Physics Documentation

Magnetic Nozzle Physics

Helicon simulates plasma expansion through a diverging magnetic field. The nozzle converts upstream thermal energy into directed exhaust momentum via two mechanisms:

1. Ion detachment — ions have large Larmor radii (r_L >> L_B at the throat exit) and decouple from the field ballistically
2. Electron detachment — electrons are light and magnetized; their demagnetization drives the hard physics

The Detachment Problem

The frozen-in flux theorem (ideal MHD) predicts that magnetized plasma follows field lines indefinitely. Real detachment occurs when the approximations break down:

Mechanism	Condition	Physics
Inertial detachment	r_L,i > L_B	Ion Larmor radius exceeds field scale → ballistic
Resistive detachment	Ω_e τ_e ~ 1	Electron collision rate comparable to gyrofrequency
Pressure anisotropy	A = P_⊥/P_∥ - 1 ≠ 0	Fire-hose instability → anomalous transport
Ambipolar field	E_r coupling	Electron pressure drives radial electric field

Detachment Efficiency Definitions

Helicon computes and reports three definitions (all different in the literature):

\[\eta_d^{\text{momentum}} = \frac{\dot{p}_{\text{exit, axial}}}{\dot{p}_{\text{injected}}}\]

\[\eta_d^{\text{particle}} = \frac{N_{\text{exit, axial}}}{N_{\text{injected}}}\]

\[\eta_d^{\text{energy}} = \frac{E_{k,\text{axial, exit}}}{E_{k,\text{injected}}}\]

Always specify which definition when comparing with literature.

Analytical Models

Paraxial Thrust Coefficient (Little & Choueiri 2013)

For a polytropic electron gas (index γ) expanding through mirror ratio R_B:

\[C_T = \sqrt{\frac{2(\gamma+1)}{\gamma-1}} \cdot \sqrt{\eta_T}\]

\[\eta_T = 1 - \left(\frac{1}{R_B}\right)^{(\gamma-1)/\gamma}\]

\[\theta_{\text{div}} = \arcsin\left(\frac{1}{\sqrt{R_B}}\right)\]

Hall Parameter (Resistive Detachment)

The electron Hall parameter from Spitzer collision theory:

\[\Omega_e \tau_e = \frac{eB/m_e}{\nu_{ei}} \quad \text{where} \quad \nu_{ei} = \frac{n e^4 \ln\Lambda}{3 \varepsilon_0^2 \sqrt{2\pi} m_e^{1/2} (k_B T_e)^{3/2}}\]

Resistive detachment onset: Ω_e τ_e ~ 1.

Biot-Savart Field Computation

For a circular coil at (z_c, a) with current I, the on-axis field:

\[B_z(0, z) = \frac{\mu_0 I a^2}{2(a^2 + (z-z_c)^2)^{3/2}}\]

Off-axis (exact, using elliptic integrals K(k²) and E(k²)):

\[B_z = \frac{\mu_0 I}{2\pi \alpha} \left[ K(k^2) + \frac{a^2 - r^2 - \delta z^2}{\beta^2} E(k^2) \right]\]

where α² = (r+a)² + δz², β² = (r-a)² + δz², k² = 4ar/α².

References

1. Breizman & Arefiev (2008), PoP 15, 057103 — paraxial nozzle model
2. Little & Choueiri (2013), PoP 20, 103501 — thrust coefficient
3. Merino & Ahedo (2016), PoP 23, 023506 — 2D collisionless nozzle
4. Moses, Gerwin & Schoenberg (1991), AIP CP 246 — resistive detachment
5. Olsen et al. (2015), IEEE TPS 43, 252 — VASIMR experimental data