The Parsec
The previous lesson established that a star's distance is inversely proportional to its parallax angle. Astronomers turned that relationship into a unit of distance in its own right — one built directly out of the geometry of the measurement itself, rather than out of how far light happens to travel in a year.
Defining the parsec
A parsec (pc) is defined as the distance at which a star would show a parallax angle of exactly one arcsecond, when the observation uses a baseline of one astronomical unit — the mean Earth–Sun distance. The name itself is a contraction of "parallax of one arcsecond", which is a useful way to remember exactly what it means: it isn't an arbitrary round number, it's a distance built directly out of the observing geometry from the previous lesson.
Because $d = 1\,\text{AU}/p$, setting $p = 1$ arcsecond by definition fixes what one parsec is equal to in metres. Working that out from the small-angle geometry gives a precise conversion factor between the parsec and the metre.
The working relation: distance from parallax angle
Because the parsec is defined this way, it slots directly into a strikingly simple general formula. If a star has a measured parallax angle $p$, expressed in arcseconds, then its distance $d$ in parsecs is just the reciprocal of $p$:
This is the practical, working version of the $d \propto 1/p$ relationship from the previous lesson, with the units chosen specifically so no other conversion factors are needed. A star with a measured parallax angle of exactly 1 arcsecond is, by definition, exactly 1 parsec away; a star with a parallax angle of 0.5 arcseconds is 2 parsecs away, and so on.
Worked example
A star is observed to have a parallax angle of $p = 0.25$ arcseconds. Find its distance in parsecs and in light years.
Converting to light years using $1\,\text{pc} \approx 3.26$ ly:
Parsecs versus light years
The parsec and the light year (the distance light travels through a vacuum in one year) are the two standard units used for astronomical distances, and it's worth being clear on why both exist rather than just settling on one.
- The light year is easy to picture and communicate — it connects distance directly to travel time for light, which makes it the natural choice for general or more popular description of astronomical scales.
- The parsec is the more natural unit for professional work that involves parallax measurements directly, since it turns the awkward, tiny-angle parallax data straight into a distance with no extra conversion factors — hence the working relation $d(\text{pc}) = 1/p(\text{arcsec})$ above.
- Both units describe the same underlying distances; converting between them just uses the fixed factor $1\,\text{pc} \approx 3.26$ ly derived above.
The parsec isn't an arbitrary round number — it's defined so that the parallax-to-distance formula becomes as simple as possible: $d(\text{pc}) = 1/p(\text{arcsec})$. This is precisely why professional parallax work uses parsecs rather than light years: it removes a conversion step from every single distance calculation.
| Unit | Approx. value | Typical use |
|---|---|---|
| 1 AU | $1.496\times10^{11}$ m | Distances within the Solar System |
| 1 light year | $9.461\times10^{15}$ m $\approx 6.32\times10^4$ AU | General/popular description of stellar and galactic distances |
| 1 parsec | $3.086\times10^{16}$ m $\approx 3.26$ ly $\approx 2.06\times10^5$ AU | Direct parallax-based distance work |
Key takeaways
- A parsec is the distance at which a star would show a parallax angle of exactly 1 arcsecond, using a baseline of 1 AU.
- Deriving this from the small-angle geometry gives $1\,\text{pc} = \dfrac{1\,\text{AU}}{\tan(1'')} \approx 3.09\times10^{16}\,\text{m} \approx 3.26$ light years.
- The general working relation is $d(\text{pc}) = 1/p(\text{arcsec})$ — distance in parsecs is simply the reciprocal of the parallax angle in arcseconds.
- A star with parallax angle 1 arcsecond is 1 pc away; a star with parallax angle 0.25 arcseconds is 4 pc away, and so on.
- Light years and parsecs are the two standard astronomical distance units; light years suit general description, parsecs suit direct parallax-based calculation.
- 1 pc $\approx$ 3.26 ly $\approx 2.06\times10^5$ AU — a useful set of conversions for sense-checking astronomical distances.
- The parsec's definition is specifically engineered to keep the parallax distance formula free of extra conversion factors.