Distance & Parallax

Part 2: Angular Measure — The Language of the Sky

Before we can talk about parallax, we need a language for measuring angles.

Degrees, Arcminutes, and Arcseconds

The full sky is divided into 360 degrees ($^\circ$) — a choice inherited from the Babylonian base-60 system. The Sun and Moon each span about $0.5^\circ$. For stellar parallax we need finer divisions: one degree is 60 arcminutes ($'$), and one arcminute is 60 arcseconds ($''$), so $1'' = 1/3600$ degree. This is the scale of stellar parallax — Proxima Centauri shifts by $0.77''$ as Earth orbits.

Radians — The Natural Unit

Degrees are human convention; physics prefers the radian, which emerges directly from geometry: 1 radian is the angle subtended by an arc whose length equals the circle’s radius.

Because it’s a ratio of lengths, a radian is dimensionless: $\text{angle (rad)} = \text{arc length}/\text{radius}$. A full circle is $2\pi$ rad, so $1\ \text{rad} = 360^\circ/2\pi \approx 57.3^\circ$.

The Critical Conversion: Arcseconds to Radians

Stellar parallax is measured in arcseconds, but the small-angle formula produces radians. Converting:

$1\ \text{rad} = \frac{360^\circ}{2\pi} = \frac{360 \times 3600''}{2\pi} = \frac{1{,}296{,}000''}{2\pi} \approx 206{,}265''.$

$\boxed{1\ \text{rad} = 206{,}265'' \qquad 1'' = \frac{1}{206{,}265}\ \text{rad} \approx 4.85 \times 10^{-6}\ \text{rad}}$