Detecting chaos in particle accelerators through the frequency map analysis method

The motion of beams in particle accelerators is dominated by a plethora of non-linear effects which can enhance chaotic motion and limit their performance. The application of advanced non-linear dynamics methods for detecting and correcting these effects and thereby increasing the region of beam stability plays an essential role during the accelerator design phase but also their operation. After describing the nature of non-linear effects and their impact on performance parameters of different particle accelerator categories, the theory of non-linear particle motion is outlined. The recent developments on the methods employed for the analysis of chaotic beam motion are detailed. In particular, the ability of the frequency map analysis method to detect chaotic motion and guide the correction of non-linear effects is demonstrated in particle tracking simulations but also experimental data.Comment: Submitted for publication in Chaos, Focus Issue: Chaos Detection Methods and Predictabilit

An Overview of the 13:8 Mean Motion Resonance between Venus and Earth

It is known since the seminal study of Laskar (1989) that the inner planetary system is chaotic with respect to its orbits and even escapes are not impossible, although in time scales of billions of years. The aim of this investigation is to locate the orbits of Venus and Earth in phase space, respectively to see how close their orbits are to chaotic motion which would lead to unstable orbits for the inner planets on much shorter time scales. Therefore we did numerical experiments in different dynamical models with different initial conditions -- on one hand the couple Venus-Earth was set close to different mean motion resonances (MMR), and on the other hand Venus' orbital eccentricity (or inclination) was set to values as large as e = 0.36 (i = 40deg). The couple Venus-Earth is almost exactly in the 13:8 mean motion resonance. The stronger acting 8:5 MMR inside, and the 5:3 MMR outside the 13:8 resonance are within a small shift in the Earth's semimajor axis (only 1.5 percent). Especially Mercury is strongly affected by relatively small changes in eccentricity and/or inclination of Venus in these resonances. Even escapes for the innermost planet are possible which may happen quite rapidly.Comment: 14 pages, 11 figures, submitted to CMD

Narrow-Angle Astrometry with the Space Interferometry Mission: The Search for Extra-Solar Planets. II. Detection and Characterization of Planetary Systems

(Abridged) The probability of detecting additional companions is essentially unchanged with respect to the single-planet configurations, but after fitting and subtraction of orbits with astrometric signal-to-noise ratio $\alpha/\sigma_d\to 1$ the false detection rates can be enhanced by up to a factor 2; the periodogram approach results in robust multiple-planet detection for systems with periods shorter than the SIM mission length, even at low values of $\alpha/\sigma_d$, while the least squares technique combined with Fourier series expansions is arguably preferable in the long-period regime. The accuracy on multiple-planet orbit reconstruction and mass determination suffers a typical degradation of 30-40% with respect to single-planet solutions; mass and orbital inclination can be measured to better than 10% for periods as short as 0.1 yr, and for $\alpha/\sigma_d$ as low as $\sim 5$, while $\alpha/\sigma_d\simeq 100$ is required in order to measure with similar accuracy systems harboring objects with periods as long as three times the mission duration. For systems with all components producing $\alpha/\sigma_d\simeq 10$ or greater, quasi-coplanarity can be reliably established with uncertainties of a few degrees, for periods in the range $0.1\leq T\leq 15$ yr; in systems where at least one component has $\alpha/\sigma_d\to 1$, coplanarity measurements are compromised, with typical uncertainties on the mutual inclinations of order of $30^\circ-40^\circ$. Our findings are illustrative of the importance of the contribution SIM will make to the fields of formation and evolution of planetary systems.Comment: 61 pages, 14 figures, 5 tables, to appear in the September 2003 Issue of the Publications of the Astronomical Society of the Pacifi

Where are the Uranus Trojans?

The area of stable motion for fictitious Trojan asteroids around Uranus' equilateral equilibrium points is investigated with respect to the inclination of the asteroid's orbit to determine the size of the regions and their shape. For this task we used the results of extensive numerical integrations of orbits for a grid of initial conditions around the points L4 and L5, and analyzed the stability of the individual orbits. Our basic dynamical model was the Outer Solar System (Jupiter, Saturn, Uranus and Neptune). We integrated the equations of motion of fictitious Trojans in the vicinity of the stable equilibrium points for selected orbits up to the age of the Solar system of 5 billion years. One experiment has been undertaken for cuts through the Lagrange points for fixed values of the inclinations, while the semimajor axes were varied. The extension of the stable region with respect to the initial semimajor axis lies between 19.05 < a < 19.3 AU but depends on the initial inclination. In another run the inclination of the asteroids' orbit was varied in the range 0 < i < 60 and the semimajor axes were fixed. It turned out that only four 'windows' of stable orbits survive: these are the orbits for the initial inclinations 0 < i < 7, 9 < i < 13, 31 < i < 36 and 38 < i < 50. We postulate the existence of at least some Trojans around the Uranus Lagrange points for the stability window at small and also high inclinations.Comment: 15 pages, 12 figures, submitted to CMD

