RayleighScattering

{
  "reference": {
    "citation": "Seinfeld, J. H. and Pandis, S. N. (2006) Atmospheric Chemistry and Physics, 2nd Edition, Chapter 15. Migrated from Aerosol.jl src/mie_scattering.jl `RayleighScattering` (esm-8aj).",
    "notes": "source_ref: Aerosol.jl@c8c640bb636b22f2425460e9ed5cba50793bfee1:src/mie_scattering.jl:111\n\nUpstream `RayleighScattering` is an MTK `@component` with five `@constants` (π_val=Float64(π), eight_thirds=8/3, four=4.0, three=3.0, two=2.0) and five `@parameters` (D_p=0.05e-6 m, λ=550e-9 m, n_refr=1.5, k_refr=0.0, ρ_p=1500 kg/m³). The eleven `@variables` are all algebraic:\n  α              ~ π_val * D_p / λ                                    [size parameter, Eq. 15.6]\n  m2_factor_real ~ rayleigh_m2_factor_real(n_refr, k_refr)            [@register_symbolic]\n  m2_factor_imag ~ rayleigh_m2_factor_imag(n_refr, k_refr)            [@register_symbolic]\n  m2_factor_mag_sq ~ m2_factor_real² + m2_factor_imag²\n  Q_scat         ~ eight_thirds * α⁴ * m2_factor_mag_sq               [Eq. 15.19]\n  Q_abs          ~ four * α * m2_factor_imag                          [Eq. 15.21]\n  Q_ext          ~ Q_scat + Q_abs                                     [Eq. 15.4]\n  ω              ~ Q_scat / Q_ext                                     [Eq. 15.5]\n  E_ext          ~ three * Q_ext / (two * ρ_p * D_p)                  [Eq. 15.41]\n  E_scat         ~ three * Q_scat / (two * ρ_p * D_p)                 [Eq. 15.42]\n  E_abs          ~ three * Q_abs / (two * ρ_p * D_p)                  [Eq. 15.43]\n\ngt-p3ep gap resolution: `rayleigh_m2_factor_real(n, k)` and `rayleigh_m2_factor_imag(n, k)` compute the real and imaginary parts of the Clausius-Mossotti factor (m²-1)/(m²+2), where m = n_refr + i·k_refr is the complex refractive index. Unlike the Bohren-Huffman Mie functions (`mie_Q_ext`, `mie_Q_scat` in AerosolExtinction), these are closed-form algebraic expressions derivable from standard complex arithmetic. With m2_re = n² - k² and m2_im = 2nk:\n  denom            = (m2_re + 2)² + m2_im²  [= |m²+2|²]\n  m2_factor_real   = [(m2_re-1)(m2_re+2) + m2_im²] / denom  [Re part]\n  m2_factor_imag   = 3·m2_im / denom                        [Im part]\nDerivation of Im[(m²-1)/(m²+2)]: writing (a+bi)/(c+di) = (ac+bd)/(c²+d²) + i(bc-ad)/(c²+d²) with a=(m2_re-1), b=m2_im, c=(m2_re+2), d=m2_im gives Im = [m2_im·(m2_re+2) - (m2_re-1)·m2_im] / denom = 3·m2_im / denom. This resolves the gt-p3ep blocking gap as direct algebraic AST without `interp.*` primitives or parameter-lifting.\n\n@constants handling: all five upstream @constants migrate to `type: parameter` with fixed defaults, consistent with the rig-wide convention (aerosol_extinction.esm, rayleigh_atmosphere.esm, etc.).\n\nDependency order (all acyclic, no ODEs):\n  {n_refr, k_refr} → {m2_re, m2_im} → denom → {m2_factor_real, m2_factor_imag}\n  {D_p, λ} → α\n  {m2_factor_real, m2_factor_imag} → m2_factor_mag_sq\n  {eight_thirds, α, m2_factor_mag_sq} → Q_scat\n  {four, α, m2_factor_imag} → Q_abs\n  {Q_scat, Q_abs} → Q_ext → {ω, E_ext}\n  {Q_scat, ρ_p, D_p} → E_scat;  {Q_abs, ρ_p, D_p} → E_abs\n\nTolerance rationale (rel=1e-9): purely algebraic system, all expressions are +, -, *, /, ^ on float64; round-off is ~1e-15. rel=1e-9 leaves six orders of magnitude of headroom."
