The distance to a star cannot be measured by running out a tape. The only handle is an angle. So we look at the same star from two separated points and measure how much the direction differs.
The most widely separated pair available is the position of the Earth six months apart. The Earth goes round the Sun, so in half a year it moves by the diameter of its orbit. As it does, nearby stars alone appear to swing back and forth against the distant ones behind them.
Half the width of that swing is the annual parallax, written . The smaller it is, the farther the star. Fix the distance at which the angle is arcsecond as parsec and the distance follows from a single division, .
This method only reaches very close by, though. The farther the star, the smaller becomes, and it soon runs up against how finely an angle can be measured. Beyond that, another method has to be joined on.
The next handle is brightness. Write for the light a star actually emits and for the brightness that reaches us.
Light spreads over the surface of a sphere. Twice the distance means four times the area, three times the distance nine times the area. The same amount of light is divided over that area, so .
can be measured with a telescope. So if is known, solving this equation for is all it takes. An object whose true brightness is known is called a standard candle.
The problem is how to know . Stars come bright and faint, and looking at one tells you nothing about which it is. The next scene is the answer.
There is a kind of variable star called a Cepheid. It swells and shrinks over and over, and its brightness runs up and down with a fixed period.
These stars obey a relation: the longer the period, the brighter the star truly is. A period is a quantity a clock alone can measure. So following the rise and fall to get the period reads off .
With known, follows from . The equation of the previous scene carries over as it stands, and distances can be measured far beyond the reach of parallax.
The relation itself was calibrated using nearby Cepheids whose distances are known from parallax. A second rung has been set on top of the first.
Cepheids cannot be seen without limit either. In a distant galaxy the individual stars can no longer be told apart.
So a brighter standard candle is used: the type Ia supernova. A white dwarf explodes on passing a fixed mass, so they all reach nearly the same brightness, putting out light to rival a whole galaxy.
The calibration works the same way here. When a supernova appears in a galaxy whose distance is known from Cepheids, its brightness is recorded. A third rung is set on the second.
Joining parallax, Cepheids and supernovae this way stretches the reach by orders of magnitude. Each rung is calibrated where it overlaps the one below, which is why it is called a ladder. The horizontal axis of Hubble's law was measured by climbing it.