The celestial coordinate grid: right ascension and declination
To measure star distances, a tiny angular shift is observed against a fixed grid of background stars. This grid is defined by two coordinates, directly analogous to latitude and longitude on Earth:
Declination (dec)
— angular distance north or south of the celestial equator (the projection of Earth’s equator onto the sky). Measured in degrees, from -90° to +90°.
Right ascension (RA)
— angular distance measured eastward around the celestial equator from a fixed zero point, the vernal equinox (where the ecliptic, the Sun’s apparent yearly path, crosses the celestial equator heading north). Usually given in hours, minutes and seconds (0h to 24h) rather than degrees.
Because distant background stars are so far away that their own parallax is immeasurably small, they act as a stable reference frame. A nearby star’s position is measured relative to these background stars in an image, converted into RA/dec, and compared between observations six months apart. The difference between the two measured positions is the apparent shift used to find the parallax angle.
One technicality worth knowing for OCR: Earth’s rotational axis drifts slowly over a ~26,000-year cycle (axial precession), so the grid itself isn’t perfectly fixed. Catalogued coordinates are therefore always stated for a specific epoch (e.g. J2000).
Parallax: the geometry
As Earth orbits the Sun, a nearby star appears to shift slightly against the fixed background stars when viewed six months apart. This apparent shift is parallax.
The setup forms a right-angled triangle:
One leg is the Earth–Sun distance, 1 AU.
The other leg is the distance to the star, d (the unknown).
The angle at the star, between the two lines of sight, is the parallax angle, p.
This gives:
tan(p) = 1 AU / d, so d = 1 AU / tan(p)
One common slip: the angle measured directly from the two observations is the full swing across the 2 AU baseline. The angle p used in the triangle is half that swing, since the triangle’s base is only 1 AU (Sun to Earth), not the full Earth-to-Earth distance.
Because p is always tiny for real stars, the small-angle approximation applies: tan(p) ≈ p (in radians). This means d ≈ 1 AU / p, with p in radians — the basis for the parsec shortcut in the next section.
Converting parallax to parsecs
Rather than working in AU and radians each time, astronomers define the parsec as the distance at which a star would show a parallax of exactly 1 arcsecond. This gives a direct shortcut:
d (parsecs) = 1 / p (arcseconds)
Worked example: Proxima Centauri
Measured parallax: p = 0.7687 arcseconds
d = 1 / 0.7687 ≈ 1.30 parsecs
For comparison, doing it the long way (through the triangle) gives the identical result:
0.7687″ ÷ 3600 = 0.0002135°
0.0002135° × (π/180) ≈ 0.0000037258 rad
d = 1 AU / tan(p) ≈ 1 / 0.0000037258 ≈ 268,400 AU ≈ 1.30 pc
The parsec definition is just this calculation done once and built into the unit, which is why d = 1/p works with no further conversion needed.
Converting parsecs to light-years
1 parsec ≈ 3.26 light-years
Worked example: Proxima Centauri
d = 1.30 pc × 3.26 ly/pc ≈ 4.24 light-years
This matches the accepted distance to Proxima Centauri, and it’s the same answer reached earlier by converting AU directly (268,400 AU ÷ 63,241 AU/ly ≈ 4.24 ly) — parsecs and light-years are just two different units describing the same distance.
Exam notes (OCR A Level Physics)
Always check whether a question wants the answer in AU, parsecs, or light-years — read the command word and unit requested carefully.
Show the small-angle approximation step (tan(p) ≈ p) explicitly if asked to derive d = 1/p; don’t just quote the formula.
Keep angle units consistent: convert arcseconds → degrees → radians before using tan(), unless using the parsec shortcut directly.
State the two key relationships clearly: d(pc) = 1/p(arcsec) and 1 pc ≈ 3.26 ly.
