Chapter 6.1, Problem 31PE

To approximate the distance from the Earth to stars relatively close by. astronomers often use the method of parallax. Parallax is the apparent displacement of an object caused by a change in the observer's point of view. As the Earth orbits the Sun, a nearby star will appear to move against the more distant background stars. Astronomers measure a star's position at times exactly $6$ months apart when the Earth is at opposite points in its orbit around the Sun. The Sun, Earth, and star form the vertices of a right triangle with $\angle PSE=90°$ . The length of is the distance between the Earth and Sun. approximately $92,900,000\text{ mi}$ . The parallax angle (or simply parallax) is denoted by $p$ . Use this information for Exercises 31-32.

a. Find the distance between the Earth and Proxima Centauri (the closest star to the Earth beyond the Sun) if the parallax angle is $0.772"$ (arcseconds). Round to the nearest hundred billion miles,

b. Write the distance in part (a) in light-years. Round to $1$ decimal place. (Hint. $1$ light-year is the distance that light travels in $1\text{ yr}$ and is approximately $5.878\times {10}^{12}\text{ mi}$ .)