Question

You observe two stars over the course of a year (or more) and find that both stars have measurable parallax. (1 arc second is 1/3600 a degree) Star X appears to move 1 arcsecond. Star Y has appears to move 1/2 an arcsecond.

Answer 1

Answer:

Star X is much closer since it is at a distance 1 parsec from the Earth.

Explanation:

The angle due to the change in position of a nearby object against the background stars it is known as parallax.

The parallax angle can be used to find out the distance by means of triangulation. Making a triangle between the nearby star, the Sun and the Earth. This angle is gotten when the position of the object is measured in January and then in July according to the configuration of the Earth with respect to the Sun in those months.

The distance between the Earth and the Sun is 150000000 Km. That distance is also known as an astronomical unit (1AU).

The parallax angle can be defined in the following way:

$\tan{p} = \frac{1AU}{d}$    

Where d is the distance to the star.

$p('') = \frac{1}{d}$    (1)  

Equation (1) can be rewritten in terms of d:

$d(pc) = \frac{1}{p('')}$ (2)

Equation (2) represents the distance in a unit known as parsec (pc).

Case of Star X (p('') = 1):

Using equation 2 the distance of star X can be known:

$d(pc) = \frac{1}{1}$

$d(pc) = 1 pc$

So, star X is at 1 parsec from Earth.

Case of Star Y ($p('') = \frac{1}{2}$):

$d(pc) = \frac{1}{(\frac{1}{2})}$

$d(pc) = 2 pc$

So, star Y is at 2 parsecs from the Earth.

Hence, star X is much closer.

Reminder:

Notice that in equation 2 the distance is inversely proportional to the parallax angle, so if the parallax angle decreases, the distance increases.