Why the sky is blue

Short answer: Sunlight contains all colors, and molecules in the air scatter shorter wavelengths (blue) much more strongly than longer wavelengths (red). That scattered blue light reaches our eyes from all directions, so the sky looks blue.

Step-by-step explanation

1. Sunlight is white light made of many wavelengths.

   Sunlight contains a continuous mix of wavelengths from violet through red. You can separate them with a prism into colors, but in free space they are all mixed together.

2. Air molecules scatter sunlight.

   Particles much smaller than a wavelength of light (mainly air molecules like N2 and O2) scatter incoming sunlight. This scattering is elastic: the light changes direction but not wavelength.

3. Shorter wavelengths are scattered much more strongly (Rayleigh scattering).

   The intensity of scattering by small particles follows Rayleigh's law: roughly I ∝ 1/λ^4, where λ is wavelength. That means blue light (shorter λ) is scattered many times more than red light (longer λ).

4. Scattered blue light fills the sky.

   Because blue is scattered in all directions more than other colors, an observer looking away from the Sun sees predominantly blue light coming from the air, so the sky appears blue.

5. Why not violet, which is even shorter?

   Several reasons: the Sun emits less violet than blue; our eyes (rods and cones) are less sensitive to violet than to blue; and some violet is absorbed higher in the atmosphere. The net result is that the scattered light we perceive is seen as blue rather than violet.

6. Why the sky near the horizon is paler or whitish.

   When you look toward the horizon, sunlight travels a much longer path through the atmosphere than when you look overhead. More scattering and also scattering by larger particles (dust, water droplets, aerosols) occur. Larger particles produce Mie scattering, which is less wavelength-selective and tends to scatter all colors similarly, making the sky near the horizon look paler, whitish, or washed out.

7. Why sunsets and sunrises are red.

   At sunrise and sunset the Sun's light passes through a long layer of atmosphere, so most of the short-wavelength (blue) light is scattered out before the direct beam reaches your eyes. The remaining direct light is dominated by longer wavelengths (reds and oranges), so the Sun and the near-sky look red.

8. Polarization is evidence for Rayleigh scattering.

   Scattered light from molecules is partially polarized. If you wear polarizing sunglasses and look at the sky at a right angle from the Sun, the brightness can change significantly—this is a practical demonstration that the sky light is scattered and polarized.

Quick recap

Air molecules scatter blue light much more than red because scattering strength goes roughly as 1/λ^4. That scattered blue light reaches our eyes from all directions, so the daytime sky looks blue. Near the horizon and at sunrise/sunset, longer paths through the atmosphere and the presence of larger particles change the balance and produce paler skies or red/orange sunsets.

Further curiosity: If you want a simple experiment, look at a glass of milk diluted with water and shine white light through it. At certain concentrations the transmitted light looks redder while the scattered light looks blue — a small-scale demonstration of the same physics.