LearnStars & evolutionSome maths

Stellar parallax

The tiny annual shift in a nearby star's apparent position, caused by the Earth viewing it from opposite sides of its orbit.

Why it matters

It is the only direct distance measurement in astronomy — pure geometry, no assumptions — and every other rung of the distance ladder is calibrated against it.

Try it

Parallax
0.286″
Distance
11.4 ly
Baseline ratio
1 : 721,927
Vs good seeing
3.5× smaller
Left, the geometry: the Earth views the star from opposite ends of its orbit, and the angle between those two sight lines is twice the parallax. That panel is schematic — at 3.5 pc the star is 721,927 times further off than the orbit is wide, so drawn to scale it would be kilometres past the edge of the screen.
Right is to scale, in arcseconds. The star traces that ellipse once a year against the distant field, and the dashed circle is the blur a good night’s seeing puts around every point source. The whole path is 3.5× smaller than that blur — which is why three centuries of astronomers looked for this and found nothing, and why it took Bessel until 1838. This is roughly 61 Cygni — Bessel's star, the first ever measured at 0.286″. Gaia resolves 20 µas, some 50,000× finer than the seeing, by observing from above the atmosphere.
The path is an ellipse rather than a circle or a line because the star sits 30° off the ecliptic. A star at the ecliptic pole traces a circle; one on the ecliptic slides back and forth along a straight line.

The maths, in layers

1 · Intuition

Hold a finger up and blink alternate eyes. The nearer it is, the more it jumps. The Earth's orbit is the distance between the eyes.

2 · The equation

d [parsecs] = 1 / p [arcseconds]

p
parallax angle, half the annual shift, in arcseconds
3 · Where it comes from

This defines the parsec: the distance at which one astronomical unit subtends one arcsecond. It is 3.26 light-years. Even Proxima, the nearest star, has p = 0.77″ — smaller than the seeing on a good night.

What people usually get wrong

  • Parallax was too small to detect, so it disproved heliocentrism.

    That was the argument against Copernicus for three centuries, and it was sound reasoning from bad data. The stars are simply far further away than anyone imagined — the first parallax was not measured until Bessel in 1838.

Rests on

3 concepts in the full chain back to first principles.

Leads to