Space-charge dynamics in Orbitrap mass spectrometers

The paper deals with space-charge interactions in the ion population trapped in the Orbitrap[Formula: see text] mass analyzer where the ions perform multiple quasi-harmonic oscillations in the axial direction. The many-particle problem for interacting ions is mathematically complicated and its solution, even numerical, is obstructed by the required precision of one per million to be maintained on a large number [Formula: see text] of oscillation periods. We develop a perturbation method based on the Bogoliubov–Krylov–Mitropolsky theory and derive averaged Hamiltonian equations in perturbations, which describe the evolution of the ions’ oscillation amplitudes and phases in so-called “slow” time. This approach provides a semi-analytical comprehensive model of resonant and nonresonant space-charge effects and allows fast and accurate numerical computation. Practical mitigation strategies for most deteriorating space-charge effects like coalescence and frequency shifts are considered.

Modeling the evection resonance for Trojan satellites: application to the Saturn system

Context. The stability of satellites in the solar system is affected by the so-called evection resonance. The moons of Saturn, in particular, exhibit a complex dynamical architecture in which co-orbital configurations occur, especially close to the planet where this resonance is present. Aims. We address the dynamics of the evection resonance, with particular focus on the Saturn system, and compare the known behavior of the resonance for a single moon with that of a pair of moons in co-orbital Trojan configuration. Methods. We developed an analytic expansion of the averaged Hamiltonian of a Trojan pair of bodies, including the perturbation from a distant massive body. The analysis of the corresponding equilibrium points was restricted to the asymmetric apsidal corotation solution of the co-orbital dynamics. We also performed numerical N-body simulations to construct dynamical maps of the stability of the evection resonance in the Saturn system, and to study the effects of this resonance under the migration of Trojan moons caused by tidal dissipation. Results. The structure of the phase space of the evection resonance for Trojan satellites is similar to that of a single satellite, differing in that the libration centers are displaced from their standard positions by an angle that depends on the periastron difference ϖ2 −ϖ1 and on the mass ratio m2∕m1 of the Trojan pair. In the Saturn system, the inner evection resonance, located at ~8 RS, may capture a pair of Trojan moons by migration; the stability of the captured system depends on the assumed values of the dissipation factor Q of the moons. On the other hand, the outer evection resonance, located at >0.4 RHill, cannot exist at all for Trojan moons, because Trojan configurations are strongly unstable at distances from Saturn longer than ~0.15 RHill. Conclusions. The interaction with the inner evection resonance may have been relevant during the early evolution of the Saturn moons Tethys, Dione, and Rhea. In particular, Rhea may have had Trojan companions in the past that were lost when it crossed the evection resonance, while Tethys and Dione may either have retained their Trojans or have never crossed the evection. This may help to constrain the dynamical processes that led to the migration of these satellites and to the evection itself.

A global approach to perform large-scale configuration–interaction calculations

We present a global approach that allows one to tackle cumbersome configuration–interaction calculations. The method is based on the use of approximate configuration-averaged Hamiltonian matrix elements that can be expressed in compact form as a combination of Slater integrals. With some assumptions, we show that the Hamiltonian matrix to be diagonalized may be reduced to a size equivalent to the number of configurations in the basis set. The approach can be used to estimate shifts of configuration average energies and changes in the total strength of transition arrays. The method is also well suited to work out roughly difficult configuration–interaction calculations, to determine the minimal set of interacting configurations to be used in actual fine-structure calculations.

Stochastic Stability of Some Hamiltonian Systems

Stochastic stability plays an important role in modern theories of nonlinear structural dynamics. Recently, more realistic models based on stochastic modelling and Itoˆ calculus, like flow induced vibrations and seismic excitations have been proposed. In this paper, the almost-sure asymptotic stability of some hamiltonian systems subjected to stochastic fluctuations is investigated. Dynamical systems are reduced to Itoˆ stochastic differential equations for the averaged hamiltonian by using a new stochastic averaging method. The stability of the original system is determined approximately by examining the behavior of the averaged hamiltonian. Analytical expressions for the stochastic stability exponents are obtained. The proposed procedure is illustrated on the Rayleigh Van der Pol Oscillator.

The Galactic Disk Tidal Force: Simulating the Observed Oort Cloud Comets

In previous papers (Prȩtka and Dybczyński, 1994; Dybczyński and Prȩtka, 1996) we presented detailed analysis of selected examples of the long-term evolution of the orbit of Oort cloud comets under the influence of the galactic disk tidal force, as well as some statistical characteristics of the simulated observable comet population. This paper presents further improvements in our Monte Carlo simulation programme which allow us to represent in a better way the real processes of production of observable comets due to galactic perturbations.In our second paper (Dybczyński and Prȩtka, 1996), following some other authors (see for example Matese and Whitman, 1989), we treated a comet as observable when its osculating perihelion distance decreased below some adopted observability limit (5 AU in our case). Limiting the investigation to the evolution of osculating elements allowed us to use very fast and efficient averaged Hamiltonian equations of motion in our simulation. However, further detailed analysis of the problem showed that the adopted observability definition was insufficient: what makes a comet observable is not its osculating perihelion distance but its true distance from the Sun, smaller than some adopted threshold value. It may happen that when the osculating perihelion distance is at its smallest, the comet is around its aphelion distance.