The parallax distance formula
As Earth orbits the Sun, nearby stars appear to shift slightly against the far more distant background. Half of that yearly shift is the parallax angle. Astronomers defined the parsec so this relationship is beautifully simple: the distance in parsecs is just the reciprocal of the parallax in arcseconds.
d (parsecs) = 1 ÷ p (arcseconds)
1 parsec = 3.26156 light-years = 206,265 AU
Worked example
Proxima Centauri has a parallax of 0.7687 arcseconds. Its distance is 1 ÷ 0.7687 = 1.30 parsecs, which is about 4.24 light-years, our nearest stellar neighbor.
How parallax measurements developed
To picture how small these angles are: one arcsecond is 1/3,600 of a degree, about the width of a US dime seen from 3.7 km (2.3 miles) away.
Parallax uncertainty and distance accuracy
Because distance is 1 ÷ parallax, a small error in the angle becomes a large error in distance for faraway stars. A star with a parallax of 10 ± 0.1 mas is 100 pc away to within about 1%. A star with 1 ± 0.1 mas is about 1,000 pc away, give or take 10%. For that reason, catalogs report the parallax and its error rather than a single distance.
From parallax to true brightness
Once you know the distance, you can work out how bright a star really is. Its absolute magnitude (M) follows from its apparent magnitude (m):
- M = m + 5 + 5 log₁₀(p), with the parallax p in arcseconds.
- Sirius: m = −1.46, p = 0.379″, so M ≈ +1.4, about 25 times as luminous as the Sun.
Convert the result to other units with the light-year converter, or see how long the starlight has been traveling with the light travel time calculator.
Frequently asked questions
How do you calculate star distance from parallax?
Distance in parsecs equals 1 divided by the parallax in arcseconds: d = 1 / p. A 0.5″ parallax means 2 parsecs.
What is stellar parallax?
The small apparent shift of a nearby star against background stars as Earth orbits the Sun. Smaller shifts mean greater distances.
What is a parsec?
The distance at which a star shows a 1-arcsecond parallax, about 3.26 light-years or 3.086 × 10¹³ km.
How do I convert milliarcseconds to distance?
Divide by 1,000 to get arcseconds, then take the reciprocal. 100 mas = 0.1″ = 10 parsecs (about 32.6 light-years).
How far away is Proxima Centauri from its parallax?
Its 0.7687″ parallax gives 1 / 0.7687 = 1.30 parsecs, about 4.24 light-years.
Why does parallax only work for nearby stars?
Distant stars shift too little to measure. Ground telescopes reach a few hundred light-years; the Gaia mission extends parallax much farther with milliarcsecond precision.
Who first measured a star's parallax?
Friedrich Bessel, in 1838, for the star 61 Cygni. It was the first direct measurement of the distance to a star other than the Sun.
How accurate are Gaia parallaxes?
For bright stars, Gaia reaches a precision of a few hundredths of a milliarcsecond, enough to measure distances across a large part of the Milky Way.