  },
  "tolerance": {
    "rel": 1e-09
  },
  "variables": {
    "π_val": {
      "type": "parameter",
      "units": "1",
      "default": 3.141592653589793,
      "description": "Pi (dimensionless). Upstream MTK @constants entry π_val = Float64(π); migrated as a fixed-default parameter."
    },
    "eight_thirds": {
      "type": "parameter",
      "units": "1",
      "default": 2.6666666666666665,
      "description": "Prefactor 8/3 in the Rayleigh scattering efficiency formula (dimensionless, Seinfeld & Pandis Eq. 15.19). Upstream MTK @constants entry eight_thirds = 8.0/3.0."
    },
    "four": {
      "type": "parameter",
      "units": "1",
      "default": 4.0,
      "description": "Prefactor 4 in the Rayleigh absorption efficiency formula (dimensionless, Seinfeld & Pandis Eq. 15.21). Upstream MTK @constants entry four = 4.0."
    },
    "three": {
      "type": "parameter",
      "units": "1",
      "default": 3.0,
      "description": "Numerator factor 3 in the mass efficiency formula E = 3Q/(2ρD_p) (dimensionless, Seinfeld & Pandis Eqs. 15.41-15.43). Upstream MTK @constants entry three = 3.0."
    },
    "two": {
      "type": "parameter",
      "units": "1",
      "default": 2.0,
      "description": "Denominator factor 2 in the mass efficiency formula E = 3Q/(2ρD_p) (dimensionless). Upstream MTK @constants entry two = 2.0."
    },
    "D_p": {
      "type": "parameter",
      "units": "m",
      "default": 5e-08,
      "description": "Particle diameter (m). Default 50 nm (0.05 μm) — well within the Rayleigh regime: α = πD_p/λ ≈ 0.286 << 1 at λ=550 nm. The upstream default is 0.05e-6 m."
    },
    "λ": {
      "type": "parameter",
      "units": "m",
      "default": 5.5e-07,
      "description": "Wavelength of incident radiation (m). Default 550 nm — visible green light, near the peak of the solar spectrum. The upstream default is 550e-9 m."
    },
    "n_refr": {
      "type": "parameter",
      "units": "1",
      "default": 1.5,
      "description": "Real part of the complex refractive index m = n_refr + i·k_refr (dimensionless). Default 1.5 — a representative value for many inorganic aerosol constituents (e.g., ammonium sulfate ~1.53, sea salt ~1.54)."
    },
    "k_refr": {
      "type": "parameter",
      "units": "1",
      "default": 0.0,
      "description": "Imaginary part of the complex refractive index m = n_refr + i·k_refr (dimensionless). Default 0 — non-absorbing (purely scattering) particle. Positive values indicate light absorption (e.g., black carbon k ≈ 0.6 at 550 nm). Enters Q_abs linearly."
    },
    "ρ_p": {
      "type": "parameter",
      "units": "kg*m^-3",
      "default": 1500.0,
      "description": "Particle material density (kg/m³). Default 1500 kg/m³ — a representative value for mixed inorganic aerosol. Used only in the mass efficiency calculations E_ext, E_scat, E_abs."
    },
    "m2_re": {
      "type": "unknown",
      "units": "1",
      "description": "Real part of m² = n_refr² - k_refr² (dimensionless). Intermediate quantity for the Clausius-Mossotti factor. At default conditions (n=1.5, k=0): m2_re = 2.25."
    },
    "m2_im": {
      "type": "unknown",
      "units": "1",
      "description": "Imaginary part of m² = 2 n_refr k_refr (dimensionless). Zero for non-absorbing particles (k_refr=0); positive for absorbing particles."
    },
    "denom": {
      "type": "unknown",
      "units": "1",
      "description": "Shared denominator |m²+2|² = (m2_re+2)² + m2_im² (dimensionless) for the Clausius-Mossotti factor (m²-1)/(m²+2). At default conditions (n=1.5, k=0): denom = 4.25² = 18.0625."
    },
    "α": {
      "type": "unknown",
      "units": "1",
      "description": "Size parameter α = π D_p / λ (dimensionless, Seinfeld & Pandis 2006, Eq. 15.6). Characterises the scattering regime: Rayleigh (α << 1), resonance (α ~ 1), geometric (α >> 1). At default conditions α = π/11 ≈ 0.286."
