Answer:
When light passes through the atmosphere, it undergoes refraction: the ray of light is deviated and changes speed.
The amount of the deviation is given by Snell's Law:
[tex]n_1 sin \theta_1 = n_2 sin \theta_2[/tex] (1)
where
[tex]n_1,n_2[/tex] are the index of refraction of the first (vacuum, in the space) and second (air, in the atmosphere) medium
[tex]\theta_1,\theta_2[/tex] are the angle that the incident and refracted ray of light make with the normal to the interface between the two mediums
The index of refraction is related to the wavelength of the light, according to:
[tex]n=\frac{\lambda_0}{\lambda}[/tex]
where
[tex]\lambda_0[/tex] is the wavelength in a vacuum
[tex]\lambda[/tex] is the wavelength in the medium
This means that the refractive index is higher for a shorter wavelength (blue light), while it is smaller for a longer wavelength (red light).
From eq.(1),
[tex]sin \theta_2 = \frac{n_1}{n_2}sin \theta_1[/tex]
However, the first medium is vacuum, so its index of refraction is 1. So we can write:
[tex]sin \theta_2 = \frac{sin \theta_1}{n_2}[/tex]
We said that the index of refraction in air is smaller for red light: since [tex]n_2[/tex] is a number slightly larger than 1, this means that for red light [tex]n_2[/tex] is closer to 1, and so the difference between [tex]\theta_1[/tex] and [tex]\theta_2[/tex] is smaller compared to other colors: and this means that red light is deviated less by the Earth's atmosphere, compared to other colors.
