Hold a finger up at arm's length and look at it with one eye, then the other. It jumps against the background. Bring it closer and it jumps more. That shift is parallax, and it is the one direct way to measure the distance to a star: no assumptions about the star, just geometry. The baseline is Earth's own orbit, measured in AU, the unit from Kepler's third law.
Why it works
Photograph a nearby star in January, then again in July, from the other side of Earth's orbit, $2$ AU away. Against much more distant stars, the nearby one seems to shift. Half of that shift is the parallax angle $p$: the angle the $1$ AU radius of Earth's orbit spans as seen from the star. For such tiny angles, angle $\approx \frac{\text{size}}{\text{distance}}$ (in radians), so the distance is $d = \frac{1\ \text{AU}}{p}$. Astronomers made this neat by defining a unit to match: one parsec (parallax-second) is the distance at which $1$ AU spans $1$ arcsecond ($\frac{1}{3600}$ of a degree, about the angle a quarter makes seen from $5$ km away). With $p$ in arcseconds, $d = \frac{1}{p}$ parsecs. Since $1$ arcsecond is $\frac{1}{206{,}265}$ radian, $1$ pc $= 206{,}265$ AU $= 3.26$ light-years.
Key idea
$d\,(\text{parsecs}) = \dfrac{1}{p\,(\text{arcseconds})}$, and $1$ pc $= 3.26$ ly. A larger parallax means a closer star. Every star's parallax is under $1$ arcsecond.
Worked examples
Worked example: The nearest star
Proxima Centauri has a parallax of $0.768$ arcseconds. How far is it, in parsecs and light-years?
- $d = \frac{1}{0.768} = 1.30$ pc.
- $1.30 \times 3.26 = 4.24$ light-years.
- Even the nearest star's parallax is less than an arcsecond, which is why it took until 1838 (Friedrich Bessel, with the star 61 Cygni) to measure one.
Answer: $1.30$ pc, or $4.24$ ly.
Worked example: Milliarcseconds
The Gaia spacecraft measures a star's parallax as $5$ milliarcseconds. How far away is it?
- $5$ mas $= 0.005$ arcseconds.
- $d = \frac{1}{0.005} = 200$ pc, about $650$ light-years.
Answer: $200$ pc.
Common mistake
Because big numbers often go with big distances, it is easy to think a larger parallax means a farther star. It is the reverse: $d = 1/p$, so a big shift means a close star, just as your finger jumps more when it is near your eye. Check: Proxima, the nearest star, has the largest parallax of all, $0.77$ arcseconds.
Earth's motion makes all the stars trace tiny yearly ellipses on the sky, aligned with the celestial coordinate grid. Gaia has measured parallaxes for over a billion stars, mapping our part of the galaxy in 3-D.