Chapter 19, Problem 1E

Explain how parallax measurements can be used to determine distances to stars. Why can we not make accurate measurements of parallax beyond a certain distance?

To determine

Toexplain:

The way the distances to the stars can be determined by measuring the parallax angle.

The reason that the accurate measurements of parallax cannot be done beyond a certain distance.

Explanation of Solution

Introduction:

The changes or shifts in the position of a star, when it is being viewed from two dissimilar locations, is known as parallax. It is used to measure distances of the stars that are within a particular distance.

Distance of star is inversely proportional to the parallax.

i.e. $\text{D=}\frac{\text{1}}{\text{p}}$

A triangle is formed between groundline, star and position at two different times,from two positions of Earth from the groundline. The ultimate angular distance between two positions gives an accurate parallax angle.

Thus, the total distance to a star is inverse in relation like above.

There is a procedure to find the distance to a star by using parallax in which $\text{D}$ is Distance to a star, $\text{P}$ is the parallax angle as shown in the diagram below:

According to the figure,

$\begin{array}{l}\text{tan p = }\frac{\text{1}}{\text{d}}\text{,}\\ \text{tanp}\approx \text{p}\\ \text{so, p=}\frac{\text{1}}{\text{D}}\end{array}$ ,

Rearrange for D,

$\text{D=}\frac{\text{1}}{\text{p}}$

Conclusion:

As the distance between the star and the earth surface increases, there is a decrease inthe parallax angle as a result of which it is quite difficult to measure small angles.