Chapter 8.1, Problem 59AYU

Parallax One method of measuring the distance from Earth to a star is the parallax method. The idea behind computing this distance is to measure the angle formed between the Earth and the star at two different points in time. Typically, the measurements are taken so that the side opposite the angle is as large as possible. Therefore, the optimal approach is to measure the angle when Earth is on opposite sides of the Sun, as shown in the figure.

(a) Proxima Centauri is 4.22 light-years from Earth. If 1 light-year is about 5.9 trillion miles, how many miles is Proxima Centauri from Earth?

(b) The mean distance from Earth to the Sun is 93,000,000 miles. What is the parallax of Proxima Centauri?

To determine

To find:

a. Proxima Centauri is $4.22$ light-years from Earth. If 1 light-year is about $\text{5}\text{.9}$ trillion miles, how many miles is Proxima Centauri from Earth?

Answer to Problem 59AYU

Distance from earth to Proxima centauri $=2.4898*{10}^{13}\text{ miles}$.

The parallax of proxima centuari is ${0.000214}^{\circ }$.

Explanation of Solution

Given:

One method of measuring the distance from Earth to a star is the parallax method. The idea behind computing this distance is to measure the angle formed between the Earth and the star at two different points in time. Typically, the measurements are taken so that the side opposite the angle is as large as possible. Therefore, the optimal approach is to measure the angle when Earth is on opposite sides of the Sun, as shown in the figure.

Formula used:

$\sin \theta \text{ = }\frac{\text{opposite side}}{\text{hypotenuse}}$

Calculation:

a. Proxima Centauri $=4.22$ light-years from Earth.

1 light-year $=\text{ 5}\text{.9}$ trillion miles $=5.9*{10}^{12}$,

Distance from earth to Proxima centauri $4.22*5.9\text{ trillion miles }=2.4898*{10}^{13}\text{ miles}$.

To determine

To find:

b. The mean distance from Earth to the Sun is $\text{93,000,000}$ miles. What is the parallax of Proxima Centauri?

Answer to Problem 59AYU

Distance from earth to Proxima centauri $=2.4898*{10}^{13}\text{ miles}$.

The parallax of proxima centuari is ${0.000214}^{\circ }$.

Explanation of Solution

Given:

One method of measuring the distance from Earth to a star is the parallax method. The idea behind computing this distance is to measure the angle formed between the Earth and the star at two different points in time. Typically, the measurements are taken so that the side opposite the angle is as large as possible. Therefore, the optimal approach is to measure the angle when Earth is on opposite sides of the Sun, as shown in the figure.

Formula used:

$\sin \theta \text{ = }\frac{\text{opposite side}}{\text{hypotenuse}}$

Calculation:

b. Let us construct a right triangle as shown below.

$a=9.3*{10}^7$, $c=2.4898*{10}^{13}$

$\sin \theta =\frac{a}{c}=9.3*{10}^7/2.4898*{10}^{13}$

$\sin \theta =3.74*{10}^{-6}$

$\theta ={0.000214}^{\circ }$

The parallax of proxima centuari is $\theta ={0.000214}^{\circ }$.