Astrospheres

\( \renewcommand{\div}{{\vec{\nabla} \cdot}} \newcommand{\del}{{\dfrac{\partial}{\partial t}}} \)
Interplanetary shocks

In the solar system various spacecraft observe travelling interplenatary shocks which are generate by solar coronal mass ejections (CME) or solar flares and the like. We are not going to discuss further how interplanetary shocks are generated, but want take spacecraft observations to determine the MHD shock charateritics. We will only concentrate on slow and fast shocks.

A databasis to find shocks and its parameter is also find the "Omni data base" or the "CfA database"

We use the Geocentric Solar Ecliptic (GSE) coordinate system, which is centered at Earth, with the x-y plane aligned with the ecliptic plane of Earth's motion about the Sun. The x-axis points towards the Sun, the y-axis is opposite the orbital motion of Earth, and the z-axis points "north" out of the ecliptic plane. Units are Earth radii, where the radius of the Earth is 6378 km.

Solutions
Shock type: It is a shock.

The indices "1" and "2" correspond to the region before and after the shock passage. "Upstream" and "downstream" depend on the type of the shock, for a forward shock "2" is upstream and "1" downstream and vice versa for a reverse shock.

All parameters are given in the shock rest frame, except the shock speed. More information are given below.

For vectors the first three rows are the components and the last row its modulus.

\(\vec{n}\)
\(v_{n,1}, v_{n,2}\)
\(u, \Theta, \delta\)
\(\vec{v}_{1}\)
\(\vec{v}_{2}\)
\(c_{s,1}, c_{s,2}\)
\( \theta_{1}, \theta_{2}\)
\(v_{A,1}, v_{f,1}, v_{s,1} \)
\(v_{A,2}, v_{f,2}, v_{s,2} \)
\(M_{1},M_{2} \)
\(\vec{v}_{t,1}\)
\(\vec{v}_{t,2}\)
\( B_{n,1}, B_{n,2}, |\vec{B}_{1}|, |\vec{B}_{2}|\)
\(\vec{B}_{t,1}\)
\(\vec{B}_{t,2}\)
\(s, \xi, \chi, \zeta\)
\(\psi, \Psi \)
\(P_{1}, P_{2}, \beta_{1}, \beta_{2}\)

Additinal information and explanations

Attention: The indices "1" and "2" correspond now to "upstream" and "downstream", respectilevly.

\(\vec{n}\) normal vector \(\vec{N} = (\vec{B}_{1} - \vec{B}_{2}) \times (\vec{B}_{1} \times \vec{B}_{2}), \qquad \vec{n} = \vec{N}/N \) "coplanarity theroem"
\(v_n, B_n \) normal components \(v_n = \vec{n}\cdot\vec{v}\qquad B_n = \vec{n}\cdot\vec{B}\)
\(\vec{v}_t, \vec{B}_t \) tangential vectors \(\vec{v}_t = \vec{v}-v_{n}\vec{n}\qquad \vec{B}_t = \vec{B}- B_{n}\vec{n}\)
\(u\) Shock speed \(u = \frac{n_{1}v_{n,1}-n_{2}v_{n,2}}{n_{1}-n_{2}}\)
\(s\) density compression ratio taken from data
\(\chi\) temperature compression ratio taken from data
\(P \) thermal pressure \(P = n k T \) assumption: ideal gas law holds
\(\xi\) pressure compression ratio \(\xi = \frac{\chi}{s}\)
\(c_{s}\) sound speed \( c_{s}^{2} = \frac{\gamma P}{ ho}= \frac{\gamma k T}{m}\) k is the Boltzmann constant, m the (proton) mass, \(\rho\) the density (for an ideal gas)
\(M\) Mach number \(M = \frac{v}{c_s}\)
\(\Theta\) Shock angle \( \Theta = \arcsin \sqrt{\frac{\xi (\gamma+1)+\gamma-1}{2\gamma M_{1}^{2}}} \) Form the hydrodynamic RH relations
\(\delta\) Flow deflection angle \( \delta = \mathrm{arcctg} \left(\tan\Theta \left[\frac{(\gamma+1)M_{1}^{2}}{2(M_{1}^{2}\sin^{2}\Theta-1)}-1 \right] \right) \) Form the hydrodynamic RH relations
\(\theta\) Angle between velocity and magentic field \(\theta = \arccos\frac{\vec{v}\cdot\vec{B}}{v\,B}\)
\(v_{A} \) Alfvén speed \(v_{A}^{2} = \frac{B^{2}}{4\pi \rho}\)
\(v_{f,s} \) fast and slow magnetosonic speeeds \(v_{f,s}^{2} = \frac{1}{2}\left[\left(v_{A}^{2}+c_{s}^{2}\right)\pm\sqrt{\left(v_{A}^{2}+c_{s}^{2}\right)^{2}-4\,v_{A}^{2}\,c_{s}^{2}\cos^{2}\theta }\right] \) the + and - sign for the fast and slow magnetosonic, respectively
\(\beta \) Plasma beta \(\beta = \frac{8\pi P}{B^{2}} \)
\(\psi \) ratio of tangential magnetic field amplitude \(\psi = \frac{B_{t,2}}{B_{t,1}} \)
\(\Psi\) ratio of magnetic field amplitude \( \Psi = \frac{B_{2}}{B_{1}} \)
\(\zeta \) entropy compression ratio \(\zeta = \xi s^{-\gamma}\le 1 \)

RH = Rankine-Hugoniot relations

TP IV AIRUB