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594.


Problem 43.20 (RHK)


The index of refraction of the Earth’s atmosphere decreases monotonically with height from its surface value (about 1.00029) to the value in space (about 1.000000 at the top of the atmosphere. This continuous (or graded) variation can be approximated by considering the atmosphere to be composed of three (or more) plane parallel layers in each of which the index of refraction is constant. Thus, in the figure, . We will consider a ray of light from a star S that strikes the top of the atmosphere at an angle with the vertical. (a) We have to show that the apparent direction of the star with the vertical as seen by an observer at the Earth’s surface is obtained from

(b) We have to calculate the shift in position of a star observed to be from the vertical.


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(a)

We will use Snell’s law successively for answering this problem.

The angle of refraction in the layer with refractive index of the ray from the star incident on the top of the atmosphere at incident angle is given by the Snell’s law

.

The angle of refraction of the ray incident on top of the layer with refractive index at angle is given by the Snell’s law

.

The angle of refraction of the ray incident on top of the layer with refractive index at angle is given by the Snell’s law

.

Combining these relations, we note that

(b)

We have to calculate the shift in position of a star observed to be from the vertical.

For

,

and

,