Multiple scattering in wind models of gamma‐ray bursts
In wind models of γ‐ray bursts, it has been assumed that soft x‐ray photons are scattered only once by relativistic electrons moving away from the star; this assumes that the scattering optical depth is small. Using the observational data from the bursts, we consider three methods for estimating this optical depth: (i) the luminosity of the burst, (ii) the ratio of the γ‐ray flux (≳10 keV) to x‐ray flux (<10 keV), (iii) the presence or lack of cyclotron lines. Within the context of the wind models, we find that multiple scattering is likely to be important in the formation of the spectrum in γ‐ray bursts.
Abstract This study contributes to the literature on socially responsible investing by examining the diversification potential of commodities, specifically oil, gold and clean energy together with the Brazilian Corporate Sustainability Index (ISE). Multivariate GARCH models are used to model volatility spillovers and conditional correlation in pairs of stocks containing ISE. Specifically, A-BEKK and A-DCC models with spillovers are estimated. The models’ results are used to compute and analyze the optimal weights and hedge ratios for stock portfolio holdings. The greatest benefit from diversification is obtained through the acquisition of gold and then OVX.