In 1963, astronaut Gordon Cooper orbited the Earth 22 times....

In $1963,$ astronaut Gordon Cooper orbited the Earth 22 times. The press stated that for each orbit, he aged two-millionths of a second less than he would have had he remained on the Earth. (a) Assuming Cooper was $160 \mathrm{km}$ above the Earth in a circular orbit, determine the difference in elapsed time between someone on the Earth and the orbiting astronaut for the 22 orbits. You may use the approximation $$\frac{1}{\sqrt{1-x}} \approx 1+\frac{x}{2}$$ for small $x$. (b) Did the press report accurate information? Explain.

Key Concepts

Special Relativity and Velocity Time Dilation

Time dilation due to relative motion is a key result of special relativity. For an observer moving at speed v relative to an inertial frame, time runs slower by a factor given by ? = 1/?(1 - v²/c²). In many practical cases, when v is much smaller than the speed of light, approximations like 1/?(1-x) ? 1 + x/2 (with x = v²/c²) are used to simplify calculations. This concept is essential when analyzing how the moving astronaut’s clock ticks slightly differently compared to a stationary clock on Earth.

Gravitational Time Dilation

General relativity predicts that clocks in a gravitational field run at different rates depending on their position; specifically, clocks deeper in a gravitational field tick more slowly. Although the astronaut in orbit is subject primarily to velocity-induced time dilation, his position in a weaker gravitational potential compared to an Earthbound observer can also affect elapsed time. Understanding the interplay between gravitational time dilation and velocity time dilation is crucial for analyzing high-precision time differences in orbits.

Orbital Mechanics in Circular Orbits

Orbital mechanics describes the motion of objects under the influence of gravity, and a circular orbit is a common idealization where an object maintains a constant altitude. For an astronaut orbiting at a fixed altitude, the orbital speed can be determined via the balance between centripetal force and gravitational pull. This orbital speed is a necessary component in calculating the relativistic time dilation effects experienced by the moving astronaut.

Approximation Techniques in Relativistic Calculations

When dealing with relativistic equations, the quantities involved are often so small that applying series expansion approximations can simplify the calculations without significantly impacting accuracy. In the context of time dilation, approximations like 1/?(1-x) ? 1 + x/2 are instrumental in reducing complex expressions into more manageable forms. This technique is particularly useful in estimating small differences in elapsed time, such as those experienced by orbiting astronauts.

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