A reverse KAM method to estimate unknown mutual inclinations in exoplanetary systems

The inclinations of exoplanets detected via radial velocity method are essentially unknown. We aim to provide estimations of the ranges of mutual inclinations that are compatible with the long-term stability of the system. Focusing on the skeleton of an extrasolar system, i.e., considering only the two most massive planets, we study the Hamiltonian of the three-body problem after the reduction of the angular momentum. Such a Hamiltonian is expanded both in Poincar\'e canonical variables and in the small parameter $D_2$, which represents the normalised Angular Momentum Deficit. The value of the mutual inclination is deduced from $D_2$ and, thanks to the use of interval arithmetic, we are able to consider open sets of initial conditions instead of single values. Looking at the convergence radius of the Kolmogorov normal form, we develop a reverse KAM approach in order to estimate the ranges of mutual inclinations that are compatible with the long-term stability in a KAM sense. Our method is successfully applied to the extrasolar systems HD 141399, HD 143761 and HD 40307.Comment: 19 pages, 3 figure

On the convergence of an algorithm constructing the normal form for lower dimensional elliptic tori in planetary systems

We give a constructive proof of the existence of lower dimensional elliptic tori in nearly integrable Hamiltonian systems. In particular we adapt the classical Kolmogorov's normalization algorithm to the case of planetary systems, for which elliptic tori may be used as replacements of elliptic keplerian orbits in Lagrange-Laplace theory. With this paper we support with rigorous convergence estimates the semi-analytical work in our previous article (2011), where an explicit calculation of an invariant torus for a planar model of the Sun-Jupiter-Saturn-Uranus system has been made. With respect to previous works on the same subject we exploit the characteristic of Lie series giving a precise control of all terms generated by our algorithm. This allows us to slightly relax the non-resonance conditions on the frequencies.Comment: 45 page

Improved convergence estimates for the Schr\"oder-Siegel problem

We reconsider the Schr\"oder-Siegel problem of conjugating an analytic map in $\mathbb{C}$ in the neighborhood of a fixed point to its linear part, extending it to the case of dimension $n>1$. Assuming a condition which is equivalent to Bruno's one on the eigenvalues $\lambda_1,\ldots,\lambda_n$ of the linear part we show that the convergence radius $\rho$ of the conjugating transformation satisfies $\ln \rho(\lambda )\geq -C\Gamma(\lambda)+C'$ with $\Gamma(\lambda)$ characterizing the eigenvalues $\lambda$, a constant $C'$ not depending on $\lambda$ and $C=1$. This improves the previous results for $n>1$, where the known proofs give $C=2$. We also recall that $C=1$ is known to be the optimal value for $n=1$.Comment: 21 page

Secular orbital dynamics of the innermost exoplanet of the υ\upsilon-Andromed{\ae} system

We introduce a quasi-periodic restricted Hamiltonian to describe the secular motion of a small-mass planet in a multi-planetary system. In particular, we refer to the motion of $\upsilon$-And $b$ which is the innermost planet among those discovered in the extrasolar system orbiting around the $\upsilon$-Andromedae A star. We preassign the orbits of the Super-Jupiter exoplanets $\upsilon$-And $c$ and $\upsilon$-And $d$ in a stable configuration. The Fourier decompositions of their secular motions are reconstructed by using the Frequency Analysis and are injected in the equations describing the orbital dynamics of $\upsilon$-And $b$ under the gravitational effects exerted by those two external exoplanets (expected to be major ones in such an extrasolar system). We end up with a $2+3/2$ degrees of freedom Hamiltonian model; its validity is confirmed by the comparison with several numerical integrations of the complete $4$-body problem. Furthermore, the model is enriched by taking into account also the relativistic effects on the secular motion of the innermost exoplanet. We focus on the problem of the stability of $\upsilon$-And $b$ as a function of the parameters that mostly impact on its orbit, i.e. the initial values of its inclination and the longitude of its node. We study the evolution of its eccentricity, crucial to exclude orbital configurations with high probability of (quasi)collision with the central star in the long-time evolution of the system. Moreover, we also introduce a normal form approach, that further reduces our Hamiltonian model to a system with $2$ degrees of freedom, which is integrable because it admits a constant of motion related to the total angular momentum. This allows us to quickly preselect the domains of stability for $\upsilon$-And $b$, with respect to the set of the initial orbital configurations that are compatible with the observations

Quasi-periodic motions in a special class of dynamical equations with dissipative effects: a pair of detection methods

We consider a particular class of equations of motion, generalizing to n degrees of freedom the "dissipative spin--orbit problem", commonly studied in Celestial Mechanics. Those equations are formulated in a pseudo-Hamiltonian framework with action-angle coordinates; they contain a quasi-integrable conservative part and friction terms, assumed to be linear and isotropic with respect to the action variables. In such a context, we transfer two methods determining quasi-periodic solutions, which were originally designed to analyze purely Hamiltonian quasi-integrable problems. First, we show how the frequency map analysis can be adapted to this kind of dissipative models. Our approach is based on a key remark: the method can work as usual, by studying the behavior of the angular velocities of the motions as a function of the so called "external frequencies", instead of the actions. Moreover, we explicitly implement the Kolmogorov's normalization algorithm for the dissipative systems considered here. In a previous article, we proved a theoretical result: such a constructing procedure is convergent under the hypotheses usually assumed in KAM theory. In the present work, we show that it can be translated to a code making algebraic manipulations on a computer, so to calculate effectively quasi-periodic solutions on invariant tori. Both the methods are carefully tested, by checking that their predictions are in agreement, in the case of the so called "dissipative forced pendulum". Furthermore, the results obtained by applying our adaptation of the frequency analysis method to the dissipative standard map are compared with some existing ones in the literature

Alternative Basic Income Mechanisms: An Evaluation Exercise with a Microeconometric Model

We develop and estimate a microeconometric model of household labour supply in four European countries representative of different economies and welfare policy regimes: Denmark, Italy, Portugal and the United Kingdom. We then simulate, under the constraint of constant total net tax revenue (fiscal neutrality), the effects of various hypothetical tax-transfer reforms which include alternative versions of a Basic Income policy: Guaranteed Minimum Income, Work Fare, Participation Basic Income and Universal Basic Income. We produce indexes and criteria according to which the reforms can be ranked and compared to the current tax-transfer systems. The exercise can be considered as one of empirical optimal taxation, where the optimization problem is solved computationally rather than analytically. It turns out that many versions of the Basic Income policies would be superior to the current system. The most successful policies are those involving non means-tested versions of basic income (Universal or Participation Basic Income) and adopting progressive tax-rules. If – besides the fiscal neutrality constraint – also other constraints are considered, such as the implied top marginal top tax rate or the effect on female labour supply, the picture changes: unconditional policies remain optimal and feasible in Denmark and the UK; instead in Italy and Portugal universal policies appear to be too costly in terms of implied top marginal tax rates and in terms of adverse effects on female participation, and conditional policies such as Work-Fare, emerge as more desirable.tax reforms, models of labour supply, universal basic income, participation basic income, work fare, minimum guaranteed income, welfare evaluation, optimal taxation