A new shock fitting procedure for the MHD Rankine-Hugoniot relations for the case of small He²⁺ slippage

To study MHD shocks in space, it is important to find the shock frame of reference from the observed plasma and magnetic field parameters. These shock parameters have to satisfy the Rankine-Hugoniot relations. In this study we present a novel procedure for shock fitting of the one-fluid anisotropic Rankine-Hugoniot relations and of the time difference between two spacecraft observations in the case of small He²⁺ slippage. Here, a Monte-Carlo calculation and a minimization technique are used. The observed variables including the upstream and downstream magnetic fields, plasma densities, plasma betas, plasma anisotropies, W (the difference between the downstream and upstream velocities, W = V₂ - v₁, and Δt (the time difference between two spacecraft observations) are used in our procedure where V is defined as the center of mass velocity of plasmas. A loss function based on a difference between the calculated and the observed values is defined, and the best fit solution is found by searching for the minimum loss function value. For shocks that cannot be fitted well, we introduce two new parameters in the modified RH relations, one in the normal momentum flux and the other in the energy flux equations. These two parameters are interpreted as the equivalent "normal momentum" and "heat" fluxes needed in the RH relations. They provide two degrees of freedom in the system, and their amounts can be estimated from our procedure. Several synthetic shocks are given to verify our procedure. We also apply this procedure to two interplanetary shocks observed by both the WIND and Geotail spacecraft. The results demonstrate that our method works for both the synthetic and the real shocks. We have shown that our method can provide accurate shock normal estimations for perpendicular and parallel shocks as well. Given that our model is based on the RH relations that do not include the effect of alpha particle (He²⁺) slippage, it can only be applied to the cases with an ignorable slippage pressure tensor. We have investigated the pressure tensor due to alpha particle slippage using the WIND spacecraft data. It is found that in general the slippage pressure is small in comparison with the thermal pressure of the system and can be ignored. Thus our model can be applied to most interplanetary shocks observed near the ecliptic plane. However, when the slippage pressure is large, the magnetic coplanarity theorem is not valid any more. A more general model that involves slippage pressure tensor is a major and important development that is beyond the scope of the present study.