How do you measure the distance to a star you can never reach? Astronomers use a beautifully simple trick you can test with your own thumb: hold it up, close one eye then the other, and watch it jump against the background. That apparent shift is parallax, and by watching a nearby star shift as the Earth orbits the Sun, we can measure its distance using nothing but geometry. This page covers the cosmic distance units and how parallax turns an angle into a distance.
📚 What you need to know
Big distances use three units: astronomical unit (AU), light-year (ly), and parsec (pc)
1 AU = mean Earth–Sun distance = 1.5 × 1011 m
1 light-year = distance light travels in one year ≈ 9.46 × 1015 m
1 parsec ≈ 3.1 × 1016 m ≈ 3.26 ly
Stellar parallax is the apparent shift of a nearby star against distant stars as Earth orbits
The relationship is p = 1/d where p is in arcseconds and d is in parsecs
Parallax works reliably only for distances up to about 100 pc
Astronomical distance units
Space is so vast that metres become useless, so astronomers use three larger units, each suited to a different scale.
Astronomical unit (AU)
The astronomical unit is the mean distance from the Earth to the Sun. Since Earth’s orbit is slightly elliptical, we average the closest and furthest distances to get 1 AU ≈ 1.5 × 1011 m. It’s handy for distances within the solar system.
Light-year (ly)
A light-year is the distance light travels in one year — found simply from distance = speed × time. With light at 3 × 108 m s−1 and a year being about 3.15 × 107 s, one light-year ≈ 9.46 × 1015 m. It’s used for interstellar distances.
Parsec (pc)
The parsec is the natural unit for parallax. It’s defined as the distance at which 1 AU subtends an angle of one arcsecond (1/3600 of a degree). Using trigonometry, 1 parsec ≈ 3.1 × 1016 m ≈ 3.26 light-years.
You don’t need to memorise the conversion factors — they’re in the data booklet. What you do need is to use them confidently. A common exam task is converting a distance between all three units, so practise chaining: parsecs → metres → light-years. Keep everything in scientific notation and the powers of ten will look after themselves.
WE 1
Alpha Centauri is about 4.35 light-years from Earth. Convert this distance into (a) astronomical units and (b) parsecs. (1 ly ≈ 9.5 × 1015 m, 1 AU = 1.496 × 1011 m, 1 pc ≈ 3.1 × 1016 m)
Step 1 — convert to metresd = 4.35 × 9.5×10¹⁵ = 4.13×10¹⁶ mStep 2 — (a) into AU4.13×10¹⁶ ÷ 1.496×10¹¹ = 2.8×10⁵ AUStep 3 — (b) into parsecs4.13×10¹⁶ ÷ 3.1×10¹⁶ = 1.3 pc(a) 2.8 × 105 AU (b) 1.3 pcAlways go via metres as the common currency. From there, dividing by the right conversion factor gets you into any unit you need.
How stellar parallax works
The principle of parallax is that a nearby object appears to shift against a distant background when you view it from two different positions. For stars, our “two positions” are opposite ends of the Earth’s orbit — say January and July, six months apart.
A nearby star appears to shift slightly against the fixed backdrop of much more distant stars. The distant stars are so far away they don’t appear to move at all. That apparent shift of the nearby star is the stellar parallax.
Viewed six months apart, a nearby star shifts against distant stars. The half-angle of that shift is the parallax anglep; a bigger baseline (1 AU) and closer star give a larger p.
The parallax equation
Applying trigonometry to the thin right-angled triangle (baseline 1 AU, distance d), the parallax angle p satisfies tan p = 1 AU / d. For the tiny angles involved, tan p ≈ p (in radians), which simplifies things enormously.
When we measure p in arcseconds and d in parsecs, the relationship becomes beautifully simple — this is exactly why the parsec was invented:
The parallax–distance relationshipp = 1 / dp = parallax angle in arcseconds (“) • d = distance in parsecs (pc)
So a star with a parallax of exactly 1 arcsecond is exactly 1 parsec away. The smaller the parallax angle, the further the star. This is accurate up to about 100 pc — beyond that, the angles become too tiny to measure reliably.
WE 2
Proxima Centauri, the nearest star to Earth, has a parallax of 0.768 arcseconds. Calculate its distance in (a) parsecs and (b) light-years. (1 pc ≈ 3.1 × 1016 m, 1 ly ≈ 9.5 × 1015 m)
Step 1 — (a) use p = 1/dd = 1/p = 1 ÷ 0.768 = 1.30 pcStep 2 — convert to metres1.30 × 3.1×10¹⁶ = 4.03×10¹⁶ mStep 3 — (b) into light-years4.03×10¹⁶ ÷ 9.5×10¹⁵ = 4.2 ly(a) 1.30 pc (b) 4.2 lyThe p = 1/d shortcut makes part (a) instant. Just remember p must be in arcseconds and the answer comes out in parsecs — then convert if needed.
⚛ Parallax & distance conversions
Given parallax? Use d = 1/p (p in arcsec → d in pc).
Convert to metres: × 3.1×1016 (from pc).
To light-years: ÷ 9.5×1015.
To AU: ÷ 1.496×1011.
Check range: parallax reliable only up to ~100 pc.
💡 Top tips
p = 1/d needs p in arcseconds and d in parsecs — nothing else.
Smaller parallax angle = further star.
Always convert via metres as the common unit.
Conversion factors are in the data booklet — use, don’t memorise.
Parallax works only up to about 100 pc.
⚠ Common mistakes
Using p = 1/d with p in degrees — it must be arcseconds
Thinking a bigger angle means further — it’s the opposite
Forgetting arcseconds (“) vs arcminutes (‘) — 1° = 3600”
Mixing up light-years and parsecs in conversions
Applying parallax beyond 100 pc, where it’s unreliable
Quick recap: Cosmic distances use the AU (1.5×1011 m), light-year (9.46×1015 m), and parsec (3.1×1016 m ≈ 3.26 ly). Stellar parallax is the apparent shift of a nearby star as Earth orbits; the distance follows p = 1/d (p in arcseconds, d in parsecs). Smaller angle = further star, reliable up to about 100 pc.
Parallax tells us how far a star is. Combine that distance with the star’s brightness and temperature, and we can work out something we could never measure directly — its actual size. Next page: Finding Stellar Radii.
Parallax and distance units tripping you up?
Book a free meeting and we’ll drill the p = 1/d equation, the three distance units, and the conversion chains examiners set.
