{
  "title": "Radio paths: distance, delay and an ideal plasma limit",
  "kind": "Analytical teaching sweeps, not measured reception or operational forecasts",
  "constants": {
    "c_m_s": 299792458.0,
    "elementary_charge_C": 1.602176634e-19,
    "vacuum_permittivity_F_m": 8.8541878188e-12,
    "electron_mass_kg": 9.1093837139e-31
  },
  "formulas": {
    "free_space_loss_dB": "20 log10(4*pi*d*f/c), SI units",
    "delay_s": "d/c in vacuum",
    "critical_frequency_Hz": "sqrt(ne*e\u00b2/(epsilon0*me))/(2*pi)",
    "secant": "f_oblique_limit=fc/cos(theta), theta is the launch zenith angle in the sub-ionospheric region (refractive index approximately 1), measured from the upward layer normal; not the local angle at the turning point"
  },
  "row_counts": {
    "free_space": 303,
    "plasma_frequency": 101,
    "secant": 228
  },
  "domains": {
    "distance_km": [
      1,
      100000
    ],
    "frequency_MHz": [
      10,
      100,
      1000
    ],
    "electron_density_m3": [
      10000000000.0,
      1000000000000.0
    ],
    "launch_zenith_angle_deg": [
      0,
      75
    ],
    "vertical_fc_MHz": [
      3,
      6,
      9
    ]
  },
  "validation": {
    "isotropic_flux_aperture_max_difference_dB": 2.842170943040401e-14,
    "doubling_distance_and_frequency_add_6_0206dB": true,
    "quadruple_density_doubles_plasma_frequency": true,
    "angle0_multiplier1_angle60_multiplier2": true,
    "one_light_second_distance_gives1000ms": true,
    "rounded_8_98_plasma_rule_relative_difference": 0.00014892960052836912,
    "out_of_domain_angles_rejected": true
  },
  "limitations": [
    "Free-space model requires an unobstructed far-field path, matched polarisation and antenna gains handled separately. Distances are hypothetical path lengths, not ground coverage.",
    "Frequency comparison holds dimensionless antenna gains fixed; fixed physical antenna apertures can change the frequency dependence.",
    "Delay is vacuum one-way travel time only, excluding equipment, processing and dispersive plasma group delay.",
    "Plasma model is cold, collisionless, unmagnetized with stationary ions and prescribed electron density.",
    "Critical frequency uses peak layer density for the ideal vertical limit.",
    "Secant relation assumes plane stratified geometry. Theta is the sub-ionospheric launch zenith angle at refractive index approximately 1, not local turning-point incidence; 75 degrees is a teaching cap, not a certified validity boundary. Near-horizon extrapolation diverges and cannot represent curved Earth and finite ionosphere.",
    "No ray paths, geographic ranges, absorption, magnetic splitting, weather variation or guaranteed usable frequencies are computed."
  ],
  "sources": [
    {
      "title": "ITU-R P.525-5 (2024), free-space attenuation",
      "url": "https://www.itu.int/dms_pubrec/itu-r/rec/p/R-REC-P.525-5-202411-I!!PDF-E.pdf"
    },
    {
      "title": "Richard Fitzpatrick, University of Texas, Plasma Frequency eq1.5",
      "url": "https://farside.ph.utexas.edu/teaching/plasma/Plasma/node5.html"
    },
    {
      "title": "NBS Monograph80, Ionospheric Radio Propagation, section4.1 plane Earth and plane ionosphere, secant law eq4.2b",
      "url": "https://nvlpubs.nist.gov/nistpubs/Legacy/MONO/nbsmonograph80.pdf"
    },
    {
      "title": "NIST CODATA2022 constants",
      "url": "https://physics.nist.gov/cuu/pdf/all.pdf"
    }
  ]
}