Final answer:
Ground-based trigonometric parallax is limited to stars within about 500 light-years due to small parallax angles and atmospheric disturbance, but space observatories have expanded this range significantly.
Step-by-step explanation:
The limitations of using ground-based trigonometric parallax to measure the distance of a star include the fact that it can only be used for stars up to about 500 light-years away. Parallax measurements involve observing the apparent shift of a star's position relative to distant background stars from opposite sides of Earth's orbit, six months apart. This method works well for nearby stars, but for distances greater than this, the parallax angles become too small to measure accurately from the ground due to the limitations imposed by the atmosphere, which spreads out the points of starlight into fuzzy disks. Additionally, there is a limit to the distances that can be reached, even with the most precise ground-based instruments, because the parallax angle diminishes as the distance increases. Space-based observatories such as the Hipparcos and Gaia have extended the range of precise parallax measurements, with Gaia expected to measure accurate parallaxes for stars out to about 30,000 light-years.