Stellar Parallax and Distance Scales

Observe a nearby star against distant background stars at opposite points of Earth’s orbit, about six months apart. The full apparent angular displacement is twice the parallax angle p in the standard symmetric geometry. The parallax angle corresponds to a baseline of 1 AU, not the full orbital diameter.

An astronomical unit is approximately the Earth–Sun distance, $1.50\times10^{11}$ m. A light year is the distance light travels in a year, about $9.46\times10^{15}$ m. A parsec is the distance at which 1 AU subtends 1 arcsecond; 1 pc is about $3.09\times10^{16}$ m or 3.26 light years.

When parallax is measured in arcseconds, the distance in parsecs is simply its reciprocal. This convenient expression already contains the angular-unit conversion and the astronomical-unit baseline.

One arcsecond is 1/3600 of a degree, approximately $4.85\times10^{-6}$ rad. If using a distance relation in SI units, use the small angle in radians; if using d=1/p in parsecs, use arcseconds. Mixing these two conventions gives a very large error.

Parallax becomes smaller for more distant stars. A nearly fixed absolute angular uncertainty therefore becomes a larger fractional uncertainty in p and hence in the inferred distance. Repeated observations can reduce random error and separate parallax from a star’s proper motion.

Space-based measurements avoid atmospheric blurring, but instrument calibration and angular precision still limit results. A parallax consistent with zero within uncertainty does not imply zero distance: it means the distance is too large for a reliable positive parallax measurement by that method.

Try each problem before opening the worked solution. State the physical model, show substitutions with consistent units, and explain what your answer means.