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Space Plasma Formulary

50 formulas, 10 categories. KAW diagnostics, instability thresholds, reconnection, turbulence, and spacecraft presets for MMS, PSP, Wind, and Aditya-L1.

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References

[1] Huba, J.D. (2016). NRL Plasma Formulary. Naval Research Laboratory.

[2] Stasiewicz, K. et al. (2000). Small scale Alfvenic structure in the aurora. Rev. Geophys. 38, 291.

[3] Gary, S.P. (1993). Theory of Space Plasma Microinstabilities. Cambridge University Press.

[4] Goldreich, P. & Sridhar, S. (1995). Toward a theory of interstellar turbulence. ApJ 438, 763.

[5] Hollweg, J.V. (1999). Kinetic Alfven wave revisited. JGR 104, 14811.

[6] Hasegawa, A. (1976). Particle acceleration by MHD surface wave. JGR 81, 5083.

[7] Chettri, M.K. et al. (2025). arXiv:2603.14436.

Acknowledgements

PLASMAform was developed by Mani K. Chettri, Department of Physics, Sikkim University, under the supervision of Dr. Hemam Dinesh Singh (NSUT) and Dr. Rupak Mukherjee (Sikkim University, IUCAA Visiting Associate).

© 2025 Mani K Chettri. If this tool contributes to your research, please cite:
Chettri, M.K. (2025). PLASMAform: An interactive space plasma formulary. mkchettri.in/plasmaform
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Benchmarking & Validation

Formula Validation Report

All PLASMAform formulas verified against three independent authoritative sources. Unit consistency and edge-case behaviour documented.

—
Passing (<0.3%)
—
Total Verified
3
Independent Sources
—
Pass Rate
ParameterFormulaPLASMAformNRL [1]B&T [2]G&R [3]Δ maxStatus
Same physical inputs in CGS and SI must agree to floating-point precision (<0.01%). This verifies that unit-system switching introduces no conversion errors.

✓ All CGS ↔ SI conversions verified — no unit inconsistencies detected

Behaviour at parameter extremes and physical boundary conditions. A reliable tool handles these gracefully without returning NaN or unphysical values.
TestInput ConditionExpectedPLASMAformStatus

Documented Scope Restrictions

FluidAll formulas assume quasi-neutral, collisionless or weakly collisional plasma. Results are not valid for dense laboratory plasmas where νei > Ωci.
Non-relativisticValid for Te ≪ 511 keV (electrons) and vA ≪ c. Relativistic corrections to gyrofrequency and Debye length become significant above these limits.
Single speciesAssumes single-ion-species plasma (H⁺ or user-selected μ). Multi-species corrections (He²⁺, O⁺) for frequencies and speeds are not included.
KAW regimeKinetic Alfvén wave diagnostics valid for k⊥ρi ~ 1, βi ∈ [0.01, 10]. Results outside this range should be interpreted with caution.
Instability thresholdsGary (1993) parameterised threshold fits are valid for βi ∈ [0.01, 10]. Extrapolation beyond this range may be inaccurate.
Temperature isotropyFormulas using T assume isotropic temperature (T⊥ = T‖). For anisotropic plasmas, refer to the β–anisotropy diagram.
Landau dampingγ/ω formula is the linear, collisionless, weak-damping approximation. Valid only when vph/vTe ≲ 1. Satisfied for KAW in solar wind where vph ~ vA ≪ vTe.
Ponderomotiveδn/n₀ formula applies in the weakly nonlinear KAW regime. CGS and SI forms differ: denominator is 16π n₀kT in CGS and 2ε₀⁻¹n₀kT in SI.
Worked Example — Real Event Reproduction

MMS Magnetopause Crossing
16 October 2015

Step-by-step reproduction of published plasma parameters from an MMS dayside magnetopause crossing, demonstrating PLASMAform accuracy against peer-reviewed measurements.

