Trigonometric Parallax

Trigonometric parallax is used to measure the distance to nearby stars. As Earth orbits the Sun, a nearby star appears to move against the background of much more distant stars, which seem fixed.The star's position is measured, then measured again six months later, when Earth is on the opposite side of its orbit. The parallax angle p is half the angle through which the star appears to move. It is the angle the radius of Earth's orbit, r = 1 AU = 1.50 × 10¹¹ m, subtends at the star.From the right-angled triangle, tan p = r/d. The angles are tiny, so tan p ≈ p (in radians) andd = r ÷ pAstronomers also use the parsec (pc): the distance at which 1 AU subtends an angle of 1 arcsecond (1/3600 of a degree). 1 pc = 3.09 × 10¹⁶ m. With p in arcseconds, d in parsecs is simply d = 1/p.Worked example: a star has p = 0.20 arcseconds.d = 1 ÷ 0.20 = 5.0 pc = 5.0 × 3.09 × 10¹⁶ = 1.5 × 10¹⁷ mThe further the star, the smaller p. Beyond a few hundred parsecs from the ground (further with space telescopes) the angle is too small to measure accurately, so other methods are needed.