Context. Stellar jets are believed to play a key role in star formation, but the question of how they originate is still being debated. Aims. We derive the physical properties at the base of the jet from DG Tau both along and across the flow and as a function of velocity. Methods. We analysed seven optical spectra of the DG Tau jet, taken with the Hubble Space Telescope Imaging Spectrograph. The spectra were obtained by placing a long-slit parallel to the jet axis and stepping it across the jet width. The resulting position-velocity diagrams in optical forbidden emission lines allowed access to plasma conditions via calculation of emission line ratios. In this way, we produced a 3D map (2D in space and 1D in velocity) of the jet's physical parameters i.e. electron density n e , hydrogen ionisation fraction x e , and total hydrogen density n H . The method used is a new version of the BE-technique. Results. A fundamental improvement is that the new diagnostic method allows us to overcome the upper density limit of the standard [S ii] diagnostics. As a result, we find at the base of the jet high electron density, n e ∼ 10 5 , and very low ionisation, x e ∼ 0.02−0.05, which combine to give a total density up to n H ∼ 3 × 10 6 . This analysis confirms previous reports of variations in plasma parameters along the jet, (i.e. decrease in density by several orders of magnitude, increase of x e from 0.05 to a plateau at 0.7 downstream at 2 from the star). Furthermore, a spatial coincidence is revealed between sharp gradients in the total density and supersonic velocity jumps. This strongly suggests that the emission is caused by shock excitation. No evidence was found of variations in the parameters across the jet, within a given velocity interval. The position-velocity diagrams indicate the presence of both fast accelerating gas and slower, less collimated material. We derive the mass outflow rate,Ṁ j , in the blue-shifted lobe in different velocity channels, that contribute to a total ofṀ j ∼ 8 ± 4× 10 −9 M yr −1 . We estimate that a symmetric bipolar jet would transport at the low and intermediate velocities probed by rotation measurements, an angular momentum flux ofL j ∼ 2.9 ± 1.5 × 10 −6 M yr −1 AU km s −1 . We discuss implications of these findings for jet launch theories. Conclusions. The derived properties of the DG Tau jet are demonstrated to be consistent with magneto-centrifugal theory. However, non-stationary modelling is required in order to explain all of the features revealed at high resolution.

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In this paper, we focus on the stability of the Trojan asteroids for the planar restricted three-body problem, by extending the usual techniques for the neighbourhood of an elliptic point to derive results in a larger vicinity. Our approach is based on numerical determination of the frequencies of the asteroid and effective computation of the Kolmogorov normal form for the corresponding torus. This procedure has been applied to the first 34 Trojan asteroids of the IAU Asteroid Catalogue, and it has worked successfully for 23 of them.The construction of this normal form allows computer-assisted proofs of stability. To show this, we have implemented a proof of existence of families of invariant tori close to a given asteroid, for a high order expansion of the Hamiltonian. This proof has been successfully applied to three Trojan asteroids.PACS numbers: 05.10.−a, 45.20.Jj, 45.50.Pk, 95.10.Ce

We reconsider the original proof of Kolmogorov's theorem in\ud the light of classical perturbation methods based on expansions in some\ud parameter. With a careful analysis of the accumulation of small divisors\ud we prove that their effect is bounded by a geometrically increasing\ud numerical sequence. This allows us to achieve the proof without using\ud the so called quadratic method.

We investigate the long time stability in Nekhoroshev’s sense for the Sun–\ud Jupiter–Saturn problem in the framework of the problem of three bodies. Using computer\ud algebra in order to perform huge perturbation expansions we show that the stability for a time\ud comparable with the age of the universe is actually reached, but with some strong truncations\ud on the perturbation expansion of the Hamiltonian at some stage. An improvement of such\ud results is currently under investigation