As the Earth moves in its orbit of the Sun, our perspective on the stars changes slightly. Nearby stars show a parallax shift compared to more distant stars. In other words, the apparent position of a nearby star will appear to oscillate slightly with respect to the backdrop of more distant stars. This oscillation repeats every year as the Earth orbits the Sun — the full extent of the parallax shift is seen in observations made six months apart.
Schematic for calculating the parallax of a star. Click here for original source URL.
The small angle equation shows how to relate angular size to the apparent size of an object as seen from a certain distance. In this application of the small angle equation, the angular size (a) becomes the parallax angle ( ρ ) and the linear size (d) is the Earth-Sun distance. If we use astronomical units (A.U.) for the distance, then the small angle equation can be rearranged as:
D = 206,265 / ρ
The distance to a star in A.U. equals 206,265 divided by the parallax angle in seconds of arc. Note that in practice we measure an angle of 2 ρ because we make observations six months apart. The base of the triangle is then twice the Earth-Sun distance (d = 2 A.U.).
Remember that the parallax angle is greatly exaggerated in most textbooks to make it visible. In the astronomical application the triangle is long and extremely skinny. A typical stellar parallax for one of the closest stars is one second of arc, which is the thickness of a piece of paper as seen from across a large room. The detection of such a tiny angle required the resolution of a large telescope, which is why was not detected until over 200 years after the invention of the telescope. For most of the stars visible to the nakes eye the parallax angle is hundreds of time smaller, so very difficult to detect even with a large telescope.
Here is where the distance unit of a parsec comes from. If we define one parsec to equal 206,265 A.U., then the equation above simplifies to:
D = 1/ ρ
The distance to a star in parsecs is one divided by the parallax angle in seconds of arc. A parsec is the distance to a star whose parallax is one second of arc. This equation is useful because it relates a unit of distance measurement to the most common unit of angular measurement in astronomy. The parsec is a useful distance unit because it corresponds to the typical distance between stars (which is equivalent to saying that the parallax angles of the nearest stars are around 1 second of arc).
Note that parallax angle has an inverse relationship with distance. More distant stars have smaller parallax angles. For example Alpha Centauri has a parallax angle of 0.77 seconds of arc, which implies a distance of 1/0.77 = 1.3 parsecs, or 4.2 light years. The bright star Altair has a parallax angle of 0.20 seconds of arc, so it is more distant at 1/0.20 = 5 parsecs, or 16.3 light years. Many of the stars in the night sky are at distances of hundreds of parsecs (roughly times three if you want the equivalent in light years), so their parallax angles are hundredths of an arc second or less.
The Hipparcos satellite. Click here for original source URL.
Before the Hipparcos satellite was launched, parallax measurements below about 0.01 seconds of arc were too small to measure reliably. This meant that we could only apply the direct trigonometric technique to stars closer that about 1/0.01 = 100 parsecs, or some tens of thousands of stars. Hipparcos pushed that limit five times smaller to 0.002 seconds of arc. As a consequence, stellar distances out to 1/0.002 = 500 parsecs. This increased the number of stars whose distances we can measure directly to 100,000 stars. Gaia provides a dramatic improvement of a factor of 100 on Hipparcos, taking the limit of 2 milliarcseconds (2 × 10-3 arc seconds) down to 20 microarcseconds (2 × 10-5 arc seconds). That will provide distances out to 500 × 100 = 50,000 parcsecs, reaching a billion stars in the Milky Way.