At finite ratios of the kinetic plasma pressure to the magnetic pressure, the magnetic confinement configurations of axisymmetric plasma columns tend to acquire characteristics that hinder the onset of instabilities driven by the combined effects of magnetic curvature and pressure gradient. A simple analytical dispersion relation that contains the main physical factors affecting an important class of these modes is given.PACS numbers: 52.30.+rThe purpose of this Letter is to discuss the maximum value of the ratio /3 of the plasma kinetic pressure to the pressure associated with the confining magnetic field that can be reached without inducing loss of confinement. This ratio is important in order to assess the main characteristics, such as transport properties and rates of radiation emission, of a given magnetic confinement configuration, as well as the type of fusion reactor that can be developed out of it.Here we limit most of our attention to the ideal magnetohydrodynamic (MHD) approximation and notice that when /3 increases toward finite values, instabilities driven by the combined effects of the magnetic field curvature and the pressure gradient can be expected to develop. 1 An essential feature of the ideal MHD model is the strong interaction between the developing plasma instability and its magnetic confinement configuration. As /3 is increased, the configuration of the plasma isobaric surfaces also changes, and, since the plasma motion is tied to that of the magnetic lines, this will have a dual effect 2 : Not only will it increase the instability driving pressure gradient, but it also will enhance the stabilizing magnetic tension in that region of unfavorable magnetic curvature where the relevant modes develop.In the earliest stability analysis of these modes, the considered equilibrium models neglected the modification of the confinement configuration that takes place as j3 evolves and becomes finite. In particular, only linear terms in the plasma ,pressure gradient were retained in the relevant normal mode equation, and rather pessimistic upper limits on j3 for stable configurations were obtained. However, as we shall show in this Letter, important nonlinear terms must also be retained; as was first demonstrated for simplified equilibrium configurations, these terms can lead to production of a "second stability region/ This circumstance is illustrated by the following dispersion relation that we have derived from a consistent description of the ideal MHD equilibrium configuration in the vicinity of the magnetic axis:• or 3/^AQG 2 \ 2 / 5 SUA/ V G * 6 s G 2 " J5> 32 -£ • (1)Here we have employed familiar notations except for the dimensionless parameters s{ip) = dlxiq{Wdlnr{$), Gti>)=-8irR 0 q 0 B 0 ml r(mp(Wdip that measure the magnetic shear and the plasma pressure gradient. For G 2 < 1.2s, corresponding to the first stability region in the (G,s) plane, the pressure gradient is not strong enough to overcome the stable shear-Alfvgn oscillations. For G 2 > 6.4s the tendency of the plasma to expand is h...
An analytical theory of ideal-MHD ballooning modes that can be excited in finite-β equilibria is carried out on model configurations which include the effects of the increase of the poloidal field toward the outer edge of the plasma column and the dependence of the rate of magnetic shear on the poloidal angle. The relevant growth rates and eigensolutions are, in fact, significantly different from those derived on the basis of ‘low-β’ model configurations that omit one or both of the effects mentioned above, and provide different indications for the expected interaction between ideal-MHD and kinetic modes. For each value of the shear parameter ŝ, the normalized growth rate Γ becomes real at a critical value of the dimensionless pressure gradient parameter G. When the latter is increased at constant ŝ, Γ is found to increase only up to a saturation point, after which it decreases and tends to vanish at a second critical value of G.
The present paper studies the powered Swing-By maneuver when performed in an elliptical system of primaries. It means that there is a spacecraft travelling in a system governed by the gravity fields of two bodies that are in elliptical orbits around their center of mass. The paper particularly analyzes the effects of the parameters relative to the Swing-By (V inf À ; r p ; w), the orbit of the secondary body around the primary one (e; m) and the elements that specify the impulse applied (dV ; a) to the spacecraft. The impulse is applied when the spacecraft passes by the periapsis of its orbit around the body, where it performs the Swing-By, with different magnitudes and directions. The inclusion of the orbital eccentricity of the primaries in this problem makes it closer to reality, considering that there are many known systems with eccentricities different from zero. In particular, there are several moons in the Solar System which orbits are not circular, as well as some smaller bodies, like the dwarf planet Haumea and its moons, which have eccentricities of 0.25 or even larger. The behavior of the energy variation of the spacecraft is shown in details, as well as the cases where captures and collisions occur. The results show the conditions that optimize this maneuver, according to some given parameters, and how much can be obtained in terms of gains or losses of energy using the best conditions found by the algorithm developed here.
The present paper studies the effects of a powered Swing-By maneuver, considering the particular and important situations where there are energy gains for the spacecraft. The objective is to map the energy variations obtained from this maneuver as a function of the three parameters that identify the pure gravity Swing-By with a fixed mass ratio (angle of approach, periapsis distance and velocity at periapsis) and the three parameters that define the impulsive maneuver (direction, magnitude and the point where the impulse is applied). The mathematical model used here is the version of the restricted three-body problem that includes the Lemaître regularization, to increase the accuracy of the numerical integrations. It is developed and implemented by an algorithm that obtains the energy variation of the spacecraft with respect to the largest primary of the system in a maneuver where the impulse is applied inside the sphere of influence of the secondary body, during the passage of the spacecraft. The point of application of the impulse is a free parameter, as well as the direction of the impulse. The results
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