Model scattering strength using wavelength and angle. Add particle size and refractive indices for realism. Export results as CSV or PDF for lab notes.
| Case | λ (nm) | a (nm) | nₚ | nₘ | θ (deg) | r (m) | I₀ (W/m²) | Scaled I_s (W/m²) |
|---|---|---|---|---|---|---|---|---|
| A | 450 | 50 | 1.50 | 1.00 | 90 | 1.0 | 1200 | 3.08236e-14 |
| B | 550 | 50 | 1.50 | 1.00 | 90 | 1.0 | 1200 | 1.38128e-14 |
| C | 650 | 50 | 1.50 | 1.00 | 90 | 1.0 | 1200 | 7.08075e-15 |
This calculator provides two practical levels:
The full model is a far-field, single-particle Rayleigh estimate. It is useful for comparisons and sensitivity studies, not absolute sky radiance without additional atmospheric parameters.
Rayleigh scattering dominates when particles are much smaller than the light wavelength. In Earth’s atmosphere, molecules and very fine aerosols fall in this regime, which is why the sky looks blue and sunsets look redder. The same scaling is used in optics labs for clean, small-particle systems.
The strongest signature is the λ⁻⁴ dependence. Using only that scaling, 450 nm blue light scatters about 4.35× more than 650 nm red light, so short wavelengths are preferentially redirected toward the observer. This is also why UV light can be strongly attenuated in clear air compared with red and near‑IR wavelengths.
In the full model, intensity scales as a⁶. Doubling particle radius increases scattering by 64× (2⁶). That steep dependence means small changes in particle size can strongly affect haze brightness and laboratory measurements.
Material contrast enters through |(m²−1)/(m²+2)|², where m=nₚ/nₘ. For air, nₘ≈1.00; common dielectric particles often use nₚ≈1.40–1.55. Larger contrast increases scattering efficiency even at fixed size. If m approaches 1, scattering drops fast, which is why index‑matching fluids can visually “hide” particles.
The angular term (1+cos²θ)/2 produces a symmetric pattern about 90°. Scattering is stronger near 0° and 180° than at 90°, and real skies also show polarization that is most noticeable at right angles to the Sun.
The far-field approximation includes a 1/r² factor. If you double the observation distance, intensity drops to one quarter. In experiments, r represents the effective detector distance in your geometry. Keep r consistent when comparing scenarios.
Visible wavelengths span roughly 380–740 nm. To stay in the Rayleigh limit, choose a<λ/10 (for 550 nm, that is a < 55 nm). Keep units consistent: the calculator converts nm to meters internally. For incident intensity examples, 1000–1200 W/m² is a reasonable noon‑sun reference.
The “Full model” output is a scaled single-particle estimate intended for comparisons, sensitivity studies, and teaching. For bulk media, multiply by particle number density and include path length and multiple-scattering effects. Use “Relative scaling” when you only need dependable color ratios quickly. If you change one input at a time, the resulting ratios are easier to interpret and report.
It is elastic scattering by particles much smaller than the wavelength. It produces the classic λ⁻⁴ dependence, making shorter wavelengths scatter far more strongly than longer wavelengths.
Relative mode isolates wavelength scaling for fast color ratios. Full mode adds size, refractive index, distance, and angle to study how geometry and material contrast change intensity.
Use it for sensitivity checks, lab setups, and “what-if” comparisons where size, refractive index, angle, or distance matters. It is most useful when a is well below λ.
The full equation represents a single-particle far-field estimate. Real scenes depend on particle concentration, path length, detector optics, and multiple scattering, so absolute radiance needs additional atmospheric or experimental parameters.
A common rule is a<λ/10. For 500–600 nm light, that means radii below roughly 50–60 nm. Larger particles transition toward Mie scattering, which has different wavelength behavior.
At low Sun angles, light travels through a longer atmospheric path. Blue wavelengths scatter out of the direct beam more efficiently, leaving a redder spectrum that reaches your eyes.
Use Relative mode with λ_ref and λ values, or rely on the built-in reference: 450 nm versus 650 nm. The λ⁻⁴ rule predicts blue scatters about 4.35× more than red, all else equal.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.