Analytic MHD theory for Earth's bow shock at low Mach numbers

Abstract

A previous MHD theory for the density jump at the Earth's bow shock, which assumed the Alfven (MA) and sonic (Ms) Mach numbers are both ≫ 1, is reanalyzed and generalized. It is shown that the MHD jump equation can be analytically solved much more directly using perturbation theory, with the ordering determined by MA and Ms, and that the first‐order perturbation solution is identical to the solution found in the earlier theory. The second‐order perturbation solution is calculated, whereas the earlier approach cannot be used to obtain it. The second‐order terms generally are important over most of the range of MA and Ms in the solar wind when the angle θ between the normal to the bow shock and magnetic field is not close to 0° or 180° (the solutions are symmetric about 90°). This new perturbation solution is generally accurate under most solar wind conditions at 1 AU, with the exception of low Mach numbers when θ is close to 90°. In this exceptional case the new solution does not improve on the first‐order solutions obtained earlier, and the predicted density ratio can vary by 10–20% from the exact numerical MHD solutions. For θ ∼ 90° another perturbation solution is derived that predicts the density ratio much more accurately. This second solution is typically accurate for quasi‐perpendicular conditions. Taken together, these two analytical solutions are generally accurate for the Earth's bow shock, except in the rare circumstance that MA ≤ 2. MHD and gasdynamic simulations have produced empirical models in which the shock's standoff distance as, is linearly related to the density jump ratio X at the subsolar point. Using an empirical relationship between as and X obtained from MHD simulations, as values predicted using the MHD solutions for X are compared with the predictions of phenomenological models commonly used for modeling observational data, and with the predictions of a modified phenomenological model proposed recently. The similarities and differences between these results are illustrated using plots of X and as predicted for the Earth's bow shock. The plots show that the new analytic solutions agree very well with the exact numerical MHD solutions and that these MHD solutions should replace the the corresponding phenomenological relations in comparisons with data. Furthermore, significant differences exist between the standoff distances predicted at low MA using the MHD models versus those predicted by the new modified phenomenological model. These differences should be amenable to observational testing.