The radial distribution of radiation belt protons: Approximate solution of the steady state transport equation at arbitrary pitch angle

1984 ◽  
Vol 89 (A3) ◽  
pp. 1527 ◽  
Author(s):  
V. Jentsch
2000 ◽  
Vol 122 (4) ◽  
pp. 437-442
Author(s):  
Shigeru Aoki ◽  
Takeshi Watanabe

This paper deals with steady-state response of the piping system with nonlinear support having hysteresis damping characteristics. Considering the energy loss for contact with a support, an analytical method of approximate solution for the beam, a one-span model of the piping system, with quadrilateral hysteresis loop characteristics is presented. Some numerical results of the approximate solution for the response curves and the mode shapes are shown. [S0094-9930(00)00204-3]


2011 ◽  
Vol 73 (7-8) ◽  
pp. 785-795 ◽  
Author(s):  
Qiuhua Zheng ◽  
Mei-Ching Fok ◽  
Jay Albert ◽  
Richard B. Horne ◽  
Nigel P. Meredith

2015 ◽  
Vol 33 (5) ◽  
pp. 583-597 ◽  
Author(s):  
H. Breuillard ◽  
O. Agapitov ◽  
A. Artemyev ◽  
E. A. Kronberg ◽  
S. E. Haaland ◽  
...  

Abstract. Chorus-type whistler waves are one of the most intense electromagnetic waves generated naturally in the magnetosphere. These waves have a substantial impact on the radiation belt dynamics as they are thought to contribute to electron acceleration and losses into the ionosphere through resonant wave–particle interaction. Our study is devoted to the determination of chorus wave power distribution on frequency in a wide range of magnetic latitudes, from 0 to 40°. We use 10 years of magnetic and electric field wave power measured by STAFF-SA onboard Cluster spacecraft to model the initial (equatorial) chorus wave spectral power, as well as PEACE and RAPID measurements to model the properties of energetic electrons (~ 0.1–100 keV) in the outer radiation belt. The dependence of this distribution upon latitude obtained from Cluster STAFF-SA is then consistently reproduced along a certain L-shell range (4 ≤ L ≤ 6.5), employing WHAMP-based ray tracing simulations in hot plasma within a realistic inner magnetospheric model. We show here that, as latitude increases, the chorus peak frequency is globally shifted towards lower frequencies. Making use of our simulations, the peak frequency variations can be explained mostly in terms of wave damping and amplification, but also cross-L propagation. These results are in good agreement with previous studies of chorus wave spectral extent using data from different spacecraft (Cluster, POLAR and THEMIS). The chorus peak frequency variations are then employed to calculate the pitch angle and energy diffusion rates, resulting in more effective pitch angle electron scattering (electron lifetime is halved) but less effective acceleration. These peak frequency parameters can thus be used to improve the accuracy of diffusion coefficient calculations.


2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
U. Filobello-Nino ◽  
H. Vazquez-Leal ◽  
R. Castaneda-Sheissa ◽  
V. M. Jimenez-Fernandez ◽  
A. L. Herrera-May ◽  
...  

This article utilizes perturbation method (PM) to find an analytical approximate solution for the Quasi-Steady-State Michaelis-Menten problem. From the comparison of Figures and absolute error values, between approximate and numerical solutions, it is shown that the obtained solutions are accurate, and therefore, they explain the general behaviour of the Michaelis-Menten mechanism.


1977 ◽  
Vol 32 (7) ◽  
pp. 697-703
Author(s):  
Fr. Kaiser

Abstract The Peierls-Boltzmann transport equation for phonons, which was re-formulated and modified in a previous paper, is extended to be applicable to arbitrary interactions and phonon processes. As a rule, it turns out that only two types of steady state solutions are possible: hysteresis and threshold. These two solutions reveal the possibility of “transport phase transitions”, i. e. a transition from the “thermodynamic” branch to a “nonthermodynamic” one via a cumulative excitation. It is shown that both the threshold and the hysteresis situation exhibit pronounced analogies to phase transi­tions in thermal equilibrium. The dependence of the steady states from the relevant parameters is discussed.


Author(s):  
Shigeru Aoki ◽  
Takeshi Watanabe

This paper deals with steady-state response of a continuous system with collision characteristics. Considering the energy loss in a collision, an analytical method of approximate solution for the continuous system with symmetrical hysteresis loop characteristics is presented. The resonance curves of nonlinear response obtained from approximate solution are shown as discontinuous line, and are discussed the phenomenon.


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