    },
    "m2_factor_real": {
      "type": "unknown",
      "units": "1",
      "description": "Real part of the Clausius-Mossotti factor Re[(m²-1)/(m²+2)] = [(m2_re-1)(m2_re+2) + m2_im²] / |m²+2|² (dimensionless). For non-absorbing particles (k=0): Re = (n²-1)/(n²+2); at n=1.5: 1.25/4.25 = 5/17 ≈ 0.294."
    },
    "m2_factor_imag": {
      "type": "unknown",
      "units": "1",
      "description": "Imaginary part of the Clausius-Mossotti factor Im[(m²-1)/(m²+2)] = 3 m2_im / |m²+2|² (dimensionless). Zero for non-absorbing particles; positive for absorbing particles. Enters Q_abs = 4α × m2_factor_imag (Eq. 15.21)."
    },
    "m2_factor_mag_sq": {
      "type": "unknown",
      "units": "1",
      "description": "Squared magnitude of the Clausius-Mossotti factor |(m²-1)/(m²+2)|² = m2_factor_real² + m2_factor_imag² (dimensionless). Enters Q_scat via Eq. 15.19. At default conditions: (5/17)² = 25/289 ≈ 0.08651."
    },
    "Q_scat": {
      "type": "unknown",
      "units": "1",
      "description": "Rayleigh scattering efficiency Q_scat = (8/3) α⁴ |(m²-1)/(m²+2)|² (dimensionless, Seinfeld & Pandis 2006, Eq. 15.19). Scales as D_p⁴ and λ⁻⁴ in the Rayleigh regime. For non-absorbing particles equals Q_ext."
    },
    "Q_abs": {
      "type": "unknown",
      "units": "1",
      "description": "Rayleigh absorption efficiency Q_abs = 4 α Im[(m²-1)/(m²+2)] (dimensionless, Seinfeld & Pandis 2006, Eq. 15.21). Scales as D_p and λ⁻¹ — linear in α rather than α⁴, so absorption dominates over scattering for absorbing particles in the Rayleigh limit. Zero when k_refr=0."
    },
    "Q_ext": {
      "type": "unknown",
      "units": "1",
      "description": "Rayleigh extinction efficiency Q_ext = Q_scat + Q_abs (dimensionless, Seinfeld & Pandis 2006, Eq. 15.4)."
    },
    "ω": {
      "type": "unknown",
      "units": "1",
      "description": "Single-scattering albedo ω = Q_scat / Q_ext (dimensionless, Seinfeld & Pandis 2006, Eq. 15.5). Equal to 1 for non-absorbing particles (k_refr=0); decreases toward 0 as absorption dominates in the Rayleigh regime."
    },
    "E_ext": {
      "type": "unknown",
      "units": "m^2*kg^-1",
      "description": "Mass extinction efficiency E_ext = 3 Q_ext / (2 ρ_p D_p) (m² kg⁻¹, Seinfeld & Pandis 2006, Eq. 15.41). The power extinguished per unit incident irradiance per unit aerosol mass concentration."
    },
    "E_scat": {
      "type": "unknown",
      "units": "m^2*kg^-1",
      "description": "Mass scattering efficiency E_scat = 3 Q_scat / (2 ρ_p D_p) (m² kg⁻¹, Seinfeld & Pandis 2006, Eq. 15.42)."
    },
    "E_abs": {
      "type": "unknown",
      "units": "m^2*kg^-1",
      "description": "Mass absorption efficiency E_abs = 3 Q_abs / (2 ρ_p D_p) (m² kg⁻¹, Seinfeld & Pandis 2006, Eq. 15.43). Zero when k_refr=0."
