Additional Equilibrium Points Research Articles

Graphical method to obtain multicomponent adsorption equilibria from intermediate breakthrough curve plateaus

Adsorption mixture equilibrium data remains challenging to obtain. An established technique is the adsorption breakthrough experiment where the final adsorbed capacities are determined. A graphical method to obtain adsorbed amounts from the intermediate plateaus in the breakthrough curves is proposed in this work. The method is proposed for isothermal, diluted breakthrough tests. Via an integrated capacity curve, the intermediate regions can be extrapolated to extract additional equilibrium points beyond the final state without the need for a multicomponent model. If the column is preloaded, the extra equilibrium points can correspond to multicomponent mixtures as multiple species can have non-zero intermediate concentration plateaus. The method is demonstrated on various simulated examples and then applied on breakthrough data from non-preloaded and preloaded columns of all-silica beta zeolite for diluted methanol/ethanol and diluted ethanol/isobutanol mixtures.

Equilibrium Points in the Restricted Full Three Body Problem with Ellipsoidal Primaries

Abstract Equilibrium points in the Restricted Full Three Body Problem with ellipsoidal primaries are investigated. The approach adopted by some in the literature that approximates the primaries by means of the spherical harmonic potential is demonstrated to be problematic when identifying equilibrium points in close vicinity of the primaries. Additional equilibrium points aside from the five well-known ones analogous to those in the Restricted Three Body Problem prove to be practically non-existent in general due to the intrinsic divergence of the spherical harmonic potential inside the Brillouin surface for irregular primaries. Accurate modeling of the primaries with the ellipsoid potentials is instead carried out. Locations and stability properties of the triangular equilibrium points in the double-ellipsoid systems of varying parameters are systematically studied and are compared with results obtained for the sphere–ellipsoid systems.

LOCATIONS OF OUT-OF-PLANE EQUILIBRIUM POINTS IN THE ELLIPTIC RESTRICTED THREE-BODY PROBLEM UNDER RADIATION AND OBLATENESS EFFECTS

This study deals with the generalization of the Elliptic Restricted Three-Body Problem (ER3BP) by considering the effects of radiation and oblate spheroid primaries. This may illustrate a gas giant exoplanet orbiting its host star with eccentric orbit. In the three dimensional case, this generalization may possess two additional equilibrium points (<TEX>$L_{6,7}$</TEX>, out-of-plane). We determine the existence of <TEX>$L_{6,7}$</TEX> in ER3BP under the effects of radiation (bigger primary) and oblateness (small primary). We analytically derive the locations of <TEX>$L_{6,7}$</TEX> and assume initial approximations of (<TEX>${\mu}-1$</TEX>, <TEX>${\pm}\sqrt{3A_2}$</TEX>), where <TEX>${\mu}$</TEX> and <TEX>$A_2$</TEX> are the mass parameter and oblateness factor, respectively. The fixed locations are then determined. Our results show that the locations of <TEX>$L_{6,7}$</TEX> are periodic and affected by <TEX>$A_2$</TEX> and the radiation factor (<TEX>$q_1$</TEX>).

Motion in a modified Chermnykh’s restricted three-body problem with oblateness

In this paper, the restricted problem of three bodies is generalized to include a case when the passively gravitating test particle is an oblate spheroid under effect of small perturbations in the Coriolis and centrifugal forces when the first primary is a source of radiation and the second one an oblate spheroid, coupled with the influence of the gravitational potential from the belt. The equilibrium points are found and it is seen that, in addition to the usual three collinear equilibrium points, there appear two new ones due to the potential from the belt and the mass ratio. Two triangular equilibrium points exist. These equilibria are affected by radiation of the first primary, small perturbation in the centrifugal force, oblateness of both the test particle and second primary and the effect arising from the mass of the belt. The linear stability of the equilibrium points is explored and the stability outcome of the collinear equilibrium points remains unstable. In the case of the triangular points, motion is stable with respect to some conditions which depend on the critical mass parameter; influenced by the small perturbations, radiating effect of the first primary, oblateness of the test body and second primary and the gravitational potential from the belt. The effects of each of the imposed free parameters are analyzed. The potential from the belt and small perturbation in the Coriolis force are stabilizing parameters while radiation, small perturbation in the centrifugal force and oblateness reduce the stable regions. The overall effect is that the region of stable motion increases under the combine action of these parameters. We have also found the frequencies of the long and short periodic motion around stable triangular points. Illustrative numerical exploration is rendered in the Sun–Jupiter and Sun–Earth systems where we show that in reality, for some values of the system parameters, the additional equilibrium points do not in general exist even when there is a belt to interact with.

Equilibrium points and stability under effect of radiation and perturbing forces in the restricted problem of three oblate bodies

This paper presents a generalized problem of the restricted three body studied in Abdul Raheem and Singh with the inclusion that the third body is an oblate spheroidal test particle of infinitesimally mass. The positions and stability of the equilibrium point of this problem is studied for a model in which the primaries is the binary system Struve 2398 (Gliese 725) in the constellation Draco; which consist of a pair of radiating oblate stars. It is seen that additional equilibrium points exist on the line collinear with the primaries, for some combined parameters of the problem. Hence, there can be up to five collinear equilibrium points. Two triangular points exist and depends on the oblateness of the participating bodies, radiation pressure of the primaries and a small perturbation in the centrifugal force. The stability analysis ensures that, the collinear equilibrium points are unstable in the linear sense while the triangular points are stable under certain conditions. Illustrative numerical exploration is given to indicate significant improvement of the problem in Abdul Raheem and Singh.

Effect of oblateness, perturbations, radiation and varying masses on the stability of equilibrium points in the restricted three-body problem

This paper investigates the combined effect of small perturbations e,e′ in the Coriolis and centrifugal forces, radiation pressure q i , and changing oblateness of the primaries A i (t) (i=1,2) on the stability of equilibrium points in the restricted three body problem in which the primaries is a supergiant eclipsing binary system which consists of a pair of bright oblate stars having the appearance of a giant peanut in space and their masses assumed to vary with time in the absence of reactive forces. The equations of motion are derived and the equilibrium points are obtained. For the autonomized system, it is seen that there are more than a pair of the triangular points as κ→∞; κ being the arbitrary sum of the masses of the primaries. In the case of the collinear points, two additional equilibrium points exist on the line joining the primaries when simultaneously κ+e′ 1. Hence, there are at least nine equilibrium points of the system. The stability of these points is explored analytically and numerically. It is seen that the collinear and triangular points are stable with respect to certain conditions controlled by κ while the non-planar equilibrium points are unstable.

A model of neuronal bursting using three coupled first order differential equations.

We describe a modification to our recent model of the action potential which introduces two additional equilibrium points. By using stability analysis we show that one of these equilibrium points is a saddle point from which there are two separatrices which divide the phase plane into two regions. In one region all phase paths approach a limit cycle and in the other all phase paths approach a stable equilibrium point. A consequence of this is that a short depolarizing current pulse will change an initially silent model neuron into one that fires repetitively. Addition of a third equation limits this firing to either an isolated burst or a depolarizing afterpotential. When steady depolarizing current was applied to this model it resulted in periodic bursting. The equations, which were initially developed to explain isolated triggered bursts, therefore provide one of the simplest models of the more general phenomenon of oscillatory burst discharge.