# Astrospheres

$$\renewcommand{\div}{{\vec{\nabla} \cdot}} \newcommand{\del}{{\dfrac{\partial}{\partial t}}}$$

## Shocks

For more details see for example Goedbloed, Keppens & Poedts, 2010 Advanced Magnetohydrodynamics.
##### Notations and definitions
The velocity is decomposed \begin{equation} \label{eq:ms2} \vec{u} = u_{n}\vec{e}_{n} + u_{t} \vec{e}_{t} \end{equation} in a component $$u_{n}$$ parallel to the vector normal to the shock (the normal vector $$\vec{n}$$), and a tangential component $$\vec{v}_{t}$$. Note, that there are two tangential directions perpendicular to each other. The indices $$n,t$$ are then the normal, respective tangenial components of the vectors, $$\rho,P, \vec{u},\vec{B}, \gamma$$ are the denisty, thermal pressure, plasma (bulk) velocity, magnetic field and polytropic index. The indes $$i\in\{1,2\}$$ denote the upstream, repsective the downstream direction. Further, we will use $$\vec{B} = \vec{B}' / \sqrt{\mu}$$, $$\vec{B}'$$ is the magnetic induction in physical units (SI or cgs). The following notations will be used ($$i \in \{u,d\}$$, where "u" are the parameter in upwind direction and "d" those downwind:
 $$v_{A,n,i}$$ $$\equiv$$ $$\frac{B_{n,i}}{\sqrt{\rho}}$$ normal Alvfén speed $$v_{c}$$ $$\equiv$$ $$\sqrt{\frac{\gamma P}{\rho}}$$ sound speed $$M_{A,n,i}$$ $$\equiv$$ $$\frac{u_{n,i}}{V_{A,n,i}}$$ normal Alvfénic Mach number $$M$$ $$\equiv$$ $$M_{A,n,u}$$ shorthand notation $$M_{A,t,i}$$ $$\equiv$$ $$\frac{B_{t,i}}{u_{n,i}\sqrt{\rho_{i}}}$$ tangential'' Alfv\'enic Mach number $$M_{s,n,i}$$ $$\equiv$$ $$\frac{u_{n}}{v_{c}}$$ normal sonic Mach number $$s$$ $$\equiv$$ $$\frac{\rho_{d}}{\rho_{u}}=\frac{u_{n,u}}{u_{n,d}}$$ the compression ratio $$\beta_{t,i}$$ $$\equiv$$ $$2P_{i} /B^{2}_{t,i}$$ tangential plasma beta $$=\frac{2}{\gamma}\frac{\gamma \rho_{i}u_{n,i}^{2}P_{i}}{\rho_{i}u_{n,i}^{2}B^{2}_{t,i}}$$ $$=\frac{2}{\gamma}\, \frac{v_{c,i}^{2}}{v_{A,t,i}^{2}} =\frac{2}{\gamma}\, \frac{M_{A,t,i}}{M_{s,n,i}}$$ alternative represnetation $$\beta_{n,i}$$ $$\equiv$$ $$2P_{i} /B^{2}_{n,i}$$ normal plasma beta $$=\frac{2}{\gamma}\frac{\gamma \rho_{i}u_{n,i}^{2}P_{i}}{\rho_{i}u_{n,i}^{2}B^{2}_{n,i}}$$ $$=\frac{2}{\gamma}\, \frac{v_{c,i}^{2}}{v_{A,n,i}^{2}} =\frac{2}{\gamma}\, \frac{M_{A,n,i}}{M_{s,n,i}}$$ alternative represnetation $$\beta_{i}$$ $$\equiv$$ $$\beta_{n,i}$$ shorthand notation

