in the 20th century. Except for the most recent warming, all identified warm events showed distinct regional expressions and none were uniform over the entire reconstruction area. The main finding of the study is that spatial temperature patterns have, on centennial time-scales, varied greatly over the last millennium. Moreover, seven climate model simulations, from the Coupled Model Intercomparison Project Phase 5 (CMIP5), over the same region of Asia, are all consistent with the temperature index reconstruction at the 99 % confidence level. Only spatial temperature patterns extracted as the first empirical orthogonal function (EOF) from the GISS-E2-R and MPI-ESM-P model simulations are significant and consistent with the temperature field reconstruction over the past millennium in Asia at the 90 % confidence level. This indicates that both the reconstruction and the simulations depict the temporal climate variability well over the past millennium. However, the spatial simulation or reconstruction capability of climate variability over the past millennium could be still limited. For reconstruction, some grid points do not pass validation tests and reveal the need for more proxies with high temporal resolution, accurate dating, and sensitive temperature signals, especially in central Asia and before AD 1400.     

Tidal evolution of hierarchical and inclined systems

We investigate the dynamical evolution of hierarchical three-body systems under the effect of tides, when the ratio of the orbital semi-major axes is small and the mutual inclination is relatively large (greater than 20°). Using the quadrupolar non-restricted approximation for the gravitational interactions and the viscous linear model for tides, we derive the averaged equations of motion in a vectorial formalism which is suitable to model the long-term evolution of a large variety of exoplanetary systems in very eccentric and inclined orbits. In particular, it can be used to derive constraints for stellar spin-orbit misalignment, capture in Cassini states, tidal-Kozai migration, or damping of the mutual inclination. Because our model is valid for the non-restricted problem, it can be used to study systems of identical mass or for the outer restricted problem, such as the evolution of a planet around a binary of stars. Here, we apply our model to various situations in the HD 11964, HD 80606, and HD 98800 systems.     

Single-binary star separation by ultraviolet color index diagrams

Detection of a composite flux in photometry can serve as an indication of a photometrically unresolved binarity and can contribute to the parameterization of the components of binary systems. A main goal of the present study is to develop a method of automatic photometric detection of binaries, based on multi-color photometry, theoretical stellar spectral energy distributions and general understanding of binary evolution. In particular, we consider an ultraviolet photometry where, in combination with optical and infrared photometry, interstellar reddening can be easier distinguished from temperature reddening.     

Characteristics of Type-II Radio Bursts Associated with Flares and CMEs

We present a statistical study of the characteristics of type-II radio bursts observed in the metric (m) and deca-hectometer (DH) wavelength range during 1997–2008. The collected events are divided into two groups: Group I contains the events of m-type-II bursts with starting frequency ≥ 100 MHz, and group II contains the events with starting frequency of m-type-II radio bursts < 100 MHz. We have analyzed both samples considering three different aspects: i) statistical properties of type-II bursts, ii) statistical properties of flares and CMEs associated with type-II bursts, and iii) time delays between type-II bursts, flares, and CMEs. We find significant differences in the properties of m-type-II bursts in duration, bandwidth, drift rate, shock speed and delay between m- and DH-type-II bursts. From the timing analysis we found that the majority of m-type-II bursts in both groups occur during the flare impulsive phase. On the other hand, the DH-type-II bursts in both groups occur during the decaying phase of the associated flares. Almost all m-DH-type-II bursts are found to be associated with CMEs. Our results indicate that there are two kinds of shock in which group I (high frequency) m-type-II bursts seem to be ignited by flares whereas group II (low frequency) m-type-II bursts are CME-driven.     

The effect of screening factors and thermonuclear reaction rates in the pre-main sequence evolution of low mass stars

In understanding the nucleosynthesis of the elements in stars, one of the most important quantities is the reaction rate and it must be evaluated in terms of the stellar temperature T, and its determination involves the knowledge of the excitation function σ(E) of the specific nuclear reaction leading to the final nucleus. In this paper, the effect of thermonuclear reaction rates to the pre-main sequence evolution of low mass stars having masses 0.7, 0.8, 0.9 and 1M_⊙ are studied by using our modified Stellar Evolutionary Program.     

Sixty Years in Solar Physics

I recount my career in solar physics beginning at Ondřejov Observatory in 1948 and ending with my ∼30 year stay at the Laboratory of Space Research in Utrecht.