Measuring distances in space.

Step 1 of 6

Parallax

Hold a finger in front of your eyes, and close first one eye and then the other: the finger jumps against the room behind it. The closer it is, the more it jumps. This is parallax. The Earth does the same on a large scale: in January and in July it is on opposite sides of the Sun, $2\ \mathrm{AU}$⁠ apart, and a nearby star seems to move against the distant stars.

The parallax $p$⁠ is half the angle through which the star moves, and it is very small: it is measured in arcseconds, $\dfrac{1}{3600}$⁠ of a degree. In the Scene the star is $890$⁠ light years away, like Delta Cephei.

Step 2 of 6

Parsecs

The distance at which the parallax is exactly one arcsecond is called a parsec, from “parallax of one second”. One parsec is about $3.26$⁠ light years, about $3.09 \times 10^{16}\ \mathrm{m}$⁠. In parsecs the formula is simple:

$d = \frac{1}{p}$⁠

with $d$⁠ in parsecs and $p$⁠ in arcseconds. Star distance is locked. The reading beside the Scene shows the parallax.

Step 3 of 6

The nearest star

In 1838 Friedrich Bessel measured the parallax of a star for the first time. Today the Gaia satellite measures the parallaxes of more than a billion stars, out to thousands of light years. Star distance is now that of Alpha Centauri, our neighbour.

Step 4 of 6

Standard candles

Farther out, parallax is lost. What is left is brightness. The light of a source giving out a power $L$⁠ spreads over a sphere, so at a distance $d$⁠ each square metre receives

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

If you know $L$⁠, the brightness $b$⁠ gives $d$⁠. A source like that is called a standard candle. Climb the ladder has reached Cepheids: stars that swell and shrink, their brightness rising and falling with a fixed period.

Step 5 of 6

The inverse-square law

The brightest standard candle is a Type Ia supernova: a white dwarf that has pulled matter from a companion star until it reached a critical mass and blew up. All these supernovae reach almost the same luminosity at their peak, billions of suns. Climb the ladder has reached the supernova stage, with a calibrating galaxy $30$⁠ million light years from us, whose distance is known from its Cepheids. Galaxy distance is locked.

Step 6 of 6

A hundred times fainter

Each rung of the ladder is calibrated by the rung below it: parallax calibrates the Cepheids, and the Cepheids calibrate the supernovae. Star distance and Cepheid period are locked, leaving only Galaxy distance.