MHD/Shocks: Rankine-Hugoniot Graphs

The Rankine-Hugoniot equations are a set of conservation relations based on the equations of motion and Maxwell’s equations that govern the changes of a medium that has passed through a shock wave. This module solves for the properties of the plasma downstream of the shock, based on the properties of the upstream plasma. Here the plasma is assumed to be isotropic and Maxwellian both upstream and down. The downstream state can vary based on four main properties of the upstream solar wind plasma. These parameters are:

Mach number
The ratio between the upstream flow velocity along the shock normal direction and the velocity at which fast magnetosonic compressional waves propagate. This parameter ranges from 1 to ∞.
β
The ratio between the thermal pressure and the magnetic pressure of the plasma. This parameter ranges from 0 (magnetic pressure dominates) to ∞ (thermal pressure dominates).
γ
The ratio of specific heats of the plasma. This parameter describes the number of degrees of freedom of the plasma, and generally ranges from 1 (infinite degrees of freedom) to 3 (one degree of freedom).
θBn
The angle between the magnetic field and shock normal directions. This parameter ranges from 0° (parallel orientation) to 90° (perpendicular orientation). When θBn is 90° the magnetic field is in the shock plane. When θBn is 0° the magnetic field is along the shock normal.

In this module, you can determine how the jumps in the magnetic field strength, number density, temperature, and plasma β vary as these parameters vary. The jumps are defined as the ratio of the downstream value of the quantity to the upstream value. To see this, select one of the options Mach number, β, γ, θBn. Then enter the minimum and maximum values for the varied parameter; the parameter will be varied within the specified range. Enter the values for the other three parameters in the boxes, and then click Calculate to plot the graphs.

The upper two graphs have linear vertical scales. The lower two graphs have logarithmic vertical scales.

If you want to find the exact solution for a given set of upstream parameters, you can do this by selecting None for the varied parameter. When you enter values for the four upstream parameters and click Calculate, the exact values for the jumps in magnetic field strength, number density, temperature, and plasma β are displayed next to the input parameters.

The ratio of criticality is also calculated. The ratio of criticality indicates whether the shock is subcritical (< 1), marginally critical (~ 1), or supercritical (> 1). This ratio becomes important in describing the physical processes observed at collisionless shocks. It is defined as the Mach number at which the downstream sound speed matches the downstream bulk velocity.

If you want to calculate downstream parameters from the upstream parameters for a standing shock, then use the MHD/Shocks: Rankine-Hugoniot Case Studies module.