Question

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 observers 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 stars 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 ^ { \circ } \\). the length of side \\( \overline { s e } \\) is the distance between the earth and sun, approximately 92,900,000 mi. the parallax angle (or simply parallax) is denoted by \\( p \\).

(a) find the distance between the earth and vega if the parallax angle is \\( 0.130 ^ { \prime \prime } \\) (arcseconds). round your answer to the nearest hundred billion miles.
the distance between the earth and vega is approximately \\( \square \\) mi.

Explanation:

Step1: Convert the parallax angle from arc - seconds to radians

We know that \(1^{\circ}=\frac{\pi}{180}\) radians and \(1^{\circ} = 3600''\). So, if \(p = 0.130''\), then \(p=\frac{0.130}{3600}\times\frac{\pi}{180}\) radians.

$$p=\frac{0.130\pi}{3600\times180}\text{ radians}$$

Step2: Use the formula for distance in the parallax method

The formula for the distance \(d\) (in miles) from the Earth - Sun distance \(a = 92900000\) miles and parallax angle \(p\) (in radians) is \(d=\frac{a}{p}\).
Substitute \(a = 92900000\) and \(p=\frac{0.130\pi}{3600\times180}\) into the formula:

$$d=\frac{92900000}{\frac{0.130\pi}{3600\times180}}$$

$$d=\frac{92900000\times3600\times180}{0.130\pi}$$

$$d=\frac{92900000\times648000}{0.130\pi}$$

$$d=\frac{6.029\times10^{13}}{0.4084}$$

$$d\approx1.48\times 10^{13}\text{ miles}$$

Answer:

The distance between the Earth and Vega is approximately \(14800000000000\) mi.