Interesting dynamics at high mutual inclination in the framework of the Kozai problem with an eccentric perturber

We study the dynamics of the 3-D three-body problem of a small body moving under the attractions of a star and a giant planet which orbits the star on a much wider and elliptic orbit. In particular, we focus on the influence of an eccentric orbit of the outer perturber on the dynamics of a small highly inclined inner body. Our analytical study of the secular perturbations relies on the classical octupole hamiltonian expansion (third-order theory in the ratio of the semi-major axes), as third-order terms are needed to consider the secular variations of the outer perturber and potential secular resonances between the arguments of the pericenter and/or longitudes of the node of both bodies. Short-period averaging and node reduction (Laplace plane) reduce the problem to two degrees of freedom. The four-dimensional dynamics is analyzed through representative planes which identify the main equilibria of the problem. As in the circular problem (i.e. perturber on a circular orbit), the "Kozai-bifurcated" equilibria play a major role in the dynamics of an inner body on quasi-circular orbit: its eccentricity variations are very limited for mutual inclination between the orbital planes smaller than ~40^{\deg}, while they become large and chaotic for higher mutual inclination. Particular attention is also given to a region around 35^{\deg} of mutual inclination, detected numerically by Funk et al. (2011) and consisting of long-time stable and particularly low eccentric orbits of the small body. Using a 12th-order Hamiltonian expansion in eccentricities and inclinations, in particular its action-angle formulation obtained by Lie transforms in Libert & Henrard (2008), we show that this region presents an equality of two fundamental frequencies and can be regarded as a secular resonance. Our results also apply to binary star systems where a planet is revolving around one of the two stars.Comment: 12 pages, 9 figures, accepted for publication in MNRA

Aspects of the planetary Birkhoff normal form

The discovery in [G. Pinzari. PhD thesis. Univ. Roma Tre. 2009], [L. Chierchia and G. Pinzari, Invent. Math. 2011] of the Birkhoff normal form for the planetary many--body problem opened new insights and hopes for the comprehension of the dynamics of this problem. Remarkably, it allowed to give a {\sl direct} proof of the celebrated Arnold's Theorem [V. I. Arnold. Uspehi Math. Nauk. 1963] on the stability of planetary motions. In this paper, using a "ad hoc" set of symplectic variables, we develop an asymptotic formula for this normal form that may turn to be useful in applications. As an example, we provide two very simple applications to the three-body problem: we prove a conjecture by [V. I. Arnold. cit] on the "Kolmogorov set"of this problem and, using Nehoro{\v{s}}ev Theory [Nehoro{\v{s}}ev. Uspehi Math. Nauk. 1977], we prove, in the planar case, stability of all planetary actions over exponentially-long times, provided mean--motion resonances are excluded. We also briefly discuss perspectives and problems for full generalization of the results in the paper.Comment: 44 pages. Keywords: Averaging Theory, Birkhoff normal form, Nehoro{\v{s}}ev Theory, Planetary many--body problem, Arnold's Theorem on the stability of planetary motions, Properly--degenerate kam Theory, steepness. Revised version, including Reviewer's comments. Typos correcte

A Semi-Analytic Algorithm for Constructing Lower Dimensional Elliptic Tori in Planetary Systems

We adapt the Kolmogorov's normalization algorithm (which is the key element of the original proof scheme of the KAM theorem) to the construction of a suitable normal form related to an invariant elliptic torus. As a byproduct, our procedure can also provide some analytic expansions of the motions on elliptic tori. By extensively using algebraic manipulations on a computer, we explicitly apply our method to a planar four-body model not too different with respect to the real Sun--Jupiter--Saturn--Uranus system. The frequency analysis method allows us to check that our location of the initial conditions on an invariant elliptic torus is really accurate.Comment: 31 pages, 4 figure

Measurement of global polarization of {\Lambda} hyperons in few-GeV heavy-ion collisions

The global polarization of {\Lambda} hyperons along the total orbital angular momentum of a relativistic heavy-ion collision is presented based on the high statistics data samples collected in Au+Au collisions at \sqrt{s_{NN}} = 2.4 GeV and Ag+Ag at 2.55 GeV with the High-Acceptance Di-Electron Spectrometer (HADES) at GSI, Darmstadt. This is the first measurement below the strangeness production threshold in nucleon-nucleon collisions. Results are reported as a function of the collision centrality as well as a function of the hyperon transverse momentum (p_T) and rapidity (y_{CM}) for the range of centrality 0--40%. We observe a strong centrality dependence of the polarization with an increasing signal towards peripheral collisions. For mid-central (20--40%) collisions the polarization magnitudes are (%) = 6.0 \pm 1.3 (stat.) \pm 2.0 (syst.) for Au+Au and (%) = 4.6 \pm 0.4 (stat.) \pm 0.5 (syst.) for Ag+Ag, which are the largest values observed so far. This observation thus provides a continuation of the increasing trend previously observed by STAR and contrasts expectations from recent theoretical calculations predicting a maximum in the region of collision energies about 3 GeV. The observed polarization is of a similar magnitude as predicted by 3D fluid dynamics and the UrQMD plus thermal vorticity model and significantly above results from the AMPT model.Comment: 8 pages, 4 figure

Impact of the Coulomb field on charged-pion spectra in few-GeV heavy-ion collisions

