How to calculate the Stellar distance

For the star in Figure 1, the parallax angle – P is half the distance moved by the star between photos.

parallax angle – P
The angle formed by the Earth’s radius as measured from a star. Half the apparent angle moved by the star against the background stars when seen from opposite sides of the Earth’s orbit.

Therefore
P = 0.5 / 2 = 0.25 seconds of arc. (1 second of arc (1″) = 1 / 3600) degrees

Figure 1

From Figure 2, the distance between the Sun and the star is:
d = r / tan P
If P is 1 second of arc:
d = 150 000 000 / tan 1″ = 30 million million km
This distance is called one parsec and is a basic unit for measuring astronomical distances.
Distance in parsecs = 1 / P in seconds of arc

 

Figure 2

For the star in Figure 1:
d = 1 / P = 1 / 0.25 = 4
Therefore the star is four parsecs away.

If parallax is given to calculate distance d

d=1/P =1/0.723 = 1.38 parsecs

There is a simple relationship between a star’s distance and its parallax angle:

d = 1/p

The distance d is measured in parsecs and the parallax angle p is measured in arcseconds.

This simple relationship is why many astronomers prefer to measure distances in parsecs.

Limitations of Distance Measurement Using Stellar Parallax

Parallax angles of less than 0.01 arcsec are very difficult to measure from Earth because of the effects of the Earth’s atmosphere. This limits Earth based telescopes to measuring the distances to stars about 1/0.01 or 100 parsecs away. Space based telescopes can get accuracy to 0.001, which has increased the number of stars whose distance could be measured with this method. However, most stars even in our own galaxy are much further away than 1000 parsecs, since the Milky Way is about 30,000 parsecs across.

Parallax is the observed apparent change in the position of an object resulting from a change in the position of the observer. Specifically, in the case of astronomy it refers to the apparent displacement of a nearby star as seen from an observer on Earth.

The parallax of an object can be used to approximate the distance to an object using the formula:

D=1/P

Where p is the parallax angle observed, and D is the actual distance measured in parsecs. A parsec is defined as the distance at which an object has a parallax of 1 arcsecond. This distance is approximately 3.26 light years.

Example:

The bright star Vega has a parallax of 0.129″, how far is it?

Solution:

P = 0.129

D = 1/0.129 = 7.75 parsecs

So Vega is about 7.75 parsecs from Earth, or about 25 light years.