How Does Stellar Parallax Measure Distance to Stars?

Stellar parallax measures a nearby star's distance by observing its apparent change in position against distant background stars as Earth moves around the Sun. The smaller the parallax angle, the farther away the star is.

The Mechanism

This is part of E.5 Fusion and stars and is required at both SL and HL.

1. Astronomers observe a nearby star from Earth at one point in Earth's orbit.
2. They observe it again six months later, when Earth is on the opposite side of its orbit.
3. The nearby star appears to shift relative to much more distant background stars.
4. The parallax angle $p$ is half the star's total apparent angular shift. It is the angle subtended by the EarthSun distance, equal to one astronomical unit.
5. The star's distance is calculated using

$$d\text{ (pc)}=\frac{1}{p\text{ (arcsec)}}.$$

Here, $d$ is the distance in parsecs and $p$ is measured in arcseconds. One parsec is the distance at which one astronomical unit subtends an angle of one arcsecond.

The equation is a small-angle result. The baseline is $1\text{ AU}$, not the full $2\text{ AU}$ separation between the two observing positions, because $p$ is half the measured shift. In practice, angular uncertainty produces a larger percentage uncertainty when $p$ is very small.

For example, if a star has a parallax angle of $0.20$ arcseconds,

$$d=\frac{1}{0.20}=5.0\text{ pc}.$$

A common misconception is that $p$ equals the star's complete apparent movement. It does not: the complete angular displacement measured six months apart is $2p$.

Exam Technique

For a Calculate question, show the equation, substitution, and answer with the unit pc. For an Explain question, connect Earth's changing observation position to the star's apparent angular shift, then state that distance is inversely proportional to parallax angle.