\( \renewcommand{\div}{{\vec{\nabla} \cdot}} \newcommand{\del}{{\dfrac{\partial}{\partial t}}} \)
Solution of the Rankine Hugoniot equations

The notations and Rankine Hugoniot equations are defined here.

The solutions for the tangential, contact and rotational discontuity are given here. The solution of the Rankine-Hugomiot shock equatios are in the HD case linear, for perpendicular shocks quadratic, and for general shocks cubic in \(s\). The latter can be found after a tedious but straightforward algebraic manipulation. Thus we have (with \(M^2 = M^2_{A,n,u}\) \begin{align} a_{3}s^{3}+a_{2}s^{2}+a_{1}s^{1}+a_{0}= 0 \end{align} The coefficients for the HD case are: \begin{align} a_{3} &= 0\\ a_{2} &= 0\\ a_{1} &= 2 + (\gamma - 1) M_{s,n,1}^2\\ \ a_{0} &= -(\gamma+1) M_{s,n,1}^{2} \end{align} for the perpendicular shocks: \begin{align} a_{3} &= 0\\ \ a_{2} &= 2M_{s,n,1}^{2}\\ \ a_{1} &= (2+(\gamma-1)M_{s,n,1}^{2})M_{A,t,1}^{2} + 2\gamma M_{s,n,1}^{2}\\ \ a_{0} &= -(\gamma+1) M_{A,t,1}^{2}M_{s,n,1}^{2} \end{align} and for the general shocks: \begin{align} a_{3} &= \left[2-(\gamma-1)\left(2 D - 1+3\tan^{2}\vartheta_{1}\right)M_{s,n,1}^{2}\right]M^2 -2 (\gamma-1)(D-2\tan^{2}\vartheta_{1}) M_{s,n,1}^{2}\\ a_{2} &= \left[-4+\left((2D -2-\tan^{2}\vartheta_{1})(\gamma-1)+\tan^{2}\vartheta_{1}\right)M_{s,n,1}^{2} \right]M^{4} - \left[(\tan^{2}\vartheta_{1}+1)(\gamma+1) - 2D(\gamma-1)\right]M_{s,n,1}^{2}M^2\\ a_{1} &= \left[2 + (\gamma -1) M_{s,n,1}^{2}\right]M^{6} + \left[\gamma(\tan^{2}\vartheta_{1}+2)+2\right]M_{s,n,1}^{2}M^{4}\\ a_{0} &= -(\gamma+1)M_{s,n,1}^{2}M^{6} \end{align} with \begin{align} D= \frac{\vec{u}_{t,1}\cdot\vec{B}_{t,1}}{u_{n,1}B_{n}} \end{align}

In principle all the solutions can be calculated analytically, which leads in the general case to very complicated expression. Thus the latter is better numerically. In the following it is assumed that the compression ratio was determined from the above. Then the expressions for the downstream values lead to relatively simple expressions.

Downstream values for hydrodynamic (\( M_{A,t,1}^2\to\infty \)) and perpendicular shocks.
In the HD case the last three lines are not defined.
exact\( \lim\limits_{M_{A,t,1}\to\infty} \)
\(\frac{\rho_{d}}{\rho_{u}}=\frac{u_{n,u}}{u_{n,d}} =s \) s
\(\vec{u}_{t,d} = \vec{u}_{t,u}\)--
\(\frac{P_{d}}{P_{u}} = \gamma \left(1-\frac{1}{s}\right)M_{s,n,u}^{2} +\frac{\gamma}{2}\frac{M_{s,n,u}^{2}}{M_{A,t,1}^2}(1-s^{2}) + 1\) \(\gamma \left(1-\frac{1}{s}\right)M_{s,n,u}^{2}\)
\(\frac{B_{t,u}}{B_{t,d}} = s\)--
\( \frac{v_{A,t,d}^{2}}{v_{A,t,u}^{2}}= s \)--

Downstream values for general shocks.
exact\( \lim\limits_{M^2\to\infty} \)
\(\frac{\rho_{d}}{\rho_{u}}=\frac{u_{n,u}}{u_{n,d}} =\frac{u_{n,u}}{u_{n,d}} =s \) s
\( \vec{u}_{t,d} = \vec{u}_{t,u} +\frac{(1-s)}{s-M^2}\,\frac{u_{n,1}}{B_{n}}\vec{B}_{t,u} \) \(\vec{u}_{t,u}\)
\( \frac{P_{d}}{P_{u}}=\gamma M_{s,n,u}^{2}\frac{s-1}{s}+ 1 + \frac{(1-s)M^2}{\beta_{t,u}} \frac{(s+1)M^2-2s}{(M^2-s)^{2}} \) \(\gamma M_{s,n,u}^{2}\frac{s-1}{s}+ 1 - \frac{1-s^{2}}{\beta_{t,u}}\)
\(\vec{B}_{t,d}= s \left[ \frac{M^2-1}{M^2-s}\right]\vec{B}_{t,u}\) \(s\vec{B}_{t,u}\)
\( B_{d}^{2}= s^{2} \frac{(M^2-1)^{2}}{(M^2-s)^{2}}B_{u}^{2}-(s-1)M^2\frac{(s+1)M^2-2s}{(s-M^2)^{2}}B_{n}^{2}\) \(s^{2}B_{t,u}^{2}+B_{n}^{2}\)
\(\frac{B_{d}^{2}}{B_{u}^{2}}=s^{2} \frac{(M^2-1)^{2}}{(M^2-s)^{2}} -(s-1)M^2\frac{(s+1)M^2-2s}{(s-M^2)^{2}}\cos^{2}\vartheta_{u}\) \(s^{2} \sin^{2}\vartheta_{u}+\cos^{2}\vartheta_{u}\)
\( \frac{v_{A,n,d}^{2}}{v_{A,n,u}^{2}} = \frac{1}{s} \) \(\frac{1}{s}\)
\( \frac{v_{A,t,d}^{2}}{v_{A,t,u}^{2}} = s\frac{(M^2-1)^{2}}{(M^2-s)^{2}}\) \(s\)

With the help of the above forulas one can estimate all shock parameter. For astrospheres (and "arbitrary" upstream parameter) a "shock"-calculator can be found here.