Zenith sky brightness (V-band / SQM, mag/arcsec²) — how residual summer twilight compares with your site's artificial skyglow.
Twilight curve. Zenith V/SQM brightness vs Sun altitude a: for 0° ≥ a ≥ −12°, m = 6.7489 − 1.057·a; for −12° ≥ a ≥ −17.3°, m = −0.0744a² − 2.5768a − 0.5845 (blended near −12°, capped at the 21.73 vertex). Empirical fit by Han Kleijn (hnsky.org) to SQM measurements cross-calibrated with ESO-Paranal V-band twilight photometry. Agrees with the Patat, Ugolnikov & Postylyakov (2006, A&A 455, 385) polynomial to ≲0.1 mag over −12°…−15° (e.g. both give ≈21.3 at −15°).
Combining components. All quantities are converted to linear flux f = 10−0.4m. Twilight contribution = f(mtwi) − f(21.75); LP contribution = f(SQMsite) − f(natural), clamped ≥ 0. Combined sky = natural + LP + twilight, converted back to mag/arcsec². The crossover Sun altitude solves twilight contribution = LP contribution; since both share the same natural baseline, this is simply where the twilight-only sky equals your site's SQM.
Bortle ↔ SQM (approx., Bortle 2001 / Wikipedia): 1: 21.76–22.0 · 2: 21.6–21.75 · 3: 21.3–21.6 · 4: 20.8–21.3 · 4.5: 20.3–20.8 · 5: 19.25–20.3 · 6: 18.5–19.25 · 7: 18.0–18.5 · 8–9: <18.
Sun position: NOAA solar algorithm (declination + equation of time), accurate to ~0.1°. No refraction term (negligible for depression angles).
Moonlight: Krisciunas & Schaefer (1991, PASP 103, 1033) model evaluated at the zenith: Moon magnitude m = −12.73 + 0.026|α| + 4×10⁻⁹α⁴ (α = phase angle), scattering f(ρ) = 105.36(1.06 + cos²ρ) + 106.15−ρ/40 with ρ = Moon zenith distance, optical path X(Z) = (1 − 0.96 sin²Z)−1/2, V-band extinction k = 0.20 mag/airmass; nanolambert result converted via B = 34.08 e20.7233−0.92104V. Checks: full Moon at 60° alt → zenith ≈ 17.9, quarter Moon ≈ 20.4 mag/arcsec². Model accuracy is 8–23% (K&S); contribution is zeroed once the Moon is ≳1° below the horizon. Moon position from a truncated Schlyter/Meeus ephemeris (~0.2°, plenty for brightness). Moonlight adds to the flux sum like the other components.
Assumptions / caveats. Zenith values only — twilight is substantially brighter toward the sunward (northern, in summer) horizon, and moonlight toward the Moon's sky quadrant, so targets near those directions fare worse. Clear, aerosol-typical sky assumed; night-to-night scatter is ±0.2–0.4 mag (Patat 2006). Deep-twilight brightness scales with atmospheric pressure: sea-level sites ≈ 0.27 mag brighter than 2600 m Paranal — use the twilight offset. SQM (broad, ~V+B) vs Johnson V differences are within these tolerances.