Refer to Quick Quiz 35.4 . By how much does the duration of...

Refer to Quick Quiz $35.4 .$ By how much does the duration of an optical day exceed that of a geometric day? Model the Earth's atmosphere as uniform, with index of refraction $1.000293,$ a sharply defined upper surface, and depth $8614 \mathrm{m} .$ Assume that the observer is at the Earth's equator, so that the apparent path of the rising and setting Sun is perpendicular to the horizon.

Key Concepts

Optical Day vs. Geometric Day

This concept differentiates between the two ways of defining the day. The geometric day is determined solely by the actual rotation of the Earth with respect to distant stars, while the optical day accounts for the apparent motion of the sun, which is affected by atmospheric refraction. The optical day is longer because the bending of light causes the sun to appear above the horizon even when it is geometrically below it.

Atmospheric Refraction

Atmospheric refraction is the bending of light as it passes through the Earth’s atmosphere, which has a varying density and index of refraction. This phenomenon causes the apparent position of celestial bodies, such as the sun, to shift from their true geometric positions. The effect is most noticeable during sunrise and sunset, contributing to the difference between optical and geometric days.

Uniform Atmosphere Model

Modeling the Earth’s atmosphere as uniform with a specific index of refraction simplifies the analysis of how light is bent when passing through it. In this model, the atmosphere is assumed to have a constant refractive index and a defined boundary, which makes it possible to calculate the precise amount of refraction and its influence on the duration of the optical day.

Solar Disk Geometry and Earth's Curvature

This concept involves understanding how the size of the sun’s disk and the curvature of the Earth interact with atmospheric refraction. As the Earth rotates, the apparent sunrise and sunset times are influenced by these factors—the sun’s disk is observed to be extended in time beyond its geometric rising and setting due to both the curvature of the Earth and the bending of light in the atmosphere.

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