Calculating Astronomical Parallax

The angles involved are very small, typically less than 1 arc second. (Remember that 1 arc second = 1/3600 of a degree). To determine the distance to a star we can write (for small angles):

d = b / 2p = ½ b / p

where d is the distance to the star, p is the parallax angle expressed in radians (see diagram), and b is the baseline. In the case of heliocentric parallax, the base b is equal to 2 Astronomical Unit (AU) -- the diameter of the Earth's orbit.

Since there are 206,265 arc seconds per radian, the formula can be re-written as:

d (in AU) = 206,265 / p

with p now measured in arc seconds.

Or, if we write the distance of one parsec as 206,265 AU, we get:

d (in parsecs) = 1 / p (in arc seconds)

Thus one parsec is the distance for which the heliocentric parallax is 1 arcsecond.

This is the distance unit astronomers use most frequently, and it is equivalent to 3.26 light years.
This unit was chosen because it makes the parallactic formula very simple.

Use this calculator to convert lightyear to parsec or the other way around.

Accuracy

How can the measurement of parallax be improved?

• Make the base larger. For this reason heliocentric parallax is more accurate than geocentric parallax. With heliocentric parallax you can measure distance to stars that are further away. The practical limit is a distance of about 500 ly (150 pc).
• Measure the angle more accurately. This is most of all limited by the angular resolution of the telescope, which is related to atmospheric conditions and lack of all-sky visibility. To vastly increase this accuracy, measurements have been carried out in space. Examples are the Hipparcos and GAIA spacecraft.

Hipparcos (for High Precision Parallax Collecting Satellite) was an astrometry mission of the European Space Agency (ESA) dedicated to the measurement of stellar parallax and the proper motions of stars. The satellite was launched on 8 August 1989.
Accurate distance determinations have been obtained to about some 500 light-years distance.

More here

ESA’s Gaia spacecraft was launched in December 2013 and is located in the L2 Lagrangian Point on the shadow side of the Earth. Gaia performs three main types of observations: astrometry (stellar position, parallax, and proper motion), photometry (magnitude) and spectroscopy (for radial velocity and astrophysics). So Gaia does much more than measuring distance from parallax, but that is one of the key objectives.

The GAIA distance catalogue  now already contains 1.33 billion stars throughout our Milky Way galaxy.

More here

In the first part of this movie we can see the effect of the annual parallax as observed by the Gaia spacecraft.