MMS1 — Dayside Magnetopause Crossing · 16 Oct 2015 · ~13:07 UT
During MMS Phase 1 science operations, all four spacecraft crossed the dayside magnetopause in a tightly-separated tetrahedron configuration. This event is among the most-studied in MMS literature for electron-scale magnetic reconnection. We reproduce the magnetosheath-side plasma parameters from published MMS burst-mode data.
Burch et al. (2016) Science 352, aaf2939  ·  doi:10.1126/science.aaf2939
MMS Burst Mode Magnetosheath Reconnection Dayside MP
1
Load Published Plasma Parameters
Input values from MMS1 FPI burst-mode measurements. Magnetic field from FGM; densities and temperatures from FPI-DIS/DES. Adjust inputs to explore the published parameter ranges.
Magnetosheath Inputs
Published: ~10–20 cm⁻³
Published: ~30–50 nT
Published: ~200–400 eV
Published: ~20–50 eV
Magnetosphere Reference
Published: ~0.5–2 cm⁻³
Published: ~40–60 nT
Published: ~800–2000 eV
Published: ~50–200 eV
2
Computed Characteristic Parameters
PLASMAform computes the following from input values using standard NRL Formulary formulas. Published reference values shown for comparison. ✓ = within accepted uncertainty range.
Magnetosheath — PLASMAform vs Published
ParameterPLASMAformPublished✓
Magnetosphere — PLASMAform vs Published
ParameterPLASMAformPublished✓
3
Reconnection Scale Diagnostic
In Burch et al. (2016), the observed current sheet thickness was ~5–10 km. Ion inertial length di (magnetosheath) = — km The electron diffusion region thickness (~1 km) is comparable to the electron inertial length Electron inertial length de (magnetosheath) = — km
4
KAW Regime Verification
For kinetic Alfvén waves to operate: βi ~ 1 and k⊥ρi ~ 1. βi (magnetosheath) = —   [KAW active when βi ~ 0.5–5] ρi (magnetosheath) = — km
5
Alfvén Speed Asymmetry
Reconnection outflow speed scales as the geometric mean: vA (magnetosheath) = — km/s  ·  vA (magnetosphere) = — km/s Geometric mean vA = — km/s   [Observed jet ~500–800 km/s ✓]
Scale Comparison: Magnetosheath vs Magnetosphere (log scale)

✓ Validation Verdict

PLASMAform successfully reproduces all characteristic plasma parameters for the MMS 16 October 2015 magnetopause crossing event to within 5–35% of published values. The computed ion inertial length (— km), ion gyroradius (— km), and proton beta (—) are all consistent with the kinetic reconnection regime reported in Burch et al. (2016).

Formula Reference — PLASMAform v1.0

Space Plasma Formulary

Every formula implemented in PLASMAform with symbolic derivation, CGS and SI numerical coefficients verified against CODATA 2018, validity regimes, and live computed values.

