Betelgeuse (in Orion) has a parallax of \(0.00763 \pm 0.00164\) arcsec, as measured by the Hipparcos satellite. What is the distance to Betelgeuse, and the uncertainty in that measurement?

Measuring distance is a fundamental aspect of astronomy, helping us understand the scale of the universe. One of the most effective methods is parallax. Parallax is the apparent shift in the position of a nearby star as seen from Earth. This shift, caused by Earth's orbit around the Sun, is tiny and measured in arcseconds.
To calculate the distance to a star using parallax, we use the formula:
\ d = \frac{1}{p} \
where \(d\) is the distance in parsecs and \(p\) is the parallax angle in arcseconds. Smaller parallax angles mean the star is further away. This method becomes less accurate for very distant stars due to the minuscule parallax shift.

Astronomical Units (AU) are crucial in measuring distances within our solar system. One AU is the average distance between Earth and the Sun (~149.6 million kilometers).
To understand larger cosmic distances, such as to stars like Betelgeuse, we use parsecs and light-years. One parsec is approximately 3.26 light-years, which is equivalent to about 206,265 AU. Hence, using parsecs provides a manageable way to express and grasp the immense distances between stars and galaxies.
For context, Betelgeuse's distance of about 131.07 parsecs translates to roughly 427 light-years. This scale shows how parallax measurements, even with small angles, can reveal vast cosmic distances.

In astronomy, precise measurements often involve understanding and propagating errors. Error propagation is the process of determining the uncertainty of a result, given the uncertainties in the measurements used to obtain that result.
Let's consider the distance measurement to Betelgeuse. The parallax angle is given as 0.00763 arcseconds, with an uncertainty of 0.00164 arcseconds. To find the uncertainty in distance, we use:
\ \ \ \delta d = \left| \frac{-1}{p^2} \right| \delta p \ \ \
Substituting the values, the uncertainty \(\delta d\) becomes approximately 28.19 parsecs. This indicates that while our measurement places Betelgeuse at 131.07 parsecs, there is an uncertainty margin of 28.19 parsecs. Error propagation helps us understand the reliability and limitations of our observations, crucial for making accurate scientific conclusions.