# Astrospheres

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

### Characteristic plasma parameters

The electron temperature is assumed to be the same as the ion temperature if not stated different. The plasma consists of electrons and protons, where charge neutrality is assumed. For a neutral gas it is assumed hat it consists of neutral hydrogen atoms (not moelcules), as in the interstellar medium. Thus care must be taken when using the derived parameters.

Parametervalueuseful formulaunitremarks
Input
Number density $$n_{p}$$Because of charge neutrality $$n_{e}=n_{p}$$ (only protons and electrons) m$$^{-3}$$
Temperature $$T$$Assumption $$T_{p}=T_{e}$$ K
Magnetic field $$B$$ T
Typical length scale $$L$$ m
Ionisation potential $$U_{p}$$default is that for hydrogen eV
Flow speed $$u$$ m/s
Neutral density $$n_{n}$$ If zero calculate after the Saha equation, not valid for dilute media m$$^{-3}$$needed for dilute media

Output
$$n_{i}/n_{n}$$ After Saha equation: $$n_{i}/n_{n}=\left(\frac{2\pi m_e k_{B}}{h^{2}}\right)^{3/2} \frac{T^{3/2}}{n_{i}} exp(-\frac{U_{i}}{k_{B}T})$$ only valid for dense media (blackbody radiation), not for interstellar media, etc. not valid for dilute media
$$n_{n}$$ Number density of neutral hydrogen $$n_{n}$$ after Saha equation or as given m$$^{-3}$$for dilute media the input neutral density is used
$$\lambda_{mfp,n}$$ Mean free path for hard spheres: $$\lambda_{mfp,n}=\sigma n_{H}$$ with Bohr radius $$a=5.29177\cdot 10^{-11}$$[m] for neutral hydrogen and $$\sigma = 2 \pi a^{2}$$ m=AU
$$\tau_{n}$$ collision time between neutrals and ions s
$$\lambda_{D}$$ Debeye length: $$\lambda_{D} = \sqrt{\frac{\epsilon_{0}k_{B}T}{e^{2}n_i}}$$ $$\approx 70 \sqrt{\frac{T}{n_{i}}}$$ m
$$N_{D}$$ Number of particles in Debeye sphere: $$N_{D}= \frac{4 \pi}{3}\lambda_{D}^{3}n_{i}$$ $$\approx 1.4\cdot 10^{6} \sqrt{\frac{T^3}{n_{i}}}$$ part.
$$v_{th,i}$$ thermal speed of protons $$v_{th} = \frac{\sqrt{2 k_{B} T}}{m_{p}}$$ $$\approx 128 \sqrt{T}$$ m/s
$$v_{th,e}$$ thermal speed of electrons $$v_{th} = \frac{\sqrt{2 k_{B} T}}{m_{e}}$$ $$\approx 5500 \sqrt{T}$$ m/s
$$\Omega_{p}$$ proton gyro frequency $$\Omega_{p} = \frac{q B}{m_{p}}$$ $$\approx 9.6 \cdot 10^{7} B$$ rad s$$^{-1}$$ $$\mathrel{\widehat{=}}$$Hz
$$\Omega_{e}$$ electron gyro frequency $$\Omega_{e} = \frac{q B}{m_{e}}$$ $$\approx 1.8 \cdot 10^{11} B$$ rad s$$^{-1}$$ $$\mathrel{\widehat{=}}$$Hz
$$R_{g,p}$$ Gyroradius for protons $$R_g= \frac{v_{th}}{\Omega}$$ $$\approx 9.5 \cdot 10^{-7} \frac{\sqrt{T}}{B}$$ m
$$R_{g,e}$$ Gyroradius for protons $$R_g= \frac{v_{th}}{\Omega}$$ $$\approx 2.2 \cdot 10^{-8} \frac{\sqrt{T}}{B}$$ m
$$\omega$$ plasma frequency $$\omega^{2} = \frac{n e^{2}}{\epsilon_{0} m_{e}}\quad n_{e} = n_{p} = n$$ $$\approx 57 \sqrt{n}$$ rad s$$^{-1}$$ $$\mathrel{\widehat{=}}$$Hz
$$v_{A}$$ Alfvén speed $$v_{A} = \frac{B}{\sqrt{\mu \rho}}$$ $$\approx 2.23\cdot 10^{5} \frac{B}{\sqrt{n}}$$ m s$$^{-1}$$
$$v_{c}$$ sound speed $$v_{s} = \sqrt{\frac{\gamma k_{B} T}{m_p}}$$ $$\approx 1.17\cdot 10^{2} \sqrt{T}$$ m s$$^{-1}$$
$$M_{A}$$ Alfvé Mach number $$M_{A} = \frac{{u}}{v_{A}}$$
$$M_{s}$$ Sound Mach number $$M_{c} = \frac{{u}}{v_{s}}$$
$$P_{therm}$$ thermal pressure (ideal gas) $$P_{therm} = n_{p} k_{B} T$$ $$N m^{-2}$$
$$P_{ram}$$ ram pressure $$P_{ram} = \rho u^{2}$$ $$N m^{-2}$$
$$P_{B}$$ magnetic field pressure $$P_{B} = \frac{B^{2}}{2 \mu}$$ $$N m^{-2}$$
$$\beta_{therm}$$ plasma beta $$\beta = \frac{P_{therm}}{P_{B}}$$
$$\beta_{ram}$$ $$\beta_{ram} = \frac{P_{ram}}{P_{B}}$$
$$Kn$$ Knudsen number: $$Kn = \frac{\lambda_{mfp}}{L}$$ only for neutrals