§ 01Frequencies
Electron Cyclotron Frequency
\(f_{ce},\;\omega_{ce}\)
— Hz
Symbolic
\( f_{ce} = \dfrac{eB}{2\pi m_e} \)
Coefficients
CGS: \(2.800\times10^6\,B\) Hz  (B in G)
SI:  \(2.800\times10^{10}\,B\) Hz  (B in T)
CGS: \(\omega_{ce}=1.759\times10^7\,B\) rad/s
Validity: Non-relativistic electrons (Te ≪ 511 keV).
Physical Significance
fce is the fastest fundamental timescale in non-relativistic plasma physics.
Ion Cyclotron Frequency
\(f_{ci},\;\omega_{ci}\)
— Hz
Symbolic
\( f_{ci} = \dfrac{ZeB}{2\pi\mu m_p} \)
Coefficients (Z=1, μ=1)
CGS: \(1.526\times10^3\,\tfrac{Z}{\mu}B\) Hz
SI:  \(1.526\times10^7\,\tfrac{Z}{\mu}B\) Hz
Physical Significance
fci defines the critical boundary below which ideal MHD is valid.
Electron Plasma Frequency
\(f_{pe},\;\omega_{pe}\)
— kHz
Symbolic
\( f_{pe} = \dfrac{1}{2\pi}\!\sqrt{\dfrac{n_e e^2}{\varepsilon_0 m_e}} \)
Coefficients
CGS: \(8.980\times10^3\sqrt{n_e}\) Hz
SI:  \(8.980\times10^6\sqrt{n_e}\) Hz
Physical Significance
fpe sets the cutoff for electromagnetic wave propagation in the plasma.
Ion Plasma Frequency
\(f_{pi},\;\omega_{pi}\)
— Hz
Symbolic
\( f_{pi} = \dfrac{Z}{2\pi\sqrt{\mu}}\!\sqrt{\dfrac{n_i e^2}{\varepsilon_0 m_p}} \)
Coefficients
CGS: \(2.095\times10^2\tfrac{Z}{\sqrt{\mu}}\sqrt{n_i}\) Hz
SI:  \(2.095\times10^5\tfrac{Z}{\sqrt{\mu}}\sqrt{n_i}\) Hz
§ 02Length Scales
Electron Debye Length
\(\lambda_{De}\)
— m
Symbolic
\( \lambda_{De}=\!\sqrt{\dfrac{\varepsilon_0 k_BT_e}{n_e e^2}} \)
Coefficients
CGS: \(7.434\times10^2\sqrt{T_e/n_e}\) cm
SI:  \(7.434\sqrt{T_e/n_e}\) m
Electron & Ion Thermal Gyroradii
\(r_e,\;\rho_i\)
— km
Electron
\( r_e = v_{Te}/\Omega_{ce} \)
Ion
\( \rho_i = v_{Ti}/\Omega_{ci} \)
Inertial Lengths
\(d_e=c/\omega_{pe},\;d_i=c/\omega_{pi}\)
— km
de Coefficients
CGS: \(5.314\times10^5 n_e^{-1/2}\) cm
di Coefficients
CGS: \(2.277\times10^7 n_i^{-1/2}\) cm
§ 03Velocities
Alfvén Speed
\(v_A\)
— km/s
Symbolic (SI)
\( v_A = B/\!\sqrt{\mu_0 n\mu m_p} \)
Coefficients
CGS: \(2.181\times10^{11}\mu^{-\frac{1}{2}}n^{-\frac{1}{2}}B\) cm/s
Sound Speed — Dual Forms
\(C_s^{\rm IA},\;C_s^{\rm MHD}\)
— km/s
Ion Acoustic
\( C_s^{\rm IA} = \!\sqrt{\gamma Zk_BT_e/\mu m_p} \)
Full MHD
\( C_s^{\rm MHD} = \!\sqrt{\gamma(ZT_e+T_i)k_B/\mu m_p} \)
Electron Thermal Speed
\(v_{Te}\)
— m/s
NRL Convention
\( v_{Te}=\!\sqrt{k_BT_e/m_e} \)
B&T Numerical
CGS: \(5.931\times10^7\sqrt{T_e}\) cm/s
§ 04Dimensionless Parameters
Plasma Beta
\(\beta_i,\;\beta_e,\;\beta_{\rm tot}\)
—
SI Form
\( \beta_s = 2\mu_0 n k_BT_s/B^2 \)
Numerical
\(\beta = 4.027\times10^{-11}\,nT/B^2\)
Electron-to-Ion Mass Ratio
\(m_e/m_i\)
5.446 × 10⁻⁴
Value (H⁺)
\( m_e/m_p = 5.4462\times10^{-4} \)
KAW Significance
KAW: βe ≫ me/mi
§ 05KAW Physics
KAW / IAW Regime Condition
\(\beta_e\) vs \(m_e/m_i\)
—
KAW Regime
\( \beta_e \gg m_e/m_i \Rightarrow \text{KAW active} \)
IAW Regime
\( \beta_e \ll m_e/m_i \Rightarrow \text{Ion Acoustic} \)
KAW Dispersion
\(\omega = k_\parallel v_A\!\sqrt{1+k_\perp^2\rho_s^2}\)
— km
Ion Acoustic Gyroradius
\( \rho_s = C_s^{\rm IA}/\Omega_{ci} \)
KAW Dispersion
\( \omega = k_\parallel v_A\!\sqrt{1+k_\perp^2\rho_s^2} \)
Taylor Break Frequency
\(f_{\rm break}=V_{sw}/(2\pi l)\)
— Hz
Symbolic
\( f_{\rm break} = V_{sw}/(2\pi l) \)
Relevant Scales
l = di, ρi, ρs
Electron Landau Damping Rate
\(\gamma/\omega\)
Regime-dep.
Formula
\( \dfrac{\gamma}{\omega}\approx-\!\sqrt{\dfrac{\pi}{8}}\dfrac{v_{ph}}{v_{Te}}\exp\!\!\left(-\dfrac{v_{ph}^2}{2v_{Te}^2}\right) \)
Condition
\(v_{ph}/v_{Te}\lesssim 1\)
Walén Relation & Obliquity Parameter
\(\delta v_\perp = \pm\delta B_\perp/\sqrt{\mu_0\rho},\;\;\chi\)
CGS↔SI
Walén Relation — SI
\( \delta v_\perp=\pm(\delta B_\perp/B_0)\,v_A \)
Obliquity Parameter
\( \chi = k_\perp\delta B/(k_\parallel B_0) \)
§ 06Physical Constants (CODATA 2018)
SymbolNameValueUnit & Status
\(e\)Elementary charge1.602 176 634 × 10⁻¹⁹C (exact)
\(m_e\)Electron mass9.109 383 7015 × 10⁻³¹kg
\(m_p\)Proton mass1.672 621 923 69 × 10⁻²⁷kg
\(\mu_0\)Vacuum permeability4π × 10⁻⁷H m⁻¹ (exact)
\(\varepsilon_0\)Vacuum permittivity8.854 187 812 8 × 10⁻¹²F m⁻¹
\(c\)Speed of light2.997 924 58 × 10⁸m s⁻¹ (exact)
\(k_B\)Boltzmann constant1.380 649 × 10⁻²³J K⁻¹ (exact)
\(m_e/m_p\)Mass ratio5.446 170 21 × 10⁻⁴dimensionless