We define the following angles: \begin{align} \vartheta_{i} &= \measuredangle \vec{B}_{i},\vec{n} \to \tan\vartheta_{i}=\frac{B_{t,i}}{B_{n}}\\ \varphi_{i} &= \measuredangle \vec{u}_{i},\vec{n} \to \tan\varphi_{i}=\frac{u_{t,i}}{u_{n}}\\ \alpha_{i} &= \measuredangle \vec{B}_{i},\vec{u}_{i}\to \cos\alpha_{i} =\frac{\vec{u}_{i}\cdot \vec{B}_{i}}{u_{i}B_{i}} \end{align} To calculate the Rankine Hugoniot equations, we replace $$\partial d/\partial t \to -U$$ where $$U$$ is the shock speed and $$\vec{\nabla}\to \vec{n}$$ and switch to the shock rest frame $$\vec{u}' = \vec{u} - \vec{U}$$. To save writings we negfelct the prime at $$\vec{u}'$$ in the following.
Further, we will use $$\vec{B} = \vec{B}' /\sqrt{\mu}$$, $$\vec{B}'$$ is the magnetic induction in physical units (SI or cgs).

##### The Rankine Hugoniot equations

Then we obtain for the Rankine Hugoniot equations in the shock rest frame: \begin{align} \left\{ \rho u_{n}\right\} &=0 &\text{(continuity)}&\\ \left\{\rho u_{n}^{2} +P + \frac{1}{2} B^{2}_{t}\right\} &=0 &\text{(normal momentum)} &\\ \rho u_{n}\left\{\vec{u}_{t}\right\}- B_{n}\left\{\vec{B}_{t}\right\} &=0 &\text{(tangential momentum)} &\\ \rho u_{n}\left\{\frac{1}{2} (u^{2}_{t}+u_{n}^{2}) + \frac{1}{\rho}\left(\frac{\gamma}{\gamma-1} P+\frac{B^{2}_{t}}{2}\right)\right\} -B_{n}\left\{(\vec{u}_{t}\cdot\vec{B}_{t})\right\} &=0 &\text{(energy)}&\\ \left\{B_{n}\right\} &=0 &\text{(normal B-flux)} &\\ \rho u_{n} \left\{\frac{\vec{B}_{t}}{\rho}\right\}-B_{n}\left\{\vec{u}_{t}\right\}&= 0 &\text{(tangential B-flux)} &\\ \end{align} where the curly brackets are the shorthand notation for $$\{a\}=a_u-a_d$$ where the indices $$u,d$$ are the upstream respctivly the downstream direction.

The above system are 8 equations for eight unkowns $$\rho, u_n, P, B_n$$ and two vector components for $$\vec{u}_t$$ and $$\vec{B}_t$$. The shocks and discontinuities are characterized as follows:
• hydrodynmic shock: $$B_n=0, \vec{B}_t=\vec{0}$$
• perpendicular shock: $$B_n = 0$$
• parallel shock: $$\vec{B}_t=\vec{0}$$
• slow shock: $$M_{A,n,2}^2\le M_{A,n,1}^2\le 1$$
• intermediate shock: $$M_{A,n,2}^2\le 1\le M_{A,n,1}^2$$
• fast shock_$$1\le M_{A,n,2}^2\le M_{A,n,1}^2$$
• switch on shock: $$B_{t,1} =0, B_{t,2}\ne 0$$
• switch off shock: $$B_{t,1} \ne 0, B_{t,2}= 0$$
• contact disconinity: $$u_n=0, B_n\ne 0\Rightarrow \{\vec{u}_t\} =0, \{\vec{B}_t\} =0, \{P\}=0$$ but $$\rho\ne 0$$
• tangential discontuity: $$u_n=0, B_n= 0 \Rightarrow \left\{P+\frac{1}{2} B_t^2\right\} =0$$, but $$\{\vec{u}_t\} \ne0, \{\vec{B}_t\} \ne0, \{P\}\ne 0$$ ,$$\rho\ne 0$$

To caluclate the donwstrema values we have to solve the above equations system with respect t $$s$$. The general case is a tedious algebrac excercise, where one has to handle the perpendicular shocks separatly, because, in general, because in the general case a divison by $$B_n$$ is required. its solutions are found here.