    }
  },
  "equations": [
    {
      "lhs": "m2_re",
      "rhs": {
        "op": "-",
        "args": [
          {
            "op": "^",
            "args": [
              "n_refr",
              2
            ]
          },
          {
            "op": "^",
            "args": [
              "k_refr",
              2
            ]
          }
        ]
      }
    },
    {
      "lhs": "m2_im",
      "rhs": {
        "op": "*",
        "args": [
          2.0,
          "n_refr",
          "k_refr"
        ]
      }
    },
    {
      "lhs": "denom",
      "rhs": {
        "op": "+",
        "args": [
          {
            "op": "^",
            "args": [
              {
                "op": "+",
                "args": [
                  "m2_re",
                  2.0
                ]
              },
              2
            ]
          },
          {
            "op": "^",
            "args": [
              "m2_im",
              2
            ]
          }
        ]
      }
    },
    {
      "lhs": "α",
      "rhs": {
        "op": "/",
        "args": [
          {
            "op": "*",
            "args": [
              "π_val",
              "D_p"
            ]
          },
          "λ"
        ]
      }
    },
    {
      "lhs": "m2_factor_real",
      "rhs": {
        "op": "/",
        "args": [
          {
            "op": "+",
            "args": [
              {
                "op": "*",
                "args": [
                  {
                    "op": "-",
                    "args": [
                      "m2_re",
                      1.0
                    ]
                  },
                  {
                    "op": "+",
                    "args": [
                      "m2_re",
                      2.0
                    ]
                  }
                ]
              },
              {
                "op": "^",
                "args": [
                  "m2_im",
                  2
                ]
              }
            ]
          },
          "denom"
        ]
      }
    },
    {
      "lhs": "m2_factor_imag",
      "rhs": {
        "op": "/",
        "args": [
          {
            "op": "*",
            "args": [
              3.0,
              "m2_im"
            ]
          },
          "denom"
        ]
      }
    },
    {
      "lhs": "m2_factor_mag_sq",
      "rhs": {
        "op": "+",
        "args": [
          {
            "op": "^",
            "args": [
              "m2_factor_real",
              2
            ]
          },
          {
            "op": "^",
            "args": [
              "m2_factor_imag",
              2
            ]
          }
        ]
      }
    },
    {
      "lhs": "Q_scat",
      "rhs": {
        "op": "*",
        "args": [
          "eight_thirds",
          {
            "op": "^",
            "args": [
              "α",
              4
            ]
          },
          "m2_factor_mag_sq"
        ]
      }
    },
    {
      "lhs": "Q_abs",
      "rhs": {
        "op": "*",
        "args": [
          "four",
          "α",
          "m2_factor_imag"
        ]
      }
    },
    {
      "lhs": "Q_ext",
      "rhs": {
        "op": "+",
        "args": [
          "Q_scat",
          "Q_abs"
        ]
      }
    },
    {
      "lhs": "ω",
      "rhs": {
        "op": "/",
        "args": [
          "Q_scat",
          "Q_ext"
        ]
      }
    },
    {
      "lhs": "E_ext",
      "rhs": {
        "op": "/",
        "args": [
          {
            "op": "*",
            "args": [
              "three",
              "Q_ext"
            ]
          },
          {
            "op": "*",
            "args": [
              "two",
              "ρ_p",
              "D_p"
            ]
          }
        ]
      }
    },
    {
      "lhs": "E_scat",
      "rhs": {
        "op": "/",
        "args": [
          {
            "op": "*",
            "args": [
              "three",
              "Q_scat"
            ]
          },
          {
            "op": "*",
            "args": [
              "two",
              "ρ_p",
              "D_p"
            ]
          }
        ]
      }
    },
    {
      "lhs": "E_abs",
      "rhs": {
        "op": "/",
        "args": [
          {
            "op": "*",
            "args": [
              "three",
              "Q_abs"
            ]
          },
          {
            "op": "*",
            "args": [
              "two",
              "ρ_p",
              "D_p"
            ]
          }
        ]
      }
    }
  ],
  "tests": [
    {
      "id": "default_non_absorbing",
      "description": "Default parameters: n_refr=1.5, k_refr=0 (non-absorbing), D_p=5e-8 m (50 nm), λ=5.5e-7 m (550 nm), ρ_p=1500 kg/m³. α = π/11 ≈ 0.2856. m2_factor_real = (n²-1)/(n²+2) = 1.25/4.25 = 5/17 ≈ 0.2941. m2_factor_imag = 0. Q_scat = (8/3)(π/11)⁴(25/289) ≈ 1.535e-3. Q_abs = 0. ω = 1 exactly. E_ext = 3Q_scat/(2ρD_p) ≈ 30.7 m²/kg.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "α",
