Suppose a luminous sphere of radius R1 (such as the Sun) is...

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 (a) when $R_{2}>n R_{1}$ and (b) when $R_{2}<n R_{1} ?$

Key Concepts

Optical Ray Tracing in Spherical Media

Optical ray tracing in spherical media involves analyzing the trajectories of light rays passing through layers with spherical symmetry, such as an atmosphere surrounding a luminous sphere. This approach is fundamental for determining how light is bent at various interfaces and for predicting the resulting apparent size and shape of the object when observed from a distance.

Apparent Angular Size

The apparent angular size of an object refers to how large the object appears from a given viewing distance. In optical systems, factors such as refraction in intervening media can alter the trajectories of light rays, effectively changing the perceived dimensions of the source. This concept is key to understanding how the physical boundaries of an object may appear enlarged or reduced when viewed through a refracting layer.

Refraction

Refraction is the phenomenon in which light changes direction when it passes from one medium to another due to a change in its propagation speed. This bending, governed by Snell’s law, is central in understanding how light rays emerging from an object are deflected, which in turn affects the observed or apparent properties of that object.

Snell’s Law

Snell’s Law provides the quantitative relationship between the angles of incidence and refraction when light encounters an interface between two media with different refractive indices. This law is essential for predicting the new path that light rays take after crossing such boundaries, and it plays a critical role in analyzing how an extended atmosphere alters the light emerging from an object.

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