Distance & Parallax

Section 4 of 7

Parallax

Part 4: Parallax — Distance from Baseline Geometry

Now we apply the small-angle formula to the most direct distance technique: parallax — use a long baseline and measure the angular shift of a nearby object against fixed background objects.

The Parallax Angle

Hold your finger at arm’s length and blink between eyes: it shifts against the background, and the farther away it is, the smaller the shift. That’s parallax. The geometry is the small-angle formula with the baseline $b$ playing the role of “size,” the parallax angle $p$ playing “angle,” and the distance $d$ what we want. Substituting $b = p\,d$ and solving:

Here $b$ is the baseline, $p$ the parallax angle (radians), and $d$ the distance in the same units as $b$ . This works for any baseline and any observer.

The distance you can probe is set by the baseline: your eyes ( $b \approx 7$ cm) give depth perception to a few meters; Earth’s orbit ( $b = 1$ AU) makes stellar parallax possible; a larger baseline reaches proportionally farther at the same angular precision. The physics is always $d = b/p$ — only the baseline changes.

Stellar Parallax: Earth’s Orbit as Baseline

As Earth moves from one side of the Sun to the other, a nearby star shifts against the distant background. We define the parallax angle $p$ as half the total shift — the angle subtended by Earth’s orbital radius (1 AU), not its diameter.

Parallax angle

The apparent angular shift of a nearby star as Earth orbits, defined as half the total six-month shift — the angle Earth’s 1 AU orbital radius subtends as seen from the star. Measured in arcseconds.

Figure 6A nearby star shifts against distant background stars between January and July; that angular shift is parallax and gives distance.cococubed.com

From General Parallax to the Parsec

Substituting Earth’s baseline $b = 1$ AU and converting to arcseconds gives $d = 206{,}265\ \text{AU} / p('')$ . Astronomers then define a unit to make this clean: one parsec is the distance at which 1 AU subtends $1''$ , so $1\ \text{pc} = 206{,}265$ AU. The relation becomes:

A star with $p = 0.5''$ is at 2 pc; $p = 0.1''$ is at 10 pc. The name “parsec” is parallax-arcsecond — it works only because Earth’s 1 AU baseline and the arcsecond are baked into the definition.

Parsec

The distance at which a star shows a parallax of $1''$ (with Earth’s 1 AU baseline): $1\ \text{pc} = 206{,}265\ \text{AU} = 3.09 \times 10^{18}\ \text{cm} = 3.26$ light-years. The natural unit for stellar distances.

For reference: $1\ \text{pc} = 3.09 \times 10^{18}$ cm $= 3.26$ light-years, and $1$ kpc $= 1000$ pc.

Worked Example 1Proxima Centauri

Problem

Proxima Centauri, the nearest known star, has parallax $p = 0.768''$ . Find its distance in parsecs and centimeters.

StepApply the parsec form

$d = \frac{1}{p} = \frac{1}{0.768''} = 1.30\ \text{pc}$

StepConvert to CGS

$d = 1.30\ \text{pc} \times 3.09 \times 10^{18}\ \text{cm/pc} = 4.02 \times 10^{18}\ \text{cm}$

Dimensional check

$1/('')$ gives pc by the definition of the parsec; pc $\times$ (cm/pc) leaves cm ✓.

Result

$d \approx 1.30$ pc $\approx 4.0 \times 10^{18}$ cm. Sanity check: nearby stars fall in the 1–5 pc range ✓. Its parallax was barely measurable from the ground — which is why parallax was historically so hard.

Observable

A nearby star's position shifts as Earth orbits

Over a year a nearby star traces a tiny ellipse against the fixed background; its semi-major axis is the parallax angle $p$ , measured in arcseconds.

Model

Parallax geometry (small-angle, 1 AU baseline)

Earth’s 1 AU orbital radius subtends $p$ at the star, so $d = b/p$ — specialized to arcseconds and parsecs, $d(\text{pc}) = 1/p('')$ .

Inference

The star's distance — purely geometric

Inverting gives the distance with no assumption about the star itself: the foundation rung of the cosmic distance ladder.

Numeric answer

Estimate the parallax angle for a star at 1000 pc.

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