In nuclear collisions the incident protons generate a Coulomb field which acts on produced charged particles. The impact of these interactions on charged-pion transverse-mass and rapidity spectra, as well as on pion–pion momentum correlations is investigated in Au + Au collisions at $\sqrt{^{S}NN}$ = 2.4 GeV. We show that the low-m$_{t}$ region (m$_{t}$ < 0.2 GeV / c$^{2}$) can be well described with a Coulomb-modified Boltzmann distribution that also takes changes of the Coulomb field during the expansion of the fireball into account. The observed centrality dependence of the fitted mean Coulomb potential energy deviates strongly from a $A_{part}^{2/3}$ scaling, indicating that, next to the fireball, the non-interacting charged spectators have to be taken into account. For the most central collisions, the Coulomb modifications of the HBT source radii are found to be consistent with the potential extracted from the single-pion transverse-mass distributions. This finding suggests that the region of homogeneity obtained from two-pion correlations coincides with the region in which the pions freeze-out. Using the inferred mean-square radius of the charge distribution at freeze-out, we have deduced a baryon density, in fair agreement with values obtained from statistical hadronization model fits to the particle yields

Inclusive e+^+e−^- production in collisions of pions with protons and nuclei in the second resonance region of baryons

Inclusive e$^+$e$^-$ production has been studied with HADES in $\pi^-$ + p, $\pi^-$ + C and $\pi^- + \mathrm{CH}_2$ reactions, using the GSI pion beam at $\sqrt{s_{\pi p}}$ = 1.49 GeV. Invariant mass and transverse momentum distributions have been measured and reveal contributions from Dalitz decays of $\pi^0$, $\eta$ mesons and baryon resonances. The transverse momentum distributions are very sensitive to the underlying kinematics of the various processes. The baryon contribution exhibits a deviation up to a factor seven from the QED reference expected for the dielectron decay of a hypothetical point-like baryon with the production cross section constrained from the inverse $\gamma$ n$\rightarrow \pi^-$ p reaction. The enhancement is attributed to a strong four-momentum squared dependence of the time-like electromagnetic transition form factors as suggested by Vector Meson Dominance (VMD). Two versions of the VMD, that differ in the photon-baryon coupling, have been applied in simulations and compared to data. VMD1 (or two-component VMD) assumes a coupling via the $\rho$ meson and a direct coupling of the photon, while in VMD2 (or strict VMD) the coupling is only mediated via the $\rho$ meson. The VMD2 model, frequently used in transport calculations for dilepton decays, is found to overestimate the measured dielectron yields, while a good description of the data can be obtained with the VMD1 model assuming no phase difference between the two amplitudes. Similar descriptions have also been obtained using a time-like baryon transition form factor model where the pion cloud plays the major role.Comment: (HADES collaboration

Feasibility studies for the measurement of time-like proton electromagnetic form factors from p¯ p→ μ+μ- at P ¯ ANDA at FAIR

This paper reports on Monte Carlo simulation results for future measurements of the moduli of time-like proton electromagnetic form factors, | GE| and | GM| , using the p¯ p→ μ+μ- reaction at P ¯ ANDA (FAIR). The electromagnetic form factors are fundamental quantities parameterizing the electric and magnetic structure of hadrons. This work estimates the statistical and total accuracy with which the form factors can be measured at P ¯ ANDA , using an analysis of simulated data within the PandaRoot software framework. The most crucial background channel is p¯ p→ π+π-, due to the very similar behavior of muons and pions in the detector. The suppression factors are evaluated for this and all other relevant background channels at different values of antiproton beam momentum. The signal/background separation is based on a multivariate analysis, using the Boosted Decision Trees method. An expected background subtraction is included in this study, based on realistic angular distributions of the background contribution. Systematic uncertainties are considered and the relative total uncertainties of the form factor measurements are presented

Precision resonance energy scans with the PANDA experiment at FAIR: Sensitivity study for width and line shape measurements of the X(3872)

This paper summarises a comprehensive Monte Carlo simulation study for precision resonance energy scan measurements. Apart from the proof of principle for natural width and line shape measurements of very narrow resonances with PANDA, the achievable sensitivities are quantified for the concrete example of the charmonium-like X(3872) state discussed to be exotic, and for a larger parameter space of various assumed signal cross-sections, input widths and luminosity combinations. PANDA is the only experiment that will be able to perform precision resonance energy scans of such narrow states with quantum numbers of spin and parities that differ from J P C = 1 - -

PANDA Phase One - PANDA collaboration

The Facility for Antiproton and Ion Research (FAIR) in Darmstadt, Germany, provides unique possibilities for a new generation of hadron-, nuclear- and atomic physics experiments. The future antiProton ANnihilations at DArmstadt (PANDA or P¯ANDA) experiment at FAIR will offer a broad physics programme, covering different aspects of the strong interaction. Understanding the latter in the non-perturbative regime remains one of the greatest challenges in contemporary physics. The antiproton–nucleon interaction studied with PANDA provides crucial tests in this area. Furthermore, the high-intensity, low-energy domain of PANDA allows for searches for physics beyond the Standard Model, e.g. through high precision symmetry tests. This paper takes into account a staged approach for the detector setup and for the delivered luminosity from the accelerator. The available detector setup at the time of the delivery of the first antiproton beams in the HESR storage ring is referred to as the Phase One setup. The physics programme that is achievable during Phase One is outlined in this paper