On the Moon's surface, lunar astronauts placed a comer...

On the Moon's surface, lunar astronauts placed a comer reflector, off which a laser beam is periodically reflected. The distance to the Moon is calculated from the round-trip time. What percent correction is needed to account for the delay in time due to the slowing of light in Earth's atmosphere? Assume the distance to the Moon is precisely $3.84 \times 10^{8} \mathrm{m}$ and Earth's atmosphere (which varies in density with altitude) is equivalent to a layer $n=1.000293$.

Key Concepts

Round-trip Time Measurement

When determining distances by timing how long it takes for a light pulse to travel to an object and back, the calculation typically assumes a constant speed c in a vacuum. Any deviation from this speed along parts of the path—such as in the atmosphere—affects the measured time and needs to be corrected for accurate distance determination.

Atmospheric Delay Correction

The delay induced by the Earth’s atmosphere results from the small reduction in the speed of light due to the refractive index. Since the refractive index is 1.000293, the light slows down by about 0.0293% in the atmosphere. This percentage represents the correction needed to account for the extra time delay when calculating distances using the round?trip time of the light pulse.

Refractive Index

The refractive index is a property of a medium that indicates by what factor the speed of light is reduced relative to its speed in a vacuum. In this context, the index of 1.000293 means that light travels slightly slower through Earth’s atmosphere compared to vacuum.

Speed of Light in a Medium

The speed of light in any medium is given by dividing the speed of light in a vacuum by the medium’s refractive index. This reduction in speed results in a slight delay in the time it takes for light to travel through that medium compared to an identical distance in a vacuum.

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