Suppose a luminous sphere of radius $R_{1}$ (such as the Sun) is surrounded by a uniform atmosphere of radius $R_{2} > R_{1}$ and index of refraction $n .$ When the sphere is viewed from a location far away in vacuum, what is its apparent radius when $R_{2} > n R_{1}$

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Solution

Assume the viewer is far away to the right. In ANS. FIG. P35.86(a), a ray directed toward the viewer comes tangentially from the edge of the glowing sphere and emerges from the atmosphere at angle $θ_{2} .$ The apparent radius of the glowing sphere is $R_{3}$ as shown. For the figure, we see that
$sin θ_{1} = \frac{R_{1}}{R_{2}} and sin θ_{2} = \frac{R_{3}}{R_{2}}$
Then, $n sin θ_{1} = 1.00 sin θ_{2}$
and $n \frac{R_{1}}{R_{2}} = \frac{R_{3}}{R_{2}} \to R_{3} = n R_{1}$

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Suppose a luminous sphere of radius R1 (such as the Sun) is surrounded by a uniform atmosphere of radius R2>R1 and index of refraction n. When the sphere is viewed from a location far away in vacuum, what is its apparent radius when R2>nR1

Suppose a luminous sphere of radius $R_{1}$ (such as the Sun) is surrounded by a uniform atmosphere of radius $R_{2} > R_{1}$ and index of refraction $n .$ When the sphere is viewed from a location far away in vacuum, what is its apparent radius when $R_{2} > n R_{1}$