Stellar Parallax Distance

As Earth swings from one side of its orbit to the other, a nearby star appears to shift against the distant background by twice the parallax angle p. Trigonometry gives d = AU/tan p, but real stellar parallaxes are so tiny — always under one arcsecond — that tan p ≈ p in radians, and the small-angle form d = AU/p is exact to better than one part in 10¹⁰. The nearest star system, Proxima Centauri, has p = 0.7685″ = 3.726 × 10⁻⁶ rad, so d = 1.496 × 10¹¹ / 3.726 × 10⁻⁶ ≈ 4.02 × 10¹⁶ m — about 4.25 light-years.

This geometry defines the parsec: the distance at which the parallax is exactly one arcsecond, 3.086 × 10¹⁶ m, so that d in parsecs is simply 1/p in arcseconds. Friedrich Bessel made the first successful measurement in 1838 (61 Cygni, p ≈ 0.31″), finally proving the Earth moves; today ESA's Gaia mission measures parallaxes to a few microarcseconds, pinning down distances for nearly two billion stars and anchoring every rung of the cosmic distance ladder.