How do we know how far a star is? Hold up your thumb

Hold your thumb out at arm's length. Close your left eye. Now swap: open the left, close the right. Your thumb jumps sideways against the background. Bring the thumb closer to your nose and the jump gets bigger. That jump is parallax, and it's the first method humans ever used to measure the distance to a star. It sounds too simple to be real science. It's the same trick, just with a much, much bigger baseline.

From two eyes to Earth's orbit

Your eyes are about 6 cm apart. That separation is your baseline. For nearby objects, the two views differ enough that your brain can work out depth. For a mountain 10 km away the difference is invisible, which is why depth perception gives up beyond some tens of metres.

For stars we need a far longer baseline, and Earth supplies one. In January we view a nearby star from one side of Earth's orbit; six months later from the opposite side. The two positions are $2$ AU apart, where 1 AU (the Earth-Sun distance) is about $1.496 \times 10^{8}$ km. The nearby star seems to shift slightly against the far-away stars behind it. Half of that total shift is called the parallax angle $p$.

The geometry is a very thin right triangle: one side is 1 AU, the angle at the star is $p$, and the long side is the distance $d$:

$\tan p = \frac{1\ \text{AU}}{d} \quad\Rightarrow\quad d \approx \frac{1\ \text{AU}}{p} \qquad (p \text{ in radians, tiny})$

I love that $\tan\theta \approx \theta$ shows up in real life and not just in a chapter on approximations.

The parsec is a definition that makes this painless

Angles in the sky are measured in degrees, arcminutes and arcseconds: 1 degree = 60 arcminutes = 3600 arcseconds. One arcsecond is absurdly small. A 1 cm coin would subtend about one arcsecond if you looked at it from 2 km away.

Put $p = 1$ arcsecond $= 4.848 \times 10^{-6}$ rad into the formula:

$d = \frac{1.496 \times 10^{8}\ \text{km}}{4.848 \times 10^{-6}} \approx 3.09 \times 10^{13}\ \text{km}$

That distance is defined as 1 parsec (the name is short for "parallax of one arcsecond"). One light year is $9.46 \times 10^{12}$ km, so 1 parsec is about 3.26 light years. And now the payoff, with $p$ in arcseconds:

$d\ (\text{in parsec}) = \frac{1}{p\ (\text{in arcsec})}$

No constants to remember. That's the entire reason the unit exists.

Worked example

Proxima Centauri, the nearest star after the Sun, has a parallax of about 0.77 arcsec. So $d \approx 1/0.77 \approx 1.3$ pc, which is about 4.2 light years. A star with $p = 0.05''$ would sit at $1/0.05 = 20$ pc, about 65 light years.

Notice that no star has a parallax of even 1 arcsecond, because the nearest one is already 1.3 pc away. These shifts are tiny. The idea was understood for ages, but the first successful measurement only came in 1838, because it needed telescopes and patience that earlier astronomers didn't have.

Why it stops working

From the ground the atmosphere blurs everything, so ground-based parallax is only good for stars up to roughly a hundred parsecs away. Space fixes that. A satellite called Hipparcos measured about a hundred thousand stars in the 1990s. Its successor, Gaia, has done it for well over a billion stars and reaches out thousands of parsecs. The Milky Way's disc is about 30,000 parsecs across, so we can now map a big chunk of our own galaxy this way. Other galaxies are far too distant for parallax.

The ladder

So astronomers use a cosmic distance ladder: each rung is calibrated using the one below it.

Rung	Method	Idea
1	Parallax	Pure geometry, nearby stars
2	Cepheid variables	Pulsation period tells you true brightness
3	Type Ia supernovae	Explosions of roughly standard brightness, visible across billions of light years
4	Redshift / Hubble's law	Expansion speed of far galaxies

Rung 2 is my favourite. In 1912 Henrietta Leavitt noticed that among variable stars in the Small Magellanic Cloud (all at roughly the same distance from us), the ones that pulsed more slowly were intrinsically brighter. So if you measure a Cepheid's period, you know its true luminosity $L$. Measure how bright it looks, $b$, and the inverse-square law gives the distance:

$b = \frac{L}{4\pi d^{2}} \quad\Rightarrow\quad d = \sqrt{\frac{L}{4\pi b}}$

And how do we calibrate the true luminosity of Cepheids in the first place? Using the few that are close enough to have a measured parallax. That's why this stuff matters to me: the thumb trick is the bottom rung. If it's wrong, everything above it is wrong.

Try it with your own thumb

You can estimate the distance to something far away using nothing but your hand. Hold your thumb at arm's length. Your arm is roughly ten times your eye separation, so when you swap eyes, the thumb's apparent jump at the distance of your target is about one tenth of the distance to that target. So: line the thumb up with a faraway object (say a bus), swap eyes, and estimate how far the thumb jumped measured against things at that distance. If it jumped across about three metres worth of background, the object is around 30 m away. Multiply the jump by ten. It's rough, but it works, and the similar-triangles reasoning is the same as the star version.

Next time I'm bored in a bus, I'm measuring things with my thumb. You'll find me weird. I accept this.