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Celestial Mechanics and Astrodynamics
Published in K.T. Chau, Applications of Differential Equations in Engineering and Mechanics, 2019
where r is the distance between the two masses denoted by mS (smaller mass) and mL (larger mass). The accuracy of this formula for the Earth-Moon and the Sun-Earth system are considered in Problems 12.11 and 12.12. Physically, the Hill sphere or Roche sphere is the gravitational sphere of influence. It is named after American astronomer G.W. Hill and French astronomer E. Roche as they derived the formula independently. Clearly, the Hill sphere extends somewhere between the L1 and L2 Lagrangian Points.
Instability analysis of a quaternary deposition slope after two sudden events of river water fluctuations
Published in European Journal of Environmental and Civil Engineering, 2023
Yue Zhou, Shun-chao Qi, Lei Wang, Ming-liang Chen, Chen Xie, Jia-wen Zhou
For the considered range of K2 values, Figure 13(d) shows that the landslides/failures mostly occurred shortly after the water level reached the elevation of about 2,954m. The variations of K2 in the 2# layer had apparent adverse influences on the safety factor during both events. However, as a result of the smaller impact sphere of influence to the potential failure mass and the permeability coefficient with a difference of more than ten times, the effect of K2 was significantly less than K1 in second event as shown in Figure 13(c,d). Figure 14 shows the variations of the thickness of failure mass as the permeability of 1# and 2# layer changed. Since most of the bank failure occurred in shallow layers, permeability coefficient K2 of deeper one had little influence on the thickness of the sliding mass while increasing K1 value would increase the thickness of failure mass. The phenomena also indicated that the upper layer with a higher permeability was more likely to fail after being submerged/saturated.
Multi-agent genetic algorithm with controllable mutation probability utilizing back propagation neural network for global optimization of trajectory design
Published in Engineering Optimization, 2019
Mingcheng Zuo, Guangming Dai, Lei Peng
Each planet is surrounded by a Sphere Of Influence (SOI), and in this SOI, movement of the spacecraft is mainly effected by the planet instead of the sun. The spacecraft escapes the planet on the hyperbola trajectory, and at the boundary of SOI, a residual velocity can drive the spacecraft to the destination planet, which can be shown as follows: where is the gravitational constant of the sun, and are the orbital radii of the departure planet and the destination planet, respectively. The position of escaping from the departure planet, the perigee, is , which can be shown by Equation (4): where is the gravitational constant of the departure planet, e is the eccentricity of the escape trajectory, and is the angular momentum of the escape trajectory, which can be obtained by Equation (5):