The evolution of the orbit distance in the double averaged restricted 3-body problem with crossing singularities

We study the long term evolution of the distance between two Keplerian confocal trajectories in the framework of the averaged restricted 3-body problem. The bodies may represent the Sun, a solar system planet and an asteroid. The secular evolution of the orbital elements of the asteroid is computed by averaging the equations of motion over the mean anomalies of the asteroid and the planet. When an orbit crossing with the planet occurs the averaged equations become singular. However, it is possible to define piecewise differentiable solutions by extending the averaged vector field beyond the singularity from both sides of the orbit crossing set. In this paper we improve the previous results, concerning in particular the singularity extraction technique, and show that the extended vector fields are Lipschitz-continuous. Moreover, we consider the distance between the Keplerian trajectories of the small body and of the planet. Apart from exceptional cases, we can select a sign for this distance so that it becomes an analytic map of the orbital elements near to crossing configurations. We prove that the evolution of the 'signed' distance along the averaged vector field is more regular than that of the elements in a neighborhood of crossing times. A comparison between averaged and non-averaged evolutions and an application of these results are shown using orbits of near-Earth asteroids.Comment: 29 pages, 8 figure

Validation of rotational thromboelastometry during cardiopulmonary bypass : a prospective, observational in-vivo study

Le ROTEM est un test de coagulation réalisable au près du malade qui permet d'objectiver la coagulopathie, de distinguer la contribution des différents éléments du système de coagulation et de cibler les produits procoagulants comme le plasma frais congelé (PFC), les plaquettes, le fibrinogène et les facteurs de coagulation purifiés ou les antifibrinolytiques. 3 des tests disponibles pour le ROTEM sont: EXTEM, INTEM, HEPTEM. Le premier test est stable sous hautes doses d'héparine alors que le deuxième est très sensible à sa présence. Dans le dernier test on rajoute de l'héparinase pour mettre en évidence l'éventuel effet résiduel de l'héparine en le comparant à l'INTEM. Idéalement, le ROTEM devrait être effectué avant la fin du bypass cardiopulmonaire (CEC), donc sous anticoagulation maximale pas héparine, afin de pouvoir administrer des produits pro¬coagulants dans les délais les plus brefs et ainsi limiter au maximum les pertes sanguines. En effet la commande et la préparation de certains produits procoagulants peut prendre plus d'une heure. Le but de cette étude est de valider l'utilisation du ROTEM en présence de hautes concentrations d'héparine. Il s'agit d'une étude observationnelle prospective sur 20 patients opérés électivement de pontages aorto-coronariens sous CEC. Méthode : l'analyse ROTEM a été réalisée avant l'administration d'héparine (TO), 10 minutes après l'administration d'héparine (Tl), à la fin de la CEC (T2) et 10 minutes après la neutralisation de l'anticoagulation avec la protamine (T3). L'état.d'héparinisation a été évalué par l'activité anti-Xa à T1,T2,T3. Résultats : Comparé à TO, la phase de polymérisation de la cascade de coagulation et l'interaction fibrine-plaquettes sont significativement détériorées par rapport à Tl pour les canaux EXTEM et HEPTEM. A T2 l'analyse EXTEM et INTEM sont comparables à celles de EXTEM et HEPTEM à T3. Conclusion: les hautes doses d'héparine utilisées induisent une coagulopathie qui reste stable durant toute la durée de la CEC et qui persiste même après la neutralisation de l'anticoagulation. Les mesures EXTEM et HEPTEM sont donc valides en présence de hautes concentrations d'héparine et peuvent être réalisés pendant la CEC avant l'administration de protamine

Existence of periodic orbits near heteroclinic connections

We consider a potential $W:R^m\rightarrow R$ with two different global minima $a_-, a_+$ and, under a symmetry assumption, we use a variational approach to show that the Hamiltonian system \begin{equation} \ddot{u}=W_u(u), \hskip 2cm (1) \end{equation} has a family of $T$-periodic solutions $u^T$ which, along a sequence $T_j\rightarrow+\infty$, converges locally to a heteroclinic solution that connects $a_-$ to $a_+$. We then focus on the elliptic system \begin{equation} \Delta u=W_u(u),\;\; u:R^2\rightarrow R^m, \hskip 2cm (2) \end{equation} that we interpret as an infinite dimensional analogous of (1), where $x$ plays the role of time and $W$ is replaced by the action functional $$J_R(u)=\int_R\Bigl(\frac{1}{2}\vert u_y\vert^2+W(u)\Bigr)dy.$$ We assume that $J_R$ has two different global minimizers $\bar{u}_-, \bar{u}_+:R\rightarrow R^m$ in the set of maps that connect $a_-$ to $a_+$. We work in a symmetric context and prove, via a minimization procedure, that (2) has a family of solutions $u^L:R^2\rightarrow R^m$, which is $L$-periodic in $x$, converges to $a_\pm$ as $y\rightarrow\pm\infty$ and, along a sequence $L_j\rightarrow+\infty$, converges locally to a heteroclinic solution that connects $\bar{u}_-$ to $\bar{u}_+$.Comment: 36 pages, 4 figure

