Editing Openai/680a25f8-9f18-8004-a42d-cd36374ba000 (section)

=== User: Expected Behavior of an Ice Crystal at 10 km Altitude in Direct Sunlight ===
Expected Behavior of an Ice Crystal at 10 km Altitude in Direct Sunlight

Setup: 20 µm spherical ice crystal at 10 km altitude (~250 hPa, -50°C = 223 K). Radius: 10⁻⁵ m. Cross-sectional area: 3.14×10⁻¹⁰ m². 
Surface area: 1.26×10⁻⁹ m². Volume: 4.19×10⁻¹⁵ m³. Mass: 3.84×10⁻¹² kg (ρ_ice = 917 kg/m³).

Equilibrium Without Sunlight: Absorbs 544 W/m² × 3.14×10⁻¹⁰ m² = 1.71×10⁻⁷ W (longwave). Emits 6.93×10⁻¹⁷ T⁴ W; at 223 K: 1.71×10⁻⁷ W, balancing absorption. Vapor pressure at 223 K: 4.02 Pa. Air at RHi 100% = 4.02 Pa; below 100%, sublimation; above, deposition.

Sunlight Introduced (400 W/m² Absorbed): Incident solar flux at 10 km ≈ 1300 W/m² (~40% near-IR, 520 W/m²). The crystal absorbs 400 W/m² (broadband α ≈ 0.308), predominantly in the near-IR (~335 W/m², α_IR ≈ 0.644), with minimal visible absorption (~65 W/m², α_vis ≈ 0.1). Near-IR absorption aligns with literature (e.g., Warren, 1984; Q_abs ~ 0.5–0.9 for 20 µm crystals at 1.5–2.5 µm). Total absorbed (solar + atmospheric): 2.97×10⁻⁷ W. Emission increases as T rises: initially 1.71×10⁻⁷ W at 223 K, rising to 2.97×10⁻⁷ W at 243 K (-30°C, vapor pressure 15 Pa) in ~1.3 s, balancing absorption. Sublimation starts immediately as T rises, accelerating as vapor pressure increases from 4.02 Pa to 15 Pa.

Sublimation Threshold in Sunlight: Air at -50°C (4.02 Pa) needs RHi = (15 / 4.02) × 100 = 373% to match 15 Pa and prevent sublimation. Below 373%, sublimation occurs (vs. 100% without sunlight).

Sublimation Rate (Coupled Process): dm/dt = (4πr D M / RT) × ΔP. D = 2.1×10⁻⁵ m²/s, M = 0.018 kg/mol, R = 8.314 J/mol·K, T ≈ 243 K, r = 10⁻⁵ m (approx. constant). Coefficient: 5.05×10⁻¹¹ kg/s·Pa. dm/dt = k m^(1/3), integrate: m^(2/3) = -kt + C.

Total Sublimation Times (RHi 100%–200%, Heating + Sublimation Overlap):
RHi 100% (4.02 Pa): ΔP = 10.98 Pa, k = 5.54×10⁻¹⁰ kg^(2/3)/s, total ~1.05 s.
RHi 110% (4.422 Pa): ΔP = 10.58 Pa, k = 5.34×10⁻¹⁰, total ~1.06 s.
RHi 120% (4.824 Pa): ΔP = 10.18 Pa, k = 5.14×10⁻¹⁰, total ~1.07 s.
RHi 130% (5.226 Pa): ΔP = 9.77 Pa, k = 4.93×10⁻¹⁰, total ~1.08 s.
RHi 140% (5.628 Pa): ΔP = 9.37 Pa, k = 4.73×10⁻¹⁰, total ~1.09 s.
RHi 150% (6.03 Pa): ΔP = 8.97 Pa, k = 4.53×10⁻¹⁰, total ~1.10 s.
RHi 160% (6.432 Pa): ΔP = 8.57 Pa, k = 4.33×10⁻¹⁰, total ~1.11 s.
RHi 170% (6.834 Pa): ΔP = 8.17 Pa, k = 4.13×10⁻¹⁰, total ~1.13 s.
RHi 180% (7.236 Pa): ΔP = 7.76 Pa, k = 3.92×10⁻¹⁰, total ~1.14 s.
RHi 190% (7.638 Pa): ΔP = 7.36 Pa, k = 3.72×10⁻¹⁰, total ~1.16 s.
RHi 200% (8.04 Pa): ΔP = 6.96 Pa, k = 3.51×10⁻¹⁰, total ~1.18 s.

Assumptions: T and D constant (ΔT <10%, D variation <5%). r fixed (mass loss alters it slightly, <5% effect). No shading or external stabilization. Air heating negligible (~0.005 K/s).

Key Insight: Sunlight, primarily via near-IR absorption, raises the ice crystal’s vapor pressure from 4.02 Pa to 15 Pa, shifting the sublimation threshold from RHi 100% (without sunlight) to 373%. A 20 µm crystal sublimates in ~1.05–1.18 s across RHi 100%–200%, so contrails should not persist in sunlight unless RHi is unrealistically high or external factors (e.g., nano-stabilization) are at play.

Can you check that's all watertight please?