          "time": 0.0,
          "expected": 0.28559933214452665,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "m2_re",
          "time": 0.0,
          "expected": 2.25,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "m2_im",
          "time": 0.0,
          "expected": 0.0,
          "tolerance": {
            "abs": 1e-15
          }
        },
        {
          "variable": "denom",
          "time": 0.0,
          "expected": 18.0625,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "m2_factor_real",
          "time": 0.0,
          "expected": 0.29411764705882354,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "m2_factor_imag",
          "time": 0.0,
          "expected": 0.0,
          "tolerance": {
            "abs": 1e-15
          }
        },
        {
          "variable": "m2_factor_mag_sq",
          "time": 0.0,
          "expected": 0.08650519031141869,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Q_scat",
          "time": 0.0,
          "expected": 0.001534757089990882,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Q_abs",
          "time": 0.0,
          "expected": 0.0,
          "tolerance": {
            "abs": 1e-15
          }
        },
        {
          "variable": "Q_ext",
          "time": 0.0,
          "expected": 0.001534757089990882,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "ω",
          "time": 0.0,
          "expected": 1.0,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "E_ext",
          "time": 0.0,
          "expected": 30.69514179981764,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "E_scat",
          "time": 0.0,
          "expected": 30.69514179981764,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "E_abs",
          "time": 0.0,
          "expected": 0.0,
          "tolerance": {
            "abs": 1e-12
          }
        },
        {
          "variable": "α",
          "time": 1.0,
          "expected": 0.28559933214452665,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "ω",
          "time": 1.0,
          "expected": 1.0,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "absorbing_particle",
      "description": "Absorbing particle: k_refr=0.1 with n_refr=1.5, D_p=5e-8 m, λ=5.5e-7 m. Q_abs = 4α × Im[(m²-1)/(m²+2)] is now nonzero; ω = Q_scat/Q_ext << 1, showing absorption dominates over scattering in the Rayleigh regime for k > 0 (Q_abs ∝ α but Q_scat ∝ α⁴).",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "k_refr": 0.1
      },
      "assertions": [
        {
          "variable": "m2_factor_imag",
          "time": 0.0,
          "expected": 0.0498129247935531,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "m2_factor_mag_sq",
          "time": 0.0,
          "expected": 0.09008390710443002,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Q_scat",
          "time": 0.0,
          "expected": 0.0015982499388173016,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Q_abs",
          "time": 0.0,
          "expected": 0.056906152212817196,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Q_ext",
          "time": 0.0,
          "expected": 0.0585044021516345,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "ω",
          "time": 0.0,
          "expected": 0.02731845604839925,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "E_ext",
          "time": 0.0,
          "expected": 1170.0880430326902,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "E_abs",
          "time": 0.0,
          "expected": 1138.123044256344,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "size_parameter_alpha_fourth_scaling",
      "description": "Double D_p from 5e-8 to 1e-7 m (n=1.5, k=0, λ=5.5e-7). α doubles → Q_scat scales as α⁴ = 2⁴ = 16×. Verifies: Q_scat(1e-7) = 16 × Q_scat(5e-8) = 16 × 0.001534757 ≈ 0.024556.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "D_p": 1e-07
      },
      "assertions": [
        {
          "variable": "α",
          "time": 0.0,
          "expected": 0.5711986642890533,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Q_scat",
          "time": 0.0,
          "expected": 0.024556113439854112,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Q_abs",
          "time": 0.0,
          "expected": 0.0,
          "tolerance": {
            "abs": 1e-15
          }
        },
        {
          "variable": "ω",
          "time": 0.0,
          "expected": 1.0,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "E_ext",
          "time": 0.0,
          "expected": 245.56113439854113,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "high_refractive_index",
      "description": "Higher refractive index n_refr=2.0, k_refr=0. m2_factor_real = (n²-1)/(n²+2) = 3/6 = 0.5 exactly. m2_factor_mag_sq = 0.25. Q_scat is enhanced relative to n=1.5 by the factor (0.25)/(25/289) ≈ 2.89.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "n_refr": 2.0
      },
