Umbra Radius (Eq. 3)
ru(x) = R⊕(1 − x/Lu)
Radius of the umbra (complete shadow) at distance x from Earth.
The umbra converges to an apex.
Penumbra Radius (Eq. 4)
rp(x) = R⊕ + x(R☉ + R⊕)/d☉
Radius of the penumbra (partial shadow) at distance x.
The penumbra diverges outward.
Lunar Position (Eq. 5)
rMoon(t) = a[cos(ωt), sin(ωt)sin(i), sin(ωt)cos(i)]
Moon's orbital position with inclination i = 5.14°.
Causes Moon to pass through shadow vertically.
Perpendicular Distance (Eq. 8)
d⊥ = √(|rrel|² − λ²)
Distance from Moon to shadow axis. Used to determine eclipse phase:
• Total: d⊥ < ru
• Partial: ru < d⊥ < rp
🔴 Rayleigh Scattering
σ(λ) ∝ 1/λ⁴
Wavelength dependence of atmospheric scattering.
Blue light (λ = 450nm) scatters 4.2× more than red light (λ = 650nm).
Result: During totality, Earth's atmosphere filters out blue,
allowing only red wavelengths to refract onto the Moon → Blood Moon 🌕
Blood Moon Color Model (Eq. 9)
RGB = (0.85, 0.08, 0.03) during totality
Red channel dominant (85%),
blue suppressed (3%).
Ratio: R:B = 28:1
This creates the characteristic copper-red glow.
Umbral Apex Distance (Eq. 2)
Lu = R⊕d☉ / (R☉ − R⊕)
Distance from Earth to the umbral apex (where umbra converges to a point).
In simulation: Lu = 350 units