Measuring the mixing efficiency in a simple model of stirring:some analytical results and a quantitative study via Frequency Map Analysis

We prove the existence of invariant curves for a $T$--periodic Hamiltonian system which models a fluid stirring in a cylindrical tank, when $T$ is small and the assigned stirring protocol is piecewise constant. Furthermore, using the Numerical Analysis of the Fundamental Frequency of Laskar, we investigate numerically the break down of invariant curves as $T$ increases and we give a quantitative estimate of the efficiency of the mixing.Comment: 10 figure

Scaling law in the Standard Map critical function. Interpolating hamiltonian and frequency map analysis

We study the behaviour of the Standard map critical function in a neighbourhood of a fixed resonance, that is the scaling law at the fixed resonance. We prove that for the fundamental resonance the scaling law is linear. We show numerical evidence that for the other resonances $p/q$, $q \geq 2$, $p \neq 0$ and $p$ and $q$ relatively prime, the scaling law follows a power--law with exponent $1/q$.Comment: AMS-LaTeX2e, 29 pages with 8 figures, submitted to Nonlinearit

Constructing the secular architecture of the solar system II: The terrestrial planets

We investigate the dynamical evolution of the terrestrial planets during the planetesimal-driven migration of the giant planets. A basic assumption of this work is that giant planet migration occurred after the completion of terrestrial planet formation, such as in the models that link the former to the origin of the Late Heavy Bombardment. The divergent migration of Jupiter and Saturn causes the g5 eigenfrequency to cross resonances of the form g5=gk with k ranging from 1 to 4. Consequently these secular resonances cause large-amplitude responses in the eccentricities of the terrestrial planets. We show that the resonances g5=g_4 and g5=g3 do not pose a problem if Jupiter and Saturn have a fast approach and departure from their mutual 2:1 mean motion resonance. On the other hand, the resonance crossings g5=g2 and g5=g1 are more of a concern as they tend to yield a terrestrial system incompatible with the current one. We offer two solutions to this problem. The first uses the fact that a secular resonance crossing can also damp the amplitude of a Fourier mode if the latter is large originally. A second scenario involves a 'jumping Jupiter' in which encounters between an ice giant and Jupiter, without ejection of the former, cause the latter to migrate away from Saturn much faster than if migration is driven solely by encounters with planetesimals. In this case, the g5=g2 and g5=g1 resonances can be jumped over, or occur very briefly.Comment: Astronomy & Astrophysics (2009) in pres

Chromospheric Variability in SDSS M Dwarfs. II. Short-Timescale H-alpha Variability

[Abridged] We present the first comprehensive study of short-timescale chromospheric H-alpha variability in M dwarfs using the individual 15 min spectroscopic exposures for 52,392 objects from the Sloan Digital Sky Survey. Our sample contains about 10^3-10^4 objects per spectral type bin in the range M0-M9, with a total of about 206,000 spectra and a typical number of 3 exposures per object (ranging up to a maximum of 30 exposures). Using this extensive data set we find that about 16% of the sources exhibit H-alpha emission in at least one exposure, and of those about 45% exhibit H-alpha emission in all of the available exposures. Within the sample of objects with H-alpha emission, only 26% are consistent with non-variable emission, independent of spectral type. The H-alpha variability, quantified in terms of the ratio of maximum to minimum H-alpha equivalent width (R_EW), and the ratio of the standard deviation to the mean (sigma_EW/), exhibits a rapid rise from M0 to M5, followed by a plateau and a possible decline in M9 objects. In particular, R_EW increases from a median value of about 1.8 for M0-M3 to about 2.5 for M7-M9, and variability with R_EW>10 is only observed in objects later than M5. For the combined sample we find that the R_EW values follow an exponential distribution with N(R_EW) exp[-(R_EW-1)/2]; for M5-M9 objects the characteristic scale is R_EW-1\approx 2.7, indicative of stronger variability. In addition, we find that objects with persistent H-alpha emission exhibit smaller values of R_EW than those with intermittent H-alpha emission. Based on these results we conclude that H-alpha variability in M dwarfs on timescales of 15 min to 1 hr increases with later spectral type, and that the variability is larger for intermittent sources.Comment: Submitted to ApJ; 20 pages, 15 figure