Linear Stability of Equilibrium Points in the Generalized Photogravitational Chermnykh's Problem

The equilibrium points and their linear stability has been discussed in the generalized photogravitational Chermnykh's problem. The bigger primary is being considered as a source of radiation and small primary as an oblate spheroid. The effect of radiation pressure has been discussed numerically. The collinear points are linearly unstable and triangular points are stable in the sense of Lyapunov stability provided $\mu< \mu_{Routh}=0.0385201$. The effect of gravitational potential from the belt is also examined. The mathematical properties of this system are different from the classical restricted three body problem

The Effect of Radiation Pressure on the Equilibrium Points in the Generalised Photogravitational Restricted Three Body Problem

The existence of equilibrium points and the effect of radiation pressure have been discussed numerically. The problem is generalized by considering bigger primary as a source of radiation and small primary as an oblate spheroid. We have also discussed the Poynting-Robertson(P-R) effect which is caused due to radiation pressure. It is found that the collinear points $L_1,L_2,L_3$ deviate from the axis joining the two primaries, while the triangular points $L_4,L_5$ are not symmetrical due to radiation pressure. We have seen that $L_1,L_2,L_3$ are linearly unstable while $L_4,L_5$ are conditionally stable in the sense of Lyapunov when P-R effect is not considered. We have found that the effect of radiation pressure reduces the linear stability zones while P-R effect induces an instability in the sense of Lyapunov

Out-of-plane equilibrium points in the restricted three-body problem with oblateness

The equations of motion of the three-dimensional restricted three-body problem with oblateness are found to allow the existence of out-of-plane equilibrium points. These points lie in the $(x-z)$ plane almost directly above and below the center of each oblate primary. Their positions can be determined numerically and are approximated by series expansions. The effects of their existence on the topology of the zero–velocity curves are considered and their stability is explored numerically.

Environmental policy implications of extreme variations in pollutant stock levels and socioeconomic costs

This paper uses a real options approach to examine the impact of abrupt increases in carbon dioxide emissions and pollutant-related socio-economic costs. It derives optimal investment rules in the form of critical values for both pollutant stock levels and social costs, above which environmental policies should be adopted. Moreover, it determines the optimal emissions abatement level. Our analysis extends the methodology of Pindyck (2000) using jump diffusion processes. We show that if the stock of pollutant is subject to extreme variations and the emissions abatement level is chosen exogenously by the policymaker, then lower levels of the pollutant stock are required to trigger policy adoption. A similar, yet more prominent, effect is observed under the assumption that pollutant-related socio-economic costs and benefits are expected to exhibit abrupt changes. However, different results are obtained when we examine simultaneously the two interrelated decisions, namely, the optimal threshold of emissions abatement and the optimal abatement level. In this case, an increase in the size and/or probability of a jump increases the critical values of both pollutant stock levels and socio-economic costs but leads to higher optimal abatement. © 2013 The Board of Trustees of the University of Illinois