Parallax

Parallax is the change in apparent position of a celestial object, relative to its background, caused by a shift in the position of the observer. The trigonometric parallax (π) of a star is defined as the angle subtended by 1 astronomical unit (AU) at the star’s distance (d) from the Sun As the Earth annually orbits the Sun, the observer’s changing position causes an annual shift in the star’s measured position, tracing a small ellipse on the sky that reflects the size and orientation of the Earth’s orbit as it might be viewed from the star. The size of this ‘parallactic ellipse’ is inversely proportional to the distance. For convenience, the astronomer’s unit of distance, the parsec (pc), is defined as the distance at which a star’s parallax would be one second of arc (arcsec), the unit in which parallaxes are traditionally measured. This gives the simple relation d = π⁻¹.   

When observed approx. six months apart from opposite sides of the Earth’s orbit around the Sun, the position of a nearby star relative to the more distant background stars, will appear to have shifted by an angle Δθ. Half of this angle, π, is the annual parallax (heliocentric parallax), which is also a measure of the angular size of the radius of the Earth as seen from that star. If a star’s parallax can be measured, then its distance can be determined. A unit of stellar distance is the parsec, the distance at which a star would have a parallax of one second of arc; it is equivalent to 3.2616 light years.

This is an example of trigonometric parallax, where a baseline of known length is used to make separate observations of a celestial object, the choice of baseline depending on the distance of the object in question. For objects in the Solar System, the radius of the Earth is used as a baseline, producing a measure of diurnal parallax (geocentric parallax). Solar parallax is the Sun’s geocentric parallax, in other words the angular size of the Earth’s equatorial radius from a distance of one astronomical unit (1 AU). Another method of determining stellar distances uses a star’s spectral type and is known as spectroscopic parallax.

The astrometric satellite Hipparcos, which operated between 1989 and 1993, extended the range of accurate parallax distances by roughly 10 times, at the same time increasing the number of stars with good parallaxes by a much greater factor. Of the 118,218 stars in the Hipparcos catalogue, the distances of 22,396 are now known to better than 10% accuracy. Prior to Hipparcos, this number was less than 1000. The companion Tycho database, from another instrument on the satellite, provides lower accuracy for 1,058,332 stars.

Questions to Ponder

• What are the practical limitations on measuring distances using parallax?
• How do we measure distances to other stars using Parallax?