Backward-looking and Forward-looking NDC Pension Schemes

In order to spread notional capital accrued at retirement by members of a cohort over their own life expectancies, pay-as-you-go notional-defined-contribution (payg-ndc) scheme uses multipliers (different by retirement age) called conversion coefficients. These are backward-looking (b.l.) in that they relay on survival rates observed for previous cohorts in the past. Under increasing longevity, b.l. coefficients undervalue life expectancies, thus preventing full implementation of actuarial fairness (benefits equivalent to contributions) which is the main objective of ndc scheme. They also engender chronic deficits. Forward-looking (f.l.) coefficients, relaying on forecast survival rates, can improve actuarial fairness. Nevertheless, they face a rather serious political difficulty in that forecasting tools are fallible. This explains why switching to f.l. coefficients is unable to gain social consensus. Apart from this, the paper shows that f.l. coefficients produce ‘overshooting’. In fact, they generate chronic surpluses. The paper also shows that frontloading pension profile helps sustainability because it reduces both surpluses and deficits generated, respectively, by f.l. and b.l. approaches

Long term dynamics for the restricted N-body problem with mean motion resonances and crossing singularities

We consider the long term dynamics of the restricted N -body problem, modeling in a statistical sense the motion of an asteroid in the gravitational field of the Sun and the solar system planets. We deal with the case of a mean motion resonance with one planet and assume that the osculating trajectory of the asteroid crosses the one of some planet, possibly different from the resonant one, during the evolution. Such crossings produce singularities in the differential equations for the motion of the asteroid, obtained by standard perturbation theory. In this work we prove that the vector field of these equations can be extended to two locally Lipschitz-continuous vector fields on both sides of a set of crossing conditions. This allows us to define generalized solutions, continuous but not differentiable, going beyond these singularities. Moreover, we prove that the long term evolution of the ’signed’ orbit distance (Gronchi and Tommei 2007) between the asteroid and the planet is differentiable in a neighborhood of the crossing times. In case of crossings with the resonant planet we recover the known dynamical protection mechanism against collisions. We conclude with a numerical comparison between the long term and the full evolutions in the case of asteroids belonging to the ’Alinda’ and ’Toro’ classes (Milani et al. 1989). This work extends the results in (Gronchi and Tardioli 2013) to the relevant case of asteroids in mean motion resonance with a planet

Orbit Determination with the two-body Integrals

We investigate a method to compute a finite set of preliminary orbits for solar system bodies using the first integrals of the Kepler problem. This method is thought for the applications to the modern sets of astrometric observations, where often the information contained in the observations allows only to compute, by interpolation, two angular positions of the observed body and their time derivatives at a given epoch; we call this set of data attributable. Given two attributables of the same body at two different epochs we can use the energy and angular momentum integrals of the two-body problem to write a system of polynomial equations for the topocentric distance and the radial velocity at the two epochs. We define two different algorithms for the computation of the solutions, based on different ways to perform elimination of variables and obtain a univariate polynomial. Moreover we use the redundancy of the data to test the hypothesis that two attributables belong to the same body (linkage problem). It is also possible to compute a covariance matrix, describing the uncertainty of the preliminary orbits which results from the observation error statistics. The performance of this method has been investigated by using a large set of simulated observations of the Pan-STARRS project.Comment: 23 pages, 1 figur

On the existence of connecting orbits for critical values of the energy

We consider an open connected set Ω and a smooth potential U which is positive in Ω and vanishes on â\u88\u82Ω. We study the existence of orbits of the mechanical system u¨=Ux(u), that connect different components of â\u88\u82Ω and lie on the zero level of the energy. We allow that â\u88\u82Ω contains a finite number of critical points of U. The case of symmetric potential is also considered