Parallax (astrometry)

Parallax was the first method used by astronomers to find the distance to nearby stars. It relies on measuring the change in angle of the star being observed against more distant background stars as a result of the motion of the Earth around the sun.

The idea of parallax can be neatly demonstrated using our own eyes. Look at a nearby object and shut one eye. Make a mental note of the object’s position, then, keeping your head still, change which eye is open and which is shut. You should see the object appear to shift. This shift in angle is caused by the differing positions of our two eyes. We rely on this effect all the time to accurately gauge distances. Try throwing and catching a ball with only one eye open to see how much we use parallax.

We can use the same principle to measure the distance to stars, using telescopes instead of eyes. However, the nearest stars to Earth, other than the sun, are over four light years away. This means that we cannot move our two “eyes” apart far enough on Earth to measure an observable change in angle.

Fortunately, the Earth is not stationary. By taking two measurements at six month intervals, when the Earth is at opposite ends of its orbit around the sun, we can record a measurable change in angle to a nearby star compared to distant background stars (whose own movement is negligible). This angle measurement can be used in combination with the (already known) distance from the Earth to the sun to calculate the distance to the star. We see that the distance = 1AU/tan(a) (where 1AU (astronomical unit) is the distance from the Earth to the sun, and a is the measured parallax angle (see diagram). Using the small angle approximation tan(a) = a, we find that the distance = 1/a.

Note that regardless of the direction of the star from the Earth, there are always two points in the Earth’s orbit where the line between them is perpendicular to the line from the sun to the star. You can visualise this more easily in three dimensions using something circular to represent the Earth’s orbit.

You are trying to measure the distance to a star. You are able to measure that over 6 months the position of the star shifts by an angle (2a - see diagram) of 0.1 arcseconds.

Taking the distance from the Earth to the sun to be \(1.5\times 10^{11} \si{meter}\), calculate the distance to the star in meters.

\[ 6.2 \times 10^{17} \] \[ 1.5 \times 10^{17} \] \[ 3.1 \times 10^{17} \] \[ 7.5 \times 10^{16} \]

Even to nearby stars, the parallax angles measured are small - the nearest star system, Alpha Centauri, has a parallax angle of just 0.0002 degrees. Astronomers use the smaller unit, the arcsecond, to measure parallax angles, where 3600 arcseconds = 1 degree. From this comes a standard unit of interstellar distances - the arcsecond. The parsec is defined as the distance to a star that has a parallax of one arcsecond.

Measuring the distance to stars by parallax requires highly accurate angle measurements. Even with modern equipment, we can only use parallax to measure the distance to stars up to 10,000 light years away, which is just 10% of the diameter of our galaxy (the Milky Way). To measure the distances to more distant objects, we are able to use different techniques, such as standard candles and redshift measurements. However, these later techniques are only possible because of parallax measurements to nearby stars.