      "assertions": [
        {
          "variable": "m2_re",
          "time": 0.0,
          "expected": 4.0,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "denom",
          "time": 0.0,
          "expected": 36.0,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "m2_factor_real",
          "time": 0.0,
          "expected": 0.5,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "m2_factor_mag_sq",
          "time": 0.0,
          "expected": 0.25,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Q_scat",
          "time": 0.0,
          "expected": 0.004435447990073648,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "E_ext",
          "time": 0.0,
          "expected": 88.70895980147297,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "blue_light_wavelength_minus4",
      "description": "Blue light λ=4e-7 m (400 nm) vs default 5.5e-7 m (550 nm), n=1.5, k=0. Q_scat ∝ λ⁻⁴: ratio (5.5/4.0)⁴ = 1.375⁴ ≈ 3.5745. Expected Q_scat ≈ 3.5745 × 0.001534757 ≈ 0.005486. Verifies the λ⁻⁴ power law that explains the blue sky.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "λ": 4e-07
      },
      "assertions": [
        {
          "variable": "α",
          "time": 0.0,
          "expected": 0.3926990816987242,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Q_scat",
          "time": 0.0,
          "expected": 0.005485932264296023,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Q_abs",
          "time": 0.0,
          "expected": 0.0,
          "tolerance": {
            "abs": 1e-15
          }
        },
        {
          "variable": "ω",
          "time": 0.0,
          "expected": 1.0,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "E_ext",
          "time": 0.0,
          "expected": 109.71864528592049,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    }
  ],
  "analyses": [
    {
      "id": "wavelength_dependence_non_absorbing",
      "description": "Rayleigh scattering efficiency Q_scat vs. wavelength λ for a non-absorbing particle (n_refr=1.5, k_refr=0, D_p=5e-8 m). Q_scat ∝ λ⁻⁴: blue light (λ≈400 nm) scatters ~(550/400)⁴ ≈ 3.6× more strongly than green (550 nm), and ~(550/700)⁴ ≈ 2.5× more than red (700 nm). This is the physical origin of the blue sky — Rayleigh scattering of sunlight by air molecules and small aerosols preferentially redirects blue photons.",
      "initial_state": {},
      "parameters": {
        "n_refr": 1.5,
        "k_refr": 0.0,
        "D_p": 5e-08,
        "ρ_p": 1500.0
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "λ",
            "range": {
              "start": 3e-07,
              "stop": 8e-07,
              "count": 51,
              "scale": "linear"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "Q_scat_vs_wavelength",
          "type": "line",
          "description": "Rayleigh scattering efficiency Q_scat vs. wavelength λ. Demonstrates the λ⁻⁴ power law (Seinfeld & Pandis 2006, Eq. 15.19): short (blue) wavelengths are scattered far more strongly than long (red) wavelengths.",
          "x": {
            "variable": "λ",
            "label": "Wavelength λ (m)"
          },
          "y": {
            "variable": "Q_scat",
            "label": "Rayleigh scattering efficiency Q_scat (dimensionless)"
          }
        },
        {
          "id": "E_scat_vs_wavelength",
          "type": "line",
          "description": "Mass scattering efficiency E_scat vs. wavelength λ. Like Q_scat, scales as λ⁻⁴.",
          "x": {
            "variable": "λ",
            "label": "Wavelength λ (m)"
          },
          "y": {
            "variable": "E_scat",
            "label": "Mass scattering efficiency E_scat (m² kg⁻¹)"
          }
        }
      ]
    },
    {
      "id": "absorbing_particle_omega_vs_k",
      "description": "Single-scattering albedo ω = Q_scat/Q_ext vs. imaginary refractive index k_refr for a Rayleigh particle (n_refr=1.5, D_p=5e-8 m, λ=5.5e-7 m). In the Rayleigh regime Q_scat ∝ α⁴ while Q_abs ∝ α, so even a small k drives ω rapidly toward zero: absorption dominates. At k=0.1, ω ≈ 0.027.",
      "initial_state": {},
      "parameters": {
        "n_refr": 1.5,
        "D_p": 5e-08,
        "λ": 5.5e-07,
        "ρ_p": 1500.0
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "k_refr",
            "range": {
              "start": 0.0,
              "stop": 0.5,
              "count": 51,
              "scale": "linear"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "omega_vs_k_refr",
          "type": "line",
          "description": "Single-scattering albedo ω vs. imaginary refractive index k_refr. Shows how rapidly absorption dominates scattering in the Rayleigh limit as k increases.",
          "x": {
            "variable": "k_refr",
            "label": "Imaginary refractive index k_refr (dimensionless)"
          },
          "y": {
            "variable": "ω",
            "label": "Single-scattering albedo ω (dimensionless)"
          }
        }
      ]
    }
  ]
}